The supercuspidal representations of p-adic classical groups Shaun Stevens School of Mathematics University of East Anglia Norwich UK Conference in honour of Phil Kutzko’s 60 th birthday Iowa City, October 2006 Shaun Stevens Supercuspidal representations
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The supercuspidal representations of p-adic classical groups
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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 .
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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 .
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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).
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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 .
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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′.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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).
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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).
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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′);
θ′ = τΛ,Λ′,β(θ) if and only if GE intertwines θ with θ′.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
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).
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
Skew semisimple characters
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′);
θ′ = τΛ,Λ′,β(θ) if and only if GE intertwines θ with θ′.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
NotationsStrataCharacters
Skew semisimple characters
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 .
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
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 ”.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
ConstructionsMaximal types
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
ConstructionsMaximal types
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
ConstructionsMaximal types
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).
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
ConstructionsMaximal types
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
ConstructionsMaximal types
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
ConstructionsMaximal types
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
ConstructionsMaximal types
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).
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
ConstructionsMaximal types
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 .
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
Main TheoremSketch Proof
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 ).
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
Main TheoremSketch Proof
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
Main TheoremSketch Proof
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
Main TheoremSketch Proof
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.
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
Main TheoremSketch Proof
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).
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
Main TheoremSketch Proof
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 .
Shaun Stevens Supercuspidal representations
The set-upBeta extensions
Exhaustion
Main TheoremSketch Proof
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