PROPERTIES OF UNIPOTENT DISCRETE

SERIES L-PACKETS FOR p-ADIC GROUPS

Mark Reeder

Boston College

1. Introduction. Let G be the rational points of a connected split group of adjoint type over a p-adic field F. A representation V of G is unipotent if there is a parahoric subgroup P < G, such that the restriction of V to P contains a cuspidal unipotent representation σ of the reductive quotient \overline{P}. By results of Moy and Prasad [MP], the pair (P, σ) is essentially unique. (Recall from Deligne-Lusztig theory that a representation of a finite reductive group M = M(\overline{𝔽}_{q})^{Fr} is unipotent if it appears in one of the virtual M-modules R_{w} := ∑_{i}(−1)^{i}H_{c}^{i}(B_{w}, ℚ̄_{ℓ}), where B_{w} is the variety of Borel subgroups B of M(\overline{𝔽}_{q}) in relative position w, for some w in the Weyl group of M(\overline{𝔽}_{q}). See [Car] for more details.)

In this talk we consider, for certain groups G, those unipotent representations which are square integrable, and we describe the structure of their L-packets in terms of formal degrees and K-types. The basic goal is to understand what properties are shared by representations in a single L-packet.

2. Unramified Langlands correspondence. We begin with Lusztig’s recent classification of unipotent representations. Roughly speaking, it says that the unipotent representations of G are precisely those whose Langlands parameters are trivial on the inertia subgroup of the Weil group of F. Such a parameter amounts to an element x ∈ Ĝ, the dual group of G, taken up to conjugacy. The precise statement of Lusztig’s classification is

Theorem[Lusztig, [L1]]. The set Irr_{unip}(G) of irreducible unipotent representations of G is partitioned as

Irr_{unip}(G) = ∐_{x∈Ĝ/conj} Π_{x},

where the disjoint union is over conjugacy classes in Ĝ, and we have a parametrization

Π_{x} = \{V_{x,ρ} : r ∈ { \atop A(x)}\},

where { \atop A(x)} is the set of irreducible characters of the finite group A(x) := Ĝ_{x}/ZĜĜ_{x}^{∘}.

Here Ĝ_{x} is the centralizer in Ĝ of x, Ĝ_{x}^{∘} is its identity component, and ZĜ is the center of Ĝ. The group A(x) is either S_{2}^{n}, S_{3}, S_{4}, S_{5}, the latter three cases occuring only in exceptional groups.

Typeset by AMS-TEX

Research partially supported by the National Science Foundation

We have

Π_{x} = ∐_{(P,σ)} Π_{x}(P, σ),

where Π_{x}(P, σ) consists of those representations in Π_{x} containing (P, σ). Representations in Π_{x} may belong to different parabolically induced representations.

To describe properties of Π_{x}, it is helpful to give names to some objects attached to x. Let x = su be the Jordan decomposition. By the Jacobson-Morozov theorem, there is a homomorphism

ϕ_{u} : SL_{2}(C) ⟶ Ĝ

such that u = ϕ_{u} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). We can arrange that ϕ_{u} maps the diagonal matrices

into a maximal torus T̂ containing s. Set

τ = sϕ_{u} ( \begin{matrix}q^{−1/2} & 0 \\ 0 & q^{1/2} \end{matrix} ) .

By the definition of the dual group, we may view T̂ as the set of unramified characters of a Borel subgroup B of the p-adic group G. Thus we have the (normalized) induced representation Ind_{B}^{G}(τ). Some of the constituents of Ind_{B}^{G}(τ) will appear in Π_{x}. More precisely, we have

Property 1. The following are equivalent (1) V_{x,ρ} is a subquotient of Ind_{B}^{G} τ.

(2) V_{x,ρ} ∈ Π_{x}(I, 1_{I}), where I is an Iwahori subgroup and 1_{I} is its trivial representation.

(3) ρ appears in the natural action of A(x) on the homology H(B^{τ} ∩ B^{u}), where B is the flag variety of Ĝ. (Here superscripts denote fixed points. )

The equivalence of (1) and (2) was proved by Borel [B] in the 70’s, and that of (2) and (3) by Kazhdan-Lusztig [KL] in the 80’s.

Property 2. The following are equivalent (1) V_{x,ρ} has a Whittaker model

(2) V_{x,ρ} contains the Steinberg representation of G(𝔽_{q}), pulled back to a hyperspecial maximal compact subgroup of G.

(3) ρ = 1 and u belongs to the dense Ĝ_{τ} -orbit on the set of v ∈ Ĝ satisfying τvτ^{−1} = v^{q}.

The second condition in (3) is automatic if it is known in advance that V_{x,ρ} is tempered. The fact that V_{x,1} is then the unique generic representation in Π_{x} is an expected feature of any tempered L-packet.

The equivalence (1)=(2) was proved for I-spherical representations by Barbasch-Moy [BM], Li [Li], and myself [R4], and [R3] contains the equivalence with (3). The validity of Property (3) for all unipotent representations was established recently, [R5], by showing that Π_{x}(P, σ) contains no generic representation unless (P, σ) = (I, 1_{I}). This is well-known for the reductive quotients, and a little Bruhat-Tits theory is used to lift non-genericity to the p-adic group.

Let Irr_{unip}^{2}(G) be the set of square integrable unipotent representations of G, up to isomorphism.

Expected Property 3. The following are equivalent.

(1) Π_{x} ⊂ Irr_{unip}^{2}(G)

(2) V_{x,1} ∈ Irr_{unip}^{2}(G)

(3) x is not contained in a proper Levi subgroup of G.

There are only finitely many conjugacy classes of x ∈ Ĝ satisfying (3). The equivalence of (2) and (3) was proved by Kazhdan-Lusztig. I have not traced through the bijection (x, ρ) ⟶ V_{x,ρ} explicitly enough to verify (1)=(2) in general. However, for specific groups G, one can enumerate Irr_{unip}^{2}(G) by counting the square integrable representations of the Hecke algebras H(P, σ) for each parahoric P and cuspidal unipotent σ on P̄. Then other properties of square integrable L-packets, to be discussed shortly, enable one to assign an (x, ρ) to a representation in Irr_{unip}^{2}(G), in such a way that Property 3 holds. This bijection (now between finite sets) is no doubt the restriction of Lusztig’s bijection.

Example: Take G = E_{8}. We have # Irr_{unip}^{2}(G) = 98, partitioned into 31 L-packets Π_{x}. We will describe one of them. Take P with \overline{P} having derived group of type E_{6}. The finite group E_{6}(𝔽_{q}) has two cuspidal unipotent representations, σ_{θ} = E_{6}[θ] (notation as in Carter), where θ is one of the two nontrivial cube roots of unity.

Consider the Hecke algebra H(P, σ), consisting of locally constant compactly supported functions f : G ⟶ End(σ) satisfying f(p_{1}gp_{2}) = σ̌(p_{1})f(g)σ̌(p_{2}), with convolution multiplication. The structure of H(P, σ) was determined by Morris [M], and in this case H(P, σ_{θ}) is an affine Hecke algebra of type G_{2}, with generators T_{s}, T_{r}, T_{0}, the usual braid relations for affine Weyl group G_{2}, and the additional relations (T_{a} −q^{c(a)}(T_{a} +1) = 0, where a ∈ {s, r, 0}, c(s) = 9, c(r) = c(0) = 1. This is not the Hecke algebra of Iwahori-spherical functions on any p-adic group.

Given an admissible G-module V containing σ upon restriction to P, we consider the H(P, σ)-module V^{σ} := Hom_{P} (σ, V). This is a functor which induces a bijection between irreducible V ’s containing (P, σ) and irreducible H(P, σ)-modules. There is a notion of square-integrability for abstract Hecke modules [Mat] and this bijection preserves square-integrability. In the present case, H(P, σ_{θ}) has exactly four L^{2} modules, hence E_{8} has exactly four L^{2} representations containing σ_{θ} upon restriction to the E_{6}-parahoric. We describe one of them. The H(P, σ_{θ})-module is one-dimensional, and each of the generators T_{a} acts by −1. Let V_{x,θ} denote the corresponding E_{8}-module, where x = su ∈ E_{8}(C) = Ĝ is given as follows. The unipotent element u is E_{8}(b_{5}) according to the Bala-Carter classification. The centralizer of ϕ_{u} is S_{3}, and s is a three-cycle! in S^{3}. The promised L-packet is

Π_{x} = {V_{x,1}, V_{x,θ}, V_{x,θ̄}}.

2. Formal degrees. The following is our

Main Result. For G = G_{2}, F_{4}, E_{6} or classical of rank ≤ 4, there is a bijection (unique up to choosing roots of unity)

(x, ρ) ↦ V_{x,ρ} : \{(x, ρ) : x ∉ proper Levi of Ĝ, ρ ∈ { \atop A(x)}\} ⟶ Irr_{unip}^{2}(G),

such that the formal degree, with volume of Iwahori equal to one, is given by

(1)  deg V_{x,ρ} = \frac{ρ(1)}{#A(x)#ZĜ} \frac{q^{ν} ∏′ α(τ) − 1}{∏′ qα(τ) − 1},

where ν is the number of positive roots of Ĝ, and ∏′ denotes the product of nonzero terms over all roots of Ĝ.

Remarks

i) We have deg V_{x,ρ} = ρ(1)V_{1,ρ}. Assuming certain conjectures about L-packets and stable distributions, Shahidi has shown that the formal degrees in an L-packet should be proportional. Our formula indicates that the proportionality constant should be ρ(1).

ii) Let B^{u} be the variety of Borel subgroups of Ĝ containing u. One can show that, as a power series in q, the right side of (1) is

deg V_{x,ρ} = \frac{ρ(1) det(I − Ads|_{𝔪_{u}})}{#A(x)#ZĜ}q^{dim B^{u}} + higher powers,

where 𝔪_{u} is the Lie algebra of the centralizer of the image of ϕ_{u}. The determinant is nonzero when su is not in a proper Levi subgroup of Ĝ. For exceptional groups, those u which are part of L^{2} Langlands parameters are determined by their dim B^{u}. Thus, the formal degree of a L^{2}-module V, as a rational function in q, determines the unipotent part of the Langlands parameter of V. A similar phenomenon was observed for finite Hecke algebras by Lusztig [L4], and predicted by him for affine Hecke algebras with equal parameters.

Recall that the value at 0 of the L-function L(x) of the Langlands parameter corresponding to x is

L(x, 0) = det(I − Ad(τ)|_{𝔤^{u}})^{−1}.

If we grade 𝔤^{u} = 𝔤_{0} +𝔤_{1} +…, we see that det(I −Ads|_{𝔪_{u}})^{−1} is the factor of L(x, 0) coming from 𝔤_{0}.

iii) Our formulation of (1) was inspired by the following result of Heckman and Opdam [HO], valid for any split G. They assert that

deg(V_{x,1}) = c\frac{q^{ν} ∏′ α(τ) − 1}{∏′ qα(τ) − 1},

where c is a nonzero rational number, independent of q, but otherwise unknown. As with previous formal degree calculations by Shalika, Borel and myself, this relies on the existence of a nice K-type, namely the Steinberg, which exists for ρ = 1 by property 2. Formula (1) indicates (and shows, in the cases for which (1) is proved) that

c = \frac{1}{#A(x)#ZĜ} .

iv) Since the conference, formula (1) has been verified for all non-Iwahori-spherical unipotent L^{2} representations of E_{8}.

4. How to compute formal degrees We verify (1) by computing each formal degree separately. The main tool, that avoids the dependence on nice K-types, is a theorem of Schneider and Stühler, which says, when applied to unipotent representations, that

(2)  deg V = ∑_{c}(−1)^{dim c} \frac{dim V^{U_{c}}}{vol(P_{c}^{+})} ,

where the sum is over G-orbits of facets in the building, P_{c}^{+} is the stabilizer of facet c, and U_{c} is the pro-unipotent radical of P_{c}^{+}. There are about 2^{ℓ+1} such facets, where ℓ is the rank of G. Using the duality theory for finite reductive groups developed by Alvis, Curtis and Kawanaka, one can reduce to a sum of about ℓ + 1 terms (essentially a sum over vertices). Moreover, if V contains some cuspidal unipotent (σ, P), then we only get a contribution from facets c containing the facet of P in their closure. (Note: Schneider and Stühler require charF = 0 in their general formula, but at least for unipotent representations, formal degrees can be expressed in terms of Hecke algebra formal degrees. It follows that formula (2) holds for any F.)

Thus, the main problem is to compute V^{U_{c}}. First assume V is Iwahori-spherical. Here all facets contribute to the formal degree, but there are additional tools from algebraic geometry at our disposal. It is enough to let q → 1 and determine [V]_{q=1} as a module over the affine Weyl group W̃. Then V^{U_{c}} is found by restricting to the appropriate parahoric subgroup of W̃.

The following result was proved by Lusztig for s = 1, and without the tildes, but is easily extended to the version stated here. Some notation: Let W̃_{s} be the full inverse image in W̃ of the stabilizer W_{s} of s in W. We can view s as a character of W̃_{s}. Let B_{s}^{u} be the fixed points of u in the flag manifold of Ĝ_{s}. And let H(B_{s}^{u})_{ρ} be the ρ-isotypic component in homology, with grading ignored, viewed as the Springer representation of W_{s}, and with translations acting by cap-product with the Chern classes of their corresponding line bundles.

Proposition. Assume V_{x,ρ} has Iwahori fixed vectors. Then

[V_{x,ρ}]_{q=1} = sgn ⊗ IndW̃ ; W̃_{s}(s ⊗ H(B_{x}^{u})_{ρ}).

The W_{s}-modules H(B_{s}^{u})_{ρ} were determined by the Green polynomial calculations of Beynon-Spaltenstein [BS] and Shoji [Sh], and the decomposition of the induced representations is greatly facilitated by the tables of Alvis [A].

This result does not apply to non-Iwahori spherical representations, but in compensation, these involve smaller Hecke algebras (at most rank 4, for exceptional G), and we can use the elementary theory of weight diagrams, as developed in [R2].

5. Leading K-types. The above calculations lead to the following (as yet imprecise) observation, namely a relation between Langlands parameters, and Lusztig’s families of unipotent representations.

First, some recollections on truncation (see [L3]). Let W_{0} be a finite Coxeter group, and let R denote its reflection representation. Let ℍ = ⊕ℍ_{d} denote the W_{0}-harmonic polynomials on R. For any W_{0} module U, let b(U) be the minimum d for which ⟨U, ℍ_{d}⟩ ≠ 0. Define

J_{0}(U) := ∑_{\begin{matrix}ψ∈Ŵ_{0} \\ b(ψ)=b(U) \end{matrix}}⟨ψ, U⟩ψ.

Now suppose V is a unipotent representation of our p-adic group G, containing the cuspidal unipotent pair (P, σ). Let W_{0} be the finite Coxeter group underlying H(P, σ). Then we define

J(V) := J_{0}([V^{σ}]_{q=1}),

where [V^{σ}]_{q=1} is viewed as a W_{0}-module by restricting from the affine Weyl group underlying H(P, σ). We call J(V) the “leading K-type” of V, even though it could, at first glance, be reducible.

However, we have observed: For the groups and representations mentioned above, J(V) is irreducible, and almost determines the L-packet of V.

To explain this, recall that Lusztig has partitioned the unipotent representations of G(𝔽_{q}) into families F_{u}, one family for each special unipotent class [u] in Ĝ. Furthermore, he showed there is a canonical quotient Γ_{u} of A(u), with the following property: Let M(u) be the set of conjugate pairs (s, ρ), where s ∈ Γ_{u} and ρ is a representation of the centralizer of s. Then we have a parametrization

F_{u} := {χ_{s,ρ}^{u} : (s, ρ) ∈ M(u)},

with certain nice properties, and which Lusztig has explicitly calculated [L3].

The connection with leading K-types is as follows. It is only verified in the examples for which (1) is proved, E8, and a few others. To keep things simple, assume

(i) The canonical quotient A(u) ⟶ Γ_{u} is an isomorphism (this rarely fails, for exceptional groups). Identify A(u) = Γ(u).

(ii) The unipotent element u does not live in a proper Levi subgroup of Ĝ. (Again, this holds with few exceptions, if u is part of an L^{2} parameter). Then we can identify A(su) with the centralizer of s in Γ_{u}.

Then we have

J(V_{su,ρ}) = χ_{s,ρ}^{u}.

Thus, the Langlands parameter can often be read off from Lusztig’s parameter for the leading K-type.

References

[A]. D. Alvis, Induce/restrict matrices for Weyl groups, unpublished tables. [BM]. D. Barbasch, A. Moy, Whittaker models with an Iwahori-fixed vector, Representation

theory and analysis on Homogeneous spaces, vol. 177, Contemp. Math., 1994, pp. 153– 215.

[BS]. M. Beynon, N. Spaltentstein, tables of Green Polynomials for exceptional groups, War-

wick computer science centre report no. 23, 1986. [B]. A. Borel, Admissible representations of a semisimple group over a local field with vectors

fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 133–159. [Car]. R. Carter, Finite groups of Lie type: Conjugacy classes and characters, Wiley, 1985. [HO]. G. Heckman, E. Opdam, Yang’s system of particles and Hecke algebras, Ann. Math.

(1996). [KL]. D. Kazhdan, G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras,

Invent. Math. 87 (1987), 153–215. [Li]. J.-S. Li, Some results on the unramified principal series of p-adic groups, Math. Ann.

292 (1992), 747-761. [L1]. G. Lusztig, Some examples of square integrable representations of semisimple p-adic

groups, Trans. A.M.S 277 (1983), 623–653. [L2]. G. Lusztig, Classification of unipotent representations of simple p-adic groups, IMRS

11 (1995), 517-589. [L3]. G. Lusztig, Representations of reductive groups over a finite field, vol. 107, annals of

math studies, 1984. [L4]. G. Lusztig, Leading coefficients of character values of Hecke algebras, proc. symp. pure

math., vol. 47, AMS, 1987, pp. 235–262. [Mat]. H. Matsumoto, Analyse harmonique dans les systems de Tits bornologiques de type

affine, vol. 590, Springer lecture notes, 1977. [M]. L. Morris, Tamely ramified intertwining algebras, Invent. Math. 114 (1993), 233–274.

[MP]. A. Moy, G. Prasad, Jacquet functors and unrefined minimal K-types, Comm. Math.

Helv. 71 (1996), 98–121. [R1]. M. Reeder, Nonstandard intertwining operators and the structure of unramified princi-

pal series representations of p-adic groups, Forum. Math. 9 (1997), 457-516. [R2]. M. Reeder, On the Iwahori spherical discrete series of p-adic Chevalley groups; formal

degrees and L-packets, Ann. Sci. Ec. Norm. Sup. 27 (1994), 463–491. [R3]. M. Reeder, Whittaker models and unipotent representations of p-adic groups, Math.

Ann. (1997). [R3]. M. Reeder, Whittaker functions, prehomogeneous vector spaces and standard represen-

tations of p-adic groups, J. Reine. Angew. Math. 450 (1994), 83-121. [R4]. M. Reeder, p-adic Whittaker functions and vector bundles on flag manifolds, Composi-

tio. Math. 85 (1994), 9-36. [R5]. M. Reeder, Whittaker models and unipotent representations of p-adic groups, to appear,

Math. Ann. (1997). [SS]. M. Reeder, Representation theory and sheaves on the Bruhat-Tits building, preprint

(1995). [Sh]. F. Shahidi, A proof of Langland’s conjecture on Plancherel measures; Complementary

series for p-adic groups, Ann. Math. 132 (1990), 273-330. [Sho]. T. Shoji, Green polynomials for Chevalley groups of type F4, Comm. Alg. 10 (1982),

505-543.

Chestnut Hill, MA 02167

E-mail address: reederma@bc.edu