ON MATRIX COEFFICIENTS OF THE SATAKE ISOMORPHISM: COMPLEMENTS TO THE PAPER OF M. RAPOPORT

THOMAS J. HAINES

Abstract. We consider the matrix for the Satake isomorphism with respect to natural bases. We give a simple proof in the case of Chevalley groups that the matrix coefficients which are not obviously zero are in fact positive numbers. We also relate the matrix coefficients to Kazhdan-Lusztig polynomials and to Bernstein functions.

Introduction

M. Rapoport’s paper ([10]) is concerned with a matrix C_{λ,μ} which expresses the Satake isomorphism in terms of natural bases (see below for a precise definition of C_{λ,μ}). The main theorem is that all of the matrix coefficients C_{λ,μ} which are not obviously zero are in fact positive numbers. The present article is intended to complement that of Rapoport, and consists of three parts. In the first section we present the main result, which is a short proof of the positivity property in the case of p-adic Chevalley groups. The key tools are a theorem of Dabrowski ([2]) which gives a criterion for the intersection of cells in the Iwahori and Iwasawa decompositions of a p-adic Chevalley group, and Macdonald’s formula for the Satake transform.

In the second section we relate the matrix C_{λ,μ} to the Kazhdan-Lusztig polynomials for the affine Weyl group attached to a split p-adic group. We use the work of S. Kato ([7]), which is itself an extension of work of G. Lusztig ([8]). For the affine Weyl group of type Ã_{1} the Kazhdan-Lusztig polynomials are very simple, and using this observation we identify the coefficients C_{λ,μ} for the group Gl_{2}. This provides a direct verification of the positivity property in this case, and indicates that Kazhdan-Lusztig polynomials might provide an explanation of the positivity phenomenon in general.

In the third section we show that the matrix C_{λ,μ} is also the one that relates Bernstein functions to spherical functions, relative to the natural isomorphism between the center of the Iwahori-Hecke algebra and the spherical Hecke algebra of a split p-adic group.

1. A simple proof of Positivity for Chevalley groups

Let F be a p-adic field, with algebraic closure \overline{F}. Let q = p^{j} denote the cardinality of the residue field of F. Let O denote the ring of integers in F, with ϖ a uniformizer of the maximal ideal. Let G be a Chevalley group, or more generally a split connected reductive group over F whose adjoint group is the Chevalley group corresponding to an irreducible and reduced root system R. The group G is then defined over O. Fix a choice of positive roots R^{+} and simple positive roots Δ. For any root α ∈ R^{+}, let Φ_{α} denote the corresponding group homomorphism Φ_{α} : Sl_{2} → G. For any a, c ∈ \overline{F}, c ≠ 0, let

x_{α}(a) = Φ_{α} (\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) ,

x_{−α}(a) = Φ_{α} (\begin{matrix}1 & 0 \\ a & 1 \end{matrix}) ,

t_{α}(c) = Φ_{α} (\begin{matrix}c & 0 \\ 0 & c^{−1} \end{matrix}) ,

ω_{α} = Φ_{α} (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .

Let Z denote the center of G. Let U, T, and N be respectively the subgroups of G generated by the sets \{x_{α}(a) | a ∈ \overline{F} , and α ∈ R^{+}\}, Z ∪\{t_{α}(c) | c ∈ \overline{F} , c ≠ 0, and α ∈ R^{+}\}, and T ∪ {ω_{α} | α ∈ R^{+}}. Then T is a maximal torus of G, contained in the Borel subgroup B = T U, and N is the normalizer of T in G. Let K = G(O); it is a special, good, maximal compact open subgroup of G(F). Let T_{sc} = T ∩ G_{sc}, the inverse image of T under G_{sc} → G. Let T_{0}, U_{0} and N_{0} denote, respectively, the intersections T (F)∩K, U(F)∩K and N(F)∩K.

Let W = N(F)/T (F) be the finite Weyl group of G, and let W̃ = N(F)/T_{0} denote the extended affine Weyl group. There is a canonical isomorphism W̃ ⟶^{∽} X_{∗}(T) ⋊ W. Let W_{a} = X_{∗}(T_{sc}) ⋊ W; this is a Coxeter group with generators consisting of the simple affine reflections S_{a} = {s_{α} | α ∈ Δ}∪{s_{a} = t_{−α̌_{0}}s_{α_{0}}}, where t_{λ} ∈ W̃ is the translation corresponding to the cocharacter λ ∈ X_{∗}(T), and α_{0} is the highest root of R^{+}. Let Ω denote the subgroup of W̃ which preserves the set of simple reflections S_{a} ⊂ W̃ under conjugation. Then there is a canonical isomorphism W̃ ⟶^{∽} Ω ⋉ W_{a}. The length function l : W_{a} → ℕ is extended to a length function on W̃ by setting the length of ωx (ω ∈ Ω, x ∈ W_{a}) to be l(x).

We fix an Iwahori subgroup for G(F) as follows. Let I denote the subgroup of K generated by U_{0}, T_{0}, and the subgroup V generated by the set {x_{−α}(a) | a ∈ ϖO, α ∈ R^{+}}. Then I is an Iwahori subgroup of G(F) in good position relative to K, meaning that IW I = K. Note that both I and K contain T_{0}, so for any w ∈ W̃ it makes sense to write wI = nI (resp. wK = nK), where n ∈ N(F) maps to w under the projection N(F) → W̃. For λ ∈ X_{∗}(T), let ϖ^{λ} denote the image of λ ⊗ ϖ under the canonical isomorphism X_{∗}(T) ⊗ F^{×} ⟶^{∽} T (F). The element ϖ^{λ} ∈ T (F) maps to t_{λ} ∈ W̃ under the projection N(F) → W̃, so with the conventions above we can write Kt_{λ}K = Kϖ^{λ}K, It_{λ}I = Iϖ^{λ}I, and U(F)t_{λ}I = U(F)ϖ^{λ}I, etc. (The notation in the main theorem below thus differs slightly from that of Rapoport [10].)

Although the groups G(F), K, T (F), T_{0}, U(F) and I change if F is replaced by a finite extension field, the groups W, W̃, and W_{a} do not; in particular they are independent of the quantity q.

Next we will summarize some definitions and results of R. Dabrowski ([2]). If w ∈ W̃, denote by \overline{w} the image of w in the finite Weyl group W under the canonical projection W̃ → W. Fix an element τ ∈ W̃ and a reduced expression τ = ωs_{1} … s_{k}, where the s_{i} are simple affine reflections, and ω ∈ Ω. Following Deodhar ([4]) we call a sequence of elements {σ_{j}}_{j=0}^{k} a subexpression if σ_{0} = ω and for each j ≥ 1, σ_{j} ∈ {σ_{j−1}, σ_{j−1}s_{j}}. Following Dabrowski ([2]) we call the subexpression good if the following hold:

• \overline{σ}_{j−1}s_{j} ≤ \overline{σ}_{j}for all j such that s_{j} ∈ S,

• \overline{σ}_{j−1}s_{α_{0}} ≥ \overline{σ}_{j} for all j such that s_{j} = s_{a}.

Here α_{0} denotes the highest root of R^{+} and s_{a} = t_{−α̌_{0}}s_{α_{0}} is the corresponding simple affine reflection. The ordering ≤ denotes the Bruhat order on the Coxeter group (W, S), where S is the set W ∩ S_{a}. Let G denote the set of all good subexpressions (relative to a fixed reduced expression of a fixed τ). Denote by G(τ) the set {σ_{k} | {σ_{j}}_{j=0}^{k} ∈ G}. We will use the following result of Dabrowski:

Proposition 1.1. Let τ and ω be arbitrary elements of W̃. Then the following are equivalent:

(1) U(F)ωI ∩ IτI ≠ ∅,

(2) ω ∈ G(τ). In particular, the set G(τ) is independent of the choice of reduced expression for τ.

Proof. This is Proposition 3.2 of [2].

□

Let b : C_{c}(K\G(F)/K) ⟶^{∽} ℂ[X_{∗}(T)]^{W} denote the Satake isomorphism, as defined in ([1]). In particular, this means that Haar measure on G(F) is normalized so that K has volume 1. For μ ∈ X_{∗}(T) a dominant coweight, let f_{μ} = char(Kt_{μ}K) and m_{μ} = |W_{μ}|^{−1} ∑_{w∈W} w(μ) (here W_{μ} is the stabilizer of μ in W). We define the matrix C_{λ,μ} by the following equation

b(f_{λ}) = ∑_{μ} C_{λ,μ}m_{μ}.

For μ and λ in X_{∗}(T), we write μ ≤^{!} λ if λ − μ is a (possibly empty) sum of simple coroots.

When in addition μ ≠ λ, we write μ <^{!} λ. The main result of this paper is the following theorem.

Theorem 1.2. Let μ and λ be dominant coweights in X_{∗}(T). Then the following statements are equivalent:

(1) C_{λ,μ} > 0,

(2) U(F)t_{μ}K ∩ Kt_{λ}K ≠ ∅,

(3) μ ≤^{!} λ,

(4) U(F)t_{μ}I ∩ IτI ≠ ∅, for some τ ∈ W t_{λ}W,

(5) t_{μ} ∈ G(τ), for some τ ∈ W t_{λ}W.

Proof. The equivalence of (1) and (2) is well-known ([1]). The equivalence of (4) and (5) is a special case of Proposition 1.1 above. The implication (4) ⇒ (2) holds because I ⊂ K and KτK = Kt_{λ}K (the group K ∩N(F) contains a set of representatives of W). The implication (2) ⇒ (4) follows from Kt_{λ}K = IW It_{λ}IW I = IW t_{λ}W I. We now prove (5) ⇒ (3): It is clear from the definition that G(τ) ⊂ {x ∈ W̃ | x ≤ τ} , where ≤ denotes the Bruhat order on W̃. Therefore if (5) holds, then t_{μ} ≤ τ and so W t_{μ}W ≤ W τW = W t_{λ}W in the Bruhat order

on W \W̃/W. It is well-known that this implies μ ≤^{!} λ. Finally, we show (3) ⇒ (4): Suppose

μ ≤^{!} λ. Then by the Lemma below we see that (2) holds for sufficiently large q. Then as above it follows that (4) holds for sufficiently large q. But (5) and hence (4) is independent of q; hence (4) holds for all q. □

Lemma 1.3. If μ ≤^{!} λ, then U(F)t_{μ}K ∩ Kt_{λ}K ≠ ∅ for q ≫ 0.

Proof. The intersection is nonempty if and only if the matrix coefficient C_{λ,μ}(q) > 0, where

b(f_{λ}) = ∑_{μ} C_{λ,μ}(q)m_{μ}.

We write C_{μ,λ}(q) to emphasize the dependence of this matrix on q.

Now recall Macdonald’s formula (see Theorem 2.4 of [7]):

b(f_{λ}) = \frac{q^{l(λ)/2}}{W_{λ}(q^{−1})} ∑_{w∈W} e^{w(λ)} ∏_{α>0} \frac{1 − q^{−1}e^{−w(α)}}{1 − e^{−w(α)}} .

As q → ∞, the sum approaches

∑_{w∈W} e^{w(λ)} ∏_{α>0}(1 − e^{−w(α)})^{−1}.

By Weyl’s character formula, this last expression is the character of the highest weight module for λ:

χ_{λ} = ∑_{μ} m_{λ}(μ)m_{μ}.

One knows that m_{λ}(μ) > 0 if and only μ ≤^{!} λ (see [5], p.204), and if so it follows that (W_{λ}(q^{−1})/q^{l(λ)/2})C_{μ,λ}(q) and therefore C_{μ,λ}(q) is > 0 for q ≫ 0. □

2. The Relation with Kazhdan-Lusztig polynomials

Let G be as above (or more generally an almost simple, connected reductive group, defined and split over F). Define a matrix M(q) by

M(q)_{μ,λ} = q^{−l(λ)/2}P_{w_{0}μ,w_{0}λ}(q).

Here μ and λ range over dominant coweights in X_{∗}(T) (in a suitable ordering compatible with

≤^{!}), w_{0} denotes the longest element of W, and P_{x,y}(q) are the Kazhdan-Lusztig polynomials for the group W̃.

Note first that Theorem 1.5 of [7] with q = 1 gives

χ_{λ} = ∑_{μ} M(1)_{μ,λ}m_{μ}.

On the other hand, Lemma 2.7, (3.5), and Theorem 1.8 of (loc. cit.) imply that

χ_{λ} = ∑_{μ} M(q)_{μ,λ}b(f_{μ}).

It follows that

Proposition 2.1. C_{λ,μ}(q) = [M(1)M(q)^{−1}]_{μ,λ}.

We can use this to illustrate the positivity property in the case of Gl_{2}. Since Gl_{2} is of type A_{1}, we know that the Kazhdan-Lusztig polynomials for W̃ are particularly simple: P_{x,y}(q) = 1 if x ≤ y and 0 otherwise. It follows that

M(1)_{μ,λ} = \begin{cases}1, & if μ ≤^{!} λ, \\ 0, & otherwise, \end{cases}

and thus

M(1)_{μ,λ}^{−1} = \begin{cases}−1, & if μ = λ − α̌, \\ 1, & if μ = λ, \\ 0, & otherwise. \end{cases}

Here α̌ = (1, −1) is the unique simple coroot. Moreover M(q) = M(1)diag(…, q^{−l(λ)/2}, …). This implies

M(q)_{σ,λ}^{−1} = \begin{cases}−q^{(l(λ)/2)−1}, & if σ = λ − α̌, \\ q^{l(λ)/2}, & if σ = λ, \\ 0, & otherwise, \end{cases}

and thus

C_{λ,μ}(q) = ∑_{μ ≤^{!} σ ≤^{!} λ} M(1)_{μ,σ}M(q)_{σ,λ}^{−1} = \begin{cases}q^{l(λ)/2} − q^{(l(λ)/2)−1}, & if μ <^{!} λ, \\ q^{l(λ)/2}, & if μ = λ, \\ 0, & otherwise. \end{cases}

The positivity of C_{λ,μ}(q) for μ ≤^{!} λ is therefore evident in this case.

3. The Relation with Bernstein Functions

We retain the notation of the previous section. Consider the following diagram:

[figure: ℂ[X_{∗}(T)]^{W}; C_{c}(K\G(F)/K); b; ∽; ℂ[X_{∗}(T)]^{W}; Bern.; ∽; Z(C_{c}(I\G(F)/I)); e_{K}∗ −]

Here the right vertical map is multiplication by e_{K} = char(K). The convolution product in C_{c}(K\G(F)/K) (resp. C_{c}(I\G(F)/I)) is defined using the Haar measure on G(F) for which vol(K) = 1 (resp. vol(I) = 1). The Satake isomorphism is defined using the same normalizations as in [1]. The map Bern was discovered by Bernstein and is constructed as follows (see [8], [9]). For any coweight ν ∈ X_{∗}(T), write ν = ν_{1} − ν_{2} where the elements ν_{i} are dominant. Then define an element of the Iwahori-Hecke algebra

Θ_{ν} = q^{−(l(ν_{1})−l(ν_{2}))/2}T_{t_{ν_{1}}} T_{t_{ν_{2}}}^{−1},

where T_{w} denotes the generator of the Iwahori-Hecke algebra C_{c}(I\G(F)/I) given by the characteristic function of the double coset IwI, for any w ∈ W̃. Then the element Θ_{ν} is independent of the choice of ν_{1} and ν_{2}. Moreover it is proved in [8] that if μ ∈ X_{∗}(T) is dominant and we set

z_{μ} = ∑_{ν∈W (μ)} Θ_{ν},

then the element z_{μ} is in Z(C_{c}(I\G(F)/I), and moreover the map which associates to m_{μ} ∈ ℂ[X_{∗}(T)]^{W} the element z_{μ} defines an isomorphism of ℂ-algebras Bern : ℂ[X_{∗}(T)]^{W} ⟶^{∽} Z(C_{c}(I\G(F)/I). We call the elements z_{μ} the Bernstein functions.

The next proposition seems to be well-known to the experts. It implies that the center of the Iwahori-Hecke algebra is naturally isomorphic to the spherical Hecke algebra. A proof can be found in a paper of J.-F. Dat ([3]). It can also be deduced easily from known results of Lusztig ([8]) and S. Kato ([7]), as is explained in [6].

Proposition 3.1. The Satake and Bernstein isomorphisms are compatible, i.e., the diagram above is commutative.

We deduce that the matrix C_{λ,μ} relates the Bernstein functions to the spherical functions.

Proposition 3.2. f_{λ} = ∑_{μ} C_{λ,μ}e_{K} ∗ z_{μ}.

Proof. Apply the Satake isomorphism b to both sides, then use the commutativity of the above diagram and the definition of the matrix C_{λ,μ}. □

From this we get an interesting relation for the matrix coefficients C_{λ,μ}:

Corollary 3.3. Let ρ denote half the sum of the positive roots. Let W (q) = ∑_{w∈W} q^{l(w)}. Then for any dominant λ ∈ X_{∗}(T),

W (q)^{−1} ∑_{w∈W t_{λ}W} q^{l(w)} = ∑_{μ ≤^{!} λ} C_{λ,μ}(q)( ∑_{ν∈W (μ)} q^{⟨ρ,ν⟩}).

Proof. Consider the ℂ[q^{1/2}, q^{−1/2}]-algebra homomorphism χ : C_{c}(I\G(F)/I) → ℂ[q^{1/2}, q^{−1/2}] defined by χ(T_{w}) = q^{l(w)}, where T_{w} denotes the generator of the Iwahori-Hecke algebra corresponding to w ∈ W̃. We have f_{λ} = ∑_{w∈W t_{λ}W} T_{w} and e_{K} = ∑_{w∈W} T_{w}. Moreover, if ν ∈ W (μ) and ν = ν_{1} − ν_{2} (ν_{i} dominant), one has χ(Θ_{ν}) = q^{−(l(ν_{1})−l(ν_{2}))/2}χ(T_{ν_{1}})χ(T_{ν_{2}})^{−1} = q^{(⟨2ρ,ν_{1}⟩−⟨2ρ,ν_{2}⟩)/2} = q^{⟨ρ,ν⟩}. The result now follows by applying χ to the equation in the preceeding proposition. □

4. Acknowledgements

This paper was written during visits to the Max-Planck-Institut-fuer-Mathematik in Bonn and to the IHES in Bures-sur-Yvette. I thank the Instituts for their support and for providing excellent working environments. I thank R. Dabrowski for sending me some comments regarding the proof of his Proposition 3.2, cited above. Finally, I thank M. Rapoport for many interesting conversations about this question, and for bringing the paper of R. Dabrowski to my attention.

References

[1] P. Cartier, Representations of p-adic groups, Proc. of Symp. in Pure Math., vol. 33 (1979), part 1, pp. 111-155.

[2] R. Dabrowski, Comparison of the Bruhat and the Iwahori Decompositions of a p-adic Chevalley Group, Journal of Algebra 167, 704-723 (1994).

[3] J.-F. Dat, Caractères à valeurs dans le centre de Bernstein , J. Reine Angew. Math. 508 (1999), 61-83.

[4] V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79, 499-511 (1985).

[5] W. Fulton and J. Harris, Representation Theory, Springer GTM 129 (1991).

[6] T. Haines, The Combinatorics of Bernstein functions, preprint, 1998.

[7] S. Kato, Spherical Functions and a q-Analoque of Kostant’s Weight Multiplicity Formula, Inv. Math. 66, 461-468 (1982).

[8] G. Lusztig, Singularities, character formulas and a q-analog of weight multiplicities, Astérisque 101-102 (1983), 208-229.

[9] G. Lusztig, Affine Hecke Algebras and Their Graded Version, J. of the Amer. Math. Soc. 2 No. 3 (1989) 599-635.

[10] M. Rapoport, A positivity property of the Satake isomorphism, preprint 1999.

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email:haines@math.toronto.edu