SYMPLECTIC LOCAL ROOT NUMBERS, CENTRAL CRITICAL L … › ~dprasad › ggp5.pdf · for the orthogonal groups. We verify these conjectures in certain low rank cases and for depth zero

SYMPLECTIC LOCAL ROOT NUMBERS, CENTRAL CRITICALL-VALUES, AND RESTRICTION PROBLEMS IN THE

REPRESENTATION THEORY OF CLASSICAL GROUPS

WEE TECK GAN, BENEDICT H. GROSS AND DIPENDRA PRASAD

Abstract. We give a conjectural description of the restriction of an irreducible repre-sentation of a unitary group U(n) to a subgroup U(n − 1) over a local or global field.We formulate analogous conjectures for the restriction problem from U(n) to a subgroupU(m) (m < n) using Bessel and Fourier-Jacobi models, and also similar restriction prob-lems for symplectic groups. The conjectures are analogs of those in [GP1] and [GP2]for the orthogonal groups. We verify these conjectures in certain low rank cases and fordepth zero supercuspidal representations, and prove that the conjectures about Besseland Fourier-Jacobi models follow from the conjectural description of the restriction of anirreducible representation of a unitary group U(n) (resp. SO(n)) to a subgroup U(n− 1)(resp. SO(n− 1)).

Contents

Introduction 2

Part 1. CONJECTURES 51. Hermitian Spaces and Unitary Groups 52. Self-Dual Representations 63. Langlands Parameters for Unitary Groups 84. The Langlands conjecture for U(M) 115. Parameters for U(M)× U(M0) 126. The Local Conjecture 147. The refined local conjecture –the definition of χ 168. χ is a character of Aϕ 209. Alternative Description of Recipe 2210. Unramified parameters 2611. Discrete series parameters 2812. The Global Conjecture 32

Part 2. EVIDENCE 3613. U(1,1) 3613.1. Parameter for U(1, 1) 3714. Trilinear forms for U(2) 38

1

2 WEE TECK GAN, BENEDICT H. GROSS AND DIPENDRA PRASAD

15. Using the theta correspondence for U(2, 1) 4015.1. Completing the branching from U(2, 1) to U(2) 4316. Branching laws for GL(n,Fq) 4417. Branching laws for U(n,Fq) via base change 4618. Depth zero supercuspidals 4819. Global implies local: some partial results, I 50

Part 3. BESSEL and FOURIER-JACOBI MODELS 5420. Bessel Models for Unitary Groups 5621. Compatibility of Various Conjectures 5822. Fourier-Jacobi Models for Symplectic Groups 62

23. Local Langlands Conjecture for Sp(2n) 6824. Conjectures for Fourier-Jacobi Models 7125. Fourier-Jacobi Models for Unitary Groups 72References 77

Introduction

In this paper, we will consider several problems about the restriction of irreducible complexrepresentations of unitary, orthogonal, and symplectic groups to subgroups of similar type(possibly fattened by a unipotent group when there is room left). The basic question isto determine the irreducible representations of the subgroup which occur together withtheir multiplicities. We formulate the general problem as follows. Let π1 be an irreduciblerepresentation of a group G1, and let π2 be an irreducible representation of a subgroupG2 ⊂ G1. Then π = π1 ⊗ π2 is an irreducible representation of the product groupG = G1 × G2, which contains the subgroup H = G2 embedded diagonally. We saythat the dual representation π∨2 occurs in the restriction of π1 with multiplicity d if thecomplex vector space HomH(π,C) of H-invariant linear forms on π has dimension d.

We first consider the case when k is a local field, and K is an etale quadratic k-algebra,and G1 = U(W ) is the unitary group of a non-degenerate Hermitian space W of rankn over K. We let G2 ⊂ G1 be the subgroup fixing a non-isotropic line in W , which isisomorphic to the unitary group U(W0) of the orthogonal complement, of dimension (n−1).Thus we are considering a generalization of the classical problem of restricting irreduciblerepresentations of the compact Lie group U(n) to the subgroup U(n− 1).

It has recently been shown by Aizenbud-Gourevitch-Rallis-Schiffmann in [AGRS], thelong-awaited result (for k non-Archimedean) that for any irreducible, complex representationπ = π1⊗π2 of G = U(W )×U(W0), the vector space HomH(π,C) ofH = U(W0)-invariantlinear forms on π has dimension d ≤ 1. The problem, then, is to determine for which pairs(π1, π2), it is non-trivial.

We propose a general conjecture which answers this problem in two important cases ofnumber theoretic interest:

LOCAL ROOT NUMBERS AND BRANCHING LAWS 3

• when k is local and the representation π of G(k) lies in a generic L-packet, and

• when k is global with ring of adeles A and π is an automorphic tempered representationof G(A).

Our method assumes the Langlands parameterization of irreducible representations π ofG into finite L-packets. (For a general discussion of the local Langlands conjecture, see[GR]. For unitary groups, where it is almost established, see [M].) We conjecture that thereis a unique representation π′ of a pure inner form G′ = U(W ′) × U(W ′

0) of G in eachgeneric L-packet, such that HomH′(π′,C) is non-zero.

The irreducible representations in an L-packet should correspond to irreducible repre-sentations of the component group Aϕ of the centralizer of the Langlands parameter ϕ.When G = U(W ) × U(W0), the component group is an elementary abelian 2-group. Weconstruct a homomorphism

χ : Aϕ → 〈±1〉

using the local root numbers of symplectic summands of a natural symplectic representationV of the L-group. We conjecture that the unique representation π′ in the L-packet with anH ′-invariant linear form corresponds to the character χ. This is similar to our conjectures[GP1] in the orthogonal case; in the unitary case, V has dimension 2n · (n − 1) and isinduced from a tensor product of standard representations.

As in the orthogonal case [GP2], one can define the character χ and formulate a moregeneral conjecture on restriction from U(W ) to U(W0), where W0 is a non-degeneratesubspace of odd codimension in W . This conjecture also includes a generic character of aunipotent subgroup of U(W ), when the codimension of W0 in W is greater than one, andcorresponds to the classical theory of Bessel models. We show that the general conjectureabout Bessel models follows from the original one (when W0 has codimension 1 in whichcase the subgroup H is reductive, unlike higher codimension case when the subgroup hasa unipotent part) in both unitary and orthogonal cases.

We then turn to the restriction problem for Hermitian spaces W0 ⊂ W over a globalfield k, with quadratic etale algebra K. Let G = U(W ) × U(W0) and assume that π isan irreducible automorphic representation of G(A), where A is the ring of adeles of k. Ifthe vector space HomH(A)(π,C) is non-zero, our local conjecture predicts that the globalroot number ε(π, V, 1

2) is equal to 1. If we assume π to be tempered, then our calculation

of global root numbers, and general conjectures of Langlands and Arthur predict that πappears with multiplicity one in the discrete spectrum of G. We conjecture that the periodintegrals on the corresponding space of functions

f 7→∫H(k)\H(A)

f(h) dh

gives a non-zero element in HomH(A)(π,C) if and only if the central critical L-valueL(π, V, 1

2) is non-zero.


One case in which all of these conjectures are known to be true is when the quadratic etalealgebra K is split: K ∼= k × k.Then G1

∼= GLn(k) and G2∼= GLn−1(k). When k is local,

and π is a generic representation of G = G1 × G2, the theory of Rankin-Selberg integral,cf. [P4, Theorem 3], shows that the complex vector space HomH(π,C) has dimension 1(assuming [AGRS] which shows that the dimension is at most 1). This agrees with ourlocal conjecture, as the L-packets for G = GLn(k)×GLn−1(k) contain single element. Ifk is global and π is a tempered automorphic representation of G(A), then π appears withmultiplicity one in the discrete spectrum and the periods over H(k)\H(A) give a non-zerolinear form if and only if the tensor product L-function L(π, stdn ⊗ stdn−1, s) is non-zeroat s = 1

2[JPSS]. In the split case, V is the sum of stdn ⊗ stdn−1 and its dual, and the

local and global root numbers are all equal to 1.We also present some material on the restriction of representations of the symplectic

and metaplectic groups, using the Weil representations and the Fourier-Jacobi models. Allof the situations studied involve representations of the Weil-Deligne group of the formV = V1 ⊗ V2 with V1 symplectic and V2 orthogonal. We recall that in [GP1] and [GP2],the orthogonal representation V2 was always even dimensional, being the L-group of aneven dimensional orthogonal group. It is thus natural to look for representation theoreticcontexts which might involve representations of the Weil-Deligne group of the form V1⊗V2

with V1 symplectic and V2 orthogonal of odd dimension. This suggests looking for branchinglaws involving odd orthogonal and symplectic groups. This does not seem to lead to anymeaningful representation theoretic question. However, it is known that Sp2n(C) can betaken to be the L-group also of the metaplectic group Mp2n, and then a possible branchinglaw involves restriction from Symplectic to metaplectic group, or vice-versa. This is how theWeil representation, and the Fourier-Jacobi models for symplectic group arise in this paper,completing the picture of [GP1] and [GP2]. In fact, methods of theta correspondence allowsone to reduce these questions once again to the basic branching laws of [GP1] involving(SOn, SOn−1) as we see in this paper.

Finally, we provide evidence for our conjectures in various cases:

• unitary groups of low rank, namely U(2)× U(1) and U(3)× U(2);• discrete series representations in the real case;• tame parameters in the p-adic case.

As already mentioned, the last two authors proposed a similar conjecture in [GP1] forthe restriction of irreducible representations of SO(n, k) to SO(n−1, k). Since then, therehas been considerable progress in the local case [GR, GW] as well as in the global case[GJR], but the conjecture remains open in general. Shortly after the publication of [GP1],it was also realized that the conjectural framework for (SO(n), SO(n− 1)) should continueto hold for (U(n),U(n − 1)), but as the case of unitary groups seemed so similar to thatof the orthogonal groups, the last two authors did not write it down. However, given thecontinued interest in these restriction problems, the recent proof of the multiplicity oneresult [AGRS], the recent progress in the local Langlands conjecture for unitary groups[M] and the fact that the unitary groups U(n) are in many ways more accessible (being


related to GL(n) by base change and supporting Shimura varieties), it seems an appropriatemoment to give an account of the Gross-Prasad conjecture for unitary groups. In fact, aswe were just finishing writing the paper, we learnt of an announcement by Waldspurgerof a multiplicity 1 theorem (and not just less than or equal to one) for certain temperedL-packets for (SO(n), SO(n − 1)), assuming certain natural statements about charactersin an L-packet.

Acknowledgments: W. T. Gan is partially supported by NSF grant DMS-0801071.Dipendra Prasad was partially supported by a Clay Math Institute fellowship during thecourse of this work.

Part 1. CONJECTURES

1. Hermitian Spaces and Unitary Groups

Let k be a field, K a separable quadratic field extension of k and τ the non-trivialinvolution of K fixing k.

A Hermitian space M of rank n is an n-dimensional K-vector space, equipped with anon-degenerate Hermitian form

β : M ×M −→ K.

The Hermitian condition is that β is K-linear in the first variable and satisfies

β(v, w)τ = β(w, v) for any v, v ∈M .

The non-degenerate condition is that the map M −→ HomK(M,K) taking a vector w inM to the linear form v 7→ β(v, w) is a K-linear bijection.

We say that the Hermitian space M is split if it contains an isotropic subspace H (i.e., asubspace on which the restriction of β is trivial) of the largest possible dimension, namely:{

n/2, if n is even;

(n− 1)/2, if n is odd.

When n is even, there is a unique split Hermitian space of rank n up to isomorphism; it isgiven by M = H ⊕H∨ where H∨ is isotropic and dual to H under β. When n is odd, asplit Hermitian space of rank n has the form M = H ⊕H∨⊕ 〈α〉, where 〈α〉 is the rank 1Hermitian space determined by α ∈ k×. Thus, there are #[k×/NK/k(K

×)] split Hermitianspaces of odd rank n.

The unitary group U(M) is the subgroup of GLK(M) preserving β:

U(M) = {T ∈ GLK(M) | β(Tv, Tw) = β(v, w)} .


It is a reductive group over k of dimension n2 and is quasi-split precisely when M is split.The group U(1) = U(∧nM) is a 1-dimensional torus whose isomorphism class dependsonly on K/k and not on M . This torus is isomorphic to the center of U(M) and to theabelianization of U(M). We normalize the maps

U(1)center−−−→ U(M)

det−−−→ U(1)

so that the composite is multiplication by n.

We can now formulate one of the main questions we will study in this paper. Let M bea split Hermitian space of dimension n ≥ 1, and let M0 ⊂ M be a non-degenerate splitHermitian subspace of codimension 1 in M . Writing

M = M0 ⊕M⊥0 with dimM⊥

0 = 1,

we obtain a homomorphism of algebraic groups over k

i : U(M0) −→ U(M)

mapping a unitary transformation of M0 to one which acts by the identity on the line M⊥0

in M . This gives a homomorphism

j = i× 1 : U(M0) −→ U(M0)× U(M⊥0 ).

Our aim is to study the restriction of irreducible complex representations π⊗π0 of U(M)×U(M0) to the subgroup U(M0) when k is a local field. More precisely, we want to determineexactly when HomU(M0)(π ⊗ π0,C) is nonzero. To do this, we will need to recall theconjectural Langlands parameterization.

2. Self-Dual Representations

Let G denote a complex reductive group. We fix a pinning (epinglage) of G and let

T ⊂ B ⊂ G

be the corresponding maximal torus and Borel subgroup. Consider the semi-direct product

Go 〈1, τ〉,

where the involution τ acts on G via a pinned automorphism (possibly trivial) which maps

to the opposite involution in Out(G). Then for every complex algebraic representation V

of G, the conjugate representation V τ is isomorphic to the dual representation V ∨. Inparticular, the induced representation

Ind(V ) = Ind(V τ ) = Ind(V ∨)

of G o 〈1, τ〉 is self-dual. For example, when V = C is the trivial representation, then

Ind(C) ∼= C⊕C(χ) where χ is the non-trivial character of Go〈1, τ〉 with trivial restriction

to G.


The pinning determines a principal homomorphism

ϕ : SL2 −→ G

which is fixed by τ . Hence, we obtain a homomorphism

ϕ× 1 : SL2 × 〈1, τ〉 −→ Go 〈1, τ〉.

Let ε = ϕ(−1) in G. Then ε is in T , ε2 = 1 and ε acts trivially on each root space of

Lie(G). Hence ε lies in the center Z(G). Since ε is fixed by τ , it lies in the center Z(G)τ

of Go 〈1, τ〉.

The following result is due to Deligne and generalizes a result in Bourbaki [Bo].

Proposition 2.1. Every representation W of Go〈1, τ〉 is self-dual. If W is irreducible,the invariant pairing on W is (ε|W )-symmetric.

The proof is similar to that in Bourbaki. To determine the sign of the pairing on anirreducible W , one restricts W to SL2 × 〈1, τ〉 and observes that there is an irreduciblecomponent of multiplicity one.

The groups Go 〈1, τ〉 which arise in this section are precisely the L-groups (in the senseof Langlands) of anisotropic groups G over R, with 〈1, τ〉 = Gal(C/R). As a special case,we have the L-group of the compact Lie group G = U(n) associated to a positive definiteHermitian space of rank n. Here

G ∼= GLn(C) and τ(A) = J · tA−1 · J−1

with

J =

1−1

1−1

··

·

.

In this case, the center of the L-group is

(C×)τ = 〈±1〉and

ε = J2 = (−1)n−1.

Indeed, the principal SL2 −→ GLn is given by the representation Symn−1. The transferhomomorphism

V er :(LG

)ab= 〈1, τ〉 −→ GLn(C)ab = C×


is trivial, as LG is a semi-direct product.

Proposition 2.2. Let W be the irreducible faithful representation of dimension 2n ofLU(n) = GLn(C) o 〈1, τ〉

which is induced from the standard representation V = Cn of GLn(C). Then W =Ind(V ) = Ind(V ∨) is symplectic when n is even and orthogonal with det = χ when nis odd.

Proof. This follows from our calculation of ε and the fact that

det(Ind(V )) = V er(detV ) · χdimV .

�

Next, consider the compact Lie group G = U(n) × U(n − 1). The L-group LG isisomorphic to (GL(W ) × GL(V )) o 〈1, τ〉, with dimW = n − 1 and dimV = n. ThisL-group has two faithful irreducible representations of dimension 2 · n · (n− 1):{

U = Ind(W ⊗ V ) = Ind(W∨ ⊗ V ∨),

U ′ = Ind(W ⊗ V ∨) = Ind(W∨ ⊗ V ).

They are both symplectic, as the principal SL2 acts on the tensor products by the repre-sentation Symn−2 ⊗ Symn−1. We have the decomposition

U ⊕ U ′ = Ind(W )⊗ Ind(V ).

3. Langlands Parameters for Unitary Groups

We now assume that K/k are locally compact fields. Let M be a split Hermitian spaceof rank n over k. The L-group of G is isomorphic to

LG ∼= GL(V ) o Gal(K/k),

where V is a complex vector space of dimension n.

Let W ′k denote the Weil group Wk of k if k = R, and the product Wk × SL2(C) if k is

non-Archimedean. A Langlands parameter for G = U(M) is a homomorphism

ϕ : W ′k −→ LG

satisfying some conditions. In particular, the restriction ϕK of ϕ to the subgroup W ′K of

index 2 is a complex representation

ϕK : W ′K −→ GL(V ).

Let s ∈ W ′k be a representative of the non-trivial coset of W ′

K . Then

σ 7→ sσs−1


gives an outer automorphism of W ′K and the representation V given by ϕK satisfies

V s ∼= V ∨.

Let V = ⊕imiVi be the decomposition of the representation ϕK into irreducible repre-sentations Vi of W ′

K , occurring with multiplicities mi. The centralizer of the image of ϕKin GL(V ) is then isomorphic to the product

∏i GLmi

(C) by Schur’s lemma. Our aim inthis section is to determine the centralizer Cϕ in GL(V ) of the image of ϕ, as a subgroupof

∏i GLmi

(C), and to examine the group Aϕ of connected components of Cϕ.

Since V s ∼= V ∨, there are two possibilities for each irreducible factor Vi:{V si∼= V ∨

j with j 6= i;

V si∼= V ∨

i .

The first case implies that mi = mj. The second implies that there is a non-degeneratebilinear form (unique up to scaling)

B : Vi × Vi −→ C

which satisfies:

B(σv, sσs−1w) = B(v, w).

Since the form

B′(v, w) := B(w, s2v)

enjoys the same properties, we have B = ci · B′ for some ci ∈ C×. We find that c2i = 1and the induced representation Ind(Vi) of W ′

k is ci-symmetric. One could define ci = ±1using the sign of the self-duality of Ind(Vi) if that representation were irreducible. However,when Vi = V s

i = V ∨i is both conjugate self-dual and self-dual, Ind(Vi) is reducible and has

a pairing of either of the signs.

Proposition 3.1. The centralizer in GL(V ) of the image of ϕ is determined by theirreducible decomposition V = ⊕miVi of the representation ϕK and the signs ci of theirreducible conjugate self-dual factors Vi as follows:

(1) If V si∼= V ∨

j with i 6= j, then mi = mj and the centralizer of ϕ is the diagonalsubgroup GLmi

(C) ↪→ GLmi(C)×GLmi

(C).

(2) If V si∼= V ∨

i and ci = (−1)n, then mi is even and the centralizer of ϕ is thesymplectic subgroup Spmi

(C) ⊂ GLmi(C).

(3) If V si∼= V ∨

i and ci = (−1)n−1, then the centralizer of ϕ is the orthogonal subgroupOmi

(C) ⊂ GLmi(C).

Proof. This is proved in [P1]. �


As a consequence, the full centralizer of ϕ in GL(V ) is a product

Cϕ =∏

(i, i′) of type 1

GLmi(C)×

∏j of type 2

Spmj(C)×

∏k of type 3

Omk(C)

over the factors of the 3 types in the proposition above. Observe that we have the congru-ence

n = dimV =∑i

mi dimVi ≡∑

i of type 3

mi dimVi mod 2.

From the above discussion, it is easy to determine the component group

Aϕ = π0(Cϕ)

of the centralizer of ϕ in GL(V ):

Corollary 3.2. Letϕ : W ′

k −→ GL(V ) o Gal(K/k)

be a Langlands parameter for U(M) where M is a split Hermitian space of rank n.Then we have:

(1) The parameter ϕ is a discrete parameter, i.e. C0ϕ is trivial, if and only if only

representations of type 3 occur, and all the multiplicities mi are 1.(2) Aϕ is an elementary abelian 2-group of rank ≤ n;(3) Aϕ has a distinguished basis over Z/2Z, indexed by the irreducible factors Vi of

the representation ϕK which are of type 3 in the sense of proposition 3.1. Foreach i of type 3, the corresponding basis element is given by the image εi ≡ giof an element gi ∈ Omi

(C) with det(gi) = −1.

Remark : We have noted that the L-group LU(n) has an irreducible representationof dimension 2n which is either orthogonal or symplectic. Thus given a parameter ϕ :W ′k −→ LU(n) for a unitary group, we get a parameter ϕ′ : W ′

k −→ O(2n,C), or ϕ′ :W ′k −→ Sp(2n,C) depending on whether n is odd or even. This induces a mapping on

the group of connected components: Aϕ → Aϕ′ . But because of the possibility of anirreducible representation ϕ : Wk → GL(m,C) such that ϕ 6∼= ϕ∨, but ϕ ∼= ϕ∨ ⊗ ωK/k,the mapping on the component groups is not injective (such a parameter restricted toWK remains irreducible); and because of the possibility of an irreducible representationϕ : WK → GL(m,C) such that ϕ ∼= ϕ∨, but ϕ 6∼= ϕs, the mapping on the componentgroups is not surjective (such a parameter induced to Wk is self-dual and irreducible). Wenote, however, that for K/k = C/R, there are no such representations, and therefore themapping of the component groups is an isomorphism in this case.

Remark : Since at some places later, we will consider the parameter ϕ of a unitary grouponly through its restriction to K, denoted here by ϕK , it is nice to note, cf. [BC, propositionA.11.3] that a conjugate-self-dual representation ϕK of W ′

K extends to a parameter of theunitary group in a unique way in either of the two situations:


• ϕK is a sum of irreducible representations, each occurring with multiplicity 1.• ϕK is an abelian representation.

Now one has a natural inclusion

Z(LG) = 〈±1〉 ↪→ Cϕ

which induces a homomorphism of groups

Z(LG) −→ Aϕ.

The following lemma determines the image of this homomorphism:

Lemma 3.3. The image of −1 is the element∑

imiεi, where the summation is overthe representations Vi of type 3, and mi is the multiplicity of Vi in V .

When n is odd, the image of −1 is always non-trivial in Aϕ, as∑i of type 3

mi dimVi ≡ n mod 2.

When n is even, −1 is trivial in Aϕ precisely when all Vi of type 3 occur with even multiplicityin V .

We conclude this section by defining a distinguished character η (possibly trivial when nis even) of Aϕ. First define η as a homomorphism Cϕ −→ 〈±1〉, using the formula

η(a) = (−1)dimV a=−1

.

It is easy to see that this descends to a character of Aϕ with

η(−1) = (−1)n.

On the basis elements εi, one has

η(εi) = (−1)dimVi

4. The Langlands conjecture for U(M)

Let M be a split Hermitian space of rank n ≥ 1 and let G = U(M). In this section, webriefly recall the local Langlands conjecture for G and its pure inner forms.

The pointed setH1(k,U(M)) parameterizes the isomorphism classes of rank n Hermitianspaces (M ′, β′). When k = R, there are n+ 1 classes, determined by the signature of theHermitian form β′(v, v). When k is non-Archimedean,

H1(k,U(M))det−−−→ H1(k,U(1)) = k×/NK×

is a group of order 2, which Kottwitz has shown is dual to Z(LG). The two Hermitianspaces M and M ′ are determined by their Hermitian discriminant, and when n is odd, theyare both split.


Recall that G = U(M) is quasi-split over k. Let B be a Borel subgroup and N itsunipotent radical. A homomorphism

f : N −→ Ga

of unipotent algebraic groups over k is called generic if its stabilizer in T = B/N is equalto the center U(1). Fix a T -orbit on the set of generic characters of N ; there are two orbitswhen n is even and one orbit when n is odd. Then, relative to this choice of T -orbit ofgeneric characters of N , one has:

The Langlands conjecture for G = U(M):

(i) The irreducible admissible complex representations π of G(k) can be parametrized bypairs (ϕ, ρ), where ϕ is a Langlands parameter for G and ρ is an irreducible representationof the finite group Aϕ which is trivial on the image of Z(LG).

(ii) Two pairs (ϕ, ρ) and (ϕ′, ρ′) correspond to the same irreducible representation π if and

only if they are conjugate by an element of G = GLn(C).

(iii) If ϕ is a generic parameter, in the sense that its adjoint L-function L(s, ϕ,Ad) isregular at s = 1, then the trivial representation ρ = 1 should correspond to the uniquerepresentation in the L-packet on which N has an f -generic linear form.

More generally, a pair (ϕ, ρ) (up to G-conjugacy) with ρ an arbitrary representation ofAϕ should parameterize an irreducible representation π(ϕ, ρ) of a pure inner form U(M ′)of U(M). When k is non-Archimedean, the group G′ = U(M ′) will be determined by therestriction of ρ to the image of Z(LG). The distinguished character η of Aϕ defined inthe previous section should correspond to a further generic representation in the L-packet,which will be on a quasi-split pure inner form U(M ′) when n is odd.

By the recent work of Arthur and Moeglin [M], the local Langlands conjecture for G =U(M) is now essentially known.

5. Parameters for U(M)× U(M0)

We now consider the Langlands parameters for the group G = U(M) × U(M0). LetK/k be a separable quadratic extension of local fields. Let M be a split Hermitian spaceof dimension n ≥ 1, and let M0 ⊂ M be a non-degenerate split Hermitian space ofcodimension 1. Then we have a natural homomorphism of quasi-split groups over k:

j : U(M0) −→ U(M0)× U(M).

By the local Langlands conjecture discussed in the previous section, the irreducible rep-resentations π of G(k) and its pure inner forms are parametrized by pairs (ϕ, ρ) where

ϕ : W ′k −→ LG = (GL(V0)×GL(V )) o Gal(K/k)


andρ : Aϕ −→ 〈±1〉

is an irreducible representation of Aϕ. We have seen in Section 2 that LG has two irreduciblesymplectic representations of dimension 2 · n · (n− 1):

U = Ind(V0 ⊗ V ) and U ′ = Ind(V0 ⊗ V ∨).

We will need to distinguish them, which we can do using the embedding of Hermitian spacesM0 ↪→M .

Let M0 = ⊕n−1i=1 Li be an orthogonal decomposition into non-isotropic lines, and complete

this to an orthogonal decomposition of M by taking Ln = M⊥0 . These decompositions

give maximal tori T0 and T in the unitary groups. The corresponding characters of thesetori on the lines Li give basis

〈e1, ..., en−1〉 and 〈e1, ..., en〉of X∗(T0) and X∗(T ) respectively. We take the dual bases

〈e∨1 , ..., e∨n−1〉 and 〈e∨1 , ..., e∨n〉

in X∗(T0) and X∗(T ), and let V0 and V be the irreducible representations of the dualgroups with these weights. This gives an isomorphism

G ∼= GL(V0)×GL(V )

and allows us to select the representation U = Ind(V0 ⊗ V ) of LG.

The embedding M0 ↪→ M also selects one of the two orbits of generic f : NG −→ Ga.We have

NabG = Nab

0 +Nab

with dimNab0 = n−2 and dimNab = n−1. Assume that n is even, so that dimM0 is odd

and there is a unique generic orbit of f0 : No −→ Ga. Since a maximal isotropic subspaceH ⊂M determines a maximal isotropic subspace H0 = H ∩M0 ⊂M0, we find that

Nab = Nab0 + L

where L is a line on which Z(U(M0)) acts non-trivially. We define f : Nab −→ Ga as thesum f0 + l where l is the linear form on L given by

λ 7→ β(λv, v)

β(v, v), where 〈v〉 = M⊥

0 .

This determines an orbit of generic characters on NG when n is even. When n is odd, wecomplete M⊥

0 to a hyperbolic plane P and use the codimension one subspace P⊥ ⊂M0 ⊂M to give an orbit of generic characters on NG.

Having chosen a generic f forNG, the local Langlands conjecture predicts that for genericparameters ϕ of G = U(M0)×U(M), the pairs (ϕ, ρ) parameterize irreducible representa-tions of G(k) and its pure inner forms G′(k), in such a way that the trivial character ρ = 1


corresponds to the unique f -generic representation in the L-packet associated to ϕ. Thegroups G′ have the form U(M ′

0) × U(M ′), where M ′0 is a Hermitian space of rank n − 1

and M ′ is of rank n. If M ′0 embeds as a Hermitian subspace of M ′ and M ′/M ′

0∼= M/M0

as 1-dimensional Hermitian spaces, we call the pair (M ′0,M

′) relevant.

The center of LG is now the group

Z(LG) = (C× × C×)τ = 〈±1〉 × 〈±1〉.

In the non-Archimedean case, a pure inner form corresponds to a character

ρ : Z(LG) −→ 〈±1〉.

The pair (M ′0,M

′) associated to ρ is relevant if and only if

ρ(−1,−1) = 1.

In the real case, this is a necessary condition, but the relevancy also depends on thesignatures of M ′

0 and M ′.

6. The Local Conjecture

After the preparation of the previous sections, we are now ready to return to the problemformulated in Section 1. Here is our local conjecture:

Conjecture 6.1. Let ϕ be a generic Langlands parameter for G = U(M0) × U(M).There is a (unique) character χ : Aϕ −→ 〈±1〉 which satisfies:

(1) χ(−1,−1) = 1, so the pair (M ′0,M

′) of Hermitian spaces associated to χ isrelevant;

(2) the irreducible representation π(ϕ, χ) of G′(k) = U(M ′0)×U(M ′), associated to

(ϕ, χ) under the local Langlands correspondence, satisfies:

dim HomU(M ′0)(π(ϕ, χ),C) = 1;

(3) for any other relevant ρ 6= χ,

HomU(M ′0)(π(ϕ, ρ),C) = 0.

(4) The representation π(ϕ, χ) of G′(k) = U(M ′0) × U(M ′) which has a non-zero

linear form invariant under U(M ′0), lives on G′ such that the Hermitian space

in the even number of variables has the discriminant of a maximally split Her-mitian space (hence G′ is quasi-split if k is non-Archimedean) if and only if

ε(1

2, ϕ1 ⊗ ϕ2, ψ) = 1,

where the restriction of ϕ to K, ϕK, is of the form ϕK = ϕ1 × ϕ2, and thecharacter ψ of K is chosen to be trivial on k (the value of the epsilon factor at1/2 is independent of ψ).


In other words, the conjecture says that one has “multiplicity one for restriction in theL-packet of ϕ”:

∑relevant ρ

dim HomU(M ′0)(π(ϕ, ρ),C) = 1.

Note again that the bijection of irreducible representations with pairs (ϕ, ρ) depends on ourchoice of a generic f . However, observe that for the purpose of this conjecture, one onlyneed to have the partition of the set of irreducible representations of U(n)×U(n− 1) intoL-packets; one does not really need the parameterization of these L-packets by Langlandsparameters, or the fine parameterization of the members of an L-packet by the charactersof the component group.

To conclude this section, we shall describe a slightly refined local conjecture which as-sumes that one has a parameterization of the L-packets by Langlands parameters. It doesnot require the precise parameterization of the members of an L-packet by the charactersof the component group, which can be a very delicate issue. This conjecture is typicallywhat is checked in practice.

Conjecture 6.2.

(1) Fix a generic L-parameter for U(M):

ϕ : W ′k −→ GL(V ) o Gal(K/k)

and consider its restriction ϕK to the subgroup W ′K of index 2. Write

ϕK = n1σ1 ⊕ n2σ2 ⊕ · · · ⊕ nrσr ⊕ τ,

where the σi’s are the distinct irreducible conjugate-self-dual representations of W ′K

appearing in ϕK such that the sign c(σi) of the bilinear form Bi on σi is equal to(−1)dimV+1.

Fix an irreducible representation π of U(M ′) (a pure inner form of U(M)) in theL-packet Π(ϕ) associated to ϕ. For any generic Langlands parameter ϕ0 of U(M0) withassociated L-packet Π(ϕ0), set

Hom(π, ϕ0) :=⊕

π0∈Π(ϕ0)

HomU(M ′0)(π ⊗ π0,C).

If ϕ0 and ϕ′0 are two generic Langlands parameters of U(M0) such that

Hom(π, ϕ0) 6= 0 and Hom(π, ϕ′0) 6= 0,

then for each i, we have:

ε(1/2, σi ⊗ ϕ0,K , ψ) = ε(1/2, σi ⊗ ϕ′0,K , ψ) ∈ ±1,


where ψ is a fixed non-trivial additive character on K which is trivial in k (the valuesof the epsilon factor actually depend on ψ). In particular, π determines a characterχπ on Aϕ = (Z/2)r, whose value on the non-trivial element in the i-th copy of Z/2Z isequal to ε(1/2, σi ⊗ ϕ0,K , ψ) for any ϕ0 such that Hom(π, ϕ0) 6= 0.

(2) The association π 7→ χπ gives a bijection between the L-packet Π(ϕ) associated toϕ and the set of irreducible characters of the component group Aϕ.

(3) One can exchange the role of U(M) and U(M0) in the above statements. Moreprecisely, suppose that we are given a Langlands parameter ϕ0 of U(M0) such that

ϕ0,K = n1σ1 ⊕ n2σ2 ⊕ · · · ⊕ nrσr ⊕ τas above and an element π0 in the associated L-packet Π(ϕ0). Then the collection ofepsilon factors ε(1/2, ϕK ⊗ σi, ψ) is independent of the choice of a generic Langlandsparameter ϕ of U(M) for which

Hom(ϕ, π0) :=⊕

π∈Π(ϕ)

HomU(M ′0)(π ⊗ π0,C) 6= 0.

This determines a character χπ0 of Aϕ0 and the association π0 7→ χπ0 gives a bijectionbetween the L-packet Π(ϕ0) and the set of irreducible characters of Aϕ0.

In the conjecture of Langlands and Vogan as given in Section 4, the elements in anL-packet Π(ϕ) are parametrized by characters of the component group Aϕ. The aboveconjecture says that one can alternatively use the collection of epsilon factors describedabove to serve as parameters for elements of Π(ϕ). In the next section, we will present arefined local conjecture by specifying the distinguished character χ above. This recipe usessymplectic root numbers for the representation U = Ind(V0 ⊗ V ) of W ′

k.

7. The refined local conjecture –the definition of χ

We are now going to specify the character χ of Aϕ in Conjecture 6.1, up to a smallambiguity.

Assume that char(k) 6= 2 for simplicity, and write K = k + ke with τ(e) = −e. Then eis the unique element of order 2 in the one dimensional torus U(1) = K×/k× over k. LetωK/k : k×/NK× ∼→ ±1 be the quadratic character given by the class field theory.

Let V be a representation of W ′(K) which satisfies

V s ∼= V ∨

as in § 3. Assume that we have a non-degenerate bilinear form B : V × V → C whichsatisfies

(7.1)

{B(σv, sσs−1w) = B(v, w)B(w, s2v) = cB(v, w)


with c = c(V ) = ±1. The induced representation Ind(V ) ofW ′(k) is then c(V )-symmetric.The 1-dimensional representations V satisfying (7.1) correspond to certain characters

α : K× → C×, by local class field theory. Since ατ = α−1, the kernel of α must containthe subgroup NK×. Moreover, the restriction of α to k× is given by

Res(α) =

{1 if c(α) = 1

ωK/k if c(α) = −1.

If V satisfies (7.1), so does the 1-dimensional representation det(V ), and we have

c(det(V )) = c(V )dim(V ) in ± 1.

In particular, if c(V )dim(V ) = 1, det(V ) is a character of K×/k×, and may be evaluated onthe involution e.

We now assume that c(V ) = −1, so Ind(V ) is symplectic. In this case, we may definethe local root number [Gr, Motives]

ε(IndV ) = ε(IndV, ψ, dx,1

2)

where ψ is any non-trivial additive character of k and dx is the unique Haar measure whichis self-dual for Fourier transform with respect to ψ. Then

ε(IndV ) = ±1,

andε(Ind(V ⊕W )) = ε(IndV )ε(IndW ).

Fix the choice of an auxiliary character,{µ : K× → C×

Res(µ) = ωK/k.

Then c(V ⊗ µ) = 1 and we may define the sign

det(V ⊗ µ)(e) = ±1.

Then det((V ⊕W )⊗ µ)(e) = det(V ⊗ µ)(e) · det(W ⊗ µ)(e).If V has even dimension, the sign of det(V ⊗ µ)(e) is independent of the choice of µ.

Indeed, in this case we find

det(V ⊗ µ)(e) = detV (e) · µdimV (e)

= detV (e) · µdim V

2 (e2)

= detV (e) · µdim V

2 (−1) as e2 = −ee(7.2)

= detV (e) · ωK/k(−1)dim V

2 as − 1 ∈ k×

We now give our first (2 term) formula for χ. Let ϕ be a parameter for U(W )×U(W0).Then

ϕK = ResKϕ : W ′K → GL(V1)×GL(V2)


with V1 even dimensional and V2 odd dimensional. The representation V = V1⊗V2 satisfies(7.1) with c(V ) = −1.

Let a be an involution in the centralizer Cϕ of ϕ in GL(V1)×GL(V2). Then V decom-poses as a direct sum of eigenspaces

V = V a=1 ⊕ V a=−1.

Both summands satisfy (7.1) with c(V ) = −1, and we define

χ(a) = ε(Ind(V a=−1)) · det(V a=−1 ⊗ µ)(e).(7.3)

In the next section, we are going to show that

χ(ab) = χ(a)χ(b)

whenever a and b are commuting involutions in Cϕ, and that χ(a) depends only on theimage of a in Cϕ/C

0ϕ = Aϕ. We assume this for the rest of this section. This gives the

desired character (up to the choice of µ)

χ : Aϕ → 〈±1〉.From (7.2), we obtain,

Corollary 7.1. If W = V a=−1 has even dimension, then

χ(a) = ε(IndW ) detW (e)ωK/k(−1)12

dimW .

In this case, the value χ(a) is independent of the choice of the auxiliary characterµ : K×/NK× → C×.

In the situation of Corollary 7.1, we can find an equivalent formula for χ(a) which involvesonly local root numbers. Indeed, let W be a representation of W ′(K) satisfying (7.1) withc = −1, and assume that dimW ≡ 0 mod 2. Then detW is a character of K×/k× andInd(W ) is symplectic.

Let ψ0 be a non-trivial additive character of K which is trivial on k. Then

ψ0(x) = ψ ◦ Tr(ex)

for a non-trivial additive character ψ of k. Let dx0 be the unique Haar measure on K whichis self-dual with respect to ψ0, and define

ε0(W ) = ε(W,ψ0, dx0,1

2).

Using formal properties of local constants, as well as their inductivity in dimension 0, wecan show

ε0(W ) = ε(IndW ) detW (e)ωK/k(−1)12

dimW .(7.4)

Hence the formula in Corollary 7.1 becomes

χ(a) = ε0(Va=−1).


Before giving the proof that χ(a) depends only on the image of the involution a inAϕ = Cϕ/C

0ϕ, we present some other formulae for χ(a) using the decomposition

a = a1 × a2 in Cϕ = C1 × C2,

where C1 ⊂ GL(V1) is the centralizer of ϕK,1, and C2 ⊂ GL(V2) is the centralizer of ϕK,2.Since V = V1 ⊗ V2 is a tensor product, we have

V a=−1 = (V a1=−11 ⊗ V a2=1

2 )⊕ (V a1=11 ⊗ V a2=−1

2 )

= (V a1=−11 ⊗ V2)⊕ (V1 ⊗ V a2=−1

2 )− 2(V a1=−11 ⊗ V a2=−1

2 ).

Since all of these representations have ε and det invariants in ±1, which are homomorphismson the Grothendieck group, we find the (4 term) formula:

χ(a1 × a2) = ε(Ind(V a1=−11 ⊗ V2)) · det(V a1=−1

1 ⊗ V2 ⊗ µ)(e)

ε(Ind(V1 ⊗ V a2=−12 ) · det(V1 ⊗ V a2=−1

2 ⊗ µ)(e).(7.5)

We have the congruences:

dimV1 ≡ 0 mod 2

dimV2 ≡ 1 mod 2

dimV a1=−11 ≡ dimV a=−1 mod 2.

If dimV a=−1 is even, we have the following formula for (7.5), using the argument of (7.4)

χ(a1 × a2) = ε0(Va1=−11 ⊗ V2)ε0(V1 ⊗ V a2=−1

2 ).(7.6)

If we expand the determinants in formula (7.5), we obtain a 6-term formula:

χ(a1 × a2) = ε(Ind(V a1=−11 ⊗ V2)) · det(V a1=−1

1 ⊗ µ)(e)dimV2 detV2(e)dimV

a1=−11

ε(Ind(V1 ⊗ V a2=−12 )) · det(V1 ⊗ µ)(e)dimV

a2=−12 detV a2=−1

2 (e)dimV1(7.7)

Since detV a2=−12 (e) = ±1, we find the last term

detV a2=−12 (e)dimV1 = 1.

Also the third term is trivial when detV2(e) = 1. When detV2(e) = −1, it is given by thecharacter

η1(a1) = (−1)dimVa1=−11 .


8. χ is a character of Aϕ

Let ϕ be a Langlands parameter for U(W ) × U(W0), and let ϕK : W ′K → GL(V1) ×

GL(V2) be its restriction to the Weil-Deligne group ofK. We assume V1 has even dimensionand V2 has odd dimension, and define the representation

V = V1 ⊗ V2.

For an involution a = a1 × a2 in the centralizer Cϕ = C1 × C2 of ϕ in GL(V1)×GL(V2),we have defined the sign

χ(a) = ε(Ind(V a=−1) · det(V a=−1 ⊗ µ)(e)

in (7.3). This definition uses an auxiliary character µ : K× → C× with Res(µ) = ωK/k onk×, but when V a=−1 has even dimension, the sign χ(a) is independent of the choice of µ.

In this section, we will show that χ(a) depends only on the image of a in the quotientgroup Aϕ = Cϕ/C

0ϕ, and that the resulting function on Aϕ is a homomorphism

χ : Aϕ → 〈±1〉.We will also describe the dependence of the character χ on the choice of µ.

When the involutions a and b in Cϕ commute, we have the identity

χ(ab) = χ(a)χ(b) in 〈±1〉.This follows from the identity,

V ab=−1 = V a=−1 + V b=−1 − 2Va=−1b=−1

in the representation ring of W ′K . Since commuting involutions generate the cosets of C0

ϕ

in Cϕ, it suffices to show that χ(a) depends only on the image of a in Aϕ.Using the description of Cϕ = C1 × C2 in §3, we see that there are 6 cases to check:

(1) a = a1 × 1 with a1 ∈ GL(n).(2) a = a1 × 1 with a1 ∈ Sp(2n).(3) a = a1 × 1 with a1 ∈ O(n).(4) a = 1× a2 with a2 ∈ GL(n).(5) a = 1× a2 with a2 ∈ Sp(2n).(6) a = 1× a2 with a2 ∈ O(n).

In cases (1), (2), (4), (5) we must show that χ(a) = 1. In cases (3) and (6) we need toshow that χ(a) is a function of det(a1) or det(a2) respectively.

To check this, we will use the formulae,

χ(a1 × 1) = ε(Ind(V a1=−11 ⊗ V2)) · det(V a1=−1

1 ⊗ µ)(e)dimV2 det(V2)(e)dimV

a1=−11

χ(1× a2) = ε(Ind(V1 ⊗ V a2=−12 ) · det(V1)(e)

dimVa2=−12 .

In cases (1) and (2), we have dimV a1=−11 ≡ 0 mod 2, so

χ(a1 × 1) = ε(Ind(V a1=−11 ⊗ V2)) · det(V a1=−1

1 )(e)ωK/k(−1)12

dimVa1=−11 .


In cases (4) and (5), we have dimV a2=−12 ≡ 0 mod 2, so

χ(1× a2) = ε(Ind(V1 ⊗ V a2=−12 )).

(1) Let 0 ≤ p ≤ n be the number of −1’s in a1. Then,

V a1=−11 = p(W +W ′)

where W is the irreducible summand of V1 and W ′ = (W s)∨. Hence the ε-factoris,

ε(Ind(W ⊗ V2)) + Ind(W ′ ⊗ V2))p = det(Ind(W ⊗ V2))(−1)p

=(det(W ⊗ V2)(−1) · ωK/k(−1)dimW⊗V2

)p=

(detW (−1) · ωK/k(−1)dimW

)p.

The other terms in χ(a) are

detW · detW ′(e)p = detW (−1)p(ωK/k(−1)dimW

)p.

Hence χ(a) = 1.(2) Suppose −1 appears with multiplicity 2p, with 0 ≤ p ≤ n in a1, and

V a1=−11 = 2pW,

with W an irreducible summand of V1 of type 2. The ε-factor is

ε(Ind(W ⊗ V2))2p = det(Ind(W ⊗ V2))(−1)p

= det(W ⊗ V2)(−1)p · ωK/k(−1)p·dim(W⊗V2)

= detW (−1)p · ωK/k(−1)pdimW .

Again this cancels with other factors in χ(a).(3) Let p be the number of −1’s in a1, so a1 ∈ SO(n) if and only if p is even. Here,

V a1=−11 = p ·W

where W is an irreducible summand of V1 of type 3. The ε-factor is

ε(Ind(W ⊗ V2))p,

which depends only on the parity of p, and is 1 if p is even. The other factors are,

det(W ⊗ µ)(e)p,

det(V2)(e)pdimW .

Again these depend only on the parity of p, and are 1 when p is even.(4) The proofs in cases (4), (5), (6) are similar, and we leave them to the reader.


What is the dependence of the character χ of Aϕ on the choice of µ? If

µ′ = γ · µ,is another choice, with γ : K×/k× → C×, we find

χ′(a) = γ(e)dimV a=−1 · χ(a)

= γ(e)dimV a1=−1 · χ(a1 × a2).

Hence χ′ is either equal to χ in Hom(Aϕ,±1), or to the character χ · (η1 × 1), where

η1 : A1 : → 〈±1〉a1 7→ (−1)dimV a1=−1

is the generic character defined in §3.

9. Alternative Description of Recipe

The recipe for the character χ given in the refined local conjecture of the previous sectionsis along the lines of that given in [GP1] for the orthogonal groups. In this section, we willgive an alternative description of the recipe which is perhaps more transparent. The recipegiven in this section also depends on the choice of a character µ : K× → C× as in the lastsection.

We begin by giving this alternative description in the orthogonal case.

Orthogonal groups

We reinterpret the recipe in the paper [GP1] which, to a pair (Σ1,Σ2) consisting of anorthogonal parameter

Σ1 : W ′k → O(2m,C),

and a symplectic parameter

Σ2 : W ′k → Sp(2n,C),

attaches a character

εΣ1,Σ2 : AΣ1 × AΣ2 → {±1}.Here, AΣ1 and AΣ2 are the component groups of Σ1 and Σ2 respectively. We recall thatthe component group AΣ1 (resp. AΣ2) is a free abelian group over Z/2, with basis indexedby the distinct irreducible representations σ1 (resp. σ2) of W ′

k appearing in Σ1 (resp. Σ2)which are orthogonal (resp. symplectic).

We first define a tentative character εtentΣ1,Σ2

on the free abelian group over Z/2 generatedby such σ1’s and σ2’s. This is given by:{

εtentΣ1,Σ2

(σ1) = ε(σ1 ⊗ Σ2, ψ);

εtentΣ1,Σ2

(σ2) = ε(Σ1 ⊗ σ2, ψ).


We shall need to modify the tentative character εtentΣ1,Σ2

, which is supposed to detect thenon-vanishing of certain linear forms in the Gross-Prasad conjecture, so that it is manifestlytrivial when it is supposed to be. For this we employ the following heuristic, which shouldbe possible to prove, but we have not managed to do so:

(1) An irreducible generic principal series representation of a split orthogonal groupSO(n+ 1) always contains the generic member of a generic L-packet of SO(n) asa quotient.

(2) The generic member of a generic L-packet of SO(n + 1) contains an irreducibleprincipal series representation of the split SO(n) as a quotient.

This heuristic suggests that we define, εΣ1,Σ2 by the following modifications:{εΣ1,Σ2(σ1) = ε(σ1 ⊗ Σ2)ε(σ1 ⊗ Ps2)

εΣ1,Σ2(σ2) = ε(Σ1 ⊗ σ2)ε(Σ1 ⊗ ps2).

where Ps2 is the parameter of a principal series representation on SO(2n+ 1), and ps2 isthe parameter of a principal series representation on SO(2d2 + 1) where σ2 has dimension2d2.

With this modification, εΣ1,Σ2 becomes the trivial character when it is expected to be.Moreover, we have:

Lemma 9.1. The character εΣ1,Σ2 defined above agrees with the character defined in[GP1].

Remark : For comparing the character defined here with that in [GP1], note that for aprincipal series representation of SO(2n+1) with parameter ps = τ⊕τ∨ with det τ(−1) =1,

ε(σ ⊗ ps) = detσ(−1)n.

Unitary groups

Now we come to the unitary case, which is the subject matter of this paper. Let σ bean irreducible representation of W ′

K :

σ : W ′K → GLm(C),

where K is a quadratic extension of a local field k such that

σ ∼= σ∨

where σ denotes the representation of W ′K obtained by conjugating σ by an element of

W ′k rW ′

K . Then one knows that one of the following holds:


(1) m is even, in which case detσ is trivial on k×, and IndW ′

K

W ′kσ is a symplectic repre-

sentation of W ′k;

(2) m is odd, in which case detσ is trivial restricted to k× if and only if c(σ) = 1; ifdetσ is nontrivial, it is ωK/k.

We next turn to the local root numbers, where we will only be interested in representationsσ with σ ∼= σ∨. Let ψ be a non-trivial character on K which is trivial on k.

Lemma 9.2. For a representation σ of W ′K with σ ∼= σ∨, character χσ of K× such

that χσ = detσ on k×, and ψ a nontrivial character on K which is trivial on k,

ε(σ, ψ) · ε(χσ, ψ) = ±1,

and is independent of ψ.

Proof : This follows from the fact that such characters ψ form a principal homogeneousspace over k×, and the following generalities:

(1) ε(σ, ψa) = (detσ)(a) · ε(σ, ψ).

(2) ε(σ, ψ) = ε(σ, ψ).

(3) ε(σ, ψ) · ε(σ∨, ψ−1) = 1.

Remark : By a theorem due to Frohlich and Queyrut, for a character ψ of K which istrivial on k, ε(χσ, ψ) = 1 if χσ is trivial on k×, and therefore this can be taken to be a‘correction’ term introduced to make ε(σ, ψ) independent of ψ.

Let V1 be an even dimensional Hermitian space over K and V2 an odd dimensionalHermitian space over K. Let U(V1) and U(V2) be the corresponding unitary groups over k,and let ϕ1 and ϕ2 be generic Langlands parameters for U(V1) and U(V2). Write Σ1 and Σ2

for the restriction of ϕ1 and ϕ2 to W ′K . As described in Section 3, the component group

Aϕ1 (resp. Aϕ2) is an elementary abelian 2-group, with basis given by the set of irreducibleconjugate-self-dual representations σ1 (resp. σ2) of W ′

K appearing in Σ1 (resp. Σ2) whichhave the further property that c(σ1) = −1 (resp. c(σ2) = 1).

Let’s note that

det(Σ1 ⊗ σ2) = det(Σ1)dimσ2 · det(σ2)

dim Σ1 = ν

with ν a character of K× with trivial restriction to k× as Σ1 is even dimensional of trivialdeterminant restricted to k×. Also,

det(σ1 ⊗ Σ2) = det(σ1)dim Σ2 · det(Σ2)

dimσ1 = ν ′

with ν ′ a character of K× whose restriction to k× is the same as the restriction of detσ1

to k× as Σ2 is odd dimensional of trivial determinant restricted to k×.We define a character εtent

ϕ1,ϕ2on Aϕ1 × Aϕ2 , which has basis given by the σ1’s and σ2’s

as above, by


{εtentϕ1,ϕ2

(σ1) = ε(σ1 ⊗ Σ2, ψ)ε(χσ1 , ψ)

εtentϕ1,ϕ2

(σ2) = ε(Σ1 ⊗ σ2, ψ),

where χσ1 is the trivial character of K× if detσ1 is trivial restricted to k×, and is µ if detσ1

is non-trivial restricted to k×. (We recall having fixed a character µ : K× → C× in thelast sections whose restriction to k× is ωK/k.)By the lemma above, εtent

ϕ1,ϕ2is independent of the choice of the additive character ψ of K

trivial on k. Now, as in the orthogonal case, let us see if we need to modify our recipebased on the following suggestions about principal series representations of unitary groups.

(1) A principal series representation of the quasi-split group U(2n) contains the f -generic member of a generic L-packet on U(2n− 1).

(2) An irreducible principal series representation of the quasi-split group U(2n) is con-tained in the f -generic member of an L-packet on U(2n+ 1).

Remark : It is worth noting that these statements about principal series representationsof U(n) are sensitive to the parity of n. For example it is not true that principal seriesrepresentation of U(3) contains the generic member of an L-packet on U(2), for example,because the U(2) considered might be compact!

We now calculate our character εtentϕ1,ϕ2

for the case when ϕ1 corresponds to an irre-ducible principal series representation for U(2d). Recall that the maximal torus of the Borelsubgroup B in U(2d) can be taken to be K× × · · · ×K×, and thus a principal series rep-resentation on U(2d) is parametrized by a d-tuple of characters (χ1, · · · , χd) of K×. Theparameter ϕ1 of the corresponding principal series representation of U(2d) has restrictionto W ′

K given by:

Σ1 = χ1 ⊕ · · · ⊕ χd ⊕ χ−11 ⊕ · · · ⊕ χ−1

d = τ ⊕ τ∨.

We now calculate the local root numbers using the following lemma.

Lemma 9.3. For a representation ϕ of W ′K,

ε(ϕ, ψ) · ε(ϕ∨, ψ) = 1.

Proof: This follows from generality about epsilon factors:

ε(ϕ, ψ) · ε(ϕ∨, ψ) = ε(ϕ, ψ) · ε(ϕ∨, ψ)

= ε(ϕ, ψ) · ε(ϕ∨, ψ−1)

= detϕ(−1) · ε(ϕ, ψ) · ε(ϕ∨, ψ)

= 1.


Note now that the determinant of a representation of the form π ⊗ (τ ⊕ τ∨) with π ∼= π∨

is of the form χ/χ, and by the theorem of Frohlich-Queyrut, ε(χ/χ, ψ) = 1. Therefore forϕ1 as above, we have:

εtentϕ1,ϕ2

= 1.

This means that unlike the orthogonal case, we do not need any modification to εtentϕ1,ϕ2

, andwe simply set

εϕ1,ϕ2 = εtentϕ1,ϕ2

.

The character εϕ1,ϕ2 is the same as the one described in the refined local conjectureexcept for the possible ambiguity between χ and χ · (η1 × 1).

10. Unramified parameters

In this section, we assume that k is non-Archimedean. We determine the componentgroups Aϕ and the character χ of Aϕ for parameters ϕ whose restriction to W ′

K

ϕK : W ′K → GL(V0)×GL(V1)

is unramified. We call these unramified parameters –although the corresponding L-packetsπ contain unramified representations only when the quadratic extension K/k is also un-ramified.

First consider a single unitary group U(W ), where the split Hermitian space W hasdimension n over K. Then an unramified parameter is completely determined by theconjugacy class of the semi-simple element

ϕK(Fr) =

α1

······αn

in GL(V ) = GLn(C). The irreducible submodules Vi all have dimension 1; since ord(x) =ord(sx) for x ∈ K×, we have V s

i∼= Vi for all i. Hence an analysis of the types of

submodules, as in section 3, shows that either α2i = 1, or there is an eigenvalue αj with

αi · αj = 1. Moreover, the eigenvalues αi and αj occur with the same multiplicity in V .The component group Aϕ is determined by the multiplicities of the eigenvalues αi = ±1.

Assume first that K/k is unramified. Then the eigenvalue −1 corresponds to the un-ramified quadratic character µ of K∗, which satisfies Ver(µ) = ωK/k. The eigenvalue 1corresponds to the trivial character of K∗, which has Ver(1) = 1. These are the only un-ramified characters of K× which are trivial on NK×. Since detV is an unramified characterwith Ver(detV ) = 1, we must therefore have detV = 1.


Proposition 10.1. When K/k is unramified, the eigenvalue −1 occurs with even mul-tiplicity in V , and the eigenvalue 1 occurs with multiplicity ≡ n mod 2 in V . Wehave:

(1) When n is odd, the component group Aϕ = Z/2, and is generated by the imageof −1 in Z(LG).

(2) When n is even, the component group is given by

Aϕ =

{1 if the multiplicity of − 1 is zero in V

Z/2 if − 1 occurs as an eigenvalue in V,

and the image of −1 in Z(LG) is trivial in Aϕ.

Next assume that K/k is ramified. Then the only unramified characters of K× whichare trivial on NK× are again the trivial and quadratic character. In this case, both aretrivial on k×, so there are no unramified characters µ with Ver(µ) = ωK/k.

Proposition 10.2. When K/k is ramified and n is even, the eigenvalues ±1 bothappear with even multiplicity in V . Hence detV = 1, and Aϕ = 1.

If n is odd, one of the eigenvalues ±1 appears with even multiplicity and the otherappears with odd multiplicity in V , depending on the character detV , with (detV )2 = 1.We have

Aϕ =

{Z/2 if one of these eigenvalues appears in V

(Z/2)2 if both do

We now consider generic unramified parameters for the group G = U(W )×U(W0), andinvestigate the character χ of Aϕ = A1 × A2. Recall that χ depends, up to multiplicationby η1×1, on the choice of an auxiliary homomorphism µ : K× → C× with Ver(µ) = ωK/k.When A1 = 1, the character χ is well-defined. This is the case when K/k is ramified. Inthe unramified case, we have

Proposition 10.3. Assume that K/k is unramified and that ϕ is an unramified genericparameter for G. If we choose µ to be the unique unramified character of K× withVer(µ) = ωK/k, then χ = 1.

Proof : We have A1 of order 1 or 2, and A2 is generated by the image of −1 in the center.Since −1 has trivial image in A1, and χ(−1, 1) = χ(1,−1), we see that χ is trivial on thesubgroup 1×A2. It suffices to show it is trivial on the subgroup A1×1. If A1 is non-trivial,a basis element a corresponds to the summand of V1 which is the −1 eigenspace. Then

V a=−11 = µ, detV2 = 1,

and

χ(a, 1) = ε(Ind(µ⊗ V2))µ2(e).

Since the epsilon factor of an unramified representation is 1, and µ2 = 1, we have χ = 1.


The determination of χ (for a fixed choice of µ) in Proposition 10.3 does not specify therepresentation π in the L-packet with HomH(π,C) 6= 0, as we have not chosen a genericcharacter. Instead, we will specify π using spherical vectors. There are two conjugacyclasses of hyperspecial maximal compact subgroup in G(k) = U(W0)×U(W ), but there isa unique H-conjugacy class of such J ⊂ G(k) with the additional property that H(k)∩J ishyperspecial in H(k). If W⊥

0 = 〈en〉 and we have chosen an orthogonal basis 〈e1, · · · , en−1〉for W0 with 〈ei, ei〉 = 〈en, en〉 for all i, then J is the subgroup of U(W0) × U(W ) whichstabilizes the lattices L0 = ⊕i=n−1

i=0 Akei and L = ⊕i=ni=0Akei in W0 and W respectively. See§18 for the discussion of all of the maximal compact subgroups in G(k).

Conjecture 10.4. The unique representation π of G in the L-packet of a generic,unramified parameter with HomH(π,C) 6= 0 is J-spherical: πJ 6= 0. Moreover, thepairing HomJ(C, π)× HomH(π,C)→ C is non-degenerate.

11. Discrete series parameters

We now consider certain discrete series L-packets for G = U(W )×U(W0). In this case,all the irreducible factors Vi in the Galois representations V1 and V2 are of type 3, and allhave multiplicity mi = 1. We will assume further that each irreducible factor has dimension1, so

V1 =⊕

V (αi)

V2 =⊕

V (βi)

where each αi is a character of K×/NK× with Ver(αi) = ωK/k, and each βi is a characterof K×/k×. Since the multiplicities are 1, all of these characters are distinct, and we havethe component groups

A1 = (Z/2)dimV1

A2 = (Z/2)dimV2

as large as possible.These are the parameters of the discrete series representations for k = R, and include

the tamely ramified elliptic parameters when k is non-Archimedean, K/k is unramified, andFrobenius maps to −1 in the extended Weyl group (Sn × Sn−1) · Gal(K/k). The groupAϕ has basis

〈e1, · · · , edimV1 ; f1, · · · , fdimV2〉with

V ei=−11 = V (αi)

V fi=−12 = V (βi).

Choosing an auxiliary character µ : K× → C× with Ver(µ) = ωK/k, we find the formulae:


χ(ei) =∏k

ε(Indαiβk) · αiµ(e) ·∏k

βk(e)

χ(fi) =∏k

ε(Indαkβi) ·∏k

αk(e) · εk(−1)12

dimV1 .

From this it follows that

χ(ei)χ(ej) =∏k

ε(Indαiβk)ε(Indαjβk) · ωK/k(−1) · αiαj(e)

χ(fi)χ(fj) =∏k

ε(Indαkβi)ε(Indαkβj)

independent of the choice of µ.We now need a formula for the symplectic root number of the two dimensional represen-

tations

Ind(αβ)

where α and β are characters of K× with Ver(α) = ωK/k, Ver(β) = 1. We achieve this inthe following proposition when k is non-Archimedean.

Proposition 11.1. Let k be non-Archimedean, K/k is unramified and α and β tamelyramified characters of K×. Then α · β = γ · µ, where γ is a tamely ramified characterof K×/k× and µ is the unramified quadratic character of K× (with Ver(µ) = ωK/k).Then:

(1) If γ = 1, ε(Indαβ) = 1.(2) If γ 6= 1, ε(Indαβ) = −γ(e).

Corollary 11.2. Order the characters αi and βi in the parameter ϕ so that

α1 · β1 = α2 · β2 = · · · , αp · βp = µ,

and for no other products, αiβj = µ. Then

χ(ei)χ(ej)χ(fi)χ(fj)

}=

{1 if i, j ≤ p or i, j > p−1 otherwise.

Moreover, χ(−1, 1) = χ(1,−1) = (−1)p.

Proof. We have ε(Indγ) = γ(e) by the formula of Frohlich-Queyrut [F-Q]. Since µ isunramified, we find that

ε(Ind(γµ)) = µ(fγ) · γ(e)with fγ the conductor of γ, which is zero when γ = 1, and one when γ 6= 1, proving theproposition. �


To prove the corollary, write

αiβj = α0iβj · µ

where α0i is the restriction to the units, viewed as a character of UK/Uk = K×/k×. We

have ordered our characters so that

γij = α0iβj = 1

precisely when i = j ≤ p.For i, j ≤ p we find that

χ(ei)χ(ej) =∏k 6=i

−α0iβk(e)

∏k 6=j

−α0iβk(e) · αiαj(e)

= βj(e)βk(e) · αiαj(e) = 1

as the products have an even number of terms.For i, j > p, we find,

χ(ei)χ(ej) =∏k

−α0iβk(e)

∏k

−α0jβk(e) · αiαj(e)

= α0i (e)α

0j (e) · αiαj(e) = 1

as the products have an odd number of terms.For i ≤ p, and j > p, we find,

χ(ei)χ(ej) =∏k 6=i

−α0iβk(e)

∏k

−α0jβk(e) · αiαj(e)

= −α0j (e)βi(e) · αiαj(e) = −1.

Similar results hold good for i > p and j ≤ p, as well as for χ(fi)χ(fj), completing theproof of the corollary.

Now assume k = R, and K = C. Then e = i satisfies e2 = −1, and the characters αand β are given by the formulae

α(z) = (z/z)α = z2α/(zz)α α ∈ 12Z− Z

β(z) = (z/z)β β ∈ Z.

In particular, β(e) = (−1)β.

Proposition 11.3. We have the formulae:

ε(Ind(αβ)) =

{(−1)α+β+ 1

2 if α+ β > 0

−(−1)α+β+ 12 if α+ β < 0


To calculate χ(ei)χ(ej) and χ(fi)χ(fj), we order the characters αi and βi in the param-eter ϕ so that

α1 > α2 > α3 · · · in 12Z− Z

β1 > β2 > β3 · · · in Z.

Corollary 11.4. For i < j, we have

χ(ei)χ(ej) = (−1)#{k:αi+βk>0>αj+βk}

χ(fi)χ(fj) = (−1)#{k:βi+αk>0>βj+αk}.

Proof. The formula in the Proposition follows from Tate [Ta]. To obtain the formula inthe corollary, we recall that

χ(ei)χ(ej) =∏k

ε(Indαiβk)ε(Indαjβk) · µ2αiαj(e).

If αi + βk and αj + βk have the same sign, the kth term in the product is

(−1)αi+βk+αj+βk+1 = −(−1)αi+αj .

Otherwise, it is the negative of this. Since there are an odd number of terms k, the productis equal to

−(−1)αi+αj · (−1)#{k:αi+βk>0>αj+βk}.

Since µ2αiαj(e) = −(−1)αi+αj , this gives the first result.Similarly,

χ(fi)χ(fj) =∏k

ε(Indαkβi)ε(Indαkβj).

If βi + αk and βj + αk have the same sign, the kth term in the product is

(−1)2αk+βi+βj+1.

Otherwise, it is the negative of this. Since there are an even number of terms k in theproduct, we find

χ(fi)χ(fj) = (−1)P

2αk · (−1)#{k:βi+αk>0>βj+αk}.

But (−1)P

2αk = detV (−1)2. Since detV has trivial transfer,∑αk is an integer and

detV (−1)2 = 1, completing the proof of the corollary.�

Since we know how to describe the representations in the L-packets of discrete seriesparameters, the calculation of χ(ei)χ(ej) and χ(fi)χ(fj) allows us to say something aboutthe representation π = π(ϕ, χ) of G(k) with the (conjectural) H(k)-invariant linear form.We write π = π1 ⊗ π2 following our notation V = V1 ⊗ V2. In the case when k = R, π1

and π2 are discrete series representations of even and odd dimensional unitary groups, withinfinitesimal characters

α1 > α2 > · · ·β1 > β2 > · · ·


in X? + ρ respectively. Moreover, in the chambers defined by their Harish-Chandra param-eters, the simple root walls corresponding to

ei − ei+1 is compact ←→ χ(ei)χ(ei+1) = −1

fi − fi+1 is compact ←→ χ(fi)χ(fi+1) = −1.

More generally, the root

ei − ej is compact ←→ χ(ei)χ(ej) = (−1)i+j

fi − fj is compact ←→ χ(fi)χ(fj) = (−1)i+j.

This determines G(k), and in almost all cases π.For example, if

α1 > −βd > α2 > −βd−1 > · · · ,we find that G(R) = U(n)×U(n− 1) is compact and π is finite dimensional, determinedby its infinitesimal character. In this case, χ(ei)χ(ej) = χ(fi)χ(fj) = (−1)i+j for all i, j.

In the p-adic case, the representation π in the L-packet of ϕ all have the form

π = IndGM(R)

where M is a compact parahoric subgroup containing the unramified elliptic torus T =U(1)2n−1 ⊂ U(W ′) × U(W ′

0), and R is a Deligne-Lusztig representation of the reductivequotient M(q) of M . The cuspidal representation R = R(T, ϕ) of M(q) corresponds tothe tame characters α and β in the parameter ϕ, which give characters of T (q) by localclass field theory.

Our calculation of χ determines the root system of M(q) over Fq2 : ei − ej and fi − fjappear as roots if χ(ei)χ(ej) = χ(fi)χ(fj) = 1. From the corollary, we deduce that thederived group of M(q) is isomorphic to

(SU(p)× SU(n− p))× (SU(p)× SU(n− 1− p)) .Moreover, the representation R of M(q) has the form

(R1 ⊗R2)⊗ (R1 ⊗R′2),

where the characters αi and βi giving the representations R2 and R′2 of Un−p(q) and

Un−p−1(q) are all distinct.

12. The Global Conjecture

We now assume that k is a global field, with char(k) 6= 2. Let A denote the ring of adelesof k. Let K be a quadratic field extension of k. We write K = k+ ke with Tr(e) = 0, andfix an auxiliary Hecke character µ : A×

K/K× → C× with

Ver(µ) = ωK/k : A×/(NA×K · k

×)→ 〈±1〉.Let W0 ⊂ W be a pair of split Hermitian spaces of dimensions n − 1 and n. Let G =

U(W )× U(W0) be the corresponding quasi-split group over k, with diagonally embedded


subgroup H = U(W0). We fix a generic homomorphism f : NG → Ga, and composewith a character of A/k to get a generic character ψ : NG(A)→ C×. Finally, let π be anautomorphic, generic representation of G(A) which is tempered and appears in the discretespectrum of G.

We write π = ⊗πv as a restricted tensor product, where each πv is a ψv-generic irreduciblerepresentation of the local group G(kv). Let ϕv be its Langlands parameter, and let π′v bethe unique representation in the generic L-packet of ϕv with

HomH′v(π′v,C) 6= 0.

If v splits in K, then G(kv) = GLn(kv)×GLn−1(kv), and π′v = πv. If v is not split in K,so Kv is a field, we define the local character

χv : Aϕv → 〈±1〉

using the auxiliary character µv and Proposition 7.1. Our local conjecture predicts that π′vcorresponds to either χv or its product with (η1 × 1).

In any case, π′v is an irreducible representation of the group G′v = U(M ′

v) × U(M ′0,v),

where M ′0,v ⊂M ′

v is a pair of Hermitian spaces over Kv with M ′v/M

′0,v∼= (M/M0)⊗KKv.

A necessary condition for M ′v∼= M ⊗K Kv, is that

χv(−1, 1) = χv(1,−1) = 1.

When v is non-Archimedean, this is also sufficient.The representation πv and the local character µv are both unramified for almost all places

v. At these places, π′v = πv of G′v = G(kv), and χv = 1. This allows us to define the

tensor product representation

π′ = ⊗vπ′vof the locally compact group

G′A =

∏v

G′v.

The representation π′ is nearly equivalent to the generic representation π. It is the uniquerepresentation in this near equivalence class with

HomH′(A)(π′,C) 6= 0.

There are now three basic questions to address, each depending on the previous answer.

1. Is G′A the adelic points of a group G′ = U(M ′)× U(M ′

0) defined over k, associated toa pair of Hermitian spaces M ′

0 ⊂M ′ with M ′/M ′0∼= M/M0?

2. If the answer to 1. is yes, so G′A = G′(A) contains the discrete subgroup G′(k), does the

tempered representation π′ of G′(A) appear with multiplicity one in the discrete spectrumof G′?


3. If the answer to 2. is yes, so π′ embeds uniquely up to scaling in a space of cuspidalfunctions f on G′(k)\G′(A), is the H ′(A)-linear form on π′:

f →∫H′(k)\H′(A)

f(h) dh

non-zero in the one dimensional vector space HomH′(A)(π′,C) 6= 0?

We will provide conjectural answers to these questions in order, using

(1) the global root number ε(π, V ), associated to π and the symplectic representationV = Ind(V1 ⊗ V2) of the L-group of G of dimension 2n(n− 1).

(2) the collection of global root numbers ε(π, V a=−1), for elements a in the centralizerof the global Langlands parameter.

(3) the central critical L-value L(π, V, 12).

To answer 1., let d(M ′v) be the Hermitian discriminant of the space M ′

v in k×/NK×v . In

the non-Archimedean case, this invariant determines M ′v up to isomorphism. We have the

formula:(d(M ′

v), D)v = χv(−1,−1)(d(M), D)vwhere D is the discriminant of K/k and (, )v is the local Hilbert symbol.

The collection of local spaces (· · · ,M ′v, · · · ) comes from a global Hermitian space M ′ if

and only if the image of (· · · , d(M ′v), · · · ) is trivial in the group

A×/(NA×K · k

×) ∼= Gal(K/k).

Since∏

v(d(M), D)v = 1 by global reciprocity, a global space M ′ exists precisely when∏v

χv(−1, 1) = 1.

But our formulae for χv evaluated at (−1, 1) give,

χv(−1, 1) = ε(πv, V )detv(V1 ⊗ V2)(e)εKv(−1)12

dim(V1⊗V2).

Since ∏v

detv(V1 ⊗ V2)(e) =∏v

εKv(−1) = 1,

by global reciprocity, we see that a global space M ′ exists with localizations M ′v if and only

if ∏v

ε(πv, V ) = ε(π, V ) = 1.

In this case, the subspace M ′0 also exists globally, and is characterized by M ′/M ′

0∼= M/M0.

Hence the group H ′ ↪→ G′ exists over k when ε(π, V ) = 1. To determine if the rep-resentation π′ appears in the discrete spectrum, we use a conjectural multiplicity formulaof Langlands and Arthur. Let ϕ : Lk → LG be the global Langlands parameter of thetempered automorphic representation π, and let Aϕ be the component group of its central-izer. This is again an elementary abelian 2-group, determined by the irreducible submodulesVi of V1 ⊗ V2, which maps to the local groups Aϕv for all places v. The representation


π′ = ⊗π′v should appear in the discrete spectrum with multiplicity 1 if and only if thecharacter χ : Aϕ → 〈±1〉 defined by

χ(a) =∏v

χv(a)

is trivial. When dimV a=−1, we can use the formula for χv(a) and global reciprocity toshow that this is equivalent to the condition:∏

v

ε(πv, Va=−1) = ε(π, V a=−1) = 1.

If ε(π, V a=−1) = 1 for all a in the centralizer of the parameter, there is no obvious reasonfor the global L-function L(π, V, s) to vanish at the central critical point s = 1

2. We then

make the main global conjecture.

Conjecture 12.1. Assume that ε(π, V a=−1) = 1 for all a in Aϕ. Then the adelicrepresentation π′ appears with multiplicity 1 in the discrete spectrum of G′, and theperiod integrals

f →∫H′(k)\H′(A)

f(h) dh

of functions f in π′ give a non-zero H ′(A)-invariant linear form if and only if L(π, V, 12) 6=

0.

The recent work of Ginzburg-Jiang-Rallis [GJR] gives definitive progress towards thisglobal conjecture.

Notes (1) As in [Ichino-Ikeda], one can further conjecture an exact formula relating theseperiods and the central critical value.

(2) When ε(π, V ) = −1, the collection of local Hermitian spaces M ′0,v ⊂ M ′

v does notarise from a global space. However, in the situation where k is a totally real number fieldand the local spaces M ′

v are definite at all real places of k, one should be able to usethe arithmetic geometry of unitary Shimura varieties of rank 1 to study the central criticalderivative L′(π, V, 1

2) using the framework of [G].

Our conjectures relate central critical L-values of symplectic representations (i.e., forwhich the exterior-square L-function has a pole) to certain period integrals. It seemsreasonable to expect that these are the only L-functions which can vanish at the center ofthe critical strip. We state it explicitly in the following.

Conjecture 12.2. Let Π be a cuspidal automorphic representation of GLn(AQ) withunitary central character. Suppose

L(1

2,Π) = 0.


Then,Π ∼= Π∨,

and Π is a symplectic automorphic representation, i.e.,

L(s,2∧,Π)

has a pole at s = 1.

Remark 1. Considerations of Artin representations of the Galois group suggest that it isessential to have Q in the above conjecture, and which is of course no loss of generalitywhen dealing with L-functions. We have not found any explicit reference to such a preciseconjecture even for Dirichlet characters besides the well-known conjecture about quadraticcharacters of Q, which is, L(1

2, ω) 6= 0 for any character ω : A×

Q/Q× → ±1.

Part 2. EVIDENCE

In this part of the paper, we verify our conjectures about branching laws from U(2)to U(1), as well as for those representations of U(3) restricted to U(2) whose parameterrestricted to K becomes reducible. We should add that even for these small groups, sincethere is no natural labelling of representations in an L-packet through characters of thecorresponding component group, our verification will be for the form of the conjectures in6.1 and 6.2, and not the more precise one formulated in sections 7,8,9.

13. U(1,1)

The group U(1, 1) is a small variation on GL2, and our conjectures 6.1 and 6.2 reduceto known results in this case. We discuss this here.

Let U(1, 1) be the unitary group defined using the Hermitian form(0 11 0

).

If GU(1, 1) denotes the unitary similitude group, then it is easy to see that

GU(1, 1) ∼= [GL2(k)×K×]/∆k×,

where ∆k× sits inside GL2(k) as the scalar matrices, and inside K× as t → t−1. Thesimilitude character on GU(1, 1) in its identification as [GL2(k) × K×]/∆k× is the de-terminant character on GL2(k), and the norm on K×. Define GL+

2 (k) be the subgroupof GL2(k) consisting of elements of GL2(k) with determinant in NK×. It follows thatU(1, 1) is contained inside G = [GL+

2 (k) × K×]/∆k×. Clearly U(1, 1) and K× inside Gcommute, and generate G with U(1, 1) ∩K× = U(1). Thus a representation π of U(1, 1)can be extended to a representation of G by simply extending the central character ωπ of πfrom K1 = U(1) to a character χ of K×. The representation of G restricted to GL2(k)

+

is irreducible with central character χ|k× .


The group U(1, 1) being the subgroup of GU(1, 1) consisting of elements of similitudefactor 1, sits in the following exact sequence

1→ U(1, 1)→ [GL+2 (k)×K×]/∆k× → NK× → 1.

We have the natural embedding of U(1) inside U(1, 1) given through its action ona one dimensional subspace of the two dimensional Hermitian space on which U(1, 1)operates, such that the action of U(1) on the orthogonal complement (which is a onedimensional Hermitian space with discriminant one) is trivial. Thinking of U(1) as K×/k×,the embedding of U(1) inside U(1, 1) is given by

K×/k× ↪→ [GL+2 (k)×K×]/∆k×

where K× embeds in GL2(k)+ through one of its embeddings (there are possibly two of

them!), and the mapping from K× to K× is t→ t−1.It thus follows that for a character µ of K×/k× to appear in the restriction of a repre-

sentation π × χ of [GL+2 (k)×K×]/∆k×, it is necessary and sufficient that the character

µχ of K× appears in the representation π of GL+2 (k). This is exactly the refinement of

the theorem of Saito and Tunnell considered by the third author in [P2] which we statenow. In the statement of the theorem below, we will not consider those representations ofGL2(k) which remain irreducible when restricted to GL+

2 (k), being known from results dueto Saito and Tunnell for GL2(k).

Theorem 13.1. A representation of GL2(k) decomposes into two irreducible compo-nents when restricted to GL+

2 (k) if and only if its Langlands parameter is induced froma character, say α, of K×. Suppose that this is the case. The two representations ofGL+

2 (k) so obtained can be indexed as π+ and π− in such a way that a character β ofK× with α|k× = ωK/kβ|k× appears in π+ if and only if ε(αβ−1, ψ) = ε(αβ−1, ψ) = 1,and appears in π− if and only if ε(αβ−1, ψ) = ε(αβ−1, ψ) = −1; here ψ is a fixedadditive character of K which is trivial on k, changing which will change the orderingof π+ and π−. There is a similar statement for D×+.

13.1. Parameter for U(1, 1). For translating the above theorem about GL+2 (k) to

U(1, 1), one needs to associate L-parameters to representations of U(1, 1). Assume there-fore that π is a representation of U(1, 1) obtained by restriction of a representation π × χof [GL+

2 (k)×K×]/∆k×. Then the L-parameter of π restricted to K is σπ|K ⊗ χ−1 whereσπ is the parameter for π.

In the notation of the previous theorem, σπ|K = α ⊕ α, and therefore the parameterof the representation of U(1, 1) restricted to K is αχ−1 ⊕ αχ−1 where we recall that(αχ−1)|k× = ωK/k. Therefore conjectures 6.1 and 6.2 predict that a character µ of K×/k×

appears in π if and only if ε(αχ−1µ−1) = ±1, and ε(αχ−1µ−1) = ±1, take fixed values.On the other hand as observed before, µ appears in π if and only if the character µχappears in π. Thus the conclusion of Theorem 13.1 is exactly what the conjectures 6.1 and6.2 predict. We note for purposes here, as well as for later use, that for a character µ ofU(1) = K×/k×, its base change to K, i.e., to K× is the character µ itself.


14. Trilinear forms for U(2)

Given π1, π2, π3, three irreducible admissible representations of G = GL2(k)+ or U(2),

with the product of their central characters trivial, we calculate the dimension of the spaceof trilinear forms

dimHomG[π1 ⊗ π2 ⊗ π3,C].

These results when combined with Seesaw duality in theta correspondence will translateinto branching laws from U(3) to U(2) in the next section.

Let G be a subgroup of GL2(k) containing SL2(k). The group G is uniquely determinedby the subgroup k×G of k× consisting of determinants of elements of G. It thus makes senseto define a corresponding subgroup, say GD, inside D× containing SL1(D). Restrictingrepresentations of GL2(k) to G, or of D× to GD, one gets a notion of L-packet of repre-sentations of G, and of GD. Representations of GL2(k) restrict to G with multiplicity 1,but this need not be the case for representations of D× restricted to GD. For a represen-tation π′ of GD, let m(π′) denote the multiplicity with which it appears in an irreduciblerepresentation of D× when restricted to GD.

Theorem 14.1. Let π1, π2, π3 be three irreducible admissible representations of GL2(k)with the product of their central characters equal to 1. Let π′1, π

′2, π

′3 be the corresponding

three irreducible representations of D× associated to π1, π2, π3 by the Jacquet-Langlandscorrespondence. (If π is not essentially square-integrable representation of GL2(k), welet π′ = 0.) Then,∑

π1,π2,π3

dimHomG[π1 ⊗ π2 ⊗ π3,C]

+ m(π′1)m(π′2)m(π′3)∑

π′1,π′2,π

′3

dimHomGD[π′1 ⊗ π′2 ⊗ π′3,C] = ](k×/k∗2k×G),

where the sum is taken over irreducible representations π1, π2, π3 of G (or, π′1, π′2, π

′3 of

GD) which are contained in the representations π1, π2, π3 of GL2(k), or representationsπ′1, π

′2, π

′3 of D×.

Proof. Clearly,

HomG[π1 ⊗ π2 ⊗ π3,C] ∼=∑

χ:k×/k×G→Z/2

HomGL2(k)[π1 ⊗ π2 ⊗ π3,Cχ],

where χ′s are characters of GL2(k) trivial on G, and Cχ denotes the 1-dimensional repre-sentation of GL2(k) on which it operates by the character χ; by considerations of centralcharacter, χ’s are of order ≤ 2. Similarly,

HomGD[π′1 ⊗ π′2 ⊗ π′3,C] ∼=

∑χ:k×/k×G→Z/2

HomD× [π′1 ⊗ π′2 ⊗ π′3,Cχ].

But by [P],


dimHomGL2(k)[π1 ⊗ π2 ⊗ π3,Cχ] + dimHomGD[π′1 ⊗ π′2 ⊗ π′3,Cχ] = 1

for all characters χ of order ≤ 2 (by absorbing χ in one of the π′is).Thus by adding up the contribution of the various χ’s, we get the conclusion of the

theorem.�

Corollary 14.2. Let G = GL+2 (k). Then as m(π′) = 1 for all irreducible representa-

tions π′ of D∗+, one has,∑π1,π2,π3

dimHomG[π1 ⊗ π2 ⊗ π3,C] +∑

π′1,π′2,π

′3

dimHomGD[π′1 ⊗ π′2 ⊗ π′3,C] = 2,

where the sum is taken over irreducible representations π1, π2, π3 of G (or, π′1, π′2, π

′3 of

GD) which are contained in the representations π1, π2, π3 of GL2(k), or representationsπ′1, π

′2, π

′3 of D×.

The group U(1, 1) sits in the following exact sequence

1→ U(1, 1)→ [GL2(k)×K×]/∆k× → k× → 1.

Any irreducible representation of U(1, 1) is contained in the restriction of a representationπ×χ0 of [GL2(k)×K×]/∆k× where π is a representation of GL2(k) and χ0 is a character ofK× such that if ωπ denotes the central character of π then ωπ = χ0|k× ; in fact the restrictionof the π × χ0 of [GL2(k)×K×]/∆k× to U(1, 1) gives an L-packet of representations onU(1, 1). By an analysis done exactly as in theorem 15.1, we get the following:

Corollary 14.3. Let U′(2) and U′′(2) be the two unitary groups corresponding to the2 Hermitian space of dimension 2. Then∑π′1,π

′2,π

′3

dimHomU(2)[π′1 ⊗ π′2 ⊗ π′3,C] +

∑π′′1 ,π

′′2 ,π

′′3

dimHomU ′(2)[π′′1 ⊗ π′′2 ⊗ π′′3,C] = 2,

where the sum is taken over irreducible representations π′1, π′2, π

′3 of U′(2), and π′′1 , π

′′2 , π

′′3

of U′′(2) which belong to the same Vogan packet of representations with the product oftheir central characters trivial.

Corollary 14.4. (Multiplicity 1) For G = GL2(k)+ or U(2), let π1, π2, π3 be 3 irre-

ducible admissible representations of G, such that for one of the representations, sayπ1, its L-packet has more than 1 element (so 2). Then

dimHomG[π1 ⊗ π2 ⊗ π3,C] ≤ 1.

Proof. If HomG[π1⊗ π2⊗ π3,C] 6= 0, then so is HomG[π′1⊗ π′2⊗ π′3,C], where π′i denotesthe conjugate of πi by an element of GL2(k) which is not in GL2(k)

+. Since the sum ofthe dimensions of HomG[π1 ⊗ π2 ⊗ π3,C] and HomG[π′1 ⊗ π′2 ⊗ π′3,C], is bounded by 2,each one is bounded by 1. �

Although the next two corollaries play no role in this paper, we point them out anyway.


Corollary 14.5. (Failure of multiplicity 1) If π1, π2, π3 are three irreducible admissiblerepresentations of GL2(k) with the product of their central characters trivial, and suchthat each of the πi’s remain irreducible when restricted to GL2(k)

+, and one of themis a principal series representation, then

dimHomGL+2 (k)[π1 ⊗ π2 ⊗ π3,C] = 2.

Corollary 14.6. (High multiplicity for SL2) Since k×/k∗2 is a 2-group whose cardi-nality can be made arbitrarily large by choosing k appropriately, and since the L-packetof representations of SL2(k) is bounded by 4, it follows that the multiplicity,

m(π1, π2, π3) = dimHomSL2(k)[π1 ⊗ π2 ⊗ π3,C]

can be made arbitrarily large.

15. Using the theta correspondence for U(2, 1)

In this section we will use methods of theta correspondence to prove our conjecturesfor certain representations of U(2, 1) in both real and the p-adic case. These are therepresentations of U(2, 1) which can be obtained as a theta lift from smaller unitary groups.This method in particular covers all the discrete series representations of U(2, 1) over thereals. It should be noted here that all our conjectures about restriction of a representationof U(n) to U(n − 1) are invariant under simultaneous twisting of the representation ofU(n) and of U(n− 1) by the same character of U(1). Therefore, although the parametersof the representations of U(2, 1) obtained by theta lifting from U(2) appears to be smallerthan the set of parameters of U(2, 1) which become reducible when restricted to K, thiswill not be an issue for us.

Here is the main theorem about U(2, 1) that we shall be able to prove.

Theorem 15.1. Let σ : W ′k → LU(3) be the parameter of an irreducible representation

π of a unitary group U(3) in 3-variables over a local field. Let K be the quadratic fielddefining the unitary group U(3), and let σK : W ′

K → GL(3,C) be the restriction of theparameter σ to K. Assume that there is a character χ : K×/k× → C× such that

σK = σ1 ⊕ χ.(1) If σ1 is an irreducible 2-dimensional representation of W ′

K, the extended L-packetassociated to σ has 4 elements, 2 of which lie on this U(3), and 2 lie on the other(isomorphic) copy of U(3) defined by the non-isomorphic Hermitian form. Let σ2 :W ′k → LU(2) be the parameter of an irreducible admissible representation π2 of U(2)

(which is any unitary group in 2 variables defined by a rank 2 Hermitian form over Kcontained in the rank 3 Hermitian form defining U(3)). Then if HomU(2)[π⊗π2,C] 6= 0,ε(σ1⊗σ2,K) and ε(χ⊗σ2,K) take values in ±1 independent of π2 (as long as HomU(2)[π⊗π2,C] 6= 0). By varying π in the extended packet (with 4 elements), the characters soobtained on (Z/2)2 gives a bijection of the members of the L-packet of π with charactersof (Z/2)2.


(2) If σ1 is a sum of 2 characters, say σ1 = α⊕ β, the extended L-packet associated toσ has 8 elements, 4 of which lie on this U(3), and 4 lie on the other (isomorphic) copyof U(3) defined by the non-isomorphic Hermitian form. Let σ2 : W ′

k → LU(2) be theparameter of an irreducible admissible representation π2 of U(2) (which is any unitarygroup in 2 variables defined by a rank 2 Hermitian form over K contained in the rank 3Hermitian form defining U(3)). Then if HomU(2)[π⊗π2,C] 6= 0, ε(α⊗σ2,K), ε(β⊗σ2,K),ε(χ ⊗ σ2,K) take values in ±1 independent of π2 (as long as HomU(2)[π ⊗ π2,C] 6= 0).By varying π in the extended packet (with 8 elements), the character so obtained on(Z/2)3 gives a bijection of the members of the L-packet of π with characters of (Z/2)3.

Our proof about restriction from U(V ) to U(W ) will be based on the following seesawdiagram where W is a codimension one subspace of V such that V = W ⊕K as Hermitianspaces:

U(2, 1) U(W ′)× U(W ′)

U(W )× U(1) ∆U(W ′).

HHH

HHHH��

��

It follows from this seesaw diagram that the branching from U(V ) to U(W ) will be basedon the following two informations.

(1) Explicit theta lifting between unitary groups in 2 variables with those in 1 and 3variables.

(2) Tensor product of two irreducible representations of U(W ′) where W ′ is a twodimensional Hermitian space.

Before we discuss the theta lifting between unitary groups, let us remind ourselves that thetheta correspondence between unitary groups U(W1) and U(W2) depends crucially on fixingcharacters χ1 and χ2 on K× such that χ1|k× = ωdimW2

K/k , and χ2|k× = ωdimW1

K/k , which are

used to fix a lifting of U(W1)×U(W2) to the metaplectic group corresponding to the Her-mitian space W1⊗KW2. We will assume fixing the characters (χ1, χ2) = (µdimW2 , µdimW1)where the character µ of K× (with restriction to k× equal to ωK/k) is chosen as before. Thechoice of these characters has the advantage that for the corresponding liftings of U(W1)and U(W2), the centers of U(W1) and U(W2) get identified to each other, and as a result,the theta lifting preserves the central characters, cf. [HKS, Corollary A.8]. Furthermore,the seesaw diagram,

U(W1 ⊕W2) U(V )× U(V )

U(W1)× U(W2) ∆U(V )

HHHHHHH��

��


which exists at the level of symplectic groups, works exactly the same way in the metaplecticcovering groups, in which we will use the characters (µdimV , µdimW1) for the dual pair(U(W1),U(V )), (µdimV , µdimW1+dimW2) for the dual pair (U(W1 ⊕W2),U(V )) etc.

Suppose that π1 is an irreducible admissible representation of U(W1) and θ(π1) = π2 is itstheta lift to U(W2) which we assume is nonzero. Assume that σ1 and σ2 are the Langlandsparameters of π1 and π2 respectively, and that σ1,K and σ2,K denote their restrictions ofW ′K . We will then use the following about theta correspondence:

(1) If dimW1 = dimW2, then σ1,K = σ2,K .(2) If dimW1 = dimW2 + 1, then σ1,K = µσ2,K ⊕ µ− dimW2 .

Although one would expect these in general, we will need to use these relations forthe Langlands parameters of representations related by theta liftings only for the pairs(U(1),U(2)), and (U(2),U(3)) which are well known, such as in [GRS] for the second pair.One expects furthermore, that in the pair (W1,W2) with dimW1 = dimW2 + 1, as onevaries representations in a Vogan packet on U(W2), one gets a Vogan packet on the largerunitary group U(W1). Again, this is known when dimW2 is 1 or 2.

Given a character ν of K1 ⊂ K×, let ν be an extension of ν to K×. Define χν(z) =ν(z/z) = ν(z/z). Observe that z → z/z being the norm mapping for U(1), χν(z) =ν(z/z), is the base change of ν.

With this notation, we can write the theta lifting of the character ν of U(1) to be arepresentation of U(1, 1) which is contained in the representation

(3) Ind(µν)⊗ ν(x) of [GL2(k)×K×]/∆k×.Notice that given this representation of [GL2(k)×K×]/∆k×, from the recipe in section

13.1, the Langlands parameter of the representation of U(1, 1) after base change to K is:(4) Ind(µν)|K ⊗ ν−1(x) = [µν ⊕ µ¯ν]⊗ ν(x)−1 = µχν(x)⊕ µ−1,

which matches with the parameter as given in (2) (after noticing that the base change ofthe character ν of U(1) is the character χν(x) = ν(x/x) of K×).

Recall from the previous section that U(1, 1) is a subgroup of G = [GL+2 (k)×K×]/∆k×

such that any irreducible representation of U(1, 1) is the restriction of an irreducible rep-resentation of G, and further, an irreducible representation of G remains irreducible whenrestricted to U(1, 1). It follows that the calculation of the tensor product of irreducible rep-resentations of U(1, 1) is essentially the same as that of G. As G = [GL+

2 (k)×K×]/∆k×,clearly calculation of tensor product of irreducible representations of G is the same as thatof GL+

2 (k), something which has been essentially done by the 3rd author, cf. [P5]. We willcontent ourselves by stating the result in just one case.

Theorem 15.2. Let π1, π2 be two irreducible admissible representations of GL+2 (k),

and let π3 be one of GL2(k) such that the product of the central characters ωπ1ωπ2ωπ3 =1. Assume that the Langlands parameters σi of the representations πi are obtained fromcharacters αi of K× for i = 1, 2. Then if

HomGL+2 (k)[π1 ⊗ π2 ⊗ π3,C] 6= 0,


ε(Ind(α1α2)⊗ σ3) and ε(Ind(α1α2)⊗ σ3) take values in ±1, independent of π3 as longas

HomGL+2 (k)[π1 ⊗ π2 ⊗ π3,C] 6= 0.

Similarly for D×. The character of (Z/2)2 so obtained (by these two epsilon values)identifies the extended L-packet of GL+

2 (k) × GL+2 (k) (and D×+ × D×+) containing

π1 × π2 in which the representations π1 × π2 and π′1 × π′2 are identified where π′1 (resp.π′2) is the other member of the L-packet containing π1 (resp. π2).

15.1. Completing the branching from U(2, 1) to U(2). We look at the seesawdiagram

U(2, 1) U(W ′)× U(W ′)

U(W )× U(1) ∆U(W ′),

HH

HH

HHH�

��

��

��

starting with a representation π1 of ∆U(W ′) with Langlands parameter restricted to Kequal to σ1, and a representation π2 × ν of U(W ) × U(1) with Langlands parameterrestricted to K equal to σ2 × ν. The seesaw identity gives us,

HomU(W )×U(1)[θ(π1), π2 × ν] = HomU(∆W ′)[θ(π2)× θ(ν), π1].

Here U(1) is the unitary group of a Hermitian form in 1 variable, and U(W ),U(∆W ′)are the unitary groups of Hermitian forms in 2 variables. Noting that the direct sum ofHermitian spaces of dimension 1 and 2 creates a Hermitian space of dimension 3, and thatthis direct sum construction is a two-to-one mapping onto Hermitian spaces of dimension3, multiplicity 2 theorem about triple products of representations of U(2) (Corollary 14.3)translates to multiplicity 1 theorem for the restriction of Vogan L-packet on U(3) to aVogan L-packet on U(2).

We now carry out the comparison of relevant epsilon factors. The Langlands parameterof θ(π1) restricted to K is σ3 = µσ1 ⊕ µ−2, therefore the branching from U(3) to U(2)depends on ε([µσ1 ⊕ µ−2] ⊗ σ∨2 ), and therefore on the epsilon factors (of representationsover K):

ε(µσ1 ⊗ σ∨2 ), and(15.1)

ε(µ−2σ∨2 ).(15.2)

On the other hand, assuming that the representations π1 and π2 of U(1, 1) arise fromrepresentations of GL(2, k) with Langlands parameters τ1 and τ2 (and with base changeparameters σ1 and σ2), the non-vanishing of HomU(2)[θ(π2)× θ(ν), π1] is controlled by thetriple product epsilon factor :


ε(τ∨1 ⊗ τ2 ⊗ Ind(µν)) = ε(BC(τ1)∨ ⊗BC(τ2)⊗ µν).

(Here we have used the notation, BC(V ) to denote the restriction of a representation Vof Wk to WK , and the fact that ε(V ⊗ Indα) = ε(V |K ⊗ α) whenever 4| dimV .) Let ω1

and ω2 be the central characters of the representations π1 and π2 of U(∆W ′) and U(W )respectively. So, ω1 and ω2 are characters of U(1) = K×/k×. Comparing central characterof the representation θ(π1) of U(2, 1) with that of the representation π2×ν of U(2)×U(1),we must have

ω1 = ω2 · ν, or, ν = ω1ω−12 .

Let ω1, ω2 and ν be character of K× extending the characters ω1, ω2, ν of K1 = U(1). Weassume that these extensions are so chosen that,

ν = ω1ω−12 .

Recall that for a representation π of U(1, 1) obtained by restriction of a representationπ×χ of [GL+

2 (k)×K×]/∆k×, the L-parameter of π restricted to K is σπ|K ⊗ χ−1 whereσπ is the parameter for π. Therefore,

BC(τ1) = σ1 ⊗ ω1, and BC(τ2) = σ2 ⊗ ω2.

Therefore,

BC(τ1)∨ ⊗BC(τ2)⊗ ν = σ∨1 ⊗ σ2 ⊗ (ω−1

1 ω2ν) = σ∨1 ⊗ σ2.

Therefore,

ε(τ∨1 ⊗ τ2 ⊗ Ind(µν)) = ε(BC(τ1)∨ ⊗BC(τ2)⊗ µν)

= ε(σ∨1 ⊗ σ2 ⊗ µ)

= ε(σ1 ⊗ σ∨2 ⊗ µ−1)

= ε(σ1 ⊗ σ∨2 ⊗ µ).

Here we have used the facts that ε(V ) = ε(V ∨) if detV (−1) = 1, as is the case for us,and in such a case, ε(V ) = ε(V σ) where σ is the automorphism of K over k.

Thus we get the same epsilon factor that we encountered in (15.1) for branching fromU(3) to U(2). We note that by a theorem due to Harris-Kudla-Sweet in [HKS], the value ofthe other epsilon factor, ε(µ−2σ∨2 ) determines whether θ(π2) is zero or nonzero, therefore bythe seesaw duality, the two epsilon factors in (15.1) and (15.2) together determine whetherthe representation π2 of U(W ) appears in the representation θ(π1) of U(2, 1).

16. Branching laws for GL(n,Fq)

In this section we calculate the restriction of a representation of GL(n,Fq) to GL(n −1,Fq) where GL(n− 1,Fq) sits inside GL(n,Fq) in the natural way as

A→(A 00 1

).


These branching laws are surely known in the literature, such as in the work of Thoma;however, we have preferred to give a different independent treatment.

We begin by recalling the notion of twisted Jacquet functor. For P = MN any groupsuch that N is a normal subgroup of P and ϕ a character of N which is left invariantunder the inner-conjugation action of M on N , the twisted Jacquet functor associates toany representation V of P , the representation Vϕ which is the largest quotient of V onwhich N operates via the character ϕ; clearly Vϕ is a representation space for Mϕ, thesubgroup of M which operates trivially on ϕ. Let En−1 be the subgroup of GL(n,Fq) withlast row equal to (0, 0, · · · , 0, 1) and let Nn be the group of upper triangular matrices inGL(n,Fq) with 1’s on the diagonal. We fix a nontrivial character ψ0 of k and let ψn bethe character of Nn, given by ψn(u) = ψ0(u1,2 + u2,3 + · · ·+ un−1,n). For a representationπ of GL(n,Fq), let πi denote its i-th derivative which is a representation of GL(n− i,Fq).Recall that if Rn−i = GL(n− i,Fq)Vi is the subgroup of GL(n,Fq) consisting of(

g v0 z

)with g ∈ GL(n− i,Fq), v ∈ M(n− i, i), z ∈ Ni, and if the character ψi on Ni is extendedto Vi by extending it trivially across M(n− i, i), then πi = πψi

.If π is an irreducible cuspidal representation of GL(n,Fq), then πi = π for i = 0, and

πn = 1, the trivial representation of GL(0,Fq) which is a group with 1 element; all theother derivatives of π are 0.

The following proposition is from Bernstein-Zelevinsky [BZ], where it is done for non-Archimedean local fields, but works for finite fields as well.

Proposition 16.1. For π1 a representation of GL(n1,Fq) and π2 of GL(n2,Fq), we letπ1 × π2 denote the representation of GL(n1 + n2,Fq) induced from the correspondingrepresentation of the parabolic with Levi subgroup GL(n1,Fq)×GL(n2,Fq). Then thereis a composition series of the k-th derivative (π1 × π2)

k whose successive quotients areπi1 × πk−i2 for i = 0, · · · , k.

Here is a generality from Bernstein and Zelevinsky [BZ].

Proposition 16.2. Any representation Σ of En−1 has a natural filtration of E = En−1

modules 0 ⊂ Σ0 ⊂ Σ1 ⊂ Σ2 ⊂ · · · ⊂ Σn such that Σi+1/Σi = indERi(Σn−i ⊗ ψn−i) for

i = 0, · · · , n, where Ri = GL(i,Fq) · Vn−i is the subgroup of GL(n,Fq) consisting of(g v0 z

)with g ∈ GL(i,Fq), v ∈ M(i, n − i), z ∈ Nn−i, and the character ψn−i on Nn−i isextended to Vn−i by extending it trivially across M(i, n− i).

The following corollary is clear.

Corollary 16.3. Let n = n1 + · · ·nr be a sum of positive integers, and let πi bean irreducible cuspidal representation of GL(ni,Fq). Let Σ = π1 × · · · × πr be the


corresponding parabolically induced representation of GL(n,Fq). Then the restrictionof π1 × · · · × πr to GL(n− 1,Fq) is a sum of the following representations:

πi1 × πi2 × · · · × πis ×GG[n− 1− (i1 + · · ·+ is)]

where 1 ≤ i1 < i2 < · · · < is ≤ r (the empty sequence is allowed), i1 + · · · + is < n,GG[m] denotes the Gelfand-Graev representation of GL(m,Fq) which is the inducedrepresentation of GL(m,Fq) from a non-degenerate character of its group of upper-triangular unipotent matrices; GG[1] is the regular representation of F×q ; GG[0] is thetrivial representation of the trivial group.

Proof : Since GL(n− 1,Fq)Rn−i = En−1 for any i, it follows from proposition 16.2 thatthe restriction of Σi+1/Σi to GL(n− 1,Fq) is the representation Σn−i×GG[n− i], whereGG[n− i] denotes the Gelfand-Graev representation of GL(n− i,Fq) which is the inducedrepresentation of GL(n − i,Fq) from a non-degenerate character of its group of upper-triangular unipotent matrices (where GG[1] is the regular representation of F×q , GG[0] isthe trivial representation of the trivial group). It only remains to calculate the derivativesΣn−i of Σ which follows from proposition 16.1 (the Leibnitz rule about derivatives).

As a simple consequence of this corollary, we have the following.

Corollary 16.4. Let n = n1 + · · · + nr be a sum of positive integers, and let πi bean irreducible cuspidal representation of GL(ni,Fq). Assume that the representationsπ1, · · · , πr consist of distinct representations, so that π1 × · · · × πr, the correspondingparabolically induced representation of GL(n,Fq) is irreducible. Similarly, let n − 1 =m1 + · · · + ms be a sum of positive integers, and let µi be an irreducible cuspidalrepresentation of GL(mi,Fq). Assume that the representations µ1, · · · , µs consist ofdistinct representations, so that µ1 × · · · × µs, the corresponding parabolically inducedrepresentation of GL(n− 1,Fq) is irreducible. Then the restriction of π1 × · · · × πr toGL(n−1,Fq) contains the representation µ1×· · ·×µr of GL(n−1,Fq) with multiplicity

2d,

where d is the cardinality of the set of common representations in {π1, · · · , πr} and{µ1, · · · , µs}.

17. Branching laws for U(n,Fq) via base change

We use the method of base change, also called Shintani descent, to deduce some conclu-sions about branching laws for the restriction of a representation of U(n,Fq) to U(n−1,Fq)from the corresponding results for general linear groups obtained in the previous section. Wemake crucial use of multiplicity 1 theorem for unitary groups which has recently been provedfor non-Archimedean fields by Aizenbud, Gourevitch, Rallis and Schiffmann in [AGRS]. Webegin with a brief review of what is called the Shintani descent.


Let G be a connected reductive algebraic group over Fq. Let m ≥ 1 be a fixed integer.The group G(Fqm) comes equipped with its Frobenius automorphism F , whose fixed pointsare G(Fq). There is a natural map, called the norm mapping, from F -conjugacy classes inG(Fqm) to conjugacy classes in G(Fq) which is an isomorphism of the corresponding sets,and is furthermore an isometry:

〈χ1, χ2〉G(Fq) = 〈χ′1, χ′2〉G(Fqm ),

where χ1 and χ2 are class functions on G(Fq) which are related through the norm mappingto F -conjugacy class functions χ′1 and χ′2 on G(Fqm); the inner products are normalized sothat

〈1, 1〉G(Fq) = 〈1, 1〉G(Fqm ) = 1.

We recall that according to Deligne-Lusztig, given a maximal torus T of G defined overFq, and a character θ : T (Fq) → C∗, there is a (virtual) representation of G(Fq) denotedby R(T, θ), now called the Deligne-Lusztig representation.

Given a character θ : T (Fq) → C∗, there are characters θm : Tm = T (Fqm) → C∗

obtained by composing with the norm mapping: T (Fqm)→ T (Fq).It is a basic fact that the characters of R(Tm, θm) and R(T, θ) are related by the norm

mapping. We assume in what follows that m = 2, and that R(T2, θ2) is an irreduciblerepresentation of G(Fq2) which is invariant under the action of 〈F 〉 = Gal(Fq2/Fq), andtherefore extends to an irreducible representation of G(Fq2) o 〈F 〉.

Since one can use irreducible cuspidal representation of U(n,Fq) to construct irreduciblecuspidal representation of U(n, k) where k is the local field with maximal compact subringthe Witt ring of Fq, the following is an easy consequence of multiplicity 1 theorem for p-adicgroups; we will not give a proof.

Proposition 17.1. Let π1 be an irreducible cuspidal representation of U(n,Fq), andπ2 be an irreducible cuspidal representation of U(n − 1,Fq). Then π2 appears in π1

restricted to U(n− 1,Fq) with multiplicity at most 1.

With notation as in the proposition, let χ1 be the character of the representation π1 andχ2 that of π2. Let χ′1 and χ′2 be the characters of the corresponding representations ofGL(n,Fq2) and GL(n− 1,Fq2) obtained by base change, which we assume are irreducible,and hence extend to representations of semi-direct products: GL(n,Fq2) o 〈F 〉,GL(n −1,Fq2) o 〈F 〉. We denote the characters of these representations of GL(n,Fq2) o 〈F 〉 andGL(n− 1,Fq2) o 〈F 〉 also by χ′1 and χ′2. It follows that

〈χ′1, χ′2〉GL(n−1,Fq2 )o〈F 〉 = 〈χ′1, χ′2〉GL(n−1,Fq2 ) + 〈χ′1, χ′2〉GL(n−1,Fq2 )·F .

Therefore,

〈χ′1, χ′2〉GL(n−1,Fq2 )o〈F 〉 = 〈χ′1, χ′2〉GL(n−1,Fq2 ) + 〈χ1, χ2〉U(n−1,Fq).

Now we observe that the left hand side of this last equality is an even integer (by SchurOrthogonality, observing that the group GL(n− 1,Fq2) o 〈F 〉 has been given volume 2),


whereas by our calculations of the last section, 〈χ′1, χ′2〉GL(n−1,Fq2 ) is either an even integer

if the cuspidal supports of χ′1 and χ′2 are not disjoint, or equals 1 if the cuspidal supportsare disjoint. The term 〈χ1, χ2〉U(n−1,Fq) is either 0 or 1 by Proposition 17.1.

Therefore we get the following theorem as our only option.

Theorem 17.2. Let π1 be an irreducible cuspidal representation of U(n,Fq), and π2

an irreducible cuspidal representation of U(n− 1,Fq). Assume that both π1 and π2 areDeligne-Lusztig representations. Then π2 appears in the restriction of π1 if and only iffor the representations π′1 and π′2 which are the base change of π1 and π2 to GL(n,Fq2)and GL(n− 1,Fq2), the cuspidal supports are disjoint.

Remark : A Deligne-Lusztig representation R(T, θ) is known to be irreducible if and onlyif θ, a character of T (Fq), is not stabilized by any non-trivial element of W (Fq), where Wis the Weyl group of T . It can be seen that the base change of such a character of T (Fq)to T (Fqn) continues to have this property, and as a result, the base change of an irreducibleDeligne-Lusztig representation is an irreducible Deligne-Lusztig representation.

18. Depth zero supercuspidals

We now test the local conjecture in the p-adic case, for some tamely ramified discreteparameters, using the calculation of root numbers in §11. We will consider parameters

ϕ : Wk → Go Gal(K/k)

trivial on SL2(C), with K/k unramified. We will further assume that ϕ is tamely ramified

and that the centralizer of the image of inertia is the maximal torus T of G. Finally we willassume that the image of the Frobenius in the quotient group NLG(T )/T acts as −1 on

X ·(T ). It would be interesting to test the local conjecture on more general tamely ramifiedparameters.

Under these conditions, ϕ determines a Langlands parameter ϕ(T ) for the unramifiedanisotropic torus T = U(1)n × U(1)n−1 over k. By local class field theory, we obtain atame regular character ρ : T (k) → C×, which factors through the quotient T (Fq). If weidentify T (Fq) with the product (F×q2/F

×q )n× (F×q2/F

×q )n−1, then ρ is given by the collection

(α0i , βj) as in §11.The centralizer of our parameter ϕ is given by

Cϕ = Aϕ = T [2] = 〈±1〉n × 〈±1〉n−1.

We first show how a character λ : Aϕ → 〈±1〉 corresponds to an embedding of torus Tinto a pure inner form G′ of G. We do this analysis for each unitary space W ′ and W ′

0

separately.An embedding of the torus S = U(1)n into the unitary group U(W ) corresponds to

a decomposition of W into orthogonal lines over K, each stable under the action of S:


W = ⊕i=ni=1Kei. The conjugacy class of the embedding f : S → U(W ) depends on thediscriminants

〈ei, ei〉 in k×/NK×.

Since K/k is unramified, it depends on the signs

(−1)ord〈ei,ei〉 in 〈±1〉n.

There are 2n−1 possible conjugacy classes, as we have the relation∏〈ei, ei〉 ≡ disc(W ) mod NK×.

Since the 2 pure inner formsW andW ′ have distinct discriminants, there are 2n embeddingsof S into either U(W ) or U(W ′).

Given an embedding f : S → U(W ′), there is a unique maximal compact subgroupKf ⊂ U(W ′) which contains the image of S. This is the subgroup stabilizing the lattice,Lf = ⊕i=ni=1AKei, where the orthogonal vectors ei are chosen on the S-stable lines to satisfyord〈ei, ei〉 = 0, or 1. The reduction of Kf ( mod π) has reductive quotient

Kf∼= U(p)× U(n− p),

over Fq, where p is the number of ei with (−1)ord〈ei,ei〉 = −1. Hence Kf will be hyperspecialif and only if all of the inner products 〈ei, ei〉 have valuations of the same parity. This justifiesthe claim made in §10, before the statement of Conjecture 10.4.

From the calculation of the character χ of Aϕ in Corollary 11.2, we conclude that theassociated embedding

f : U(W ′0)× U(W ′)

has image contained in the maximal compact subgroup Kf of G′ with reduction isomorphicto

G′ = Kf = (U(p)× U(n− 1− p))× (U(p)× U(n− p))over Fq. Here p is the number of pairs (α0

i , βi) with α0iβi = 1.

The regular character ρ of T (Fq) allows us to construct an irreducible, supercuspidalrepresentation R(T, ρ) of the finite group Kf (Fq), using the method of Deligne and Lusztig.Since Kf (Fq) is a direct product, we may write R(T, ρ) as the tensor product of 4 terms

(R1 ⊗R2)⊗ (R3 ⊗R4).

Since α0iβi = 1 for 1 ≤ i ≤ p, we conclude that R1 and R3 are dual irreducible rep-

resentations of the finite group U(p). In particular, there is a U(p)-invariant linear formR1 ⊗R3 → C, unique up to scaling.

To prove the existence of (unique) H ′ = U(p) × U(n − 1 − p)-invariant linear form onR(T, ρ), we must establish the existence of a unique U(n − 1 − p)-invariant linear formR2 ⊗R4 → C. In other words, we must show that the dual of R2 occurs with multiplicity1 in the restriction of R4 from U(n− p) to U(n− p− 1). This follows from Theorem 17.2,as the tame characters α0

i and βj giving R2 and R4 satisfy α0iβj 6= 1 for all i, j.


Since π′ = IndG′

Kf(R(T, ρ)), the H ′-invariant linear form on R(T, ρ) gives a non-zero

H ′-invariant form on π′. Thus HomH′(π′,C) is non-zero, as predicted.

19. Global implies local: some partial results, I

The aim of this section is to prove the following local theorem by global methods.

Theorem 19.1. Let W0 be a 2 dimensional non-degenerate Hermitian subspace ofa Hermitian space W of dimension 3 over a non-Archimdean local field K, with Kquadratic over k. Suppose that π0 (resp. π1) is an irreducible representation of U(W0)(resp. U(W1)) which belongs to a generic L-packet. Let the Langlands parameter of π0

(resp. π1) restricted to K be σ0 (resp. σ1). Suppose that HomU(W0)(π0 ⊗ π1,C) 6= 0.Then

ε(σ0 ⊗ σ1) =

{1 if U(W0)× U(W1) is quasi− split−1 otherwise

Remark : The method that we follow to prove this theorem is pretty general, but it isbased on a global theorem of Ginzburg, Jiang, and Rallis [GJR, theorem 4.6] which assumesthat automorphic forms on unitary groups U(n) have base change to GL(n) somethingwhich is known at the moment only for generic automorphic representations on quasi-splitunitary groups. However, by Rogawski [Ro], base change is known for any unitary groupin 3 variables, which is why we have restricted ourselves to U(3) in the above theorem.Nonetheless, we have formulated some of the preliminary results below in greater generality.

We begin with the following globalization result about local fields, which will be appliedto globalize Hermitian spaces over local fields so that there is no ramification outside theplace being considered, and the unitary groups at infinity are either compact, or of rank 1.

Lemma 19.2. Let K be a quadratic extension of a non-Archimedean local field k. Thenthere exists a totally real number field F with k as its completion, and a quadratic totallyimaginary extension E of F with corresponding completion K such that E is unramifiedover F at all finite places different from K.

Proof: Except for the requirement about E being unramified except for the place K, thisis well-known. Suppose that a quadratic extension E1 over F1 with possible ramifications isconstructed. Then a well-known technique, crossing with a field, says that after a suitablebase change, one can get rid of the ramifications; we leave the details to the reader.

Lemma 19.3. Let W be a non-degenerate Hermitian space over a non-Archimedeanlocal field K, with K quadratic over k. Let F be a totally real number field withcompletion k at a place of F , and let E be a quadratic totally imaginary extension ofF with corresponding completion K. Then there is a Hermitian space V over E givingrise to W over K in such a way that the corresponding unitary group is quasi-split atall finite places of F except the one corresponding to the completion k; and at all butone infinite place the group is the compact group U(n), and at the remaining infinite


place, the group is either U(n), or U(n − 1, 1); if n is odd, we can assume that thegroup is compact at all the infinite places.

Proof : The proof of the lemma will depend on the well-known classification of a Hermitianform over a number field, according to which a Hermitian form over a number field isdetermined by

(1) the normalized discriminant, and(2) the signatures at the infinite places.

Moreover, given any normalized discriminant, and signatures at infinite places (except forobvious compatibility between normalized discriminant and signatures), there is a Hermitianform.

We also note the following exact sequence from classfield theory,

0→ F×/NE× → A×F/NA×

E → Gal(E/F )→ 0,

from which it follows that one can construct an element in F× which is trivial in F×v /NE×

v

at all the finite places except k, and which at the infinite places has the desired signs, exceptthat the product of the signs is 1 or -1, depending on whether the element in k×/NK× istrivial or non-trivial.

The proof of the lemma is now completed by observing that a Hermitian form of nor-malized discriminant 1 over a non-Archimedean local field defines a quasi-split group, andthat the normalized discriminants of the Hermitian spaces Z1Z1 +Z2Z2 + · · ·+ZnZn, andZ1Z1 + Z2Z2 + · · · + Zn−1Zn−1 − ZnZn over C are negative of each other, and if n isodd, the normalized discriminants of the Hermitian spaces Z1Z1 +Z2Z2 + · · ·+ZnZn, and−(Z1Z1 + Z2Z2 + · · ·+ ZnZn) are negative of each other.

Corollary 19.4. Let W0 be a non-degenerate Hermitian subspace of codimension 1 ofa Hermitian space W1 over a non-Archimedean local field K, with K quadratic overk. Let F be a totally real number field with completion k at a place of F , and letE be a quadratic totally imaginary extension of F with corresponding completion K.Then there is a Hermitian subspace V0 of codimension 1 of a Hermitian space V1 overE giving rise to W0 and W1 over K in such a way that the corresponding unitarygroups are quasi-split at all the finite places of F except the one corresponding to thecompletion k; assuming F 6= Q, the group U(V1) is the compact group U(n + 1) at allbut two infinite places, and at the remaining infinite places, the group is either U(n+1),or U(n, 1); the subgroup U(V0) is compact at all but one infinite place.

Proof : Let W1 = W0 ⊕ Lc where Lc is K with the Hermitian structure cZZ, c ∈ k×.Globalize W0 by the previous lemma, and globalize c so that the normalized discriminantof W1 is 1 at all the finite places of F other than k (so that U(V1) is quasi-split at all thefinite places of F other than k), and so that c has arbitary signs at infinity, with only theproduct of the signs pre-determined which allows for the desired conclusion.

We omit a proof of the following corollary of the lemma which follows exactly as in theprevious corollary.


Corollary 19.5. If n ≡ 1, 2 mod 4, then Hermitian spaces W0 ⊂ W1 of dimensionsn, n+ 1 can be globalized keeping them positive definite at infinity, and maximally splitat all finite places other than K. For n ≡ 2 mod 4, if W0 has an isotropic subspace ofdimension n/2, and for n ≡ 1 mod 4, if W1 has an isotropic subspace of dimension(n + 1)/2, then there are an even number of real places in F , else an odd number ofreal places.

Proof of Theorem 19.1: If the representation π0 is an irreducible principal series repre-sentation of U(W0), or is contained in a unitary principal series representation, then

σ0 = α⊕ α∨.It follows that

σ0 ⊗ σ1 = τ ⊕ τ∨,for τ = α⊗ σ1, and therefore by lemma 9.3,

ε(σ0 ⊗ σ1) = 1.

As the principal series representation occurs only on the quasi-split form of U(2), this provesthe conclusion desired in the theorem.

If the representation π1 is an irreducible principal series, then the theorem is a consequenceof our analysis in section 15.

It is easy to see that the Steinberg representation of U(1, 1) is a quotient of the Steibergrepresentation of U(2, 1) (by simply restricting functions on the flag variety of U(2, 1) to theflag variety of U(1, 1)), and that (denoting stn the n-dimensional irreducible representationof the SL2(C) part of W ′

K)

ε(st3 ⊗ st2) = ε(st4 ⊕ st2) = 1.

It therefore suffices to assume for the rest of the proof that both the representations π0

and π1 are supercuspidal.Since the group U(V1) is compact at infinity, it is easy to see that we can globalize the

representation π1 of U(W1) to an automorphic representation Π1 in U(V1)(A) in such away that it is unramified at all the finite places of F except k.

By Lemma 1 of [P3], we can globalize π0 to an automorphic representation Π0 such thatthe period integral ∫

U(V0)\U(V0)(A)

f0f1 6= 0,

for some f0 in Π0, and f1 in Π1.By the theorems due to Ginzburg, Jiang, and Rallis, cf. [JGR2, theorem 4.6], since the

period integral is nonzero, the central critical L-value,

L(1

2,ΠE

0 ⊗ ΠE1 ) 6= 0,

where ΠE0 and ΠE

1 denote base change of Π0 and Π1 to E.


This implies that the global root number,

ε(1

2,ΠE

0 ⊗ ΠE1 ) = 1.

Let

Π0 = ⊗wΠ0,w, and Π1 = ⊗wΠ1,w,

with Π0,v = π0, and Π1,v = π1. From the nonvanishing of the period integral, it followsthat

HomU(V0,w)(Π0,w ⊗ Π1,w,C) 6= 0.

Since the representations Π1,w for w, a finite place of F , not v, are unramified byconstruction, they are in particular quotients of principal series representations. It followsfrom section 15, or also by a direct calculation involving Mackey theory, that

εw(1

2,ΠE

0,w ⊗ ΠE1,w) = 1.

Since the global epsilon factor is a product of local epsilon factors, and since the rep-resentation Π1 is unramified at all finite places of F except that corresponding to k, wehave

ε(1

2, σ0 ⊗ σ1)ε∞(

1

2,ΠE

0,∞ ⊗ ΠE1,∞) = 1.

The following lemmas then complete the proof of the theorem on noting that there arean even number of places at infinity if U(W0) is quasi-split, and odd number of places atinfinity when U(W0) is not quasi-split.

Lemma 19.6. Let W0 be a codimension 1 Hermitian subspace of a positive definiteHermitian space W of dimension n + 1 over C. Suppose that π0 (resp. π1) is a finitedimensional irreducible representation of U(W ) (resp. U(W0)). Let the Langlandsparameter of π0 (resp. π1) restricted to K be σ0 (resp. σ1). Suppose that HomU(W0)(π0⊗π1,C) 6= 0. Then

ε(σ0 ⊗ σ1) =

{1 if n ≡ 0, 3 mod 4−1 if n ≡ 1, 2 mod 4.

Proof: The proof of this lemma is a simple consequence of the well-known branching lawfrom the compact group U(n+ 1) to U(n), combined with the value of the epsilon factorgiven by the following lemma.

Lemma 19.7. Let ψ be the additive character on C given by ψ(z) = e−2πiy wherez = x+iy. For n an integer, let χn denote the character χn(z) = eniθ for z = reiθ ∈ C×.Then for n odd,

ε(χn, ψ) =

{1 if n > 0−1 if n < 0.


Lemma 19.8. Let π0 (resp. π1) be a finite dimensional irreducible representation ofthe compact group U(n) (resp. U(n + 1)) with L-parameter restricted to C× given byan n-tuple of half-integers σ0 = {λ1 < λ2 < · · · < λn} (resp. {µ1 < µ2 < · · ·µn+1}an (n + 1)-tuple of half-integers), where all the λ′is are half-integers but not integersif n is even, and are integers in n is odd, and µ′is are all integers if n is even, andhalf-integers but not integers if n is odd. Then π0 appears in π1 restricted to U(n) ifand only if

µ1 < λ1 < µ2 < λ2 < · · · < λn < µn+1.

Corollary 19.9. With notation as in the lemma, and assuming that π0 appears in π1

ε(µk ⊗ σ0) = (−1)k−1, for all k,

and therefore,

ε(σ1 ⊗ σ0) =n+1∏k=1

(−1)k−1

= (−1)n(n+1)

2 .

Remark : Notice that ε(µk ⊗ σ0) = (−1)k−1, for all k, is independent of the representa-tion π0 of U(n) as long as it appears in the representation π1 of U(n+ 1), exactly as oneof the formulations of our local conjecture in §6.

Part 3. BESSEL and FOURIER-JACOBI MODELS

In Part 3 of this paper, we shall consider certain generalizations of the restriction prob-lem studied in Part 1 and formulated certain extensions of Conjectures 6.1, 6.2. Theseextensions are analogs of those treated in [GP2] for the case of orthogonal groups. So letus briefly revisit the case of orthogonal groups here.

We will deal exclusively with local fields in this part. There are natural global periodstoo in this context, for which the answer will depend on

(1) Existence of local Bessel or Fourier-Jacobi models.(2) Non-vanishing of a certain L-value at the center of the critical strip.

As these considerations are totally analogous to what we have discussed in [GP1] and inthis paper for U(n),U(n− 1)), we will not dwell on it at all.

In [GP1], the restriction problem for the pair (SO(2n+1), SO(2n)) or (SO(2n), SO(2n−1)) was studied and the proposed solution is expressed in terms of root numbers constructedout of the symplectic representation

W ′k −→ L(SO(2n+ 1)× SO(2n)) ⊂ Sp2n(C)×O2n(C),


or

W ′k −→ L(SO(2n− 1)× SO(2n)) ⊂ Sp2n−2(C)×O2n(C).

From the Galois theoretic point of view, it is natural to consider general symplectic repre-sentations of W ′

k of the form

W ′k −→ L(SO(2m+ 1)× SO(2n)) ⊂ Sp(2m)×O2n(C),

i.e. obtained by the tensor product of a symplectic representation and an orthogonalrepresentation of even dimension. This should correspond to a restriction problem for thepair (SO(2m+ 1), SO(2n)) corresponding to a pair

W ⊂ V

of quadratic spaces of odd codimension. As explained in [GP2], such a restriction prob-lem can be formulated in terms of Bessel models. One of our main results in Part 3 isthat the conjecture for Bessel models as formulated in [GP2] is a consequence of that for(SO(n), SO(n− 1)) as formulated in [GP1].

On the other hand, one can go one step further and consider symplectic representationsof the form

W ′k −→ Sp2m(C)× SO2n+1(C),

i.e. obtained by the tensor product of a symplectic representation and an orthogonalrepresentation of odd dimension. What restriction problem might this correspond to? Aswe shall see in Sections 22 and 24, the corresponding restriction problem is one for themetaplectic-symplectic pair

Sp(2m)× Sp(2n)

where Sp(2m) denotes the unique two-fold cover of Sp(2m). This restriction problem isformulated in terms of Fourier-Jacobi models. In particular, we are following the folklore

that the L-group of Sp(2m) is Sp2m(C). As we shall see in Section 23, this is confirmedby a recent result of Kudla-Rallis [KR] which classifies the irreducible representations of

Sp(2m) in terms of those of SO(2n+ 1).

Returning to unitary groups, we recall that the problem studied in Parts 1 and 2 is therestriction of an irreducible representation of U(n) to U(n− 1) and the proposed solutioninvolves the root number associated to a natural 2n(n− 1)-dimensional representation of

L(U(n)× U(n− 1)) = (GLn(C)×GLn−1(C)) o Gal(K/k).

As above, it is natural to consider restriction problems for a pair (U(W ),U(V )) = (U(m),U(n))where

W ⊂ V

is a pair of Hermitian spaces of odd codimension, so that the L-group

L(U(m)× U(n)) = (GLm(C)×GLn(C)) o Gal(K/k)


has a natural 2mn-dimensional symplectic representation. As we shall see in Section 20,the corresponding restriction problem is expressed in terms of the Bessel models. However,as in the orthogonal case, our conjecture for Bessel models for unitary groups turns out tobe a consequence of the conjectures formulated in Part 1 of this paper.

On the other hand, if one considers (U(m),U(n)) where m and n have the same parity,then the corresponding restriction problem is expressed in terms of Fourier-Jacobi models.In this case, the L-group of U(m)×U(n) actually has a natural orthogonal representation,but one which can be twisted slightly to create a symplectic representation as we will seein Section 25.

20. Bessel Models for Unitary Groups

As in [GP2], one can formulate a restriction problem from U(V ) to a smaller U(W )where W ⊂ V has odd codimension > 1. This proceeds by the consideration of Besselmodels. The purpose of this section is to introduce these Bessel models and to formulatea conjecture on their non-vanishing.

Let W be a Hermitian subspace of a Hermitian space V of odd codimension such thatW⊥ is a split Hermitian space. Hence, we may write

V = X ⊕W ⊕ 〈e〉 ⊕X∨

where X and X∨ are isotropic subspaces of V in duality with each other and e is a non-isotropic vector in W⊥. Let P (X) be the parabolic subgroup in U(V ) stabilizing thesubspace X, and let M(X) be the Levi subgroup of P (X) which stabilizes both X andX∨. Then

M(X) ∼= GLK(X)× U(W ⊕ 〈e〉)stabilizes Y = X ⊕X∨ and W ⊕ 〈e〉 as well. We have

P (X) = M(X) nN(X)

where N(X) is the unipotent radical of P (X) and sits in an exact sequence of M(X)-modules,

0→ Λ2X → N(X)→ X ⊗ (W ⊕ 〈e〉)→ 0

with Λ2(X) denoting the space of skew-Hermitian forms on X∨.

Let `1 : X → Ga be a nonzero homomorphism, and let

`W : W ⊕ 〈e〉 −→ Ga

be a nonzero homomorphism which is zero on the hyperplane W . This gives a map

`1 ⊗ `W : X ⊗ (W ⊕ 〈e〉) −→ Ga,

and one can consider the composite map,

m : N(X) −→ N(X)ab = X ⊗ (W ⊕ 〈e〉) −→ Ga.


The subgroup of M(X) which stabilizes the map m is

GL(X)`1 × U(W )

where GL(X)`1 is the stabilizer in GL(X) of the linear form `1. Let UX be the maximalunipotent subgroup in this mirabolic subgroup GL(X)`1 of GL(X). Define the subgroupN of P (X) by

N = UX nN(X).

Let `X : UX → Ga be a homomorphism which is nontrivial on each simple root space ofGL(X) in UX , so that `X is a generic character. There is then a unique homomorphism

` : N → Ga

which is equal to `X on the subgroup UX , and equal to m on the subgroup N(X). If ψ isa non-trivial additive character on k, then we obtain by composition a character

ψ` : N(k) −→ k −→ C×

of N .

Now we have the following proposition.

Proposition 20.1. The pair (N, `) is uniquely determined up to conjugacy in thegroup U(V ) by the pair W ⊂ V (with W⊥ split). In particular, the pair (N,ψ`) is alsodetermined up to conjugacy in U(V ) by W ⊂ V .

We define the Bessel subgroup BW,X to be

BW,X := SO(W ) nN = (SO(W )× UX) nN(X).

By the proposition, BW,X depends only on W ⊂ V up to conjugacy. Observe that U(W )fixes the character ψ` and so we may extend ψ` to BW,X so that it is trivial on U(W ). Thepair (BW,X , ψ`) only depends on W ⊂ V up to conjugacy. As a result, we may suppressthe mention of X, ` or ψ from further notation.

With the notation as above, we now come to the notion of a Bessel model.

Definition (Bessel model): Suppose that W ⊂ V are Hermitian spaces such that W⊥

is a split Hermitian space of odd dimension 2d + 1 (d ≥ 0). Let π and π0 be irreducibleadmissible representations of U(V ) and U(W ) respectively. Then the π0-Bessel model ofπ is the space

Bd(π, π0) := HomBW,X(π ⊗ (π0 ⊗ ψ`),C).

We say that π has π0-Bessel model if the above Hom space is nonzero.

Note that when d = 0, then W has codimension 1 in V and (N, `) is trivial, so that

B0(π, π0) = HomU(W )(π ⊗ π0,C)


is the space studied in Part 1 of this paper. Thus, the following conjecture should beregarded as an extension of Conjectures 6.1 and 6.2.

Conjecture 20.2.(1) For any irreducible representations π and π0 of U(V ) and U(W ),

dimBd(π, π0) ≤ 1.

(2) Fix a generic Langlands parameter ϕ for U(V ) × U(W ) with associated L-packetΠ(ϕ). Then ∑

π⊗π0∈Π(ϕ)

dimBd(π, π0) = 1,

where the space Bd(π, π0) is interpreted to be zero if π ⊗ π0 is a representation of anirrelevant U(V ′)× U(W ′).

(3) The unique representation π⊗π0 which has nonzero contribution to the sum in (2)is given by a character on the component group Aϕ, specified by the same recipe as inConjecture 6.1.

When k is non-Archimedean, we shall see in Corollary 21.3 that the multiplicity oneassertion in (1) follows from the multiplicity one theorem of [AGRS]. Moreover, we shallshow that (2) follows from Conjecture 6.1.

21. Compatibility of Various Conjectures

The aim of this section is to prove that the Bessel model conjecture formulated in theprevious section for general codimension 2d+ 1 follows from the case when d = 0, i.e. is aconsequence of Conjectures 6.1 and ??. Indeed, the same implication holds in the case oforthogonal groups, so that the expectations of [GP2] follow from those of [GP1]. Since itis easier to work with orthogonal groups than unitary groups, we shall give the details forthe orthogonal case; the proof works essentially verbatim for the unitary case.

We begin by fixing some notation. Let

V = Xm ⊕ Y ⊕X∨m

be an n-dimensional quadratic space over a non-Archimedean local field, with

Xm = 〈e1, · · · , em〉 and X∨m = 〈f1, · · · , fm〉

isotropic subspaces such that 〈ei, fj〉 = δij, and Y a non-degenerate quadratic space whichis orthogonal to Xm ⊕X∨

m. Let

emm = em − fm and fmm = em + fm,

so that with Xm−1 = 〈e1, · · · , em−1〉, we have

V = Xm ⊕ Y ⊕X∨m = Xm−1 ⊕ Y ⊕X∨

m−1 ⊕ emm ⊕ fmm.


Let P = P (Xm) be the parabolic subgroup of SO(V ) stabilizing the subspace Xm, andM the Levi subgroup of P stabilizing both Xm and X∨

m, so that

M ∼= GL(Xm)× SO(Y ).

Let τ be a supercuspidal representation of GL(Xm), and π0 an irreducible admissible rep-resentation of SO(Y ). Let

I(τ, π0) = IndSO(V )P τ � π0

be the (unnormalized) induced representation of SO(V ) from the representation τ ⊗ π0 ofP .

The aim of this section is to prove the following theorem, which computes the Homspace

B0(I(τ, π0), π) = HomSO(W )(I(τ, π0), π∨).

Theorem 21.1. Let

W = Xm−1 ⊕ Y ⊕X∨m−1 ⊕ emm

be a non-degenerate quadratic subspace of

V = Xm ⊕ Y ⊕X∨m

of codimension 1. Suppose that W has an m-dimensional isotropic subspace X ′m ⊃

Xm−1 such that

W ∼= X ′m ⊕ Y ′ ⊕X ′∨

m,

with Y ′ a codimension 1 subspace of Y . Let P (X ′m) be the parabolic subgroup of SO(W )

stabilizing X ′m with Levi subgroup GL(X ′

m)× SO(Y ′).

Let π be an irreducible admissible representation of SO(W ). Assume that π∨ does notbelong to the Bernstein component of SO(W ) associated to (GL(X ′

m)× SO(Y ′), τ ⊗ µ)for any irreducible representation µ of SO(Y ′). Then we have:

B0(I(τ, π0), π) ∼= Bm−1(π, π0).

In other words,

HomSO(W )(I(τ, π0), π∨) ∼= HomBY,Xm−1

(π, π∨0 ⊗ ψ`).

where (BY,Xm−1 , ψ`) is as defined in the previous section.

Proof. We calculate the restriction of Π := I(τ, π0) to SO(W ) by Mackey’s orbit method.For this, we begin by observing that SO(W ) has two orbits on the flag variety SO(V )/P (Xm)consisting of:

(1) m-dimensional isotropic subspaces of V which are contained in W ; a representativeof this orbit is the space X ′

m and its stabilizer in SO(W ) is the parabolic subgroupPW (X ′

m) = P (X ′m) ∩ SO(W );


(2) m-dimensional isotropic subspaces of V which are not contained in W ; a represen-tative of this orbit is the space Xm and its stabilizer in SO(W ) is the subgroupH = P (Xm) ∩ SO(W ).

By Mackey theory, this gives a filtration on the restriction of Π to SO(W ) as follows:

0 −−−→ indSO(W )H (τ ⊗ π0)|H −−−→ Π|SO(W ) −−−→ Ind

SO(W )PW (X′

m)τ ⊗ π0|SO(Y ′) −−−→ 0,

where the induction functors here are unnormalized.

By our assumption, π∨ does not appear as a quotient of the 3rd term of the above shortexact sequence and we have

HomSO(W )(indSO(W )H (τ ⊗ π0)|H , π∨) = HomSO(W )(Π, π

∨).

It thus suffices to analyze the representations of SO(W ) which appear on the open orbit.For this, we need to determine the group H = P (Xm)∩ SO(W ) as a subgroup of SO(W )and P (Xm).

Recall that V = Xm ⊕ Y ⊕ X∨m, and W is the codimension 1 subspace Xm−1 ⊕ Y ⊕

X∨m−1 ⊕ emm which is the orthogonal complement of fmm = em + fm. It is not difficult to

see that as a subgroup of SO(W ),

H = SO(W ) ∩ P (Xm) ⊂ PW (Xm−1).

Indeed, if g ∈ H, then g fixes em+ fm and stabilizes Xm, and we need to show it stabilizesXm−1. If e ∈ Xm−1, it suffices to show that 〈g · e , fm〉 = 0. But

〈g · e , fm〉 = 〈e , g−1 · fm〉 = 〈e , fm + em − g−1 · em〉 = 0,

as desired. Now we claim that

H = (GL(Xm−1)× SO(Y )) nNW (Xm−1) ⊂ PW (Xm−1).

To see this, given an element h ∈ PW (Xm−1), we need to show that h · em ∈ Xm if andonly if h belongs to the RHS above. We know that h · (em + fm) = em + fm and we maywrite

h · (em − fm) = λ · (em − fm) + y + e, with y ∈ Y and e ∈ Xm−1.

Hence, h · em ∈ Xm if and only if λ = 1 and y = 0. Thus, h fixes em − fm modulo Xm−1

and so stabilizes Y modulo Xm−1. This implies that h lies in (GL(Xm−1) × SO(Y )) nNW (Xm−1), so that we have a description of H as a subgroup of SO(W ).

Since we are restricting the representation τ � π0 of P (Xm) to the subgroup H, we alsoneed to know how H sits in P (Xm). for this, we have:


0 −−−→ N(Xm) −−−→ P (Xm) −−−→ GL(Xm)× SO(Y ) −−−→ 0x x x0 −−−→ N(Xm) ∩H −−−→ H −−−→ Em−1 × SO(Y ) −−−→ 0

where

Em−1 ⊂ GL(Xm)

is the mirabolic subgroup which stabilizes the subspace Xm−1 ⊂ Xm and fixes em. Notealso that N(Xm) ∩H ⊂ NW (Xm−1) and

NW (Xm)/(NW (Xm) ∩H) ∼= emm ⊗Xm−1.

As a consequence, one has:

(τ � π0)|H = τ |Em−1 � π0.

We now note the following well-known proposition due to Gelfand and Kazhdan.

Proposition 21.2. Let τ be a supercuspidal representation of GLm(k) and considerits restriction to the mirabolic subgroup Em−1. Then

τ |Em−1∼= ind

Em−1

Umψ

where Um is the maximal unipotent subgroup (of upper triangular unipotent matrices)in Em−1 and ψ is a generic character on Um.

By induction in stages, it follows from the proposition that

indSO(W )H (τ ⊗ π0)|H ∼= ind

SO(W )BY,Xm−1

π0 ⊗ ψ`.

Thus, by dualizing and Frobenius reciprocity, one has

HomSO(W )(Π(τ, π0), π∨) ∼= HomBY,Xm−1

(π, π∨0 ⊗ ψ`).

This completes the proof of the theorem. �

We note two corollaries of the theorem and [AGRS], both of which follow by applyingthe theorem for appropriate choices of the supercuspidal representation τ .

Corollary 21.3. The multiplicity 1 theorem holds for Bessel models of orthogonal andunitary groups. In other words, for any irreducible representations π and π0 of SO(V )and SO(W ) respectively,

dim HomU(W )·N(π, π0 ⊗ ψ`) ≤ 1.

Corollary 21.4. Conjecture 6.1 implies Conjecture 20.2(2) for tempered representa-tions.


In Theorem 21.1, we have calculated the branching from SO(V ) to SO(W ) assuming thatthe representation of the larger group SO(V ) is a principal series representation. One cansimilarly calculate the branching from SO(V ) to SO(W ) assuming that the representationof the smaller group SO(W ) is a principal series representation. We merely state the endresult.

Theorem 21.5. Let W ⊂ V be a non-degenerate quadratic subspace of codimension 1over a non-Archimedean local field k. Suppose that

W = Ym ⊕W0 ⊕ Y ∨m

and

V = Ym ⊕ V0 ⊕ Y ∨m ,

with Ym and Y ∨m isotropic subspaces and W0 ⊂ V0. Let PW (Ym) be the parabolic in

SO(W ) stabilizing Ym with Levi subgroup

M = GL(Ym)× SO(W0)

For an irreducible supercuspidal representation τ of GL(Ym) and an irreducible admis-sible representation π0 of SO(W0), let

I(τ, π0) = IndSO(W )P (Ym) τ � π0

be the corresponding (unnormalized) principal series representation of SO(W ). Let π bean irreducible admissible representation SO(V ) which does not belong to the Bernsteincomponent associated to (GL(Ym) × SO(V0), τ � µ) for any irreducible representationµ of SO(V0). Then

B0(π, I(τ, π0)) ∼= Bm(π, π0).

In other words,

HomSO(W )(π ⊗ I(τ, π0),C) ∼= HomSO(W0)·N(π ⊗ (π0 ⊗ ψ`),C).

22. Fourier-Jacobi Models for Symplectic Groups

The Bessel model conjecture addresses the restriction of representations from U(V ) toU(W ) when W is a subspace of V of odd codimension. One may ask for an analogousrestriction problem when W has even codimension in V . Such a restriction problem involvesthe so-called Fourier-Jacobi models. The natural setting for the consideration of Fourier-Jacobi models is the setting of symplectic groups. Because of the lack of a suitable referencein the literature, we shall discuss the case of symplectic groups in this section.

Let W0 be a non-degenerate symplectic subspace of a symplectic space W . Write

W = X ⊕W0 ⊕X∨


where X and X∨ are isotropic subspaces of W which are in duality with each other. LetP (X) = M(X) · N(X) be the parabolic subgroup in Sp(W ) stabilizing the subspace X,with Levi subgroup

M(X) ∼= GL(X)× Sp(W0)

stabilizing both X and X∨, and unipotent radical N(X) sitting in an exact sequence ofM -modules,

0→ Sym2X → N(X)→ X ⊗W0 → 0.

The unipotent group N(X) is thus a 2-step nilpotent group with Sym2(X) in its center.The commutator map

[−,−] : N(X)×N(X)→ N(X)

gives rise to a skew-symmetric bilinear form

Λ2(X ⊗W0)→ Sym2(X),

or equivalently by duality,

Sym2(X∨)→ Λ2(X∨ ⊗W0).

In our case, this last map is the reflection of the fact that, using the symplectic structureon W0, symmetric bilinear forms on X can be naturally embedded in the space of skew-symmetric bilinear forms on X ⊗W0.

Let

` : X → Ga

be a nonzero homomorphism. This gives rise to a linear map

Sym2(`) : Sym2(X)→ Ga,

as well as a map ` : X⊗W0 → W0. As a consequence, one has the following commutativediagram of groups, which realizes the Heisenberg group H(W0) as a quotient of N(X):

0 −−−→ Sym2(X) −−−→ N(X) −−−→ X ⊗W0 −−−→ 0

Sym2(`)

y y `

y0 −−−→ Ga −−−→ H(W0) −−−→ W0 −−−→ 0.

Therefore, given a nontrivial character ψ : k → C×, one may consider the uniqueirreducible representation ωW0,ψ of H(W0) with central character ψ and pulling back bythe above diagram, one obtains an irreducible representation ωW0,ψ`

of N(X) with centralcharacter

ψ` = ψ ◦ Sym2(`).


One knows that this representation of N(X) can be extended to an irreducible representa-

tion of Sp(W0) ·N(X), where Sp(W0) is the unique two fold cover of the symplectic groupSp(W0) (usually called the metaplectic group). The representation

ωW0,ψ,` of Sp(W0) ·N(X)

is called a Weil representation. This Weil representation depends on the central characterψ`, up to the action of k×2. Thus we see that there are #(k×/k×2) such Weil representa-tions.

Now note that the stabilizer in GL(X) of ` is a mirabolic subgroup GL(X)`. LetUX ⊂ GL(X)` be a maximal unipotent subgroup. Then we define the Jacobi groupassociated to W0 and X to be

JW0,X = (UX × Sp(W0) nN(X).

The group JW0,X depends only on the subspace W0 ⊂ W up to conjugacy in Sp(W ). IfψX be a generic character of UX , then we have the representation

ψX � ωW0,ψ`of JW0,X .

As we remarked above, the representation ωW0,ψ`depends on the central character ψ` up

to the action of k×2. Without loss of generality, we shall fix the homomorphism ` and allowψ to vary. Thus, we shall suppress ` from the notation henceforth. For example, we shallsimply write ωW0,ψ.

We now come to the notion of Fourier-Jacobi model.

Definition (Fourier-Jacobi model): Consider W = X ⊕W0 ⊕X∨ as above, whereW0 has codimension 2d in W . Fix an additive character ψ of k. Let π and π0 be irreducible

admissible representations of Sp(W ) and Sp(W0) respectively. The (π0, ψ)-Fourier-Jacobimodel of π is the space

FJψ,d(π, π0) = HomJW0,X(π ⊗ (ψX � (π0 ⊗ ωW0,ψ),C).

Observe that, in the above definition, the representation π0 is regarded as a representation

of Sp(W0)·H(W0) through the natural map Sp(W0)·H(W0)→ Sp(W0), so that π0⊗ωW0,ψ

is actually a representation of Sp(W0) ·N(X). Similarly, one has the analogous definition

if π is a representation of Sp(W ) and π0 is a representation of Sp(W0).

Note that the above definition makes sense even if d = 0. In that case, X = 0 andW0 = W , so that

FJψ,0(π, π0) = HomSp(W )(π ⊗ π0 ⊗ ωW,ψ,C).

This should be considered as the analog of the restriction problem from SO(n) to SO(n−1).


In Section 24, we shall state a conjecture concerning the non-vanishing of the spacesFJψ,d(π, π0). This conjecture is analogous to the Bessel model conjecture in the orthogonalcase [GP2]. Not surprisingly, we shall then show that the conjecture for FJψ,d follows fromthe case d = 0. This is a consequence of Theorem 22.1 below.

Before coming to Theorem 22.1, we recall that a parabolic subgroup P in Sp(W ) isnothing but the inverse image of a parabolic P in Sp(W ). It is known that the metaplecticcovering splits (uniquely) over unipotent subgroups, so for a Levi decomposition P = M ·N ,

it makes sense to talk of the corresponding Levi decomposition P = M · N in Sp(W ).Furthermore, we note that for a Levi subgroup of the form M = GL(X) · Sp(W0) inSp(W ),

M =(GL(X)× Sp(W0)

)/∆µ2

where GL(X) is a certain two fold cover of GL(X) defined as follows. As a set, we write

GL(X) = GL(X)× {±1},

and the multiplication is given by

(g1, ε1) · (g2, ε2) = (g1g2, ε1ε2 · (det g1, det g2)2),

where (−,−)2 denotes the Hilbert symbol on k× with values in {±1}.

The two fold cover GL(X) has a natural genuine 1-dimensional character

χψ : GL(X) −→ C×

defined as follows. The determinant map gives rise to a natural group homomorphism

det : GL(X) −→ GL(∧topX) = GL(1).

On the other hand, one has a genuine character on GL(1) defined by

(a, ε) 7→ ε · γ(a, ψ)−1,

where

γ(a, ψ) = γ(ψa)/γ(ψ)

and γ(ψ) is an 8-th root of unity associated to ψ by Weil. Composing this character with

det gives the desired genuine character χψ on GL(X), which satisfies:

χ2ψ(g, ε) = (det(g),−1)2.

Thus, there is a bijection between the set of irreducible representations of GL(X) and the

set of genuine representations of GL(X), given simply by

τ 7→ τψ = τ ⊗ χψ.


Note that this bijection depends on the additive character ψ of k. Now associated to a

representation τ of GL(X) and π0 of Sp(W0), one has the representation

τψ � π0 of M.

Then one can consider the (unnormalized) induced representation

Iψ(τ, π0) = IndcSp(W )bP (τψ ⊗ π0).

Here is the main result of this section, which reduces the computation of FJψ,d forgeneral d to the case d = 0. It is the analog of Theorem 21.1.

Theorem 22.1. Consider W = X ⊕W0 ⊕ X∨ as above with dimX = d and fix theadditive character ψ. Let

• τ be a supercuspidal representation of GL(X);

• π0 be a genuine representation of Sp(W0);

• π be an irreducible representation of Sp(W ),

and consider the (unnormalized) induced representation Iψ(τ, π0) of Sp(W ). Assumethat π∨ does not belong to the Bernstein component associated to (GL(X)×Sp(W0), τ�µ) for any representation µ of Sp(W0). Then

FJψ,0(Iψ(τ, π0), π) ∼= FJψ,d(π, π0).

In other words,

HomSp(W )(Iψ(τ, π0)⊗ π ⊗ ωW,ψ,C) ∼= HomJW0,X(π ⊗ (ψX � (π0 ⊗ ωW0,ψ)),C).

Proof. Let P = P (X) = M(X) · N(X) be the parabolic subgroup in Sp(W ) stabilizingthe subspace X. Recall that

M(X) ∼= GL(X)× Sp(W0))/∆µ2

and we have fixed an element

` ∈ Hom(X,Ga) ∼= X∨

which leads to a quotient mapN(X) −→ H(W0).

Now the Weil representation ωW,ψ has a convenient description as a P -module; this isthe so-called mixed model of the Weil representation. This model of ωW,ψ is realized onthe space

S(X∨)⊗ ωW0,ψ

of Schwarz-Bruhat functions on X∨ valued in ωW0,ψ. In particular, evaluation at 0 gives a

P -equivariant mapev : ωW,ψ −→ χψ|detX |1/2 � ωW0,ψ,


where N(X) acts trivially on the target space. In fact, this map is the projection of ωW,ψonto its space of N(X)-coinvariants. From this, one deduces the following short exact

sequence of P -modules:

0 −−−→ indbPbP`χψ|detX |1/2 � ωW0,ψ`

−−−→ ωW,ψev−−−→ χψ|detX |1/2 � ωW0,ψ`

−−−→ 0,

where

P` =((GL(X)` × Sp(W0))/∆µ2

)·N(X)

is the stabilizer of ` in P and its action on ωW0,ψ`is via the Weil representation of Sp(W0) ·

N(X). Moreover, the compact induction functor ind is unnormalized.

Tensoring the above short exact sequence by τψ � π0 and then inducing to Sp(W ), onegets a short exact sequence of Sp(W )-modules:

0yind

Sp(W )P`

|detX |1/2 · χ2ψ · τ |GL(X)`

⊗ (π0 ⊗ ωW0,ψ) = AyIψ(τ, π0)⊗ ωW,ψ = By

IndSp(W )P (τ · χ2

ψ|detX |1/2 ⊗ (π0 ⊗ ωW0,ψ)) = Cy0.

By our assumption on π,

HomSp(W )(C, π∨) = 0 and HomSp(W )(B, π

∨) = HomSp(W )(A, π∨).

Moreover, by Proposition 21.2,

τ |GL(X)`∼= ind

GL(X)`

UXψX ,

so that

A = indSp(W )JW0,X

(ψX � π0)⊗ ωW0,ψ).

Therefore, the desired result follows by Frobenius reciprocity.�


23. Local Langlands Conjecture for Sp(2n)

In this section, we shall discuss the local Langlands correspondence for the metaplectic

groups Sp(W ). This is necessary for the statement of the conjecture for Fourier-Jacobimodels given in the next section.

Though Sp(W ) is not a linear group, it is a folklore that were it to have a Langlandsdual group at all, the only possible candidate is the symplectic group Sp2n(C) (where

2n = dimW ). This expectation is justified for the group Sp(2) by the fundamental workof Waldspurger [W], and over R by the work of Adams-Barbasch [AB]. In fact, one has thefollowing theorem of Kudla-Rallis [KR]:

Theorem 23.1. (Kudla-Rallis) Suppose that the residue characteristic of k is odd.Relative to the choice of an additive character ψ of k, there is a natural bijectionbetween

Π(Sp(2n)) := {irreducible genuine representations of Sp(2n, k)}and the disjoint union

Π(SO(2n+ 1)) ∪ Π(SO∗(2n+ 1))

of the set of irreducible representations of the split SO(2n + 1, k) and the non-splitSO∗(2n+ 1, k).

Before giving a sketch of the proof of this theorem, we deduce the following corollary:

Corollary 23.2. Assume that the residue characteristic of k is odd. Suppose thatthe local Langlands conjecture as formulated in Section 4 holds for SO(2n + 1) andSO∗(2n+ 1). Then one has a bijection (depending on ψ)

Π(Sp(2n))←→ Φ(Sp(2n)),

where Φ(Sp(2n)) is the set of conjugacy classes of pairs (ϕ, ρ) such that

ϕ : W ′k −→ Sp2n(C)

is a Langlands parameter and ρ is an irreducible character of the component group Aϕ.

Proof. The reader will not find Theorem 23.1 in the reference [KR], so let us explain howit follows from the results there. The natural bijection of the Theorem is given by the thetacorrespondence between the dual pairs

Sp(2n)×

{O(2n+ 1);

O∗(2n+ 1).

This theta correspondence depends on the choice of ψ, but since we will be fixing ψ, wesuppress it from the notation.


Given an irreducible representation σ of Sp(2n), we shall let Θ(σ) (respectively Θ∗(σ))denote the big theta lift of σ to O(2n + 1) (resp. O∗(2n + 1)), and we let θ(σ) (resp.θ∗(σ)) be the maximal semisimple quotient of Θ(σ) (resp. Θ∗(σ)). Similarly, starting witha representation π of O(2n+1) or O∗(2n+1), one has the representations Θ(π) and θ(π)

of Sp(2n).

Since the residue characteristic of k is odd, one knows by a result of Waldspurger (prov-ing the so-called Howe’s conjecture) that the representations θ(σ), θ∗(σ) and θ(π) areirreducible or zero. This is the only place where we use the assumption on the residuecharacteristic of k.

We now divide the proof into two steps:

(i) Given an irreducible representation σ of Sp(2n), exactly one of Θ(σ) or Θ∗(σ) is nonzero.

Indeed, [KR, Thm. 3.8] shows that any irreducible representation σ of Sp(2n) participatesin theta correspondence with at most one of O(2n+ 1) or O∗(2n+ 1). We claim howeverthat σ does have nonzero theta lift to O(2n + 1) or O∗(2n + 1). To see this, note that[KR, Prop. 4.1] shows that σ has nonzero theta lift to one of O(2n+ 1) or O∗(2n+ 1) ifand only if

HomcSp(2n)×cSp(2n)(IP (0), σ � σ∨) 6= 0.

Here, IP (s) denotes the degenerate principal series representation of Sp(4n) unitarily in-duced from the character χψ · | det |s of the Siegel parabolic subgroup. We thus need toshow that this Hom space is nonzero. This can be achieved by the doubling method ofPiatetski-Shapiro and Rallis, which provides a zeta integral

Z(s) : IP (s)⊗ σ ⊗ σ∨ −→ C.The precise definition of Z(s) need not concern us here; it suffices to note that for a flatsection Φ(s) ∈ IP (s) and f ⊗ f∨ ∈ σ⊗ σ∨, Z(s,Φ(s), f ⊗ f∨) is a meromorphic functionin s. Moreover, at any s = s0, the leading term of the Laurent expansion of Z(s) gives anonzero element

Z∗(s0) ∈ HomcSp(2n)×cSp(2n)(IP (s0), σ � σ∨).

This proves our contention that σ participates in the theta correspondence with exactly oneof O(2n+ 1) or O∗(2n+ 1).

By (i), one obtains a map

Π(Sp) −→ Π(O(2n+ 1)) ∪ Π(O∗(2n+ 1)).

Moreover, this map is injective by the result of Waldspurger alluded to above.

(ii) An irreducible representation π0 of SO(2n + 1) has two extensions to O(2n + 1) =SO(2n + 1) × 〈±1〉, and exactly one of these extensions participates in the theta corre-

spondence with Sp(2n).


Suppose on the contrary that π is an irreducible representation of O(2n + 1) such that

both π and π ⊗ det participates in theta correspondence with Sp(2n), say

σ = θ(π) and σ′ = θ(π ⊗ det).

Now consider the seesaw diagram:

Sp(4n)@

@@

@@

@@@

��

��

��

��

O(2n+ 1)×O(2n+ 1)

Sp(2n)× Sp(2n) O(2n+ 1)

The seesaw identity implies that

HomcSp(2n)×cSp(2n)(Θ(det), σ′ � σ∨) ⊃ HomO(2n+1)((π ⊗ det)⊗ π∨, det) 6= 0.

This implies that Θ(det) 6= 0. However, a classical result of Rallis says that the determinant

character of O(2n + 1) does not participate in the theta correspondence with Sp(2r) forr ≤ n. This gives the desired contradiction.

We have thus shown that at most one of π or π ⊗ det could have nonzero theta lift toSp(2n). On the other hand, the analog of the zeta integral argument in (i) shows that one

of π or π ⊗ det does lift to Sp(2n). This proves (ii).

Putting (i) and (ii) together, we see that the composite map

Π(Sp(2n)) −→ Π(O(2n+ 1)) ∪ Π(O∗(2n+ 1)) −→ Π(SO(2n+ 1)) ∪ Π(SO∗(2n+ 1))

provided by theta correspondence is bijective, as desired. �

As we remarked in the proof of the theorem, the only reason for the assumption of oddresidue characteristic is that Howe’s conjecture for local theta correspondence is only knownunder this assumption. In the rest of the paper, we shall assume that Howe’s conjecture isknown for even residue characteristic as well, so that the results of Theorem 23.1 can beapplied.

We conclude this section with a brief description of the local Langlands conjecture forSp(2m). The dual group of Sp(2m) is the group SO2m+1(C), which is adjoint. Thus, oneexpects that the set Π(Sp(2m)) of irreducible representations of Sp(2n) is in bijection withthe set of conjugacy classes of pairs (ϕ, ρ) where

ϕ : W ′k −→ SO2m+1(C)


is a Langlands parameter and ρ is an irreducible character of the component group Aϕ.

Putting the above discussions together, we see that a Langlands parameter for the group

Sp(2m)× Sp(2n) is a homomorphism

ϕ : W ′k −→ SO2m+1(C)× Sp2n(C).

Since the group SO2m+1(C)×Sp2n(C) has a natural symplectic representation given by thetensor product, we may regard ϕ as a symplectic representation of dimension 2n · (2m+1).It is this symplectic representation (which depends on ψ) which controls the behavior ofFourier-Jacobi models for symplectic and metaplectic groups.

24. Conjectures for Fourier-Jacobi Models

In this section, we will give a conjecture for the non-vanishing of the Fourier-Jacobimodels FJψ,k, which is analogous to the Bessel model conjecture given earlier. We shall

make use of the Langlands correspondence for Sp(2n) and Sp(2n) described in the previoussection.

Conjecture 24.1. Suppose that W0 and W are symplectic spaces such that | dimW −dimW0| = 2d.

(1) For any irreducible (genuine) representations π and π0 of Sp(W ) and Sp(W0),

dimFJψ,d(π, π0) ≤ 1

(2) Let ϕ be a generic Langlands parameter for Sp(W )× Sp(W0). Then∑π⊗π0∈Π(ϕ)

dimFJψ,d(π, π0) = 1.

(3) The unique representation π⊗π0 which has nonzero contribution to the sum in (2)is given by a character on the component group Aϕ, specified by the same recipe as in[GP1, Conj].

In fact, Theorem 22.1 implies:

Proposition 24.2. Conjecture 24.1 for general d follows from the case d = 0.

We shall now focus on the Fourier-Jacobi conjecture for FJψ,0. In particular, we shallsee that the Fourier-Jacobi conjecture for FJψ,0 is related to the Bessel conjecture for B0,i.e. the restriction problem from SO(n) to SO(n − 1). This link is again provided by thetheta correspondence.

More precisely, consider the see-saw diagram:


O(2n+ 2)@

@@

@@

@@@

��

��

��

��

Sp(2n)× Sp(2n)

O(2n+ 1)×O(1) Sp(2n)

Let π be an irreducible representation of SO(2n+1) and σ an irreducible representationof Sp(2n). Then the local see-saw identity says that

dim HomSp(2n)(Θψ(π)⊗ ω∨ψ , σ) = dim HomSO(2n+1)(Θψ(σ), π),

where Θψ(π) denotes the big theta lift of π. As a result of this identity, and the multiplicityone theorems of [AGRS], we have:

Theorem 24.3. (i) Suppose that σ is a discrete series representation of Sp(2n). Then

for any irreducible representation σ0 of Sp(2n), we have

dim HomSp(2n)(σ ⊗ σ0 ⊗ ωψ,C) ≤ 1.

(ii) Suppose further that π is a discrete series representation of SO(2n+ 1). Then

HomSp(2n)(σ∨ ⊗ θ(π)⊗ ω∨ψ ,C) ∼= HomSO(2n+1)(θψ(σ)⊗ π∨,C).

Proof. This follows by the see-saw identity, the multiplicity one theorems of [AGRS] and aresult of Muic which says that, for the dual pairs under consideration, Θψ(π) = θψ(π) fordiscrete series representations π. �

It follows from the theorem that, modulo issues of Θψ versus θψ and assuming that thelocal theta correspondence is described in terms of the Langlands parameter in the expectedway, the conjectures for B0 and FJψ,0 are essentially equivalent.

25. Fourier-Jacobi Models for Unitary Groups

In this section, we discuss the conjecture about Fourier-Jacobi models of unitary groups.

Let K be a quadratic extension of a field k and let W be a vector space over K, equippedwith a non-degenerate skew-Hermitian form:

〈−,−〉 : W ×W −→ K.

The automorphism group U(W ) of the skew-Hermitian space is a unitary group. Indeed,if δ ∈ K× is a trace zero element, then multiplication by δ gives a bijection betweenskew-Hermitian forms and Hermitian forms on W .


Let W0 be a non-degenerate subspace of W with

W = X ⊕W0 ⊕X∨

where X and X∨ are isotropic subspaces of W in duality with each other. Let P (X) bethe parabolic subgroup in U(W ) stabilizing the subspace X, and M(X) the Levi subgroupof P (X) which stabilizes both X and X∨, so that

M(X) = GLK(X)× U(W0).

The unipotent radical N(X) of P (X) sits in an exact sequence of M(X)-modules,

0 −−−→ Sym2(X, ε) −−−→ N(X) −−−→ X ⊗W0 −−−→ 0,

where Sym2(X, ε) denotes the space of Hermitian 2-forms on X∨. Thus, N(X) is a 2-stepnilpotent group with Sym2(X, ε) as its center. The commutator map

[−,−] : N(X)×N(X) −→ N(X)

gives rise to a linear mapΛ2k(X ⊗W0) −→ Sym2(X, ε)

where Λ2k(X ⊗W0) denotes 2nd exterior power of X ⊗K W0 considered as a vector space

over k. By duality, we get a k-linear map

Sym2(X∨, ε) −→ Λ2(X∨ ⊗W0).

This last map is a reflection of the fact that, since W0 has a skew-Hermitian structure,Hermitian forms on X can be embedded in the space of skew-symmetric k-bilinear formson X ⊗W0 (by taking the K/k-trace of the form on the tensor product).

Let` : X → K

be a nonzero K-linear homomorphism. This gives rise to a k-linear map

Sym2(`) : Sym2(X, ε)→ k

by the recipe

Sym2(`)(v ⊗ v) = `(v)`(v) for v ∈ X.

The map ` : X → K also gives rise to a map

W0 ⊗X −→ W0,

and one obtains the following commutative diagram of groups, which realizes the Heisenberggroup H(W0) as a quotient of N(X):

0 −−−→ Sym2(X, ε) −−−→ N(X) −−−→ X ⊗W0 −−−→ 0y y y0 −−−→ k −−−→ H(W0) −−−→ W0 −−−→ 0.

Here, we are regarding W0 as a k-vector space with a non-degenerate symplectic form


inherited from the given skew-Hermitian form (by taking the trace from K to k of theskew-Hermitian form); this gives us an embedding

U(W0) ↪→ Sp(W0).

Fix a non-trivial additive character ψ of k. As before, we have the unique irreduciblerepresentation of H(W0) with central character ψ, and pulling this back to N(X) by theabove diagram, we obtain an irreducible representation of N(X) with central character

ψ` = ψ ◦ Sym2(`).

Further, one knows that this representation of N(X) can be extended to an irreduciblerepresentation of U(W0) nN(X). This extension is, however, non-unique since U(W0) isnot its own commutator. In fact, the choice of a character µ of K× for which

µ|k× = ωK/k

determines such an extension. Therefore we have an irreducible representation

ωW0,ψ,`,µ of U(W0) nN(X),

which we call a Weil representation.

Now note that the stabilizer in GL(X) of ` is a mirabolic subgroup GL(X)`. LetUX ⊂ GL(X)` be a maximal unipotent subgroup. Then we define the Jacobi groupassociated to W0 and X by

JW0,X = (UX × U(W0)) nN(X).

The group JW0,X depends only on the subspace W0 ⊂ W up to conjugacy in U(W ). If ψXis a generic character of UX , then we have the representation

ψX � ωW0,ψ,`,µ of JW0,X .

We now come to the notion of Fourier-Jacobi model in the unitary setting:.

Definition (Fourier-Jacobi model): Consider W = X ⊕W0 ⊕X∨ as above, whereW0 has codimension 2d in W . Fix an additive character ψ of k and a character µ ofK× with µ|k× = ωK/k. Let π and π0 be irreducible representations of U(W ) and U(W0)respectively. Then the (π0, ψ, µ)-Fourier-Jacobi model of π is the space

FJψ,µ,d(π, π0) = HomJW0,X(π ⊗ (ψX � (π0 ⊗ ωW0,ψ,µ)),C).

Note that when d = 0, so that X = 0 and W0 = W , we have

FJψ,µ,0(π, π0) = HomU(W )(π ⊗ π0 ⊗ ωW,ψ,µ,C).

This should be considered the basic Fourier-Jacobi model of unitary groups.

Before stating the conjecture for Fourier-Jacobi models in the unitary case, we note thefollowing lemma.


Lemma 25.1. The L-group of U(m)×U(n) has a natural orthogonal representation ofdimension 2mn if m ≡ n mod 2, and a symplectic representation of dimension 2mnif m 6≡ n mod 2.

Proof. Let X be a complex vector space of dimension m, and let X⊕X∨ be equipped withits canonical bilinear form which is symmetric if m is odd, and skew-symmetric if m is even.Similarly let Y be a complex vector space of dimension n. The L-group of U(m) has anatural embedding into the isometry group of this bilinear form on X⊕X∨ leaving invariantthe set X

⊔X∨. Given bilinear forms on X ⊕X∨ and Y ⊕ Y ∨, there is a natural bilinear

form on the tensor product space, (X ⊕X∨)⊗ (Y ⊕Y ∨), in which X ⊗Y ⊕X∨⊗Y ∨ is anon-degenerate subspace containing X⊗Y and X∨⊗Y ∨ as isotropic subspaces in naturalduality with each other. Clearly, X⊗Y

⊔X∨⊗Y ∨ is left invariant under GL(X)×GL(Y ),

as well as for the tensor product of any automorphism of X ⊕X∨ which interchanges Xand X∨ with an automorphism of Y ⊕ Y ∨ which interchanges Y and Y ∨.

Clearly, if both X⊕X∨ and Y ⊕Y ∨ are quadratic spaces, or are both symplectic spaces,then X⊗Y ⊕X∨⊗Y ∨ is a quadratic space, providing the natural orthogonal representationof the L-group of U(m)× U(n). �

Remark : The co-ordinate free approach to the L-group of U(n) that we have used in theproof of the previous lemma, can also be used to give an embedding of L(U(n1)×U(n2)×· · · × U(nk)) into LU(n1 + n2 + · · · + nk). This needs an introduction of a characterµ of K× such that µ|k× = ωK/k for factors U(ni) for which the parity of ni is not thesame as that of n1 + · · ·+ nk modulo 2 to flip the parity of the bilinear form on Xi ⊕X∨

i

to match that of (X1 ⊕ · · · ⊕ Xk) ⊕ (X1 ⊕ · · · ⊕ Xk)∨. Note that there is a bijection

between characters of K× trivial on k× and characters of K× whose restriction to k× isωK/k obtained by multiplication by this character µ of K× which establishes an injection ofthe set of 2 dimensional orthogonal representations of WK/k into 2 dimensional symplecticrepresentations by sending Ind(α) to Ind(αµ) where the character α of K× is trivial on k×;thus ‘multiplication by µ’ has the effect of turning parameters with values in O(Xi ⊕X∨

i )preserving Xi

⊔X∨i into parameters with values in Sp(Xi⊕X∨

i ) preserving Xi

⊔X∨i giving

rise to an embedding of L(U(n1)× U(n2)× · · · × U(nk)) into LU(n1 + n2 + · · ·+ nk).

Returning now to the Fourier-Jacobi models for the unitary group, since we are in thesituation of

dimW ≡ dimW0 mod 2,

if ϕ is a Langlands parameter for U(W )× U(W0) and

ϕK : W ′K −→ GL(V )×GL(V0)

is its restriction toW ′K , then the representation Ind(V ⊗V0) is an orthogonal representation.

However, as noted in the remark above, by a slight modification, we can create a symplecticrepresentation as follows. Choose a character µ of K× such that

µ|k× = ωK/k.


Then for

ϕK,µ = ϕK ⊗ µwhere we have regarded µ as a 1-dimensional character of WK by local class field theory,

ϕµ := Ind(ϕK,µ)

is a symplectic representation.

Here is the conjecture about Fourier-Jacobi models of unitary groups:

Conjecture 25.2. Suppose that W0 ⊂ W are skew-Hermitian spaces such that dimW−dimW0 = 2d. Fix a non-trivial additive character ψ of k and a character µ of K×

such that µ|k× = ωK/k.

(1) For any irreducible representations π and π0 of U(W ) and U(W0),

dimFJψ,µ,d(π, π0) ≤ 1

(2) Let ϕ be a generic Langlands parameter for U(W )× U(W0). Then∑π⊗π0∈Π(ϕ)

dimFJψ,µ,d(π, π0) = 1.

(3) The unique representation π⊗π0 which has nonzero contribution to the sum in (2)is given by a character on the component group Aϕµ, specified by the same recipe as inConjecture.

As in the previous cases, the following result reduces the computation of FJψ,µ,d to thatof FJψ,µ,0. Since the proof is similar to that of Theorem 22.1, we shall omit the details.

Theorem 25.3. Consider W = X ⊕W0 ⊕X∨ with dimX = d and fix ψ and µ as inConjecture 25.2. Let

• τ be a supercuspidal representation of GL(X);

• π0 be an irreducible representation of U(W0);

• π be an irreducible representation of U(W ),

and consider the (unnormalized) induced representation

I(τ, π0) = IndU(W )P (X) τ � π0.

Assume that π∨ does not belong to the Bernstein component associated to (GL(X) ×U(W0), τ � µ) for any representation µ of U(W0). Then

FJψ,µ,0(I(τ, π0), π) ∼= FJψ,µ,d(π, π0).


In other words,

HomU(W )(I(τ, π0)⊗ π ⊗ ωW,ψ,µ,C) ∼= HomJW0,X(π � (π0 ⊗ ωW0,ψ,µ)),C).

The theorem allows one to focus on the basic Fourier-Jacobi model FJψ,µ,0. As in thecase of symplectic groups, one can relate the Fourier-Jacobi model FJψ,µ,0 to the BesselB0, via the following seesaw diagram:

U(n+ 1)@

@@

@@

@@@

��

��

��

��

U(n)× U(n)

U(n)× U(1) U(n)

In particular, using the seesaw identity, the results of Harris-Kudla-Sweet [HST] and theresult of [AGRS], one concludes:

Theorem 25.4. Let π and π0 be irreducible representations of U(W ) and suppose thatπ is supercuspidal. Then

dimFJψ,µ(π, π0) ≤ 1.

Ignoring issues of Θ versus θ and assuming that the local theta correspondence behavesin an expected way with respect to the local Langlands conjecture, one can make a preciserelation between FJψ,µ,0 and B0. We leave the somewhat intricate details to the interestedreader.

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W.T.G.: Department of Mathematics, University of California at San Diego, 9500Gilman Drive, La Jolla, 92093

E-mail address: [email protected]

B.H.G: Department of Mathematics, Harvard University, Cambridge, MA 02138E-mail address: [email protected]

D.P.: School of Mathematics, Tata Institute of Fundamental Research, Colaba,Mumbai-400005, INDIA

E-mail address: [email protected]

SYMPLECTIC LOCAL ROOT NUMBERS, CENTRAL CRITICAL L … › ~dprasad › ggp5.pdf · for the orthogonal groups. We verify these conjectures in certain low rank cases and for depth zero

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