IWAHORI-HECKE ALGEBRAS

THOMAS J. HAINES, ROBERT E. KOTTWITZ, AND AMRITANSHU PRASAD

Our aim here is to give a fairly self-contained exposition of some basic facts about the Iwahori-Hecke algebra H of a split p-adic group, including Bernstein’s presentation and description of the center, Macdonald’s formula, the Casselman-Shalika formula, and the Kato-Lusztig formula. There are no new results here, and the same is essentially true of the proofs. We have been strongly influenced by the notes [1] of a course given by Bernstein.

The reader may find in [21] another survey article which proves some of the results of the present paper by different methods.

The following notation will be used throughout this paper. We work over a padic field F with valuation ring O and prime ideal P = (π). We denote by k the residue field O/P and by q the cardinality of k.

Consider a split connected reductive group G over F, with split maximal torus A and Borel subgroup B = AN containing A. We write B̄ = AN̄ for the Borel subgroup containing A that is opposite to B. We assume that G, A, N are defined over O. We write K for G(O) and I for the Iwahori subgroup of K defined as the inverse image under G(O) → G(k) of B(k). For μ ∈ X_{∗}(A) we write π^{μ} for the element μ(π) ∈ A(F). Note that μ ↦ π^{μ} gives an isomorphism from X_{∗}(A) to A/A_{O}. (We will often abbreviate A(F) to A and A(O) to A_{O}, etc.)

1. Bernstein’s presentation [17]

1.1. Extended affine Weyl group. The extended affine Weyl group W̃ is the quotient of N_{G(F)}(A) by A_{O}. Thus W̃ contains the translation subgroup A/A_{O} = X_{∗}(A), as well as the finite Weyl group W, which we realize inside W̃ as the quotient of N_{K}(A) by A_{O}. Recall that W̃ is the semidirect product of W and X_{∗}(A). In the early part of the paper, when we are thinking about a cocharacter μ as an element of the translation subgroup of W̃, we denote it by π^{μ}; later on we often denote it instead by t_{μ}.

1.2. Iwahori-Hecke algebra H. We denote by H the Iwahori-Hecke algebra C_{c}(I\G/I). The convolution product is defined using the Haar measure giving I measure 1. The elements T_{x} := 1_{IxI} (x ∈ W̃) form a C-basis for H. (Throughout the paper we write the characteristic function of a subset S as 1_{S}.)

1.3. Double cosets A_{O}N\G/I = W̃. The obvious map W̃ → A_{O}N\G/I is a bijection. How does this decomposition work? Write g ∈ G as g = π^{μ}nk. Then write k = n_{O}wi with n_{O} ∈ N_{O}, i ∈ I, w ∈ W, with w inside K. Thus g = π^{μ}nn_{O}wi, showing that g lies in the double coset of π^{μ}w ∈ W̃. In short, we read off π^{μ} from the Iwasawa decomposition and then read off w ∈ W by applying the Bruhat decomposition over the residue field to the element k produced from the Iwasawa decomposition.

1.4. Definition of the module M. Put M := C_{c}(A_{O}N\G/I). Note that M is a right H-module (since it arises as the I-fixed vectors in the smooth G-module considered in (1.5.1) below).^{1} For x ∈ W̃ we denote by v_{x} the characteristic function 1_{A_{O}NxI}. The elements v_{x} (x ∈ W̃) form a C-basis for M. Of special importance (see Lemma 1.6.1 below) is the basis element v_{1} = 1_{A_{O}NI}. Let R = C_{c}(A/A_{O}) = C[X_{∗}], the group algebra of X_{∗}, an abbreviation for X_{∗}(A). Thus the elements π^{μ} (μ ∈ X_{∗}) form a basis for the vector space R. We make M into a left R-module as follows. Let μ ∈ X_{∗} and let x ∈ W̃. Then π^{μ} · v_{x} := q^{−⟨ρ,μ⟩}v_{π^{μ}·x}, where ρ is half the sum of the roots of A in Lie(N). (Note that the scalar q^{−⟨ρ,μ⟩} is equal to δ_{B}(π^{μ})^{1/2}, where for a ∈ A, δ_{B}(a) denotes the absolute value of the determinant of the adjoint action of a on Lie(N).) The actions of R and H commute, so that M is an (R, H)-bimodule.

1.5. Second point of view on the module M. Consider the representation (by right translations) of G on C_{c}^{∞}(A_{O}N\G). It is compactly induced from the trivial representation of A_{O}N; doing the induction in stages, we see that

(1.5.1)  C_{c}^{∞}(A_{O}N\G) = i_{B}^{G}(R).

Here we are using normalized induction and R is viewed as A-module via χ_{univ}^{−1}, where χ_{univ} is the tautological character A/A_{O} → R^{×} mapping π^{μ} to π^{μ}. Thus an element in this induced representation is a locally constant R-valued function ϕ on G satisfying

ϕ(ang) = δ_{B}(a)^{1/2} · a^{−1} · ϕ(g)

for all a ∈ A, n ∈ N, g ∈ G, and the group G acts by right translations. The isomorphism (1.5.1) has the following explicit description. Let φ ∈ C_{c}^{∞}(A_{O}N\G). Then the corresponding element ϕ ∈ i_{B}^{G}(R) is defined by

(1.5.2)  ϕ(g) = ∑_{a∈A/A_{O}} δ_{B}(a)^{−1/2}φ(ag) · a

for g ∈ G. There is an obvious R-module structure on i_{B}^{G}(R), with rϕ given by (rϕ)(g) = r(ϕ(g)). The isomorphism (1.5.1) induces an (R, H)-bimodule isomorphism from M to the Iwahori fixed vectors in i_{B}^{G}(R).

Let χ denote a quasicharacter χ : A/A_{O} → C^{×}. Then χ determines a C-algebra homomorphism R → C. Using this homomorphism to extend scalars, we obtain the H-module

C ⊗_{R} M = C ⊗_{R} i_{B}^{G}(χ_{univ}^{−1})^{I} = i_{B}^{G}(χ^{−1})^{I}.

1.6. Structure of the module M. The next result is due to Chriss and Khuri-Makdisi [8], who derived it from Bernstein’s presentation of the Iwahori-Hecke algebra. Here we turn the logic around, first studying M directly, then using it to produce Bernstein’s presentation.

Lemma 1.6.1. The map h ↦ v_{1}h is an isomorphism of right H-modules from H to M. In other words M is free of rank 1 as H-module with canonical generator v_{1}. In particular we have a canonical isomorphism H ≃ End_{H}(M), which identifies h′ ∈ H with the endomorphism v_{1}h ↦ v_{1}h′h of M.

1More explicitly, the action of h ∈ H on m ∈ M is given by the convolution m · h of the two functions, where convolution is defined using the Haar measure on G giving I measure 1.

Proof. It suffices to show that the map h ↦ v_{1}h, written in terms of the bases {T_{w}}_{w} and {v_{w}}_{w}, is “a triangular matrix with non-zero diagonal”. This follows from the following claim. Claim: NxI ∩ IyI ≠ ∅ ⇒ x ≤ y in the Bruhat ordering.^{2}

Proof of Claim: Suppose nx ∈ IyI, for n ∈ N. Choose μ so dominant that π^{μ}nπ^{−μ} ∈ I. Then (π^{μ}nπ^{−μ})π^{μ}x ∈ π^{μ}IyI, hence

Iπ^{μ}xI ⊂ Iπ^{μ}IyI ⊂ ∐_{y′≤y} Iπ^{μ}y′I,

from which the claim follows.

□

The following three equalities (see [8], [23]) are also useful.

(1.6.1)  v_{1}T_{w} = v_{w}, for every w ∈ W,
(1.6.2)  v_{π^{μ}}T_{w} = v_{π^{μ}w}, for every w ∈ W and μ ∈ X_{∗}(A),
(1.6.3)  v_{1}T_{π^{μ}} = v_{π^{μ}}, for μ ∈ X_{∗}(A) dominant.

Recall the Iwahori factorization I = (I ∩ N)A_{O}(I ∩ N̄). The first equality uses A_{O}NI · IwI = A_{O}NwI (a consequence of the Iwahori factorization) as well as A_{O}NI ∩ wIw^{−1}I = I (a consequence of A_{O}NI ∩ K = I), and the second equality follows from the first (using the left R-module structure on M). The third equality uses A_{O}NI · Iπ^{μ}I = A_{O}Nπ^{μ}I, a consequence of the Iwahori factorization and the dominance of μ, which implies that

π^{μ}(I ∩ N)π^{−μ} ⊂ I ∩ N and π^{−μ}(I ∩ N̄)π^{μ} ⊂ I ∩ N̄,

and also uses A_{O}NI ∩ π^{μ}Iπ^{−μ}I = I, which we leave as an exercise for the reader.

1.7. Rough structure of the algebra H. The finite dimensional Hecke algebra H_{0} = C(I\K/I) is a subalgebra of H. Moreover, elements in R can be viewed as endomorphisms of M, and hence by the previous lemma can be considered as elements in H. In this way we embed R as a subalgebra of H. We will denote by Θ_{λ} ∈ H the image of the basis element π^{λ} of R under the embedding R ↪ H. Unwinding the definitions, one finds the basic identity

(1.7.1)  v_{1}Θ_{λ} = π^{λ}v_{1},

which says that v_{1} is an eigenvector for the right action of the subalgebra R of H.

Lemma 1.7.1. Multiplication in H induces a vector space isomorphism

R ⊗_{C} H_{0} ⟶^{≈} H,

sending π^{μ}⊗h to Θ_{μ}h. Composing this isomorphism with the isomorphism h ↦ v_{1}h considered above, we get a vector space isomorphism from R ⊗_{C} H_{0} to M, sending π^{μ} ⊗ T_{w} to q^{−⟨ρ,μ⟩}v_{π^{μ}w}.

2The Iwahori subgroup I determines in a canonical way the Bruhat order on W̃, but only when W̃ is viewed as N_{G}(A)/A_{O}. When W̃ is viewed as the semidirect product of W and X_{∗}(A), the Bruhat order depends on the normalization of the isomorphism between X_{∗}(A) and A/A_{O}. Our normalization is μ ↦ π^{μ}, and therefore our Bruhat order on X_{∗}(A) ⋊ W is the one determined by the simple affine reflections about the walls of the unique alcove in X_{∗}(A) whose closure contains the origin and lies in the negative Weyl chamber. See section 7.

Proof. Using (1.6.1) and the definitions, one checks that the composition

R ⊗_{C} H_{0} → H → M

sends π^{μ} ⊗ T_{w} to q^{−⟨ρ,μ⟩}v_{π^{μ}w} and is hence an isomorphism. Since H → M is an isomorphism by Lemma 1.6.1, the map R ⊗_{C} H_{0} → H is also an isomorphism.

□

Remark 1.7.2. It follows from (1.6.3) that Θ_{λ} agrees with the element denoted by this symbol in Lusztig’s work: namely, Θ_{λ} = q^{⟨ρ,−λ_{1}+λ_{2}⟩}T_{π^{λ_{1}}} T_{π^{λ_{2}}}^{−1} , where λ = λ_{1} − λ_{2}, and λ_{1}, λ_{2} are dominant cocharacters.

1.8. Involutions on R and H. Recall that in order to pass back and forth between left and right G-modules one uses the anti-isomorphism g ↦ g^{−1} from G to itself. The corresponding way of passing from left to right H-modules uses the standard anti-involution ι on H given by ι(h)(x) = h(x^{−1}).

Moreover there is also an involution ι_{A} on R (which is the Iwahori-Hecke algebra for A); thus ι_{A} sends π^{μ} to π^{−μ}.

1.9. A sesquilinear form on M. There is an R-valued pairing on i_{B}^{G}(χ_{univ}^{−1}), defined by

(1.9.1)  (ϕ_{1}, ϕ_{2}) := ∮_{B\G} ι_{A}(ϕ_{1}(g))ϕ_{2}(g).

What is the meaning of ∮_{B\G}? Consider the induced representation i_{B}^{G}(δ_{B}^{1/2}), which consists of locally constant functions F on G satisfying

F (ang) = δ_{B}(a)F (g).

The space of G-invariant linear functionals on i_{B}^{G}(δ_{B}^{1/2}) is 1-dimensional; we denote by ∮_{B\G} the unique such functional that takes the value 1 on the function F_{0} ∈

i_{B}^{G}(δ_{B}^{1/2}) defined by F_{0}(ank) = δ_{B}(a).

This pairing is sesquilinear, in the sense that

(1.9.2)  (r_{1}ϕ_{1}, r_{2}ϕ_{2}) = ι_{A}(r_{1})r_{2} · (ϕ_{1}, ϕ_{2}).

Moreover it satisfies

(1.9.3)  (ϕ_{2}, ϕ_{1}) = ι_{A}(ϕ_{1}, ϕ_{2})

and is G-invariant.

Note that ϕ ↦ ι_{A} ∘ ϕ is an ι_{A}-linear isomorphism from i_{B}^{G}(χ_{univ}^{−1}) to i_{B}^{G}(χ_{univ}). Therefore our sesquilinear form can also be thought of as an R-bilinear pairing

(1.9.4)  i_{B}^{G}(χ_{univ}) ⊗_{R} i_{B}^{G}(χ_{univ}^{−1}) → R.

After extending scalars R → C using a quasicharacter χ : A/A_{O} → C^{×}, the pairing (1.9.4) becomes the standard pairing

(1.9.5)  i_{B}^{G}(χ) ⊗_{C} i_{B}^{G}(χ^{−1}) → C.

Recall that we have identified M with the Iwahori fixed vectors in i_{B}^{G}(χ_{univ}^{−1}). Thus, by restriction, we get a perfect sesquilinear form on M, which we denote by (m_{1}, m_{2}). It satisfies the Hecke algebra analog of G-invariance, namely

(1.9.6)  (m_{1}h, m_{2}) = (m_{1}, m_{2}ι(h))

for all h ∈ H.

1.10. Generalities on intertwiners. For each w ∈ W we would like to define an intertwiner I_{w} from one suitable completion of M to another. To this end it is best to let the Borel subgroup vary (and then recover I_{w} by bringing the second Borel back to the first by an element of the Weyl group). In this discussion the maximal torus A, the Iwahori subgroup I, and the maximal compact K will remain fixed. We let B(A) denote the set of Borel subgroups containing A. For B = AN ∈ B(A), put M_{B} = C_{c}(A_{O}N\G/I).

First let’s discuss the completions that will come up. Let J be a set of coroots which is a subset of some system of positive coroots. As usual R denotes the group algebra of X_{∗}(A). We denote by C[J] the C-subalgebra of R generated by J, and by C[J]ˆ the completion of C[J] with respect to the (maximal) ideal generated by J. Finally, we denote by R_{J} the R-algebra C[J]ˆ ⊗_{C[J]}R, a completion of R that can be viewed as the convolution algebra of complex valued functions on X_{∗}(A) supported on a finite union of sets of the form x + C_{J}, with x ∈ X_{∗}(A) and where C_{J} is the submonoid of X_{∗}(A) consisting of all non-negative integral linear combinations of elements in J.

Given B = AN ∈ B(A) and given J as above, we then denote by M_{B,J} the module R_{J} ⊗_{R} M_{B}, which can be thought of as consisting of functions f on A_{O}N\G/I satisfying the following support condition: there exists a finite union S of sets of the form x + C_{J} such that the support of f is contained in the union of the sets A_{O}Nπ^{ν}K for ν ∈ S. It is clear that M_{B,J} is a left R_{J}-module and a right H-module.

Now let B = AN, B′ = AN′ be two Borel subgroups in B(A). As usual we write B̄ = AN̄ for the Borel subgroup in B(A) opposite to B. Let J be the set of coroots that are positive for B′ and negative for B. We are going to define an intertwiner I_{B′,B} : M_{B,J} → M_{B′,J}. This intertwiner is an (R_{J}, H)-bimodule map, and is defined as follows (viewing elements in completions as functions, as above). Let φ ∈ M_{B,J}. Then the intertwiner I_{B′,B} takes φ to the function φ′ on A_{O}N′\G/I whose value at g ∈ G is defined by the integral

φ′(g) = ∫_{N′∩N̄} φ(n′g)dn′.

The Haar measure dn′ is normalized to give N′ ∩ N̄ ∩ K measure 1. Note that the integral makes sense since the integrand is a smooth and compactly supported function on the group N′ ∩N̄ (smoothness being trivial, compact support requiring justification, to be done in the lemma below). In fact things still work fine if we enlarge J in any way (but so that the enlarged set is still contained in some positive system, for instance, the positive system defined by B′).

Now suppose that we have three Borel subgroups B_{1} = AN_{1}, B_{2} = AN_{2}, B_{3} = AN_{3} in B(A). Let J_{ij} be the set of coroots that are positive for B_{i} and negative for B_{j}, and assume that J_{31} is the disjoint union of J_{21} and J_{32}. Write I_{ij} as an abbreviation for the intertwiner I_{B_{i},B_{j}}. Then I_{21}, I_{32} and I_{31} can all be defined using the biggest of the three sets J_{ij}, namely J_{31}, and when this is done we have the equality

I_{31} = I_{32}I_{21}.

In this formula we could also have taken J to be the set of all coroots that are positive for B_{3}.

Why do the integrals make sense? For this we need the following lemma, in which we return to B, B′ as above.

Lemma 1.10.1. For ν ∈ X_{∗}(A) define a subset C_{ν} of the group N′ ∩ N̄ by C_{ν} := N′ ∩ N̄ ∩ π^{ν}NK. Then:

(1) If C_{ν} is non-empty, then ν is a non-negative integral linear combination of coroots that are positive for B and negative for B′.

(2) The subset C_{ν} is compact.

Proof. We begin by recalling the definition of the retraction r_{B} : G → X_{∗}(A). Let g ∈ G and use the Iwasawa decomposition to write g = π^{μ}nk for some μ ∈ X_{∗}(A), n ∈ N, k ∈ K; then put r_{B}(g) := μ. It is well-known that r_{B′}(g) − r_{B}(g) is a non-negative integral linear combination of coroots that are positive for B′ and negative for B. (It is enough to prove this for adjacent B, B′, for which a simple computation in SL(2) does the job.)

To prove the first statement we consider an element g ∈ C_{ν}. It is clear from the definition of C_{ν} that r_{B′}(g) = 0 and r_{B}(g) = ν. Therefore ν = r_{B}(g) − r_{B′}(g) is a non-negative integral linear combination of coroots that are positive for B and negative for B′.

Now we turn to the second statement. It is enough to prove that N̄ ∩ NC is compact for any compact subset C of G, which is equivalent to proving that the map N̄ → N\G is proper (in the topological sense). But in fact N̄ → N\G is a closed immersion (in the algebraic sense), as follows from the fact that NN̄ is closed in G. Recall the proof of this: For any dominant weight λ there exists a unique regular function f_{λ} on the algebraic variety G such that f_{λ}(nan̄) = λ(a)^{−1} for all n ∈ N, a ∈ A, n̄ ∈ N̄. Then NN̄ is the closed subvariety defined by the equations f_{λ} = 1 (one for every dominant λ). □

Next we need to understand how the intertwiners behave with respect to the sesquilinear form on M_{B}. Denote by −J the set of negatives of the coroots in J. The involution ι_{A} on R extends to an isomorphism, still denoted ι_{A}, between R_{J} and R_{−J}, and the sesquilinear form (·, ·) on M_{B} extends to our completions in the following sense: given m_{1} ∈ M_{B,−J} and m_{2} ∈ M_{B,J} our old definition of (m_{1}, m_{2}) still makes sense and yields an element of R_{J}. The extended form (·, ·) still satisfies (1.9.2).

Consider the intertwiner I_{B′,B} : M_{B,J} → M_{B′,J}, where J denotes (as before) the set of coroots that are positive for B′ and negative for B. We also have the intertwiner I_{B,B′} : M_{B′,−J} → M_{B,−J}. Let m ∈ M_{B,J} and m′ ∈ M_{B′,−J}. Then we claim that

(1.10.1)  (m′, I_{B′,B}m) = (I_{B,B′}m′, m).

Indeed, let ϕ, ϕ′ be the elements of i_{B}^{G}(χ_{univ}^{−1})⊗_{R}R_{J}, i_{B′}^{G}(χ_{univ})⊗_{R}R_{J} corresponding to m, m′ respectively. Put H := A(N ∩ N′). Then one sees easily that both sides of the last equality are equal to

(1.10.2)  ∮_{H\G} ϕ′(g)ϕ(g),

where ∮_{H\G} is the unique G-invariant linear functional on

{f ∈ C^{∞}(G) : f(hg) = δ_{H}(h)f(g) (∀h ∈ H), compactly supported mod H}

that takes the value 1 on the function f_{0} supported on HK whose values on HK are given by f_{0}(hk) = δ_{H}(h).

1.11. Intertwiners I_{w}. We return now to the earlier notation, where B = AN is a fixed Borel subgroup. For each w ∈ W, we define an intertwiner

I_{w} : M_{B,w^{−1}J} → M_{B,J}

as the composition I_{B,wB}L(w). Here L(w) is the isomorphism M_{B,w^{−1}J} ˜→ M_{wB,J} given by (L(w)ϕ)(g) = ϕ(ẇ^{−1}g), where ẇ is a representative for w taken in K. Thus I_{w} is defined by the integral

(1.11.1)  I_{w}(φ)(g) = ∫_{N_{w}} φ(ẇ^{−1}ng) dn,

where N_{w} denotes N ∩ wN̄w^{−1}.

From the discussion above, the following properties are immediate.

Lemma 1.11.1. We have (i) I_{w} ∘ π^{μ} = π^{wμ} ∘ I_{w}, ∀μ ∈ X_{∗}(A), (ii) I_{w_{1}w_{2}} = I_{w_{1}} ∘ I_{w_{2}}, if l(w_{1}w_{2}) = l(w_{1}) + l(w_{2}), (iii) I_{w} is a right H-module homomorphism.

1.12. Intertwiners in the rank 1 case. We suppose for the moment that G has semisimple rank 1. We write α for the unique positive root of A, and s_{α} for the corresponding simple reflection, in this case the unique non-trivial element in W.

Now we compute φ′ = I_{s_{α}}(φ) for φ = v_{1} = 1_{A_{O}NI}. We write J(j, w) (j ∈ Z, w ∈ W) for the value of φ′ at the element π^{jα^{∨}}w. Note that other values of φ are 0 and also that J(j, w) = 0 unless j ≥ 0, which we now assume. At this point we may as well take G = SL(2). To simplify notation we temporarily write μ for jα^{∨}.

First suppose that j = 0. Note that s_{α}nw ∈ A_{O}NK iff n ∈ N_{O}. For n ∈ N_{O} the element s_{α}nw belongs to K and hence belongs to A_{O}NI iff its lower left entry is in the prime ideal in O. We conclude that J(0, 1) = 0 and that J(0, s_{α}) = q^{−1}.

Suppose j > 0. We have s_{α} = [\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}], n = [\begin{matrix}1 & x \\ 0 & 1 \end{matrix}], π^{μ} = [\begin{matrix}π^{j} & 0 \\ 0 & π^{−j} \end{matrix}], so that

s_{α}nπ^{μ} = [\begin{matrix}0 & −π^{−j} \\ π^{j} & xπ^{−j} \end{matrix}] .

For s_{α}nπ^{μ}w to lie in A_{O}NK, we must have x ∈ π^{j}O^{×}. We now assume this and

write x = π^{j}u for some unit u. Then s_{α}nπ^{μ} = [\begin{matrix}u^{−1} & −π^{−j} \\ 0 & u \end{matrix}] [\begin{matrix}1 & 0 \\ u^{−1}π^{j} & 1 \end{matrix}], the first

factor lying in A_{O}N, the second factor lying in K. Therefore s_{α}nπ^{μ} ∈ A_{O}NI iff the second factor lies in I, which is always the case. Therefore J(j, 1) is the measure of π^{j}O^{×}, namely q^{−j}(1 − q^{−1}).

Moreover s_{α}nπ^{μ}s_{α} ∈ A_{O}NI iff the product of the second factor and s_{α}, namely

[\begin{matrix}0 & −1 \\ 1 & −u^{−1}π^{j} \end{matrix}], lies in I, which never happens. Therefore J(j, s_{α}) = 0.

We have proved:

Lemma 1.12.1. φ′ = q^{−1}v_{s_{α}} + (1 − q^{−1}) ∑_{j=1}^{∞} q^{−j}v_{π^{jα^{∨}}}.

Even easier:

Lemma 1.12.2. The intertwiner sends 1_{A_{O}NK} to

q^{−1}1_{A_{O}NK} + ∑_{j=0}^{∞} q^{−j}(1 − q^{−1})1_{A_{O}Nπ^{jα^{∨}} K} = \frac{1 − q^{−1}π^{α^{∨}}}{1 − π^{α^{∨}}} 1_{A_{O}NK}.

1.13. Consequences of the calculations above. Now we return to the general case. In the next lemma the calculations reduce easily to the rank 1 case treated above, so we just record the results.

Lemma 1.13.1. Let α be a simple root and s_{α} the corresponding simple reflection. Then

I_{s_{α}}(v_{1}) = q^{−1}v_{s_{α}} + (1 − q^{−1}) ∑_{j=1}^{∞} π^{jα^{∨}}v_{1}. (i)

I_{s_{α}}(v_{1} + v_{s_{α}}) = (\frac{1 − q^{−1}π^{α^{∨}}}{1 − π^{α^{∨}}} )(v_{1} + v_{s_{α}}). (ii)

I_{s_{α}}(1_{A_{O}NK}) = (\frac{1 − q^{−1}π^{α^{∨}}}{1 − π^{α^{∨}}} )1_{A_{O}NK}. (iii)

We now introduce the following notation. For w ∈ W we denote by R_{w} the set of positive roots α such that w^{−1}α is negative.

Corollary 1.13.2 (Gindikin-Karpelevich formula). For w ∈ W we have

I_{w}(1_{A_{O}NK}) = ( ∏_{w∈R_{w}} \frac{1 − q^{−1}π^{α^{∨}}}{1 − π^{α^{∨}}} )1_{A_{O}NK}.

1.14. Intertwiners J_{w} without denominators. To eliminate denominators we define a new intertwiner J_{w} (w ∈ W) by J_{w} := (∏_{α∈R_{w}}(1 − π^{α^{∨}})) · I_{w}. Note that J_{w} preserves the subspace M of M_{B,w^{−1}J} and M_{B,J} and hence can be regarded as an element of H, via our identification of H with End_{H}(M). For a simple root α, the element of H corresponding to J_{s_{α}} is (by Lemma 1.13.1(i)) equal to

(1.14.1)  (1 − q^{−1})π^{α^{∨}} + q^{−1}(1 − π^{α^{∨}})T_{s_{α}}^{3}.

1.15. Bernstein’s relation. Equation (1.14.1), together with the equality

(1.15.1)  J_{w} ∘ π^{μ} = π^{w(μ)} ∘ J_{w}

(for w = s_{α}), yields Bernstein’s relation:

(1.15.2)  T_{s_{α}}π^{μ} = π^{s_{α}(μ)}T_{s_{α}} + (1 − q)\frac{π^{s_{α}(μ)} − π^{μ}}{1 − π^{−α^{∨}}} .

Using Bernstein’s relation one can calculate the square of J_{s_{α}}, viewed as element in H; it turns out to be the element (1 − q^{−1}π^{α^{∨}})(1 − q^{−1}π^{−α^{∨}}) in the subalgebra R of H.

Lemma 1.7.1 together with Bernstein’s relation (1.15.2) gives Bernstein’s presentation of H.

2. The center of H

2.1. A preliminary result. We are going to prove that the subalgebra R^{W} is the center of H, but we start by proving something weaker.

Lemma 2.1.1. The subalgebra R^{W} is contained in the center of H.

3Here we abuse notation and write π^{λ} in place of its image Θ_{λ} under our embedding R ↪ H.

We will often do this (e.g. in (1.15.2) and again in section 2.1), leaving context to dictate what is really meant by π^{λ}.

Proof. Let r ∈ R^{W}. Then r commutes with all elements in R, so by Lemma 1.7.1 it suffices to show it commutes with T_{s_{α}} for all simple α. By the intertwining property (1.15.1) of J_{s_{α}}, it does commute with (1 − q^{−1})π^{α^{∨}} + q^{−1}(1 − π^{α^{∨}})T_{s_{α}}. So r commutes with (1 − π^{α^{∨}})T_{s_{α}} and hence the bracket of r and T_{s_{α}} is annihilated by 1 − π^{α^{∨}}. Since H is a free R-module, the bracket vanishes. □

2.2. The normalized intertwiners K_{w}. Let L denote the field of fractions of the integral domain R. Then L^{W} is the field of fractions of R^{W}. We now consider the algebra H_{gen} := L^{W} ⊗_{R^{W}} H and the module M_{gen} := L ⊗_{R} M = L^{W} ⊗_{R^{W}} M, which is an (L, H_{gen})-bimodule.

We define the normalized intertwiners by

(2.2.1)  K_{w} := ( ∏_{α∈R_{w}} \frac{1}{1 − q^{−1}π^{α^{∨}}} ) · J_{w} = ( ∏_{α∈R_{w}} \frac{1 − π^{α^{∨}}}{1 − q^{−1}π^{α^{∨}}} ) · I_{w}.

Each K_{w} is an endomorphism of the H_{gen}-module M_{gen} and fixes the spherical vector 1_{A_{O}NK}, as one sees from Corollary 1.13.2. For simple α we have K_{s_{α}}^{2} = 1. It follows from this and Lemma 1.11.1 that

(2.2.2)  K_{w_{1}w_{2}} = K_{w_{1}}K_{w_{2}}

for all w_{1}, w_{2} ∈ W.

The involution ι_{A} extends to L, and our sesquilinear pairing form (·, ·) on M extends to a sesquilinear L-valued form, still denoted (·, ·), on M_{gen}. It follows from (1.10.1) that

(2.2.3)  w(K_{w^{−1}}(m), m′) = (m, K_{w}(m′))

for all m, m′ ∈ M_{gen}.

For later use we remark that it follows from (1.14.1) that for any w ∈ W one has

(2.2.4)  K_{w}(v_{1}) = ∑_{w′≤w} a_{ww′} · v_{w′}

for certain elements a_{ww′} ∈ L, with the diagonal elements given by the simple formula

(2.2.5)  a_{ww} = ∏_{α∈R_{w}} \frac{1 − π^{−α^{∨}}}{1 − qπ^{−α^{∨}}} .

2.3. Calculation of the center of H. Since the endomorphism ring of the H_{gen}- module M_{gen} is H_{gen}, we can view the endomorphisms K_{w} as elements of H_{gen}. The map w ↦ K_{w} is a group homomorphism from W to H_{gen}^{×} and therefore induces an algebra homomorphism from the twisted group algebra L[W] to H_{gen}.

Lemma 2.3.1. The homomorphism L[W] → H_{gen} is an isomorphism. The center of H_{gen} is L^{W}. The center of H is R^{W}.

Proof. The twisted group algebra is a matrix algebra over L^{W}, and is therefore simple, which implies our map is injective. Comparing dimensions, we see that the map is an isomorphism. Therefore H_{gen} is a matrix algebra over L^{W}, and its center is L^{W}. It follows easily that the center of H is R^{W}. (Use along the way the obvious fact that H is torsion-free as R^{W} -module.) □

3. Application: Restriction of two involutions to the center

3.1. Restriction of ι to the center. Recall from before the anti-involution ι : H → H given by ι(h)(x) = h(x^{−1}). We are going to see that the restriction of ι to the center of H is very simple.

Lemma 3.1.1. There are two involutions on R^{W}, one obtained by restricting ι_{A} to R^{W}, the other obtained by restricting ι to the center of H, which we have identified with R^{W}. The two involutions on R^{W} coincide.

Proof. This follows from (1.9.2), (1.9.6), the non-degeneracy of our sesquilinear form, and the basic identity

rφ = φz_{r} ∀ r ∈ R^{W}, ∀ φ ∈ M

where z_{r} denotes the element of the center of H that corresponds to r. □

3.2. Restriction of the Kazhdan-Lusztig involution to the center. Now consider the affine Hecke algebra H associated to G. This is an algebra over the ring ℤ[v, v^{−1}] (v an indeterminate), generated by symbols T_{w} (w ranging over the extended affine Weyl group for G), which satisfy the usual braid and quadratic relations. If q = p^{n} denotes the cardinality of the residue field of F, the map v ↦ q^{1/2} determines a ring homomorphism ℤ[v, v^{−1}] → C. There is a canonical isomorphism H = H ⊗_{ℤ[v,v^{−1}]} C (see section 7.2).

The Kazhdan-Lusztig involution h ↦ h̄ of H is determined by v ↦ v^{−1} and T_{w} ↦ T_{w^{−1}}^{−1} (beware that this does not descend to an involution of H). There is also an anti-involution on H given by v ↦ v and T_{w} ↦ T_{w^{−1}}. On specializing v → q^{1/2}, this does descend to H and gives precisely the anti-involution of H denoted ι above; therefore we denote the anti-involution of H by the same symbol.

For each dominant coweight μ ∈ X_{∗}(A), we let z_{μ} = ∑_{λ∈W μ} Θ_{λ}, where Θ_{λ} is the element of H defined by Θ_{λ} = v^{⟨2ρ,−λ_{1}+λ_{2}⟩}T_{t_{λ_{1}}} T_{t_{λ_{2}}}^{−1}, where λ = λ_{1} − λ_{2}, and λ_{i} is dominant (i = 1, 2); see Remark 1.7.2. A result of Bernstein says that the elements z_{μ} form a ℤ[v, v^{−1}]-basis for the center of H, as μ ranges over dominant coweights in X_{∗}(A) (see also section 7.5).

These considerations yield a simple proof of Corollary 8.8 in [16]:

Lemma 3.2.1. \overline{z_{μ}} = z_{μ}.

Proof. First of all we can relate the two involutions by the easily-checked formula ι(Θ_{−λ}) = \overline{Θ}_{λ}. It follows that ι(z_{−w_{0}μ}) = \overline{z_{μ}}, where w_{0} is the longest element of W. On the other hand, the previous lemma says that, at least after v is specialized to q^{1/2}, the elements ι(z_{−w_{0}μ}) and z_{μ} coincide as elements in H. Since this is true for every power of q by the same token, we must have the equality ι(z_{−w_{0}μ}) = z_{μ} in H as well, which proves that \overline{z_{μ}} = z_{μ}. □

4. Satake isomorphism [26]

4.1. Definition of H_{K} and M_{K}. Let e_{K} be the idempotent 1_{K}/ meas(K) in H. Put H_{K} := C_{c}(K\G/K), which we identify with the subring e_{K}He_{K} of H (so that 1_{K} ↦ e_{K}). We also put M_{K} := C_{c}(A_{O}N\G/K), which we identify with the H_{K}-submodule Me_{K} of M. Then M_{K} is an (R, H_{K})-bimodule, with R-module structure inherited from the one on M. Concretely, the action of the function h ∈ H_{K} on m ∈ M_{K} is given by m ∗ h, where ∗ denotes convolution using the Haar measure on G giving K measure 1.

4.2. The Satake transform. Since M_{K} is free of rank 1 as R-module (with basis element the spherical vector 1_{A_{O}NK}), we get a C-algebra homomorphism H_{K} → R, denoted h ↦ h^{∨} and called the Satake transform, characterized by the property that

(4.2.1)  m ∗ h = h^{∨} · m

for all h ∈ H_{K} and all m ∈ M_{K}.

Taking m to be the spherical vector, we get the equation

(4.2.2)  1_{A_{O}NK} ∗ h = h^{∨} · 1_{A_{O}NK}.

In fact h^{∨} lies in the subalgebra R^{W}, as one sees by applying the normalized intertwiners (which fix the spherical vector) to equation (4.2.2). Thus the Satake transform actually maps H_{K} into R^{W}.

Recall that π^{ν} (for ν ranging through X_{∗}) form a C-basis for R. Evaluating both sides of equation (4.2.2) on the element π^{ν} and using the usual G = ANK integration formula (see [4]), one sees that the coefficient of π^{ν} in h^{∨} is equal to

δ_{B}(π^{ν})^{−1/2} ∫_{N} h(nπ^{ν}) dn,

where the Haar measure dn is normalized so that N_{O} has measure 1.

4.3. Satake transform is an isomorphism. The elements h_{μ} := 1_{Kπ^{μ}K}, with μ a dominant coweight, form a C-basis for H_{K}. The elements s_{ν} := ∑_{λ∈W ν} π^{λ}, with ν a dominant coweight, form a C-basis for R^{W}. The coefficients c_{μν} of h_{μ}^{∨} in the basis s_{ν} are given by

(4.3.1)  c_{μν} = δ_{B}(π^{ν})^{−1/2} ∫_{N} 1_{Kπ^{μ}K}(nπ^{ν})dn.

The real number c_{μν} is non-negative and is non-zero if and only if Kπ^{μ}K meets Nπ^{ν}. It follows from [3, 4.4.4] that c_{μν} is 0 unless ν ≤ μ (by which we mean that μ − ν is a non-negative integral linear combination of simple coroots), and it is obvious that c_{μμ} is non-zero. Therefore a standard upper-triangular argument shows that the Satake transform is an isomorphism from H_{K} to R^{W}. In particular H_{K} is commutative.

We remark that in [19, Théoremè 5.3.17], [22], [12], it is shown that if ν ≤ μ (both dominant), then c_{μν} is non-zero.

4.4. Compatibility of two involutions. Recall from 1.8 the involutions ι, ι_{A} on H, R respectively. It is clear that ι preserves the subring H_{K} and that ι_{A} preserves the subring R^{W}. One sees easily (imitate the proof of Lemma 3.1.1) that the Satake isomorphism is compatible with these involutions, in the sense that

(4.4.1)  (ι(h))^{∨} = ι_{A}(h^{∨}).

4.5. Further discussion of the Satake transform. Consider a quasicharacter χ : A/A_{O} → C^{×}. Then χ determines a C-algebra homomorphism R → C. Using this homomorphism to extend scalars, we obtain an H_{K}-module C ⊗_{R} M_{K} which can be identified with the (1-dimensional) space of K-fixed vectors in the unramified principal series representation i_{B}^{G}(χ^{−1}). It is customary to work with left G-modules (and hence left modules over Hecke algebras) rather than right modules, and to switch back and forth between right and left one uses g ↦ g^{−1} on G (and hence the involution ι on H_{K}). Bearing these remarks in mind, one sees that for any h ∈ H_{K} and any K-fixed vector v ∈ i_{B}^{G}(χ) there is an equality

(4.5.1)  hv = h^{∨}(χ)v,

where for r ∈ R we write r(χ) for the image of r under the homomorphism R → C determined by χ. (We used that ι_{A}(r)(χ^{−1}) = r(χ).)

4.6. Compatibility of the Satake and Bernstein isomorphisms [9, 11, 16]. We now have canonical isomorphisms (Satake and Bernstein)

H_{K} ≃ R^{W} ≃ Z(H),

where Z(H) denotes the center of H. Let h ∈ H_{K}, r ∈ R^{W}, z ∈ Z(H) be elements that correspond to each other under these isomorphisms. We have

mh = me_{K}h = rme_{K} = me_{K}z

for all m ∈ M. It follows that

(4.6.1)  h = e_{K}z,

which is the compatibility referred to in the heading of this section.

5. Macdonald’s formula [6, 18, 19]

5.1. Preliminary remarks about unramified matrix coefficients. The contragredient of the induced representation i_{B}^{G}(χ) is i_{B}^{G}(χ^{−1}). (Recall that i_{B}^{G}(χ^{−1}) has as usual a left G-action given by right translations.) Now choose K-fixed vectors v ∈ i_{B}^{G}(χ) and ṽ ∈ i_{B}^{G}(χ^{−1}) such that ⟨v, ṽ⟩ = 1, and put

(5.1.1)  Γ_{χ}(g) := ⟨gv, ṽ⟩,

an unramified matrix coefficient, otherwise known as a zonal spherical function. Clearly Γ_{χ} is a C-valued function on K\G/K, and we have

(5.1.2)  Γ_{χ}(1) = 1.

Let h ∈ H_{K}. It follows from the definition of Γ_{χ} that (h ∗ Γ_{χ})(g) = ⟨gv, hṽ⟩, which by (4.5.1) is equal to h^{∨}(χ^{−1})Γ_{χ}(g). Thus we have

(5.1.3)  h ∗ Γ_{χ} = h^{∨}(χ^{−1})Γ_{χ}.

Similarly we have

(5.1.4)  Γ_{χ} ∗ h = h^{∨}(χ^{−1})Γ_{χ}.

The function Γ_{χ} is uniquely determined by (5.1.2) and either of (5.1.3), (5.1.4); indeed, taking h = 1_{Kπ^{−μ}K} = ι(Kπ^{μ}K) in (5.1.3) and then evaluating both sides at the identity element, we see that

(5.1.5)  meas(Kπ^{μ}K) · Γ_{χ}(π^{μ}) = (1_{Kπ^{μ}K})^{∨}(χ),

where the measure is taken with respect to the Haar measure on G that gives K measure 1. In other words, knowing the values of unramified matrix coefficients is essentially the same as knowing the Satake transforms of the elements 1_{Kπ^{μ}K} ∈ H_{K}.

5.2. Definition of Γ. It is more convenient to work with the R-valued matrix coefficient Γ defined by

(5.2.1)  Γ(g) := (1_{A_{O}NK}, 1_{A_{O}NK} · g),

where (·, ·) is our sesquilinear form on i_{B}^{G}(χ_{univ}^{−1}) (regarded as a right G-module).

Of course Γ is a function on K\G/K with values in R; applying the homomorphism R → C determined by χ to the values of Γ, we get the C-valued function Γ_{χ}. Therefore computing Γ is the same as computing Γ_{χ} for all χ.

We can rewrite (5.2.1) as

(5.2.2)  Γ(g) := (1_{A_{O}NK}, 1_{A_{O}NK} ∗ e_{KgK}),

where e_{KgK} denotes meas(KgK)^{−1} · 1_{KgK}, from which it follows that

(5.2.3)  Γ(g) = (e_{KgK})^{∨},

in agreement with (5.1.5). Equation (5.2.3) shows that Γ actually takes values in R^{W} and hence that Γ_{wχ} = Γ_{χ} for all w ∈ W.

Macdonald’s formula [6, 18, 19] is an explicit formula for Γ_{χ}, which we will now derive, following Casselman’s method [6]. As mentioned above, it is the same to give an explicit formula for Γ, and this is what we will do.

5.3. Decomposition of the spherical vector as a sum of eigenvectors. As a first step towards Macdonald’s formula, we are going to decompose the spherical vector 1_{A_{O}NK} ∈ M as a sum of eigenvectors for the action of the commutative subalgebra R of H. This can only be done in M_{gen}.

Recall that v_{1} denotes the element 1_{A_{O}NI} ∈ M. The vector v_{1} is an eigenvector for the subalgebra R of H by the very definition of that subalgebra; more precisely we have the formula

(5.3.1)  v_{1}Θ_{λ} = π^{λ} · v_{1},

where (as before) Θ_{λ} is a notation for the image of π^{λ} ∈ R under R ↪ H. Applying the normalized intertwiner K_{w} to this equation, we see that

(5.3.2)  K_{w}(v_{1})Θ_{λ} = π^{wλ} · K_{w}(v_{1}),

which shows that K_{w}(v_{1}) is an eigenvector for R with character w^{−1}(χ_{univ}).

Lemma 5.3.1. In M_{gen} we have the formula

1_{A_{O}NK} = ∑_{w∈W} w( ∏_{α>0} \frac{1 − qπ^{α^{∨}}}{1 − π^{α^{∨}}} ) · K_{w}(v_{1}).

Proof. Let w_{0} denote the longest element of W. Recall the standard basis elements v_{x} for M. Then v_{w} (w ∈ W) form an R-basis for M, hence an L-basis for M_{gen}. From (2.2.4) it is clear that the vectors K_{w}(v_{1}) also form an L-basis for M_{gen}. Write the spherical vector in this second basis:

(5.3.3)  1_{A_{O}NK} = ∑_{w∈W} d_{w} · K_{w}(v_{1}).

We can also write the spherical vector in the first basis; since the spherical vector is equal to ∑_{w∈W} v_{w}, it is clear that the coefficient of the basis element v_{w_{0}} in the spherical vector is 1; on the other hand, from (2.2.4) and (5.3.3), it is clear that this same coefficient is also equal to d_{w_{0}}a_{w_{0}w_{0}}; equating the two expressions for the coefficient and using the explicit formula (2.2.5) for a_{w_{0}w_{0}}, we see that

d_{w_{0}} = ∏_{α>0} \frac{1 − qπ^{−α^{∨}}}{1 − π^{−α^{∨}}} .

Moreover, since the normalized intertwiners K_{w} fix the spherical vector, we have

d_{w_{1}w_{2}} = w_{1}(d_{w_{2}})

for all w_{1}, w_{2} ∈ W, from which it follows that

d_{w} = w( ∏_{α>0} \frac{1 − qπ^{α^{∨}}}{1 − π^{α^{∨}}} ).

This completes the proof. □

5.4. Partial information about some more matrix coefficients. We see from Lemma 5.3.1 that in order to calculate Γ it would be enough to calculate the matrix coefficients (1_{A_{O}NK}, K_{w}(v_{1}) · g). Now this new matrix coefficient is a function on I\G/K = X_{∗} rather than K\G/K, and it is difficult to calculate all its values. Fortunately it easy to calculate them for elements g of the form π^{μ} for dominant coweights μ, and in the end this is enough since Γ is K-bi-invariant and hence determined by its values on such elements.

Lemma 5.4.1. For group elements of the form g = π^{μ} with μ dominant, we have

(1_{A_{O}NK}, K_{w}(v_{1}) · g) = [K : I]^{−1} · δ_{B}(π^{μ})^{1/2} · π^{wμ}.

Proof. For f ∈ M, the function f · g need not be right I-invariant, and so need not have a simple form. However, letting δ_{g} denote the Dirac measure concentrated at g, we have an equality of measures e_{I} · δ_{g} · e_{I} = e_{IgI}, where e_{X} is the characteristic function of a set X divided by its measure. Since g = π^{μ} with μ dominant, we have e_{IgI} = δ_{B}(π^{μ})T_{π^{μ}} = δ_{B}(π^{μ})^{1/2}Θ_{μ}. Using these considerations (and the fact that the idempotent e_{I} fixes both 1_{A_{O}NK} and K_{w}(v_{1})), we see that

(1_{A_{O}NK}, K_{w}(v_{1}) · g) = δ_{B}(π^{μ})^{1/2} · (1_{A_{O}NK}, K_{w}(v_{1})Θ_{μ}).

From (5.3.2) we have K_{w}(v_{1})Θ_{μ} = π^{wμ}K_{w}(v_{1}), and therefore

(1_{A_{O}NK}, K_{w}(v_{1}) · g) = δ_{B}(π^{μ})^{1/2} · π^{wμ} · (1_{A_{O}NK}, K_{w}(v_{1})).

Using (2.2.3), we see that

(1_{A_{O}NK}, K_{w}(v_{1})) = w(K_{w^{−1}}(1_{A_{O}NK}), v_{1}) = w(1_{A_{O}NK}, v_{1}).

Moreover (1_{A_{O}NK}, v_{1}) = [K : I]^{−1}, as follows immediately from the definitions. This completes the proof.

□

5.5. Macdonald’s formula. Combining Lemmas 5.3.1 and 5.4.1, we get

Theorem 5.5.1 (Macdonald). For any dominant coweight μ we have

Γ(π^{μ}) = [K : I]^{−1} · ∑_{w∈W} w( ∏_{α>0} \frac{1 − qπ^{α^{∨}}}{1 − π^{α^{∨}}} ) · δ_{B}(π^{μ})^{1/2} · π^{wμ}

and

Γ_{χ}(π^{μ}) = [K : I]^{−1} · ∑_{w∈W}( ∏_{α>0} \frac{1 − q(wχ)(π^{α^{∨}})}{1 − (wχ)(π^{α^{∨}})}) · δ_{B}(π^{μ})^{1/2} · (wχ)(π^{μ}).

5.6. Alternative version of Macdonald’s formula. For any finite subset X ⊂ W̃, define the polynomial X(t) := ∑_{w∈X} t^{l(w)}, where the length function l(·) is defined using the set of reflections for the B̄-positive simple affine roots, as in section 7.1. Let W_{μ} denote the stabilizer of μ in W. We write t_{μ} for the element π^{μ} of the translation subgroup of W̃.

Theorem 5.6.1. For any dominant coweight μ,

(1_{Kπ^{μ}K})^{∨} = \frac{q^{⟨ρ,μ⟩}}{W_{μ}(q^{−1})} ∑_{w∈W} w( ∏_{α>0} \frac{1 − q^{−1}π^{−α^{∨}}}{1 − π^{−α^{∨}}} ) · π^{wμ}.

Proof. Using Theorem 5.5.1, this follows easily from the identities [K : I] = W (q), W (q) = q^{l(w_{0})}W (q^{−1}), meas(Kπ^{μ}K) = W t_{μ}W (q)/W (q), and

(5.6.1)  W t_{μ}W (q) = \frac{W (q)q^{l(t_{μ})}W (q^{−1})}{W_{μ}(q^{−1})}.

To prove (5.6.1), note that any element in W t_{μ}W has a unique decomposition of the form w^{μ}t_{μ}w, where w ∈ W and w^{μ} is a minimal length representative for a coset in W/W_{μ}. Furthermore, such an element has length l(w) + l(t_{μ}) − l(w^{μ}) (as may be seen by induction on l(w^{μ}); note that μ is anti-dominant for B̄). □

6. Casselman-Shalika formula [7, 24, 27]

We are going to give an exposition of Casselman-Shalika’s proof of their formula [7] for unramified Whittaker functions.

6.1. Unramified characters ψ on N̄. Let Δ denote the set of simple roots. The abelian group ∏_{α∈Δ} N_{−α} is a quotient of N̄. Here N_{−α} is the root subgroup for −α, which we identify with the additive group G_{a} over O. Given characters ψ_{α} : N_{−α} → C^{×}, their product defines a character on ∏_{α∈Δ} N_{−α} and hence a character ψ on N̄. We say that ψ is principal if all the ψ_{α} are non-trivial. We say that ψ is unramified if all the characters ψ_{α} are trivial on O but non-trivial on 𝔭^{−1}.

6.2. Whittaker functionals. Let ψ be a principal character on N̄. Let S be a commutative C-algebra. The inclusion of C in S lets us view ψ as a character with values in S^{×}.

Let χ : A → S^{×} be an S-valued character, and form the induced representation i_{B}^{G}(χ), which is both a G-module and an S-module. A Whittaker functional on i_{B}^{G}(χ) is an S-module map

L : i_{B}^{G}(χ) → S

such that L(n̄ϕ) = ψ(n̄)L(ϕ) for all n̄ ∈ N̄ and all ϕ ∈ i_{B}^{G}(χ).

In case S = C Rodier [25] (see also [7]) proved that the space of Whittaker functionals is 1-dimensional and that there exists a unique Whittaker functional W whose restriction to the subspace of functions ϕ in i_{B}^{G}(χ) supported on the big cell BN̄ is given by the integral

(6.2.1)  W (ϕ) = ∫N̄ ϕ(n̄)ψ(n̄)^{−1} dn̄

(the integrand of which is compactly supported by our assumption on the support of ϕ). Here dn̄ denotes the Haar measure on N̄ that gives measure 1 to N̄ ∩ K. For general S the same proof shows that there again exists a unique Whittaker functional W given by (6.2.1) for functions supported on the big cell, and that the S-module of all Whittaker functionals is free of rank 1 with W as basis element.

Now let us consider the case in which S is R and χ is χ_{univ}^{−1}. We let J denote the set of negative coroots and consider the completion R_{J} of R defined in 1.10. It follows from Lemma 1.10.1 that the integral (6.2.1) makes sense as an element of R_{J} for all ϕ ∈ i_{B}^{G}(χ_{univ}^{−1}), and even for ϕ ∈ i_{B}^{G}(χ_{univ}^{−1}) ⊗_{R} R_{J}. Using the uniqueness of W for R_{J}, we see that for ϕ ∈ i_{B}^{G}(χ_{univ}^{−1}) the integral (6.2.1) actually takes values in the subring R of R_{J}. (In other words the presence of the principal character ψ causes all but finitely many of the coefficients of the Laurent power series W (ϕ) to vanish.) Therefore we will now regard the Whittaker functional W on i_{B}^{G}(χ_{univ}^{−1}) as being defined by the integral (6.2.1).

Recall that we have identified the module M with the Iwahori-fixed vectors in i_{B}^{G}(χ_{univ}^{−1}), and thus we also have the (restricted) Whittaker functional W : M → R. It is necessary to calculate W for a few very special vectors in M.

From now on, we assume the character ψ is principal and unramified.

Lemma 6.2.1. Let w_{0} denote the longest element in W, and let α be a simple root with corresponding simple reflection s_{α}. Then

W (v_{1}) = q^{−l(w_{0})} (i)

W (v_{1} + v_{s_{α}}) = q^{1−l(w_{0})} · (1 − q^{−1}π^{−α^{∨}}). (ii)

Proof. The first statement follows from the fact that N̄ ∩ BI = N̄ ∩ I, which has measure q^{−l(w_{0})}. Similarly, the second statement reduces to a calculation in SL(2), which we leave to the reader. Note that for SL(2) the second statement gives the value (namely 1 − q^{−1}π^{−α^{∨}}) of the Whittaker functional on the spherical vector.

□

6.3. Effect of intertwiners on the Whittaker functional. Earlier we defined normalized intertwiners K_{w}, normalized in the sense that they preserve the spherical vector 1_{A_{O}NK}. Now we normalize them differently. Put

(6.3.1)  K′_{w} := ( ∏_{α∈R_{w}} \frac{1 − q^{−1}π^{α^{∨}}}{1 − q^{−1}π^{−α^{∨}}} ) · K_{w} = ( ∏_{α∈R_{w}} \frac{1 − π^{α^{∨}}}{1 − q^{−1}π^{−α^{∨}}} ) · I_{w}.

Lemma 6.3.1 ([13],[7]). The newly normalized intertwiners K′_{w} preserve the Whittaker functional W in the sense that W ∘ K′_{w} = w ∘ W for all w ∈ W. On the right side of this equality w stands for the automorphism of R determined by w. Moreover K′_{w_{1}w_{2}} = K′_{w_{1}}K′_{w_{2}}.

Proof. One sees directly from the definition that K′_{w} is multiplicative in w. Therefore to prove the first statement of the lemma, it is enough to treat the case w = s_{α} for a simple root α. By uniqueness of W there exists c ∈ L^{×} such that

(6.3.2)  W ∘ K′_{s_{α}} = c(s_{α} ∘ W).^{4}

4Here, we are implicitly using normalized intertwiners K_{w} : L⊗_{R} i_{B}^{G}(χ_{univ}^{−1}) → L⊗_{R} i_{B}^{G}(χ_{univ}^{−1}).

The reader may derive the existence and basic properties of such intertwiners following the method of sections 1.10-2.2. Although the discussion there was limited to the theory of intertwiners on the

To prove that c = 1 we evaluate both sides of (6.3.2) on v_{1} + v_{s_{α}}, using Lemma 1.13.1(ii) and Lemma 6.2.1(ii). □

Lemma 6.3.2. In M_{gen} we have the formula

1_{A_{O}NK} = q^{l(w_{0})} · ( ∏_{α>0}(1 − q^{−1}π^{−α^{∨}})) · ∑_{w∈W} w( ∏_{α>0} \frac{1}{1 − π^{−α^{∨}}} ) · K′_{w}(v_{1}).

Proof. This follows from Lemma 5.3.1 and the definition of K′_{w}. □

6.4. Whittaker functions. We continue with S, ψ and χ as in 6.2. For any ϕ ∈ i_{B}^{G}(χ) we define the corresponding Whittaker function W_{ϕ} : G → S by

(6.4.1)  W_{ϕ}(g) := W (gϕ).

Then W_{ϕ} is an S-valued function satisfying the transformation law

(6.4.2)  f(n̄g) = ψ(n̄)f(g)  ∀ n̄ ∈ N̄,

and ϕ ↦ W_{ϕ} is a G-map from i_{B}^{G}(χ) to the space of functions satisfying (6.4.2).

6.5. Unramified Whittaker functions. From now on we assume that both ϕ and χ are unramified. Then inside i_{B}^{G}(χ) we have the normalized spherical vector ϕ_{χ} defined by

ϕ_{χ}(ank) = χ(a)δ_{B}(a)^{1/2}.

The Casselman-Shalika formula is an explicit formula for the Whittaker function W_{χ} := W_{ϕ_{χ}} corresponding to the spherical vector ϕ_{χ}. It is enough to consider the case in which S is R and χ is χ_{univ}^{−1}, in which case we abbreviate W_{χ_{univ}^{−1}} to W.

Since W is right K-invariant and satisfies (6.4.2), it is determined by its values on elements g ∈ G of the form π^{−μ} for μ ∈ X_{∗}. In fact W(π^{−μ}) = 0 unless μ is dominant. Indeed, for x ∈ N_{−α} ∩ K we have

W(π^{−μ}) = W(π^{−μ}x) = ψ_{α}(π^{−μ}xπ^{μ})W(π^{−μ}),

which implies that W(π^{−μ}) vanishes unless ψ_{α} is trivial on 𝔭^{⟨α,μ⟩}, which, since ψ is unramified, implies in turn that ⟨α, μ⟩ ≥ 0. Therefore it is enough to find the values of W(g) for g of the form π^{−μ} for dominant μ.

Theorem 6.5.1 (Casselman-Shalika). Let μ be a dominant coweight. Then

W(π^{−μ}) = ( ∏_{α>0}(1 − q^{−1}π^{−α^{∨}})) · δ_{B}(π^{μ})^{1/2} · E_{μ},

where E_{μ} ∈ R^{W} is the character of the irreducible representation of the Langlands dual group G^{∨} having highest weight μ.

Proof. As usual (see 1.5) we identify i_{B}^{G}(χ_{univ}^{−1}) with C_{c}^{∞}(A_{O}N\G). The spherical vector in the induced representation corresponds to 1_{A_{O}NK}. We begin by noting that

(6.5.1)  W(π^{−μ}) = W (1_{A_{O}NKπ^{μ}}) = W (1_{A_{O}NK} · e_{Iπ^{μ}I})

where e_{Iπ^{μ}I} denotes the characteristic function of Iπ^{μ}I divided by its measure. Here we used that Iπ^{μ}I = Iπ^{μ}(I ∩ N̄) (a consequence of the Iwahori factorization I = (I ∩ B) · (I ∩ N̄) and the dominance of μ), as well as the right I-invariance of 1_{A_{O}NK} and the fact that ψ is trivial on I ∩ N̄. Since e_{Iπ^{μ}I} = δ_{B}(π^{μ})^{1/2} · Θ_{μ}, we can rewrite the equation above as

Iwahori-invariants in the induced modules in question, it is possible to develop a similar theory on the induced modules themselves.

(6.5.2)  W(π^{−μ}) = δ_{B}(π^{μ})^{1/2}W (1_{A_{O}NK} · Θ_{μ}).

It then follows from Lemmas 6.2.1(i), 6.3.1, and 6.3.2 that

W(π^{−μ}) = δ_{B}(π^{μ})^{1/2} · ( ∏_{α>0}(1 − q^{−1}π^{−α^{∨}})) · ∑_{w∈W} w( ∏_{α>0} \frac{1}{1 − π^{−α^{∨}}} ) · π^{wμ}.

The Casselman-Shalika formula now follows from the Weyl character formula:

E_{μ} = ∑_{w∈W} w( ∏_{α>0} \frac{π^{μ}}{1 − π^{−α^{∨}}} ).

□

7. The Kato-Lusztig formula [14, 16]

Following the strategy of [14], we derive the formula of Kato-Lusztig and, as a corollary, another result of Lusztig [16]. The Kato-Lusztig formula relates the Satake transforms of the functions 1_{Kπ^{λ}K} to the character E_{μ} of the highest weight module of the Langlands dual group corresponding to μ. It is the function-theoretic counterpart of the geometric Satake isomorphism [10, 20], and it can also be formally deduced from that statement by using the function-sheaf dictionary.

The proof requires us to give v-analogs of several objects studied above (here v is an indeterminate which can be specialized to q^{1/2}). Most importantly, we need the v-analog of Theorem 5.6.1. The reader willing to accept that on faith may skip directly to section 7.7.

7.1. Preliminaries about affine roots. Write T for the group X_{∗}(A), viewed as the group of translations in the extended affine Weyl group W̃; thus W̃ = T ⋊W. We denote by t_{μ} the element of T corresponding to the cocharacter μ. For simplicity, we assume here that the root system underlying G is irreducible. Let α_{1}, …, α_{r} denote the B-positive simple roots, and let α̃ denote the B-highest root. Let s_{0} = t_{−α̃^{∨}}sα̃, and S_{aff} = S ∪ {s_{0}}. Here S = {s_{α_{i}} = s_{−α_{i}}}_{i=1}^{r} is the set of simple reflections corresponding to the B-positive (or B̄-positive) simple roots, but our definition of s_{0} means that S_{aff} is the set of simple affine reflections corresponding to the B̄-positive affine roots.

We have W̃ = W_{aff} ⋊Ω, where W_{aff} is the Coxeter group generated by S_{aff}, and Ω is the subgroup of W̃ which preserves the set of B̄-positive simple affine roots under the usual left action (an affine-linear automorphism acts on a functional by precomposition with its inverse). The set S_{aff} induces a length function and a Bruhat order on W̃ (the same as that mentioned in Lemma 1.6.1). The elements σ ∈ Ω are of length zero, and the algebra generated by the functions 1_{IσI} is naturally isomorphic to C[Ω]. We have a twisted tensor product decomposition H = H_{aff} ⊗ C[Ω], where H_{aff} is the algebra generated by the functions 1_{IxI}, x ∈ W_{aff} (this follows from the remarks following Lemma 7.2.1 below).

Recall our convention for embedding X_{∗}(A) into A: λ ↦ π^{λ} = λ(π). We also regard each w ∈ W as an element in K, fixed once and for all. These conventions tell us how we view elements of W̃ as elements in G. For example, for SL(2), we are identifying s_{0} with the element [\begin{matrix}0 & π^{−1} \\ −π & 0 \end{matrix}]. It is important to bear these

conventions in mind in this section.

7.2. More about the H-action on M.

Lemma 7.2.1. Fix φ ∈ M and π^{λ}w ∈ W̃, where w ∈ W. Suppose σ ∈ Ω, and that s = s_{α} ∈ S corresponds to a B-positive simple root α. Then we have

(i)  φT_{s_{α}}(π^{λ}w) = \begin{cases}q · φ(π^{λ}ws), if w(α) is B-positive \\ φ(π^{λ}ws) + (q − 1) · φ(π^{λ}w), if w(α) is B-negative \end{cases}

(ii) φT_{s_{0}}(π^{λ}w) = \begin{cases}q · φ(π^{λ}ws_{0}), if w(α̃) is B-negative. \\ φ(π^{λ}ws_{0}) + (q − 1) · φ(π^{λ}w), if w(α̃) is B-positive. \end{cases}
(iii)  φT_{σ}(π^{λ}w) = φ(π^{λ}wσ^{−1}).

Proof. To illustrate the method, we prove (ii). Let {x_{i}}_{i=0}^{q−1} denote a set of representatives for O/P taken in O^{×} ∪ {0}. For a root β, let u_{β} : G_{a} → G denote the associated homomorphism. Then we have the decomposition

Is_{0}I = ∐_{i} u_{−α̃}(πx_{i})s_{0}I.

We therefore have

φT_{s_{0}}(π^{λ}w) = ∑_{i} φ(π^{λ}w u_{−α̃}(πx_{i})s_{0}).

If w(α̃) is B-negative, then each term in the sum is φ(π^{λ}ws_{0}). If w(α̃) is B-positive, then the term for x_{i} = 0 is φ(π^{λ}ws_{0}). If x_{i} ≠ 0, then using the identity

u_{−α̃}(πx_{i})s_{0}I = uα̃(π^{−1}x_{i}^{−1})I

(which holds whenever x_{i} ∈ O^{×}), we see the term indexed by x_{i} is φ(π^{λ}w).

Part (i) can be proved in a similar way; alternatively it can be derived from (1.6.2) together with the usual relations in the Hecke algebra for W. □

The proof of (ii) above parallels the standard proof of the Iwahori-Matsumoto relations in H, which state that for x ∈ W̃, s ∈ S_{aff}, and σ ∈ Ω

T_{x}T_{s} = \begin{cases}T_{xs}, if x < xs \\ q · T_{xs} + (q − 1) · T_{x}, if xs < x \end{cases}
T_{x}T_{σ} = T_{xσ},

where < denotes the Bruhat order determined by S_{aff}. If H denotes the affine Hecke algebra over ℤ_{v} := ℤ[v, v^{−1}] associated to our root system, this means we have a canonical isomorphism H = H ⊗_{ℤ_{v}} C.

Note that R is the Iwahori-Hecke algebra for the group A, and hence it also has a v-analog over ℤ_{v}, which we denote by R. Concretely, we have R = ℤ_{v}[X_{∗}(A)].

We will use Lemma 7.2.1 as the starting point in defining v-analogs

M, iW̃ ; T (χ_{univ}^{−1}), (R, H) − actions, (·|·), K_{w}, M′e_{W} , h^{∨}

of the objects we have already studied

M, i_{B}^{G}(χ_{univ}^{−1})^{I}, (R, H) − actions, (·, ·), K_{w}, M_{K}, h^{∨}.

7.3. v-analogs of M and i_{B}^{G}(χ_{univ}^{−1})^{I}. Let us define M to be the set of functions φ : W̃ → ℤ_{v} which are supported on a finite subset. This is a free ℤ_{v}-module with basis given by the characteristic functions 1_{x}, x ∈ W̃.

Next we define δ : T → ℤ_{v}^{×} by δ(t_{λ}) := v^{−2⟨ρ,λ⟩}. This is the v-analog of the function δ_{B}. By δ^{1/2} we will mean the obvious square root of δ, namely the character t_{λ} ↦ v^{−⟨ρ,λ⟩}.

The left action of R on M is given by the formula t · 1_{x} := δ^{1/2}(t)1_{tx}. The right H-action is given by defining (following Lemma 7.2.1), for φ ∈ M,

(i)  φT_{s_{α}}(t_{λ}w) = \begin{cases}v^{2} · φ(t_{λ}ws), if w(α) is B-positive \\ φ(t_{λ}ws) + (v^{2} − 1) · φ(t_{λ}w), if w(α) is B-negative \end{cases}

(ii)  φT_{s_{0}}(t_{λ}w) = \begin{cases}v^{2} · φ(t_{λ}ws_{0}), if w(α̃) is B-negative. \\ φ(t_{λ}ws_{0}) + (v^{2} − 1) · φ(t_{λ}w), if w(α̃) is B-positive. \end{cases}
(iii)  φT_{σ}(t_{λ}w) = φ(t_{λ}wσ^{−1}).

These rules determine a right H-module structure on M ^{5}, and moreover M is an (R, H)-bimodule. Indeed, it suffices to observe that by Lemma 7.2.1 this statement holds after every specialization v ↦ q^{1/2}. Specialization arguments like this will be used repeatedly below to prove v-analogs of statements known for M.

Now define iW̃ ; T (χ_{univ}^{−1}) to be the set of functions ϕ : W̃ → R which satisfy

ϕ(tx) = δ^{1/2}(t)ϕ(x) · t^{−1},

for t ∈ T and x ∈ W̃. As in section 1.5, there is a canonical isomorphism

(7.3.1)  M = iW̃ ; T (χ_{univ}^{−1}).

Explicitly, we associate to φ ∈ M the function ϕ given by

ϕ(x) = ∑_{t∈T} δ^{−1/2}(t) φ(tx) · t.

The left action of R on iW̃ ; T (χ_{univ}^{−1}) is defined by (t·ϕ)(x) = t(ϕ(x)). The right action of H is defined by requiring the isomorphism M → iW̃ ; T (χ_{univ}^{−1}) to be H-linear (one could also write out an explicit rule, again in the spirit of Lemma 7.2.1). Then M = iW̃ ; T (χ_{univ}^{−1}) is an isomorphism of (R, H)-bimodules.

7.4. v-analog of the sesquilinear pairing. We will define an R-valued sesquilinear pairing (·|·) on iW̃ ; T (χ_{univ}^{−1}) (thus on M) which is almost the v-analog of (·, ·) (they differ by a constant). Hence, it will automatically satisfy the analogs of (1.9.2), (1.9.3), and (1.9.6). We write ι_{R} for the v-analog of ι_{R}, namely the involution on R = ℤ_{v}[X_{∗}(A)] induced by the identity on ℤ_{v} and the map μ ↦ −μ on X_{∗}(A).

For ϕ_{1}, ϕ_{2} ∈ iW̃ ; T (χ_{univ}^{−1}), define

(7.4.1)  (ϕ_{1}|ϕ_{2}) = ∑_{w∈W} v^{2l(w)} ι_{R}ϕ_{1}(w) ϕ_{2}(w).

Lemma 7.4.1. The pairing (·|·) on M induces the pairing W (q) (·, ·) on M = M ⊗_{ℤ_{v}} C.

5Matsumoto gives a similar definition for a left H-action in [19, section 4.1.1].

Proof. Consider the R-basis {1_{w}}_{w∈W} for M, and the corresponding R-basis {v_{w}}_{w∈W} for M. From the definitions, we easily see

(1_{w} | 1_{w′}) = v^{2l(w)} δ_{w,w′}.

It is therefore enough to prove

(v_{w}, v_{w′}) = q^{l(w)} W (q)^{−1} δ_{w,w′}.

The orthogonality is clear, and then one can easily check that

(v_{w}, v_{w}) = (1_{A_{O}NK}, v_{w}) = (1_{A_{O}NK}T_{w^{−1}}, v_{1}) = q^{l(w)}W (q)^{−1}.
□

7.5. v-analogs of normalized intertwiners. For a simple reflection s = s_{α}, define J_{s} : M → M by setting

J_{s}(1_{1}) = v^{−2}(1 − t_{α^{∨}}) · 1_{s} + (1 − v^{−2})t_{α^{∨}} · 1_{1},
J_{s}(1_{1}h) = J_{s}(1_{1})h, for h ∈ H.

This makes sense because M is the free H-module generated by 1_{1} (by the same upper-triangular argument we used to prove that M is the free H-module generated by v_{1}). Further, for any w ∈ W, choose a reduced expression w = s_{1} ⋯ s_{n}, and set

J_{w} := J_{s_{1}} ∘ ⋯ ∘ J_{s_{n}}.

(The usual specialization argument shows that the right hand side is independent of the choice of reduced expression.)

Next define L to be the fraction field of R; note that L^{W} is the fraction field of R^{W} and that L = L^{W} ⊗_{R^{W}} R. Imitating what we did before, we see that R^{W} embeds into the center of H, so that we can form the algebra H_{gen} := L^{W} ⊗_{R^{W}} H and the right H_{gen}-module M_{gen} := L^{W} ⊗_{R^{W}} M = L ⊗_{R} M. Finally we define the normalized intertwiner K_{w} : M_{gen} → M_{gen} by

K_{w} := ( ∏_{α∈R_{w}} \frac{1}{1 − v^{−2}t_{α^{∨}}} ) · J_{w}.

It is clear that

(i) K_{w} is H_{gen}-linear; (ii) K_{w} ∘ t_{λ} = t_{wλ} ∘ K_{w}; (iii) K_{w} fixes 1_{W},

where 1_{W} := ∑_{w∈W} 1_{w} is the v-analog of 1_{A_{O}NK}. It is also clear that w ↦ K_{w} defines a homomorphism W → H_{gen}^{×}, and that the v-analogs of (2.2.3-2.2.5) hold (using (·|·) in (2.2.3)). Moreover, the v-analog of Lemma 5.3.1 holds. Finally, we recover Bernstein’s result that R^{W} is the center of H by going through all the same steps we did before.

7.6. v-analog of the Satake isomorphism. Let ℤ′_{v}, R′, H′, and M′ denote the localizations of ℤ_{v}, R, H, and M at the element W (v^{2}) ∈ ℤ_{v}. Let T_{W} = ∑_{w∈W} T_{w} and e_{W} = W (v^{2})^{−1}T_{W}, an element in H′. Further, define H_{0} = e_{W} H′e_{W}, and M_{0} = M′e_{W}. Then H_{0} is a ℤ′_{v}-algebra with product ∗ := W (v^{2})^{−1}· and identity element T_{W}, where · denotes the usual product in H. Similarly M_{0} is an H_{0}-module with product ∗ := W (v^{2})^{−1}·, where · now denotes the usual H-action on M. It is clear that ∗ makes M_{0} an (R′, H_{0})-bimodule.

The R′-module M_{0} is free of rank 1, so there is a homomorphism

∨ : H_{0} → R′

characterized by

m_{0} ∗ h_{0} = h_{0}^{∨}m_{0},

for all h_{0} ∈ H_{0} and all m_{0} ∈ M_{0}.

We have the formula

(7.6.1)  h_{0}^{∨} = W (v^{2})^{−1}(1_{W} | 1_{W} ∗ h_{0}) = (1_{1} | 1_{1}h_{0}).

We can now easily derive the v-analog of Theorem 5.6.1. We apply the first equality of (7.6.1) to the function

(7.6.2)  h_{μ} := ∑_{w∈W t_{μ}W} T_{w} = \frac{W (v^{2})W (v^{−2})}{W_{μ}(v^{−2})} e_{W} T_{t_{μ}}e_{W}

to get

(h_{μ})^{∨} = \frac{W (v^{−2})}{W (v^{2})W_{μ}(v^{−2})} (1_{W} | 1_{W} · e_{W} T_{t_{μ}}e_{W})

= \frac{v^{−2l(w_{0})}v^{2(l(t_{μ})/2)}}{W_{μ}(v^{−2})} (1_{W} | 1_{W} Θ_{μ}).

Now using the v-analog of Lemma 5.3.1 as in the proof of Theorem 5.5.1, we find

(7.6.3)  (h_{μ})^{∨} = \frac{v^{2(l(t_{μ})/2)}}{W_{μ}(v^{−2})} ∑_{w∈W} w( ∏_{α>0} \frac{1 − v^{−2}t_{−α^{∨}}}{1 − t_{−α^{∨}}} ) · t_{wμ}.

7.7. The Satake isomorphism commutes with the Kazhdan-Lusztig involution. The compatibility of the Bernstein and Satake isomorphisms (4.6) is the commutativity of the following diagram:

\xymatrix{R^{′W} \ar[d]_{B} & e_{W} H′e_{W} \ar[l]_{b} \\ Z(H′), \ar[ur]_{−·e_{W}}}

where b(h_{0}) := W (v^{2})h_{0}^{∨} and where the Bernstein isomorphism B sends ∑_{λ∈W μ} t_{λ} to z_{μ}. By Lemma 3.2.1, B commutes with the Kazhdan-Lusztig involution. Since \overline{e_{W}} = e_{W} ^{6} , the diagonal map does as well. We thus have the following lemma which is implicit in [16], section 8.

Lemma 7.7.1. For every h_{0} ∈ H_{0},

b(\overline{h_{0}}) = \overline{b(h_{0})}.

Equivalently,

(\overline{h_{0}})^{∨} = v^{−2 l(w_{0})} \overline{h_{0}^{∨}}.

6Via the function-sheaf dictionary, the Kazhdan-Lusztig involution corresponds to taking the Verdier dual. The equality can then be derived from the fact that if the constant sheaf on the smooth variety G/B is placed in degree −l(w_{0}) and Tate-twisted by l(w_{0})/2, the resulting complex is Verdier self-dual.

Note that the Kazhdan-Lusztig involution on the commutative ring R′ is simply the map sending ∑_{λ} z_{λ}(v, v^{−1})t_{λ} to ∑_{λ} z_{λ}(v^{−1}, v)t_{λ}.

7.8. The Kato-Lusztig formula. We now switch notation and let q^{1/2} play the role of the indeterminate v used in sections 7.1-7.7. In this section we will use some elementary properties of the Kazhdan-Lusztig polynomials P_{x,y}(q) attached to x, y ∈ W̃, all of which may be found in [15].

Recall that throughout this article, the Bruhat order ≤ and the length function l(·) on W̃ are defined using the B̄-positive affine reflections S_{aff} := S ∪ {s_{0}}. For any dominant coweight λ, the element w_{λ} := t_{λ}w_{0} is the unique longest element in W t_{λ}W, and l(t_{λ}w_{0}) = l(w_{0}) + l(t_{λ}) = l(w_{0}) + 2⟨ρ, λ⟩. It is known that

{x ≤ w_{μ}} = ∪_{λ⪯μ}W t_{λ}W,

where λ ranges over dominant coweights such that μ − λ is a sum of B-positive coroots.

Theorem 7.8.1 (Kato, Lusztig). For any dominant coweight μ, let E_{μ} denote the character of the corresponding highest weight module of the Langlands dual group G^{∨}. Let h_{μ} denote the function ∑_{w ∈ W t_{μ}W} T_{w}. Then we have

E_{μ} = ∑_{λ⪯μ} q^{−l(t_{μ})/2}P_{w_{λ},w_{μ}}(q) (h_{λ})^{∨}.

Proof. We have the identity

\overline{q^{−l(y)/2} ∑_{x≤y} P_{x,y}(q)T_{x}} = q^{−l(y)/2} ∑_{x≤y} P_{x,y}(q)T_{x}.

Applying this to y = w_{μ} and using P_{ww_{λ}w′,w_{μ}}(q) = P_{w_{λ},w_{μ}}(q) for every w, w′ ∈ W, we get

\overline{q^{−⟨ρ,μ⟩−l(w_{0})/2} ∑_{λ⪯μ} P_{w_{λ},w_{μ}}(q) h_{λ}} = q^{−⟨ρ,μ⟩−l(w_{0})/2} ∑_{λ⪯μ} P_{w_{λ},w_{μ}}(q) h_{λ}.

Applying the Satake isomorphism to both sides and using Lemma 7.7.1, we have

q^{⟨ρ,μ⟩} ∑_{λ⪯μ} P_{w_{λ},w_{μ}}(q^{−1}) \overline{h_{λ}^{∨}} = q^{−⟨ρ,μ⟩} ∑_{λ⪯μ} P_{w_{λ},w_{μ}}(q) h_{λ}^{∨}.

By (7.6.3), this gives

∑_{λ⪯μ} q^{⟨ρ,μ−λ⟩}P_{w_{λ},w_{μ}}(q^{−1})W_{λ}(q)^{−1} ∑_{w∈W} t_{wλ} ∏_{α>0} \frac{1 − q t_{−wα^{∨}}}{1 − t_{−wα^{∨}}}

= ∑_{λ⪯μ} q^{−⟨ρ,μ−λ⟩}P_{w_{λ},w_{μ}}(q)W_{λ}(q^{−1})^{−1} ∑_{w∈W} t_{wλ} ∏_{α>0} \frac{1 − q^{−1}t_{−wα^{∨}}}{1 − t_{−wα^{∨}}} .

Now deg P_{w_{λ},w_{μ}}(q) ≤ ⟨ρ, μ − λ⟩ − 1/2 if λ < μ and so Lemma 7.8.2 below implies that the right hand side is a polynomial in q^{−1} (with coefficients in ℤ[X_{∗}]^{W}). Similarly, the left hand side is a polynomial in q. The result now follows since the constant terms are equal to

∑_{w∈W} t_{wμ} ∏_{α>0}(1 − t_{−wα^{∨}})^{−1} = E_{μ}.

□

Lemma 7.8.2. We have

W_{λ}(q^{−1})^{−1} ∑_{w∈W} t_{wλ} ∏_{α>0} \frac{1 − q^{−1}t_{−wα^{∨}}}{1 − t_{−wα^{∨}}} ∈ ℤ[q^{−1}][X_{∗}]^{W} .

Proof. It is obvious that the expression belongs to ℤ[[q^{−1}]][X_{∗}]^{W}, so it is enough by (7.6.3) to show that h_{λ}^{∨} belongs to R. But this follows from (7.6.1). □

Taking q = 1 we immediately recover Theorem 6.1 of [16]:

Theorem 7.8.3 (Lusztig). For any dominant coweight μ,

E_{μ} = ∑_{λ⪯μ} P_{w_{λ},w_{μ}}(1) ( ∑_{w∈W/W_{λ}} t_{wλ}).

References

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