The supercuspidal representations of p-adic classical groups

The supercuspidal representations of p-adicclassical groups

Shaun Stevens

School of MathematicsUniversity of East Anglia

Norwich UK

Conference in honour of Phil Kutzko’s 60th birthdayIowa City, October 2006

Shaun Stevens Supercuspidal representations

The set-upBeta extensions

Exhaustion

NotationsStrataCharacters

Notations

F a locally compact non-archimedean local field,oF its ring of integers,pF its maximal ideal,kF = oF/pF the residue field, of characteristic p.

Assumptionp 6= 2

F equipped with a (possibly trivial) galois involution x 7→ x ,with fixed field F0.ψ0 an additive character of F0 with conductor pF0 , andψF = ψ0 ◦ trF/F0 .



Exhaustion


Notations

V an N-dimensional F -vector space equipped with anon-degenerate ε-hermitian form h (where ε = ±1):

h(λv ,w) = λh(v ,w) = ελh(w , v), for v ,w ∈ V , λ ∈ F .

Assumption (for this talk)If F = F0 then ε = −1 (i.e. no orthogonal groups).

Adjoint (anti-)involution a 7→ a on A = EndF (V ), given by

h(av ,w) = h(v ,aw), for v ,w ∈ V .



Exhaustion


Notations

A = EndF (V ) ∼= M(N,F )

G = AutF (V ) ∼= GL(N,F )

Involution σ:

σ(a) = −a, for a ∈ Aσ(g) = g−1, for g ∈ G

G = Gσ, the points of a symplectic or unitary group over F0

A = Aσ ∼= Lie G



Exhaustion


Lattice sequences

An oF -lattice sequence in V is a function Λ : Z → LattoF (V )such that

Λ is decreasing: if i > j then Λ(i) ⊆ Λ(j);there exists e such that pF Λ(k) = Λ(k + e), for all k ∈ Z.

An oF -lattice sequence Λ is self-dual if, for all k ∈ Z,

{v ∈ V : h(v ,Λ(k)) ⊂ pF} = Λ(1− k).



Exhaustion


Filtrations

For Λ an oF -lattice sequence, we have a filtration on A:

an(Λ) = {a ∈ A : aΛ(k) ⊂ Λ(k + n) for all k ∈ Z}.

Put P(Λ) = a0(Λ) ∩ G, a parahoric subgroup of G, with filtration

Pn(Λ) = 1 + an(Λ), for n ≥ 1.



Exhaustion


Filtrations

If Λ is self-dual then the filtrations are stable under theinvolution σ. Put:

an = an ∩ A, giving a filtration of A;

P = P ∩G, a compact open subgroup of G;

Pn = Pn ∩G, a filtration of P.

Note

G = P/P1 is a possibly-disconnected reductive group over kF0 .The inverse image Po in P of its connected component is aparahoric subgroup of G.



Exhaustion


Strata

Definition

A stratum in A is a 4-tuple [Λ,n, r , β], whereΛ is an oF -lattice sequence;n > r ≥ 0 are integers;β ∈ a−n(Λ).

A stratum [Λ,n, r , β] is skew if Λ is self-dual and β ∈ A.

If 2r ≥ n, a stratum corresponds to the character ψβ of Pr/Pn

given by

ψβ(p) = ψF ◦ trA/F (β(p − 1)), for p ∈ Pr ;

and a skew stratum corresponds to the character ψβ = ψβ|Pr .



Exhaustion


Simple strata

DefinitionA stratum [Λ,n, r , β] is simple if

β 6∈ a1−n(Λ);E = F [β] is a field and E× normalizes Λ;k0(β,Λ) < −r .

If [Λ,n, r , β] is simple then we set GE = AutE(V ), the centralizerof E in G.

If the simple stratum is also skew, the involution on F extendsto an involution on E , with fixed field E0. We set GE = GE ∩G,a unitary group over E0.



Exhaustion


Simple strata

Attached to a simple stratum [Λ,n,0, β] are compact opensubgroups

H1(β,Λ) ⊂ J1(β,Λ) ⊂ J(β,Λ).

If the simple stratum is also skew, these groups are invariantunder σ and we set

H1 = H1 ∩G, J1 = J1 ∩G, J = J ∩G.

As before, J/J1 is a possibly-disconnected reductive groupover kE0 and we denote by Jo the inverse image in J of itsconnected component.



Exhaustion


Simple characters

Attached to a simple stratum [Λ,n,0, β], there is also a setC(β,Λ) of simple characters of H1. Among their properties are:

Intertwining for θ ∈ C(β,Λ), we have IeG(θ) = J1GE J1;

Transfer if [Λ′,n′,0, β] is another simple stratum, there isa canonical bijection

τΛ,Λ′,β : C(β,Λ) → C(β,Λ′);

θ′ = τΛ,Λ′,β(θ) if and only if GE intertwines θ with θ′.



Exhaustion


Semisimple strata

Suppose we have a decomposition V =⊕l

i=1 V i . Then we put

Ai = EndF (V i), M =l⊕

i=1

Ai , M = M×.

If Λ is an oF -lattice sequence in V , we define lattice sequencesΛi in V i by

Λi(k) = Λ(k) ∩ V i , for k ∈ Z.



Exhaustion


Semisimple strata

DefinitionA stratum [Λ,n,0, β] is semisimple if there is a decompositionV =

⊕li=1 V i such that:

Λ(k) =⊕l

i=1 Λi(k), for all k ∈ Z;

β ∈M, and we write β =∑l

i=1 βi , with βi ∈ Ai ;either [Λi ,ni ,0, βi ] is simple or βi = 0 (and at most oneβi = 0);[Λi ⊕ Λj ,max {ni ,nj},0, βi + βj ] is not equivalent to a simplestratum.

We write E = F [β] =⊕l

i=1 Ei , with Ei = F [βi ], andGE =

∏li=1 AutEi (V

i).



Exhaustion


Semisimple characters

Attached to a semisimple stratum [Λ,n,0, β] with splittingV =

⊕li=1 V i , we have compact open subgroups

H1(β,Λ) ⊂ J1(β,Λ) ⊂ J(β,Λ).

These have the property that H1 ∩ M =∏l

i=1 H1(βi ,Λi), etc.

We also have a set C(β,Λ) of semisimple characters of H1. Ifθ ∈ C(β,Λ) then

θ|eH1∩eM =l⊗

i=1

θi ,

with θi a simple character in C(βi ,Λi).



Exhaustion


Semisimple characters

Moreover, semisimple characters have the same intertwiningand transfer properties as simple characters:

Intertwining for θ ∈ C(β,Λ), we have IeG(θ) = J1GE J1;

Transfer if [Λ′,n′,0, β] is another semisimple stratum,there is a canonical bijection

τΛ,Λ′,β : C(β,Λ) → C(β,Λ′);




Exhaustion


Skew semisimple characters

Definition

A semisimple stratum [Λ,n,0, β] with splitting V =⊕l

i=1 V i isskew if the decomposition is orthogonal with respect to the formh, and each stratum [Λi ,ni ,0, βi ] is skew.

In this case, all the groups H1 etc. are invariant under σ, as isthe set of semisimple characters. We put H1 = H1 ∩G, etc.

We also put GE = GE ∩G =∏l

i=1 GEi , a product of unitarygroups (and at most one symplectic group).



Exhaustion



Definition

A skew semisimple character is the restriction to H1 of aσ-invariant semisimple character θ ∈ C(β,Λ).

Write C(β,Λ) for the set of skew semisimple characters. Then:

Intertwining for θ ∈ C(β,Λ), we have IG(θ) = J1GEJ1;

Transfer if [Λ′,n′,0, β] is another skew semisimplestratum, there is a canonical bijection

τΛ,Λ′,β : C(β,Λ) → C(β,Λ′);




Exhaustion



TheoremLet π be an irreducible supercuspidal representation of G. Thenthere exists a skew semisimple character θ ∈ C(β,Λ) such thatπ|H1 contains θ.

Theorem (Dat)Let M be a Levi subgroup of G and let π be an irreduciblesupercuspidal representation of M. Then there exists aself-dual semisimple character θ ∈ C(β,Λ) such that

θ is decomposed with respect to (M,P) (for P anyparabolic with Levi M);π|H1∩M contains θ|H1∩M .



Exhaustion

ConstructionsMaximal types

Beta extensions

Let θ ∈ C(β,Λ) be a skew semisimple character.

It is straightforward to pass from H1 to J1: there is a uniqueirreducible representations η of J1 which contains θ.

The problem is to extend η to a “nice” representation of J.

In the case of simple characters of G, “nice” means “intertwinedby all of GE ”.



Exhaustion


Extension to a Sylow pro-p subgroup

Let [Λm,nm,0, β] be a skew semisimple stratum such thatPo(Λm) ∩GE is an Iwahori subgroup of GE contained in P(Λ).Put

J1 = (P1(Λm) ∩GE)J1,

a Sylow pro-p subgroup of J.

Using the transfer property of semisimple characters, we prove:

Lemma

There is a unique extension η of η to J1 which is intertwined byall of GE . Moreover,

dim Ig(η) =

{1 if g ∈ J1GE J1,

0 otherwise.



Exhaustion


The maximal case

Suppose P(Λ) ∩GE is a maximal compact open subgroupof GE . The same methods as for GL(N,F ) show:

PropositionThere is an extension κ of η to J.

We call such an extension a β-extension.



Exhaustion


The general case

Let [ΛM,nM,0, β] be skew semisimple such that P(ΛM) ∩GE is amaximal compact subgroup of GE containing P(Λ) ∩GE .Let θM be the transfer of θ and κM a β-extension to J(β,ΛM).Put JΛ,M = (P(Λ) ∩GE)J1(β,ΛM)

If P(ΛM) ⊃ P(Λ) then there is a unique (β-)extension κ of ηsuch that κ and κM|JΛ,M induce equivalent irreduciblerepresentations of P(Λ).If not, then we pass from Λ to ΛM via intermediate steps Λi ,with P(Λi) ∩GE = P(Λ) ∩GE , and

either P(Λi) ⊃ P(Λi+1) or P(Λi) ⊂ P(Λi+1).



Exhaustion


The general case

Remarks1 The definition of β-extension depends (a priori) on the

choice of ΛM. There is a standard choice for ΛM.2 Different choices of κM do not necessarily give differentβ-extensions κ. But different choices of κM|Jo

Mdo give

different extensions κ|Jo .3 If J1 is a Sylow pro-p subgroup of J then κ|bJ1 ' η.



Exhaustion


Intertwining

Let κ be a β-extension defined relative to ΛM.

1 GE ⊂ IG(κ|bJ1);

2 P(ΛM) ∩GE ⊂ IG(κ);

3 If ΛM is the standard choice, there is an affine Weyl groupin GE which intertwines κ.



Exhaustion


Maximal types

DefinitionA maximal type in G is a pair (J, λ), where

J = J(β,Λ) and P(Λ) ∩GE is a maximal compact opensubgroup of GE .λ = κ⊗ ρ, where κ is a β-extension and ρ is the inflation toJ of an irreducible representation of J/J1 whose restrictionto Jo/J1 contains a cuspidal representation.

Theorem

Let (J, λ) be a maximal type. Then c-Ind GJ λ is an irreducible

supercuspidal representation of G.



Exhaustion


Maximal types

The crucial lemma in the proof is essentially due to Morris:

LemmaLet H be a connected reductive group over F ,

Q be a maximal parahoric subgroup of H, andB1 be the pro-p radical of an Iwahori subgroup in Q.

Let ρ be the inflation to Q of a cuspidal representation of thereductive quotient Q/Q1. Then

IH(ρ|B1) ⊂ NH(Q).



Exhaustion


Non-maximal case

One can also define “types” on Jo when P(Λ) ∩GE is notmaximal:

λ = κ|Jo ⊗ ρ,

for ρ an irreducible cuspidal representation of Jo/J1. The samemethods bound the intertwining of λ in terms of theGE -intertwining of ρ|Jo∩GE .



Exhaustion

Main TheoremSketch Proof

The main theorem

Theorem (S.)Let π be an irreducible (positive level) supercuspidalrepresentation of G. Then there exists a maximal type (J, λ)such that π|J contains λ. Moreover,

π ' c-Ind GJ λ

and (J, λ) is a [G, π]G-type.



Exhaustion


Illustration: Sp(4, F )

For G = Sp(4,F ), there are five classes of semisimple strata[Λ,n,0, β] giving rise to supercuspidals:

1 β = 0, ρ a cuspidal representation of Sp(4, kF ) orSL(2, kF )× SL(2, kF ) (level zero).

2 E/F a quadratic extension, ρ a cuspidal representation ofa 2-dimensional unitary, symplectic or orthogonal group.

3 E/F a quartic extension, ρ = 1.4 β = β1 + β2 and each F [βi ] is a quadratic extension, ρ = 1.5 β = β1 + β2 with F [β1] a quadratic extension and β2 = 0, ρ

a cuspidal representation of SL(2, kF ).



Exhaustion


A similar theorem

For the group GL(m,D), where D is a central F -divisionalgebra, Secherre has constructed simple characters andsimple types (in particular, maximal types).

Theorem (Secherre, S.)Let π be an irreducible (positive level) supercuspidalrepresentation of GL(m,D). Then there exists a maximal type(J, λ) such that π|J contains λ. Moreover,

π ' c-Ind GJ Λ,

where Λ is an extension of λ to J = NG(λ), and (J, λ) is a[GL(m,D), π]G-type.



Exhaustion


The main theorem

Theorem (S.)Let π be an irreducible supercuspidal representation of G. Thenthere exists a maximal type (J, λ) such that π|J contains λ.Moreover,

π ' c-Ind GJ λ

and (J, λ) is a [G, π]G-type.

Sketch proof

First, π contains some representation of Jo of the form κ⊗ ρ,with κ a β-extension and ρ the inflation of a cuspidalrepresentation of Jo/J1.



Exhaustion


Sketch proof

Sketch proofFor a contradiction, suppose J ∩GE is not maximal. There aretwo cases:

1 The cuspidal representation ρ is not self-dual.

In this case the bound on intertwining of λ easily gives us acover and so a non-zero Jacquet module.

2 The cuspidal representation ρ is self-dual.



Exhaustion


Sketch proof

We consider a special case (with David Goldberg, Phil Kutzko):

Suppose the underlying stratum [Λ,n,0, β] is simple andPo(Λ) ∩GE is the standard Siegel parahoric subgroup of GE .

Let P = LU be the standard Siegel parabolic subgroup of Gand put

JoP = H1(Jo ∩ P).

There is a unique irreducible representation λP of JoP which is

trivial on Jo ∩ U and such that

λ = Ind Jo

JoPλP .

(JoP , λP) should be a cover of (Jo

P ∩ L, λP |L).



Exhaustion


Sketch proof

There are two choices of maximal compact subgroup of GEcontaining Jo

P , which we write as P(Λ1) ∩GE , P(Λ2) ∩GE .

(If GE were symplectic, these would be the two good maximalparabolics.)

Each contains a Weyl group involution; with respect to asuitable basis, they are

w1 =

ε1

1

, w2 =

$E−1

1ε$E

,

where $E is a uniformizer of E .



Exhaustion


Sketch proof

Each Hecke algebra H(P(Λi), λP) is 2-dimensional and anyfunction fi with support Jo

PwiJoP is invertible.

The convolution f = f1 ∗ f2 is supported on the singledouble-coset Jo

PζJoP , where

ζ = w1w2 =

$E1

$E−1

.

Then f e(E/F ) is an invertible element of H(G, λP) supported ona strongly (P, Jo

P)-element in the centre of L.


Happy Birthday

Happy Birthday Phil


The supercuspidal representations of p-adic classical groups

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The supercuspidal representations of p-adic classical groups