Hecke Algebras of Classical Groups over p-adic field and Supercuspidal Representations

Julee Kim Yale University

Abstract

Let G be a classical group defined over p-adic field k. In order to understand the admissible representations of G, we construct a family of Hecke algebras on G. As a corollary, we also get new supercuspidal representations. When G is a symplectic group in 2n variables over k, we describe some of these algebras explicitly in terms of generators and relati ons. Our constructions are valid when the residual characteristic of k is large.

Part of this work is from my thesis for Ph.D at Yale University. I would like to thank Professor R. Howe for his patient guidance and support.

INTRODUCTION

k : p-adic field with char(\overline{k}) = p, \overline{k} = 𝔽_{q} G : (connected) reductive group over k,

Jacquet, H-C ⇒ all irreducible admissible representation of G can

be realized as a subrepresentation of the Ind_{P}^{G}σ with P = MN and σ supercuspidal representation of M.

(e.g) Spherical principal series can be understood by Iwahori Hecke algebras (Casselman, Borel)

P = B, σ = ν: unramified char of B/N Ind_{B}^{G}ν can be understood via H(G//I)

Irreducible subquotients of { Ind_{B}^{G}ν | ν:unramified }

⟷ finite dimensional simple representations of H(G//I)

W ⟷ W^{I}

[KL] classifies the simple modules of H(G//I) when G is split wit h connected center.

Definition. Let (J, ρ) be a pair of open compact subgroup J and its representation ρ. Define a spherical function algebra associated to (J, ρ) :

H(G//J, ρ) = \{ \begin{matrix}f ∈ C_{c}^{∞}(G, End(ρ)) ∥ \\ f(jgj′) = ρ(j)f(g)ρ(j′), ∀j, j′ ∈ J, g ∈ G \end{matrix} \}

This is a Hilbert algebra with convolution and *-operation;

f^{∗}(g) = f(g^{−1})^{∗}, ⟨f_{1}, f_{2}⟩ = Tr(f_{1} ∗ f_{2}^{∗}(1)).

Irr admissible repns τ of G ρ̌ ⊂ τ|J

⟷ finite dim simple repns of H(G//J, ρ)

V ⟷ (V_{ρ} ⊗ V)^{J}

Goal. Construct as many as spherical Hecke algebras to describe as many as admissible representations. Those are usually isomorphic to (generalized) affine Hecke algebras. Moreover, for some (M, σ) (M : a Levi subgroup, σ : supercuspidal representation of M), they d escribe

{ Ind_{P}^{G}σ ⊗ ψ | ψ : unramified }.

Let W̃ = NT/T_{0} be an affine Weyl group and let v : S/ ∼⟶ ℤ^{+} be the weight function. Then a (generalized) affine Hecke algebra H associated to W̃ is linearly spanned by elements parametrized by w ∈ W̃

generators  f_{s_{i}}
length preserving  \begin{matrix}f_{w} ∗ f_{w′} = f_{ww′}, \\ l(ww′) = l(w) + l(w′) \end{matrix}
quadratic relation  f_{s_{i}} ∗ f_{s_{i}} = q^{v(s_{i})}f_{1} + (q^{v(s_{i})} − 1)f_{s_{i}}
braid relation  f_{s_{i}} ∗ f_{s_{j}} ∗ ⋯ = f_{s_{j}} ∗ f_{s_{i}} ∗ ⋯

Introduce Indexed Affine Dynkin diagram reflecting the quadratic relation ⟹ Affine Dynkin diagram + weights :

Optimal result : Construct the family of (J, ρ) such that (i) H(G//J, ρ) is an Affine Hecke algebra. (ii) For any τ ∈ Ĝ^{adm}, there exists unique (J, ρ) from the family such that ρ ⊂ τ|J. (iii) If ρ ⊂ Ind_{P}^{G}σ ⊗ ψ, all subquotients of Ind_{P}^{G}σ ⊗ ψ′ contain ρ.

This result is established for GL_{n} by [HM] and [BK].

Main Result

k : p-adic field with involution σ with k^{σ} = k_{0}

V : finite dimensional vector space over k with nondegenerate sesquilinear form f_{V}

G : G(V, f_{V}), group of isometry (or similitude) on (V, f_{V})

Assumption on k.

(1) \frac{1}{dim_{k}(V)} > \frac{2ord(p)}{p−1} \frac{p}{p−1} + \frac{1}{p−1} → \{ \begin{matrix}convergence of exp on some “big” lattice, R_{p} \\ and its image(exp R_{p}) to be a group. \end{matrix}

(2) k contains enough roots of 1 ⇒ Maximal tori are split over tamely ramified abelian extensions.

Basic Data. Σ = (Γ, P_{0}, ϱ), Γ ∈ 𝔊 : s.s. of certain “truncated form” P_{0} ⊂ C_{G}(Γ) : parahoric subgroup ϱ : cuspidal representation of reductive quotient of P_{0}

Here, C_{G}(Γ) is a (central) product of G(W, f_{V} |W) with W ⊂ V and general linear groups, unitary groups defined over some finite extension of k.

Theorem 1. For a given triple Σ = (Γ, P_{0}, ϱ) as above, we construct a pair (J_{Σ}, ρ_{Σ}) of an open compact subgroup J_{Σ} and its representation ρ_{Σ} such that the Hecke algebra H = H(G//J_{Σ}, ρ_{Σ}) associated to (J_{Σ}, ρ_{Σ}) is spanned by functions f_{w} (uni que up to constant) whose support is a single double coset of J_{Σ}wJ_{Σ} represented by an element of the affine Weyl group of C_{G}(Γ).

Corollary. If G′ = C_{G}(Γ) is compact mod Z_{G}, Ind_{Z′_{G}·J_{Σ}}^{G} ρ̃_{Σ} is a supercuspidal representation, where ρ̃_{Σ} is any representat ion on Z′_{G} · J_{Σ} extending ρ_{Σ} on J_{Σ}.

More generally, if G′ does not have any GL factors and if P′_{0} in Σ is maximal parahoric in G′, c-Ind_{Z_{G′} ·J_{Σ}}^{G} ρ̃_{Σ} is irreducible and supercuspid al. where Z_{G′} is the center of G′ and where ρ̃_{Σ} is the representation of on Z · J_{Σ} extending (ρ_{Σ}, J_{Σ}).

Remark.

(J_{Σ}, ρ_{Σ}) in Theorem 1 is a Bushnell-Kutzko type ;

BK-types for principal series for split reductive groups are constructed b y A. Roche.

Remark and Conjecture.

(1) We get some new supercuspidal representations, which are not necessarily as sociated to a character of a tamely ramified torus [Morris], nor can be detected by elliptic unrefined K-types [Adler].

We have constructed all known supercuspidal representations under the above assumptions on k. Moreover, we can conjecture they give all supercuspidal representations.

(2) The above Hecke algebras parametrize all admissible representations of G in the sense of (ii) in the introduction.

Fix Ω an additive character of k_{0} with conductor O_{k_{0}}. Then

\begin{matrix}θ : 𝔊 ⟶^{∼} 𝔊̂ \\ θ(x)(y) = Ω(⟨x, y⟩) \end{matrix}

Especially, θ(Γ) ∈ 𝔊̂. We construct R_{p} ⊂ 𝔊 ; a “big” lattice where exp is defin ed. Then θ(Γ)|R_{p} defines a character. Sketch. Let 𝔗 be a maximal and maximally split torus in 𝔊 containing Gamma. Decompose 𝔊 as k_{0}-rational root subspaces of 𝔗 ;

𝔊 = 𝔗 ⊕ ∑ 𝔊_{ε}.

We find lattice structures on each 𝔊_{ε} and

R_{p} = P_{1} + ∑ L_{ε}.

(e.g) If 𝔗 is a “principal” torus, R_{p} is the O_{k_{0}}-L ie algebra of the 1-st congruence subgroup of some parahoric subgroup.

R = P_{0} · R_{p}, R_{p}  θ(Γ)|R_{p}
P_{0} · J_{p}  J_{p} = radJ′_{p}

|

|  Ad g(Γ + R_{p}^{♯})
|  = (Γ + R_{p}^{♯})

↓

\begin{matrix}J_{Γ} = P_{0} · (J_{Γ})_{p} \\ (J_{Γ})_{p} = exp(J_{Γ})_{p} \end{matrix} . . . . . . . . . . . . . . . . . . .  θ(Γ)|(J ′_{Γ})_{p}
↑
ρ_{Γ}  |
\begin{matrix}J_{p} = J′_{p} \\ → character \end{matrix}  |

J_{p} ⊋ J′_{p} → Heisenberg

|
exp(J′_{p})  X → θ(Γ)(log X)  J′_{p}
∪  ∪

Stab_{R}(θ(Γ)|R_{p}) = H_{Γ} = P_{0} · (H_{Γ})_{p}

(ℌ_{Γ})_{p}

ρ_{Σ} = ρ_{Γ} ⊗ ϱ

Ad R-invariant [J′_{p}, J′_{p}] ⊂ ker(θ(Γ))

θ(Γ)|(ℌ_{Γ})_{p} ≠ 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Crucial Lemma.

Supp(H(G//J_{Σ}, ρ_{Σ})) ⊂ J_{Σ}C_{G}(Γ)J_{Σ} = J_{Σ} W̃′J_{Σ}.

Theorem 2. Let G be a symplectic group in 2n variables over k. Let P_{0} is minimal parahoric and ϱ = 1 in Σ = (Γ, P_{0}, 1). Then we have an L^{2}-isomorphism

η : H(G//J_{Σ}, ρ_{Σ}) ⟶^{∼} H(C_{G}(Γ)//P_{0}, χ)

where χ is a suitable tamely ramified character of P_{0}.

Remark. H(C_{G}(Γ)//I_{0}, χ) is a generalized affine Hecke algebra (not necessarily Iwahori Hecke Algebra).