REDUCTION TO REAL INFINITESIMAL

CHARACTER IN AFFINE HECKE ALGEBRAS

DAN BARBASCH AND ALLEN MOY

1. INTRODUCTION

The main result of this paper is to show that the problem of the determination of the unramified unitary dual of split p–adic group is equivalent to the problem of determining the unitary dual of the corresponding graded Hecke algebra.

In [BM], the authors established this equivalence in the case of Iwahori spherical representations under a certain restriction; namely it was essential for the infinitesimal character to be real (in the terminology of [BM]). In terms of the Langlands–Deligne–Lusztig parameters (s, u, ρ) ([KL]), the restriction is that s ∈ ^{L}G, be a purely hyperbolic element. The technique used in [BM] was to combine the notion of the signature of a K–character in [V] with some facts which follow from [KL], namely that the H_{W} –characters of tempered representations are linearly independent. This is essentially true precisely when the infinitesimal character is real; for if not, s has an elliptic part s_{e} such that ^{L}G(s_{e}) ≠ ^{L}G. Then the H_{W} characters of tempered representations behave like induced characters from this smaller group and there is no a priori reason why they should be independent; in general they are not.

The removal of the real infinitesimal character restriction is Theorem 8.1. The proof of Theorem 8.1 is entirely different from [BM]. What we prove, based on ideas of Lusztig ([Ls2]), is that a certain Jantzen type filtration for standard modules for H is equivalent to the same kind of filtration for a corresponding Hecke algebra H(s_{e}). In this equivalence, representations with infinitesimal characters with elliptic part s_{e} correspond; but since s_{e} is a central element in ^{L}G(s_{e}) we are reduced to the setting of [BM].

We give a few more details. Consider the polar decomposition s = s_{e}s_{h} of s into its elliptic and hyperbolic parts. Let C(G, O) be the category of representations of G of finite length and with infinitesimal character having elliptic part in the Weyl group orbit O of s_{e}. Since G has connected center, the centralizer C_{^{L}G}(s_{e}) of s_{e} in ^{L}G is connected. Let G′ be the split p–adic group whose L–group is C_{^{L}G}(s_{e}). Then Theorem 4.3 states that there is a category equivalence between C(G, O) and C(G′, {s_{e}}). The category C(G′, {s_{e}}) in turn is naturally equivalent to C(G′, {1}), the category of representations with real infinitesimal character. The idea of the proof as well as the statement is due to Lusztig. In [Ls2], he first constructs a graded Hecke algebra ℍ corresponding to the Hecke algebra H(G//I) of Iwahori spherical functions. Then he shows for fixed matching infinitesimal characters χ and \overline{χ} that there is an isomorphism between ℍ and H(G//I)_{χ} and ℍ\overline{χ}.

Typeset by AMS-TEX

In order to apply these ideas to questions of unitarity we need two additional ingredients. The first is a slight generalization of Lusztig methods to yield an isomorphism between H(G//I)_{χ} and ℍ\overline{χ} in an analytic family setting. This is done in section 4. The differences to [Ls2] are technical, but they are essential for our argument. When C_{^{L}G}(s_{e}) is a Levi subgroup of ^{L}G, the group G′ is a Levi subgroup of G. In this case the equivalence of categories is effectively the irreducibility of certain unitarily induced representations from G′ to G and is of course much simpler. This is analogous to reduction to real infinitesimal character in real reductive groups [B,V]. However, the group C_{^{L}G}(s_{e}) need not be a Levi subgroup. Here, the equivalence of categories between C(G, O) and C(G′, 1) should be thought of as a strong realization, in a limited situation, of endoscopic transfer of representations between G′ and G. The transfer has the important property that certain data, related to the signature character of a hermitian representation, are preserved.

The second ingredient needed for unitarity is defining a natural ^{∗} operation on the graded Hecke algebra ℍ_{O} and relating it to the ^{∗} operation on H(G//I). This is done in section 5. In section 8, the unitarity of Iwahori spherical representations is reduced to the question of the determination of the unitary dual of a corresponding graded Hecke algebra at real infinitesimal character. In this respect Theorem 5.6 for the * operation is essential. We hope to pursue the problem of determining this dual in future work.

We also note Theorem 6.3 which basically describes the Hecke algebra module of an induced module in terms of the action of the affine part of the Hecke algebra. Undoubtedly this is known to the specialist.

Finally we comment on the validity of our results for the cases when G is a nonsplit group. In this setting, the Hecke algebra with respect to an Iwahori subgroup is a Hecke algebra associated to a parameter set. The allowable parameter sets can be found in [T]. When these algebras are graded at the orbit O = {1}, one obtains a graded Hecke algebra with parameter as described in section 2.13 of [Ls3]. For these graded Hecke algebras, it is known that there are only finitely many real tempered representations. The fact that the W characters of these real tempered representations are linearly independent, follows (presumably) from [Ls4] (generalization of Theorem 4.4 in [BM]). Then the techniques of [BM] apply, so the unitarity of real representations for these groups can be detected on the Iwahori fixed vectors.

ACKNOWLEDGEMENTS. The authors thank George Lusztig for conversations on his work [Ls2], [Ls3] and [Ls4], and Andrei Zelevinski for some conversations on graded Hecke algebras. The authors were supported in part by NSF grants.

2. REVIEW OF BASIC RESULTS

We use the notation in [BM]. Fix a p–adic field 𝔽, and let G be the 𝔽–rational points of a split reductive group. We do not require G to be a specific type in its isogeny class. Thus, for example in the extremes, G can be adjoint or simply connected. The Hecke algebra H(G//I) of compactly supported Iwahori–spherical functions has a vector space decomposition

(2.1)  H(G//I) = H_{K} ⊗ A,

where H_{K} and A are subalgebras of H(G//I).

The subalgebra H_{K} = H(K//I) is the finite Hecke algebra of functions whose support lies in K = G(R_{𝔽}) the integral points of G.

The subalgebra A is abelian. To describe it explicitly, choose a Borel subgroup B = AN compatible with K. Let X = Hom(G_{m}, A) be the algebraic homomorphisms of G_{m} = GL(1, 𝔽) into A. Also, let Π ⊆ R^{+} ⊆ R ⊆ X be the simple coroots, the positive coroots and the coroots determined by B and A respectively. Similarly, let Y = X̌ = Hom(A, G_{m}) and let Π̌ ⊆ Ř^{+} ⊆ Ř ⊆ X̌ be the simple roots the positive roots and the roots.

We mention for later use that ^{L}G is the (connected) complex alegbraic group attached to the data (Y, X , Ř, R, Π̌) dual to (X , Y, R, Ř, Π). In particular, ^{L}A = X̌ ⊗_{ℤ} ℂ^{×} is a maximal torus in ^{L}G, and the group algebra ℂX is the representation algebra of ^{L}A.

The double cosets of I in G are naturally parametrized by the semidirect product W^{a} = W ⋉ X of the Weyl group W and X. Let s_{α} ∈ W denote the reflection corresponding to the simple root α, and let s_{o} ∈ W^{a} be the affine reflection. A basis for the algebra H_{K} is given by the elements

(2.2)  T_{w} = characteristic function of Iw̃I,

where w ∈ W and w̃ ∈ K is a representative of w. The algebra H_{K} is generated by the elements T_{s_{α}}, α ∈ Π. Of course, we have T_{ww′} = T_{w} · T_{w′} if ℓ(ww′) = ℓ(w) + ℓ(w′) and (T_{s_{α}} + 1)(T_{s_{α}} − q) = 0. Here, q is the order of the residue field.

Let X^{+} = {x ∈ X | α̌(x) ≤ 0 ∀ α̌ ∈ Π̌}. For x ∈ X^{+} and ϖ a prime element in 𝔽, set

(2.3)  θ_{x} = characteristic function of Ix(ϖ)I .

An element L ∈ X can be written as the difference L = x − y with x and y in X^{+}. Then the element

(2.4)  θ_{L} = θ_{x}θ_{y}^{−1}

does not depend on the particular choice of x and y used in writing L = x−y. The algebra A is the subalgebra generated by the θ_{L}′s. Identify ℂX with A via the isomorphism L ⟼ θ_{L} for L ∈ X. Each L is a character of ^{L}A, hence we can view θ_{L} as a regular function on the torus ^{L}A and A as the algebra of regular functions on ^{L}A.

In order to describe the cross multiplication between the subalgebras H_{K} and A, it is useful to recall ([Ls2]) the generic affine Hecke algebra H based on the datum (X , Y, R, Ř, Π) with parameter set c, c^{∗}. A parameter set is a pair of maps c : Π ⟶ ℕ and c^{∗} : {α ∈ Π, α̌ ∈ 2Y} ⟶ ℕ satisfying c(α) = c(α′) whenever (α, α̌′) = (α′, α̌) = −1. We write c_{α}, c_{α}^{∗} for the parameters. The special case of H(G//I) in (2.1) corresponds to all the c_{α}’s, and the c_{α}^{∗}’s being equal to 1.

The generic affine algebra with parameters is an algebra H over ℂ[z, z^{−1}], where z is an indeterminant. It has generators T_{α} = T_{s_{α}} (α ∈ Π), θ_{L} (L ∈ X) and relations

(T_{α} + 1)(T_{α} − z^{2c(α)}) = 0 for α ∈ Π, (2.5a)

(2.5b) T_{α}T_{α′}T_{α} ⋯ = T_{α′}T_{α}T_{α′} … α ≠ α′, m factors on both sides, where m is the order of s_{α}s_{α′} in W,

θ_{L}θ_{L′} = θ_{L+L′}, L, L′ ∈ X , (2.5c) (2.5d) θ_{L} = 1 when L is the trivial element of X , θ_{L}(T_{s_{α}} + 1) − (T_{s_{α}} + 1)θ_{s_{α}(L)} = (θ_{L} − θ_{s_{α}(L)})G_{α}, (2.5e) where by Proposition 3.9 in [Ls2]

(2.5f)  G_{α} = \{ \begin{matrix}\frac{θ_{α}z^{2c_{α}} − 1}{θ_{α} − 1}, & if α̌ ∉ 2Y, \\ \frac{(θ_{α}z^{c_{α}+c_{α}^{∗}} − 1)(θ_{α}z^{c_{α}−c_{α}^{∗}} + 1)}{θ_{2α} − 1} & if α̌ ∈ 2Y. \end{matrix}

Let H_{W} be the subalgebras of H generated by the T_{α}′s (α ∈ Π), A the ℂ–subalgebra generated by the θ_{L}′s (L ∈ X) and A[z, z^{−1}] = ℂ[z, z^{−1}] ⊗_{ℂ} A. As a ℂ[z, z^{−1}]–module, H has the decomposition

(2.6)  H = H_{W} ⊗_{ℂ[z,z^{−1}]} A[z, z^{−1}]

When the indeterminant z is specialized to q, the order of the residue field, the algebras H, H_{W} and A[z, z^{−1}] specialize to H(G//I), H_{K} and A respectively.

We identify the algebra A[z, z^{−1}] with the algebra of regular functions on ℂ^{×} ×^{L}A. In this identification, the algebra A is the algebra of regular functions on ^{L}A, while z and z^{−1} generate the algebra of regular functions on ℂ^{×}. The center of H is characterized as follows.

Theorem 2.1. (Bernstein) The center of H is A^{W} ⊗_{ℂ} ℂ[z, z^{−1}].

An immediate consequence of Theorem 2.1 is that a character χ of the center, also called an infinitesimal character, is given by a semisimple

(2.7)  W –orbit in ℂ^{×} ×^{L}A .

Let O be the corresponding orbit in^{L}A and s a representative. The element s has a canonical polar decomposition s = s_{e} · s_{h} into an elliptic and hyperbolic element.

Definition 2.2. An infinitesimal character χ is called real if the corresponding W –orbit of s_{e} consists of a single element.

The Weyl group W acts as automorphisms on A[z, z^{−1}] by the formula w(θ_{L}) = θ_{w(L)} and w(z) = z. The algebra A[z, z^{−1}] has no zero divisors. Let F be its quotient field. As in [Ls2], consider the two algebras defined as follows. The first algebra is

(2.8a)  H(F) = H_{W} ⊗_{A[z,z^{−1}]} F,

with cross multiplication between H_{W} and F is given by the obvious extension of (2.5e). The second algebra is the algebra semidirect product

(2.8b)  ℂW ⋉ F,

with multiplication given by

(2.9)  (w_{1} ⊗ f_{1})(w_{2} ⊗ f_{2}) = w_{1}w_{2} ⊗ w_{2}(f_{1})f_{2} .

An important result on the two algebras is the following.

Theorem 2.3. ([Ls2], Proposition 5.2) The map

ι : ℂW ⋉ F ⟶ H(F)
ι(t_{α}) = (T_{α} + 1)G_{α}^{−1} − 1
ι(f) = f (f ∈ F)

defines an algebra isomorphism.

3. GRADED ALGEBRAS

In [Ls2], Lusztig associates a graded Hecke algebra ℍ to the Hecke algebra H. Lusztig defines the graded Hecke algebra ℍ as follows. Let I be the ideal in A[z, z^{−1}] consisting of the regular functions on ℂ^{×} ×^{L}A which vanish at the element (1, 1). For k an natural integer, Ĩ^{k} = H ·I^{k} is an ideal in H. The graded Hecke algebra ℍ is obtained from H as the graded algebra with respect to the descending sequence of ideals Ĩ^{1} ⊃ Ĩ^{2} ⊃ ⋯ ⊃ Ĩ^{k} ⊃ · · ·. We need a generalization of Lusztig’s construction. Observe that the single element 1 ∈ ^{L}A is W –invariant. Our generalization consists of replacing 1 by a finite W –invariant set O in LA. In analogy with 1, the set O defines an ideal I in A[z, z^{−1}], namely

(3.1)  I = {f : ℂ^{×} ×^{L}A ⟶ ℂ | f(1, σ) = 0 for all σ ∈ O}.

The ideal I^{k} consists of those functions which vanish to order at least k at every element in O. Let Ĩ^{k} = H · I^{k}. Exactly as in [Ls2], H · I^{k} = I^{k} · H so that

(3.2)  Ĩ^{1} ⊃ Ĩ^{2} ⊃ ⋯ ⊃ Ĩ^{k} ⊃ · · · .

is a descending sequence of ideals.

Definition 3.1. Let ℍ_{O} be the graded algebra obtained from H and the sequence of ideals in (3.2).

We summarize the properties of ℍ_{O} relevant for our needs. The proofs are slight modifications of those found in [Ls2].

Define elements in ℍ_{O} as follows.

(3.3a)  r = z − 1 (mod Ĩ)
(3.3b)  t_{α} = T_{s_{α}} (mod Ĩ)  α ∈ Π

The element r is central in ℍ_{O} and by (2.5a,b)

(3.4)  t_{α}^{2} = 1 and (t_{α}t_{α′})^{m} = 1 where m is the order of s_{α}s_{α′} in W

The subalgebra generated by the t_{α}′s is canonically the group algebra ℂW.

Let {F_{σ}}_{σ∈O} be functions on ^{L}A satisfying

(3.5)  F_{σ} − δ_{σ,σ′} vanishes to order at least 2 at each σ′ ∈ O.

In ℍ_{O}, the elements

(3.3c)  E_{σ} = F_{σ} (mod Ĩ)

are well defined and satisfy

(3.6)  ∑_{σ∈O} E_{σ} = 1 ,  E_{σ}E_{σ′} = δ_{σ,σ′}E_{σ} and E_{σ}t_{α} = t_{α}E_{s_{α}(σ)}

For θ = θ_{L} ∈ A and σ ∈ O, set

(3.3d)  E_{σ}ω_{θ} = \frac{θ − θ(σ)}{θ(σ)}F_{σ} (mod Ĩ^{2})
(3.3e)  ω_{θ} = ∑_{σ∈O} E_{σ}ω_{θ}.

Then

(3.7a)  ω_{θ_{1}θ_{2}} = ω_{θ_{1}} + ω_{θ_{2}} ,  for θ_{1} = θ_{L_{1}}, θ_{2} = θ_{L_{2}} ∈ A and
(3.7b)  ω_{s(θ)} = ω_{θ}− < L, α̌ > ω_{α} , for s = s_{α} ∈ W.

The Weyl group W acts on the algebra spanned by the ω_{θ}′s via (3.7b). This algebra is clearly canonically W –isomorphic to the symmetric algebra S of L = X ⊗_{ℤ} ℂ. Let E be the commutative ℂ–algebra generated by the E_{σ}′s and 𝔸 the commutative algebra E ⊗_{ℂ} S.

Proposition 3.2. ([Ls2], Proposition 4.4)

(1) If s = s_{α}, then

(3.8)  ω · t_{s} − t_{s} · s(ω) = r < ω, α̌ > ∑_{σ∈O} E_{σ}h_{σ,α},

where

(3.9)  h_{σ,α} = \{ \begin{matrix}0 & if sσ ≠ σ, \\ 2c_{α} & if sσ = σ, α ∉ 2Y, \\ c_{α} + c_{α}^{∗}θ_{−α}(σ) & if sσ = σ, α ∈ 2Y. \end{matrix}

(2) The generator r lies in the center of ℍ_{O} and

(3.10)  ℍ_{O} = ℂW ⊗_{ℂ} (ℂ[r] ⊗_{ℂ} 𝔸)

with the cross multiplication between ℂW and 𝔸 is given by (3.8).

(3) If O is the disjoint union of two W –invariant sets O_{1} and O_{2}, then as algebras

ℍ_{O} = ℍ_{O_{1}} ⊕ ℍ_{O_{2}}.

(4) The center of ℍ_{O} is

ℂ[r] ⊗_{ℂ} 𝔸^{W} .

The description of ℍ_{O} given in Proposition 3.2 leads immedately to a description of the characters of the center. We can assume that O = W · σ is a single orbit. Let Stab(σ) = {w ∈ W | w(σ) = σ}. Multiplication by E_{σ} defines an isomorphism between the Weyl group invariant elements of 𝔸 and the Stab(σ) invariant elements in S. Under this identification, an infinitesimal character for ℍ_{O} is a

(3.11)  Stab(σ)–orbit in ℂ × (X̌ ⊗_{ℤ} ℂ).

We recall some notation from [Ls2] describing the root datum necessary for defining a smaller graded Hecke algebra. Set

(3.12a)  Stab(σ) = {w ∈ W | w(σ) = σ}
(3.12b)  R_{σ} = \{α ∈ R | θ_{α}(σ) = \{ \begin{matrix}1 & if α̌ ∉ 2Y \\ ±1 & if α̌ ∈ 2Y \end{matrix} \}
(3.12c)  R_{σ}^{+} = R_{σ} ∩ R^{+}
(3.12d)  Π_{σ} = {α ∈ R_{σ}^{+} | α is simple for R_{σ}^{+}}
(3.12e)  W_{σ} = subgroup of W generated by s_{α} (α ∈ Π_{σ})
(3.12f)  Γ_{σ} = {w ∈ Stab(σ) | w(R_{σ}^{+}) = R_{σ}^{+}}

Let H_{σ} be the Hecke algebra associated to the root system (X , Y, R_{σ}, Ř_{σ}, Π_{σ}). The Weyl group of this root system is W_{σ}. The element σ is of course W_{σ}–invariant. Denote the graded version of H_{σ} at σ as ℍ_{σ}. By (3.10),

(3.13)  ℍ_{σ} = ℂW_{σ} ⊗_{ℂ} S[r],

where S is the algebra of polynomial functions on X̌ ⊗_{ℤ} ℂ. The group Γ_{σ} acts as algebra automorphisms of ℍ_{σ} via its actions on W_{σ} and S. Let

(3.14)  ℍ_{σ}′ = Γ_{σ} ⋉ ℍ_{σ}

denote the algebra semidirect product.

Let {w_{1} = 1, w_{2}, …, w_{n}} be coset representatives of W/Stab(σ). In particular, the elements of O are given by τ_{i} = w_{i}σ. Define, as in [Ls2],

(3.15)  E_{i,j} = t_{w_{i}^{−1}}E_{σ}t_{w_{j}}.

Clearly, E_{τ_{i}} = E_{i,i}.

Theorem 3.3.

(1) For each τ = τ_{i},

E_{τ} · ℍ_{O} · E_{τ} is naturally isomorphic to H_{σ}′.

(2) The E_{i,j}’s generate an n × n matrix algebra M_{n}. The algebra ℍ_{O} is naturally isomorphic to the matrix algebra

M_{n}(ℍ_{σ}′) = M_{n} ⊗_{ℂ} ℍ_{σ}′.

The vector space C_{n} = ∑_{i=1}^{n} ℂE_{i,1} is, up to isomorphism, the unique irreducible representation of M_{n}. The next corollary follows easily from Theorem 3.3.

Corollary 3.4.

(1) The functor V ⟶^{F} C_{n} ⊗_{ℂ} V is an equivalence between the categories of finite dimensional modules of Γ_{σ} ⋉ ℍ_{σ} and ℍ_{O}.

(2) As a ℂW module, F(V) = Ind_{W_{σ}}^{W}V.

Following [Ls2], we describe a completion of ℍ_{O} which will be useful later. For our purposes it is sufficient to work in the setting of holomorphic functions rather than formal power series.

The algebras ℂ[r], S and ℂ[r] ⊗_{ℂ} S consist of the polynomial functions on ℂ, X̌ ⊗_{ℤ} ℂ and M = ℂ × (X̌ ⊗_{ℤ} ℂ) respectively. Let ℂ̂[r], Ŝ and ℂ̂[r] ⊗_{ℂ} Ŝ denote the corresponding algebras of holomorphic functions and let K and K̂ be the fields of rational and meromorphic functions on M. Set 𝔸̂ = 𝔸 ⊗_{S} Ŝ. In analogy with (3.10), let

(3.16)  ℍ̂_{O} = ℂW ⊗_{ℂ} (ℂ̂[r] ⊗_{ℂ} 𝔸̂)

and in analogy with (2.8), set

ℍ_{O}(K) = ℂW ⊗ (E ⊗_{ℂ} K) ⊃ ℍ_{O} (3.17a)

ℍ̂_{O}(K̂) = ℂW ⊗ (E ⊗_{ℂ} K̂) ⊃ ℍ̂_{O} (3.17b)

Multiplication for these algebras are defined via (3.8). The algebra ℍ̂_{O}(K̂) contains ℍ_{O}(K). In [Ls2], Lusztig shows for O = {1} that the algebra ℍ̂_{O}(K̂) is naturally isomorphic to a semidirect product algebra (cf Theorem 2.1). Lusztig’s methods easily generalize to our setting, ie consider as in (2.8) the semidirect product algebra

(3.18)  ℂW ⋉ (E ⊗_{ℂ} K̂).

Let

(3.19)  g_{σ,α} = 1 + \frac{rh_{σ,α}}{α}

Theorem 3.5. ([Ls2], Proposition 5.2) The map

ι : ℂW ⋉ (E ⊗_{ℂ} K̂) ⟶ ℍ̂_{O}(K̂) ι(E_{σ}) = E_{σ} ι(f) = f (f ∈ F)

ι(t_{α}) = (t_{α} + 1)(∑_{σ} g_{σ,α}^{−1}E_{σ}) − 1

defines an algebra isomorphism.

4. ANALYTIC FAMILIES

Given a character χ of the center Z of H, let I_{χ} = {z ∈ Z|χ(z) = 0} be the kernel of χ. Both A[z, z^{−1}] · I_{χ} and H · I_{χ} are ideals in A[z, z^{−1}] and H respectively. Let

(4.1)  \begin{aligned}A_{χ} & = A[z, z^{−1}]/\{A[z, z^{−1}] · I_{χ}\} \\ H_{χ} & = H/\{H · I_{χ}\} \end{aligned}

be the quotient algebras. The analogues for the graded algebras 𝔸, ℍ_{O}, 𝔸̂, and ℍ̂_{O} are clear. If \overline{χ} is an infinitesimal character of ℍ_{O} set

(4.2)  \begin{aligned}𝔸\overline{χ} & = 𝔸/\{𝔸 · I\overline{χ}\} = 𝔸̂/\{𝔸̂ · Î\overline{χ}\} = 𝔸̂\overline{χ} \\ ℍ\overline{χ} & = ℍ_{O}/\{ℍ_{O} · I\overline{χ}\} = ℍ̂_{O}/\{ℍ̂_{O} · Î\overline{χ}\} = ℍ̂\overline{χ} \end{aligned}

The main result in this section is Theorem 4.3. It gives conditions under which it is possible to match an infinitesimal character \overline{χ} of ℍ_{O} to an infinitesimal character χ of H so that there are compatible natural isomorphisms from 𝔸\overline{χ} to A_{χ} and from ℍ\overline{χ} to H_{χ}. Theorem 4.3 is essentially Theorem 9.3 in [Ls2], the difference being that for applications to unitarity, we need to verify that the isomorphism is analytic in the ν parameter.

Assume O = W · σ is the Weyl group orbit of an element σ with trivial hyperbolic part. Let Stab(σ) be the centralizer of σ. By (2.7) and (3.11), infinitesimal characters for H and ℍ_{O} are parameterized by W –orbits in ℂ^{×} ×^{L}A and Stab(σ)–orbits in ℂ × (X̌ ⊗_{ℤ} ℂ) respectively.

Proposition 4.1.

(1) The map

(4.3)  \begin{matrix}ℂ × (X̌ ⊗_{ℤ} ℂ) & ⟶ ℂ^{×} ×^{L}A \\ (r, t) & ↦ (e^{r}, σ · e^{t}) \end{matrix}

is Stab(σ)–equivariant. It matches the infinitesimal character χ = W (e^{r}, σ · e^{t}) with the infinitesimal character \overline{χ} = Stab(σ)(r, t). The map ϕ defined by

(4.4)  \begin{matrix}ϕ & : A[z, z^{−1}] ⟶ ℂ̂[r] ⊗_{ℂ} 𝔸̂ \\ ϕ & (z) = e^{r} \\ ϕ & (θ_{x}) = ∑_{σ∈O} θ_{x}(σ) · E_{σ} · e^{x}  (x ∈ X ⊂ S) \end{matrix}

is a ℂ–algebra homomorphism. It maps I_{χ} to I\overline{χ} and defines, by passage to the quotients, an isomorphism between A_{χ} and 𝔸\overline{χ}.

(2) The map

(4.5)  \begin{matrix}Φ & : H ⟶ ℍ̂_{O}(K̂) \\ Φ & (a) = ϕ(a)  (a ∈ A[z, z^{−1}]) \\ Φ & (T_{s} + 1) = ∑_{σ∈O} E_{σ}(t_{s} + 1)\frac{ϕ(G_{α})}{g_{σ,α}}, \end{matrix}

where G_{α} and g_{σ,α} are defined in (2.5f) and (3.19) respectively, is an algebra homomorphism.

Proof. Clearly the map defined by (4.3) is Stab(σ)–equivariant. It matches infinitesimal characters because if σ · e^{t} and σ · e^{t′} represent χ then t and t′ must be conjugate by an element of Stab(σ). Consider now the map ϕ. That ϕ is a ℂ–algebra homomorphism follows from the definition. Furthermore, if y = ∑_{x} c_{x}θ_{x}, then

ϕ(y) = ∑_{x} c_{x} ∑_{σ∈O} θ_{x}(σ)E_{σ}e^{x} = ∑_{σ∈O} E_{σ}(∑_{x} c_{x}θ_{x}(σ)e^{x}).

Relating the coefficient of E_{σ} to the coefficient of E_{w(σ)}, we see that ϕ(y) lies in 𝔸̂^{W} if and only if y is in A^{W}. This means ϕ(I_{χ}) ⊂ I\overline{χ} which in turn implies both the existence of the quotient map from A_{χ} to 𝔸\overline{χ} and its injectivity. Since both A_{χ} and 𝔸\overline{χ} are of dimension |W | over ℂ, the homomorphism is in fact an isomorphism.

For part (2), note that Φ is the composition of \overline{ι} ∘ (id, ϕ) ∘ ι^{−1}|_{H}, where ι is defined in Theorem 2.3, \overline{ι} is defined in Theorem 3.5 and

(4.6)  \begin{aligned}(id, ϕ) & : ℂW ⋉ F ⟶ ℂW ⋉ (E ⊗_{ℂ} K̂), \\ (id, ϕ) & : (w, f) ↦ (w, ϕ(f)). \end{aligned}

That Φ is an isomorphism follows from the definitions of the cross multiplications in terms of G_{α} and g_{σ,α}. □

Definition 4.2. Suppose M is a real analytic manifold. An analytic family of ℂ–algebras depending on a parameter ν ∈ M is a finite dimensional space A equipped with an algebra multiplication ·_{ν} such that for each linear functional λ ∈ Hom_{ℂ}(A, ℂ), and any a_{1}, a_{2} ∈ A

λ(a_{1} ·_{ν} a_{2}) : M ⟶ ℂ

is real analytic. In coordinates this means that if we choose a basis {x_{1}, …, x_{m}} then

x_{i} ·_{ν} x_{j} = ∑_{k} c_{ij}^{k}(ν)x_{k}

where the c_{ij}^{k} are real analytic for ν ∈ M.

A morphism of analytic families is an algebra map

Φ_{ν} : A ⟶ B

such that when Φ_{ν} is expressed as a matrix with respect to the basis {x_{1}, …, x_{m}}, all entries are analytic in ν. Clearly these notions are independent of the basis chosen.

The algebras A[z, z^{−1}] and 𝔸̂ ⊗_{ℂ} ℂ̂[r] can be used to construct analytic families of ℂ– algebras of dimension |W | parameterized by M = ℂ × (X̌ ⊗_{ℤ} ℂ) which contain A_{χ} and 𝔸\overline{χ} = 𝔸̂\overline{χ} respectively. To see this, we choose a basis \{a_{1}, …, a_{m}\} (m = |W |) of A as a free A^{W} module. Define c_{i,j}^{k} ∈ A^{W} by

a_{i} · a_{j} = ∑_{k} c_{i,j}^{k}a_{k}.

By part (1) of Proposition 4.1, ϕ(a_{i}) is a basis of 𝔸̂ as a free 𝔸̂^{W} module.

For ν = (r, t) ∈ M and χ = W (e^{r}, σ · e^{t}), the algebra A_{χ} is just the finite dimensional algebra A_{ν} given formally as the commutative algebra with ℂ–basis a_{i} and multiplication

a_{i} ·_{ν} a_{j} = ∑_{k} c_{i,j}^{k}(χ)a_{k}.

Similarly, 𝔸\overline{χ} = 𝔸̂\overline{χ} is the commutative algebra 𝔸̂_{ν} with formal ℂ–basis ϕ(a_{i}), and multiplication

ϕ(a_{i}) ·_{ν} ϕ(a_{j}) = ∑_{k} ϕ(c_{i,j}^{k})(\overline{χ})ϕ(a_{k}).

In terms of analytic families, since c_{i,j}^{k}(χ) = ϕ(c_{i,j}^{k})(\overline{χ}), the map a_{i} ↦ ϕ(a_{i}) on the basis vectors is an isomorphism between A_{ν} and 𝔸̂_{ν}.

In a similar fashion, the three algebras H, ℍ_{O} and ℍ̂_{O} can also be viewed as analytic families of ℂ–algebras of dimension |W |^{2}, ie used to construct ℂ–algebras of dimension |W |^{2}. The parameter space is the same. Inclusion of ℍ_{O} into ℍ̂_{O} yields an inclusion of analytic families. The main result we want to show is that H_{χ} and ℍ\overline{χ} are isomorphic under suitable conditions on the parameter ν. We may of course view the three algebras as analytic families over M_{ℝ} = ℝ × (X̌ ⊗_{ℤ} ℝ^{×}) ⊂ M.

Theorem 4.3. For ν ∈ M_{ℝ}, the homomorphism

Φ_{ν} : H_{χ} ⟶ ℍ̂\overline{χ}

defined via Proposition 4.2 is an analytic isomorphism of the two families of algebras.

Proof. It is enough to show f_{σ,α} = \frac{ϕ(G_{α})}{g_{σ,α}} ∈ K̂ is analytic and nowhere zero on M_{ℝ}. Adapting the proof of Lemma 9.5 in [Ls2] to our situation, we find

f_{σ,α} = \{ \begin{matrix}\frac{e^{2rc_{α}}e^{α}θ_{α}(σ) − 1}{e^{α}θ_{α}(σ) − 1} & if sσ ≠ σ, \\ \frac{e^{2rc_{α}}e^{α} − 1}{e^{α} − 1} · \frac{α}{2rc_{α} + α} & if sσ = σ, α̌ ∉ 2Y, \\ \frac{(e^{r(c_{α}+c_{α}^{∗})}e^{α}θ_{α}(σ) − 1) (e^{r(c_{α}−c_{α}^{∗})}e^{α}θ_{α}(σ) + 1) α}{(e^{2α} − 1) \{r(c_{α} + c_{α}^{∗}θ_{−α}(σ)) + α\}} & if sσ = σ, α̌ ∈ 2Y. \end{matrix}

Note that θ_{α}(σ) is not a positive real number if s(σ) ≠ σ. Also, if s(σ) = σ then

θ_{α}(σ) = \{ \begin{matrix}1 & if α̌ ∉ 2Y, \\ ±1 & if α̌ ∈ 2Y. \end{matrix}

It is clear that f_{σ,α} is a nonzero nonvanishing function on M_{ℝ}. In particular, Φ_{ν} is an isomorphism for all \overline{χ} = W_{σ}–orbit of (r, ν). □

5. HERMITIAN STRUCTURES

An important property of H(G//I) is that it is a ∗–algebra with an inner product. For f_{1}, f_{2} ∈ H(G//I),

(5.1)  \begin{aligned}f_{i}^{∗}(g) & = \overline{f_{i}(g^{−1})} \\ ⟨f_{1}, f_{2}⟩ & = ∫_{G} f_{1}(g)\overline{f_{2}(g)}dg = f_{1} ∗ f_{2}^{∗}(1). \end{aligned}

There is a natural^{∗} operation on the generic affine Hecke algebra H compatible with the corresponding one on H(G//I). We describe ^{∗} on a set of generators. Assume R is irreducible. Recall that H is generated by the elements

(5.2)  {T_{α}, T_{o}, τ, θ_{x}}_{α∈Π,x∈X_{0}},

where the T_{α}’s and T_{0} are generators indexed by the simple reflections in the affine Weyl group and

(5.3)  X_{0} = {x ∈ X | < x, α̌ >= 0 for all α̌ ∈ Ř}.

We need to describe τ. The group A_{s} = {x(a) ∈ G | x ∈ X_{0}, a ∈ 𝔽} is the split component of the center of G. Let N_{s} be the normalizer of A_{s}I in G. Denote by Q the ℤ span of the elements in R. By [S], we have X_{0} ∩ Q = ∅ and X /[Q + X_{0}] is finite and isomorphic to N_{s}/{A_{s}I}. Because R is irreducible, these quotients are cyclic. Let tA_{s}I be a generator of N_{s}/{A_{s}I}. Then It = tI = ItI. In H(G//I), the characteristic function T_{t} of ItI is invertible with inverse the characteristic function of It^{−1}I = It^{−1} = t^{−1}I. Let τ be the element in H which goes to T_{t}. In terms of the generators (5.2), ^{∗} is the unique complex conjugate linear antiautomorphism which satisfies

(5.4a)  z^{∗} = z
(5.4b)  T_{α}^{∗} = T_{α}
(5.4c)  T_{o}^{∗} = T_{o},
(5.4d)  τ^{∗} = τ^{−1},
(5.4e)  θ_{x}^{∗} = θ_{−x} = θ_{x}^{−1}, x ∈ X_{0} ( split reductive group) .

The next lemma gives some indication of how ∗ behaves on A.

Lemma 5.1.

(5.5)  θ^{∗} = θ^{−1} (mod (z − z^{−1})H).

Proof. Observe that

(5.6) T_{w}^{−1} ≡ T_{w^{−1}} (mod (z − z^{−1})H), and T_{w}θ_{x} ≡ θ_{w(x)}T_{w} (mod (z − z^{−1})H).

In view of the relations in (5.4) it is sufficient to prove (5.5) for one 0 ≠ x ∈ Q. Let γ be the negative of the highest short root in R^{+}, and T_{γ} = T_{s_{γ}} the element in H corresponding to the reflection s_{γ} ∈ W about γ. Then

(5.7)  θ_{γ} = T_{0}T_{γ}.

Choose a simple root α and a sequence of simple reflections s_{1}, …, s_{k} so that

γ = s_{1} · . . . · s_{k}(α).

In the Hecke algebra

T_{γ} = T_{1} · . . . · T_{k} · T_{α} · T_{k} · . . . · T_{1}.

Thus

θ_{γ}^{∗} = (T_{γ})^{∗}T_{0} = T_{γ}θ_{γ}(T_{γ})^{−1}.

By (5.6) and (5.7), this gives

(θ_{γ})^{∗} ≡ (θ_{γ})^{−1} = θ_{−γ} (mod (z − z^{−1})H). □

If (π, V) is a finite dimensional representation of H(G//I), let (π^{∗}, V^{∗}) be the hermitian dual with respect to ^{∗}. To describe the hermitian dual of an irreducible representation, we review their classification. Kazhdan and Lusztig have shown in [KL] that the equivalence classes of irreducible modules for H are in 1–1 correspondence with ^{L}G conjugacy classes of pairs {ϕ, ρ}, where ϕ : ℤ × SL(2, ℂ) ⟶ ^{L}G is an admissible homomorphism, ie. ϕ(1, 1) is semisimple, and ρ are certain characters of the component group of the centralizer of ϕ. It is well known that ϕ is equivalent to a pair of elements s, u ∈ ^{L}G with s semisimple, u unipotent and sus^{−1} = u^{q}. If s ∈ ^{L}G, let s = s_{e}s_{h} be the polar decomposition of s into its elliptic and hyperbolic parts. Set

(5.8)  \overline{s} = s_{e}s_{h}^{−1}.

For n ∈ ℤ and g ∈ SL(2, ℂ), set \overline{ϕ}(n, g) = \overline{ϕ(n, 1)}ϕ(0, g).

Theorem 5.2. Let (π, V) correspond to {ϕ, ρ}. Then the hermitian dual (π^{∗}, V^{∗}) corresponds to \{\overline{ϕ}, ρ\}. In terms of the pair \{s, u\}, an irreducible module admits a hermitian structure precisely when \overline{s} is ^{L}G conjugate to s.

The proof is similar to the corresponding result for real reductive groups. We omit the details.

To proceed further, we need to understand when ℍ_{O} is a ∗–algebra. If s ∈ ^{L}A, let \overline{s} = s_{e} · s_{h}^{−1} be as in (5.8) and set

(5.9)  O^{∗} = \{\overline{σ} | σ ∈ O\}.

By Lemma 5.1, the ideal of H corresponding to O^{∗} is the ∗ of H · I. The ∗ operation on H induces an antiisomorphism between ℍ_{O} and ℍ_{O^{∗}}. The ∗ operation satisfies

(5.10a)  r^{∗} = r
(5.10b)  w^{∗} = w^{−1} (w ∈ W)

It follows from the relation

(5.11)  θ(\overline{σ}) = \overline{θ(σ^{−1})}.

that

(5.10c)  E_{σ}^{∗} = E\overline{σ}.

Definition 5.3. The set O is said to be hermitian if it is equal to O^{∗}.

In this setting, it follows from Lemma 5.1 that the ideal H · I is ∗ invariant, so that ℍ_{O} and ℍ̂_{O} are ∗ algebras. A particularly important instance in which O is hermitian is when it is the Weyl group orbit of an elliptic element. We assume for the remainder of the section that O is hermitian.

We compute a formula for ω^{∗} for ω ∈ ℂ ⊗_{ℤ} X .

Lemma 5.4. Let γ ∈ R^{+} be the highest short coroot, let s_{γ} ∈ W be the corresponding reflection and let T_{γ} = T_{s_{γ}} be the corresponding element in H_{W}. Then

(5.12)  γ^{∗} = t_{γ} · γ · t_{γ} = t_{l} · γ · t_{l},

where t_{l} is the element in the graded Hecke algebra corresponding to the long element w_{l} in the Weyl group.

Proof. As in the proof of Lemma 5.1, choose a simple root α and a sequence of simple reflections s_{1}, …, s_{k} so that

(5.13)  γ = s_{1} · . . . · s_{k}(α).

Then in the Hecke algebra

(5.14)  T_{γ} = T_{1} · . . . · T_{k} · T_{α} · T_{k} · . . . · T_{1},

and similarly in the Weyl group. By relation (2.5a),

(5.15)  T_{β}^{−1} = z^{−2c(β)}T_{β} − z^{−2c(β)}(z^{2c(β)} − 1) ≡ t_{β}( mod Ĩ^{∗} )

Thus

(5.16)  (T_{γ})^{−1} ≡ t_{γ} ( mod Ĩ^{∗} ),

Then,

(5.17)  θ_{γ} = T_{0} · T_{γ} ∈ X^{+}.

To simplify notation, set θ = θ_{γ}. If we apply ^{∗} to (5.17) and use the fact that by (5.14) (T_{γ})^{∗} = T_{γ}, we get

(5.18)  θ^{∗} = (T_{γ})^{∗}T_{0} = (T_{γ})^{∗}θT_{γ}^{−1} = T_{γ}θT_{γ}^{−1}.

Then,

(5.19)  (∑_{σ∈O} F_{σ} \frac{θ − θ(σ)}{θ(σ)})^{∗} = ∑_{σ∈O} \frac{T_{γ}θT_{γ} − \overline{θ(σ)}}{\overline{θ(σ)}}(F_{σ})^{∗}.

This expression is in Ĩ^{∗} and we must grade it modulo (Ĩ^{∗})^{2}. Thus we can take (T_{γ})^{∗}, (F_{σ})^{∗} and (T_{γ})^{−1} modulo Ĩ^{∗}. We get

(5.20)  \begin{aligned}γ^{∗} & = ∑_{σ∈O} \frac{t_{γ}θt_{γ} − \overline{θ(σ)}}{\overline{θ(σ)}} E\overline{σ} ( mod Ĩ^{2} ) = t_{γ} ∑_{σ∈O} E\overline{σ} \frac{θ − \overline{θ(t_{γ}(σ))}}{\overline{θ(t_{γ}(σ))}} t_{γ} \\ & = t_{γ} ∑_{σ∈O} E\overline{σ} \frac{θ − θ(\overline{σ})}{θ(\overline{σ})} t_{γ} = t_{γ}γ t_{γ} \end{aligned}

The second part of (5.12) follows from the fact that t_{l} = t_{γ} · t^{γ}, where t^{γ} is a product of simple roots all commuting with γ. □

We obtain a more general formula for ^{∗} on S as follows. For ω ∈ X, set

(5.21)  ω̃ = −t_{l}(ω)

and extend it to a conjugate linear automorphism of S. This map defines a permutation of the set of simple roots. The map ˜ extends to an automorphism of S.

Corollary 5.5. For f ∈ S ⊂ ℍ_{O},

(5.22)  f^{∗} = t_{l} · f̃· t_{l}.

Proof. It is enough to prove the formula for w(γ) with w ∈ W, or equivalently, it is enough to show that for α ∈ Π, and ω ∈ X ,

(5.23)  if ω^{∗} = t_{l} · ω̃ · t_{l}, then s_{α}(ω)^{∗} = t_{l} · { \atop s_{α}(ω)} · t_{l}.

By Proposition 3.2,

(5.24)  s_{α}ω = t_{α} · ω · t_{α} − r < ω, α̌ > ∑_{σ∈O} E_{σ}h_{σ,α}t_{α}.

Therefore

(5.25)  \begin{aligned}s_{α}(ω^{)}∗ & = t_{α} · ω^{∗} · t_{α} − r < ω, α̌ > ∑_{σ∈O} t_{α}h_{σ,α}^{∗}E_{σ}^{∗} \\ & = t_{α} · t_{l} · ω̃ · t_{l} · t_{α} − r < ω, α̌ > ∑_{σ∈O} t_{α}h_{σ^{∗},α}E_{σ^{∗}} \\ & = t_{l} · (tα̃ · ω̃ · tα̃) · t_{l} − r < ω, α̌ > ∑_{σ∈O} t_{α}h_{σ^{∗},α}E_{σ^{∗}}. \end{aligned}

Thus, by using the *–invariance of O, we can rewrite this as

(5.26)  \begin{aligned}s_{α}(ω)^{∗} & = t_{l} · [tα̃ · ω̃ · tα̃ − r ∑_{σ∈O} < { \atop s_{α}(ω)}, α > tα̃h_{σ,α}E_{σ}] · t_{l} \\ & = t_{l} · sα̃(ω̃) · t_{l} = t_{l} · { \atop s_{α}(ω)} · t_{l} □ \end{aligned}

We now prove a different more explicit formula for *. Recall the definition of h_{σ,α} in Proposition 3.2. We extend it to any root β ∈ R, subject to h_{wσ,wα} = h_{σ,α}. This is well defined in view of the properties of the functions c, c^{∗}.

Theorem 5.6. For ω ∈ X ,

(5.27)  ω^{∗} = −ω + r ∑_{β∈R^{+}} t_{β} < ω, β̌ > ∑_{σ∈O} E_{σ}h_{σ,β}.

Proof. For each ω, consider the expression

(5.28)  τ(ω) = ω − \frac{r}{2} ∑_{β∈R^{+}} t_{β} < ω, β̌ > ∑_{σ∈O} E_{σ}h_{σ,β}.

We claim that τ(ω) has the property that

(5.29)  t_{w}τ(ω) = τ(w(ω))t_{w}.

To prove this it is enough to consider the case w = s_{α} for α a simple root. Here, it is enough to show that

(5.30)  t_{s}τ(ω)t_{s} = τ(s(ω)).

By part (1) of Proposition 3.2

(5.31)  t_{s}ωt_{s} = s(ω) + r < ω, α̌ > ∑_{σ∈O} E_{σ}h_{σ,α}t_{s}.

On the other hand,

(5.32)  \begin{aligned}t_{s} ∑_{β∈R^{+}} t_{β} & < ω, β̌ > ∑_{σ∈O} E_{σ}h_{σ,β} \\ & = ∑_{β∈R^{+}} t_{s(β)} < ω, β̌ > ∑_{σ∈O} E_{s(σ)}h_{σ,β}t_{s} \\ & = ∑_{β∈R^{+}\α} t_{s(β)} < ω, β̌ > ∑_{σ∈O} E_{s(σ)}h_{σ,β}t_{s}+ < ω, α̌ > ∑_{σ∈O} E_{σ}h_{σ,α} \\ & = ∑_{β∈R^{+}} t_{β} < ω, β̌ > ∑_{σ∈O} E_{σ}h_{σ,β}t_{s} − 2r < ω, α̌ > ∑_{σ∈O} E_{σ}h_{σ,α} \end{aligned}

because of (5.24). Formula (5.29) follows by combining (5.30) and (5.31).

We now apply this to the highest coroot γ. In view of Lemma 5.4,

(5.33)  t_{γ}(τ(γ))t_{γ} = τ(t_{γ}(γ)) = −τ(γ)

Therefore

(5.34) γ^{∗} = −γ + \frac{r}{2}t_{γ} ∑_{β∈R^{+}} t_{β} < γ, β̌ > ∑_{σ∈O} E_{σ}h_{σ,β}t_{γ} + \frac{r}{2} ∑_{β∈R^{+}} t_{β} < γ, β̌ > ∑_{σ∈O} E_{σ}h_{σ,β}.

The proof is complete once we observe that

(5.35)  t_{γ} ∑_{β∈R^{+}} t_{β} < ω, β̌ > ∑_{σ∈O} E_{σ}h_{σ,β}t_{γ} = ∑_{β∈R^{+}} t_{β} < ω, β̌ > ∑_{σ∈O} E_{σ}h_{σ,β}

This completes the proof of (5.27) for γ. Formula (5.27) now follows for any w(γ) with w ∈ W by

τ(w(γ))^{∗} = (t_{w}τ(γ)t_{w}^{−1})^{∗} = t_{w}τ(γ)^{∗}t_{w}^{−1} = −τ(w(γ))

The general case follows from the fact that ∗ is a (complex conjugate) antiautomorphism. □

At this point we are ready to explain the relationship between the ^{∗} operation on H and the ^{∗} operation on ℍ_{O}. Recall that f_{σ,α} = \frac{ϕ(G_{α})}{g_{σ,α}}. Define

(5.36)  \begin{aligned}F_{σ,ρ} & = ∏_{α∈R^{+}} f_{σ,α},  F_{ρ} = ∑ E_{σ}F_{σ,ρ} \\ m & = Φ(T_{l}) · F_{ρ}^{−1} · t_{l} \end{aligned}

where T_{l} is the element in H corresponding to w_{l}.

Theorem 5.7. For ν real, the isomorphism Φ of Theorem 4.3 is compatible with the * structures on H_{χ} and ℍ\overline{χ} in the following sense:

(5.37a)  Φ(T_{α}^{∗}) = m · Φ(T_{α})^{∗} · m^{−1}, α ∈ Π,
(5.37b)  Φ(θ^{∗}) = m · Φ(θ)^{∗} · m^{−1}, θ ∈ A.

Proof. The fact that m is invertible follows from the fact that f_{σ,α} are analytic nowhere zero on M_{ℝ} established in the proof of Theorem 4.3.

Observe first that T_{α}^{∗} = T_{α}. Furthermore,

(5.38)  Φ(T_{α})^{∗} = [t_{l}f̃α̃t_{l}(t_{α} + 1) − 1] = [fα̃(tα̃ + 1) − 1]t_{l},

and since fα̃F_{ρ}^{−1} is tα̃ invariant,

(5.39)  F_{ρ}^{−1}[[fα̃(tα̃ + 1) − 1] = [fα̃(tα̃ + 1) − 1]F_{ρ}^{−1}.

Then (5.37a) follows by applying (5.38) and (5.39).

For (5.37b) it is enough to show the relation for θ = θ_{γ}. This is clear from Lemma 5.4 and Corollary 5.5. □

Corollary 5.8. The relation

(5.40)  m^{∗} = m

holds.

Proof. Relation (5.40) can be rewritten as

(5.41)  Φ(T_{l})^{∗} = t_{l} · F_{ρ}^{−1} · Φ(T_{l}) · F_{ρ} · t_{l}.

because F̃_{ρ} = F_{ρ}. Then (5.41) follows by repeated application of (5.37a) and formulas (5.38) and (5.39). □

Suppose (π, V) is a representation of ℍ_{O} with infinitesimal character \overline{χ} and via Φ a representation of H with infinitesimal character χ. It follows from Corollary 5.8 that (π, V) is a hermitian or unitary representation of ℍ_{O} precisely when it is a hermitian or unitary representation of H.

We conclude this section with a few words on the relationship of hermitian representations and Theorem 3.3. The * structure on the matrix algebra M_{n} in Theorem 3.3 is given by E_{i,j}^{∗} = E_{j,i}. The irreducible module C_{n} of M_{n} admits a compatible positive definite hermitian form, namely

(5.42)  < v, w >_{C_{n}}= w^{∗} · v.

The functor F of Corollary 3.4 has the property that V admits a compatible hermitian form precisely when F(V) does. The two are related as follows. If V has invariant hermitian form <, >, then F(V) has invariant hermitian form <, >_{F}=<, >_{C_{n}} · <, > . Every ∗ compatible invariant hermitian form on F(V) is a multiple of one of this type. The <, > is positive definite if and only if <, >_{F} is positive definite.

6. PARABOLIC INDUCTION

In this section, we relate the functor F of Corollary 3.4 with parabolic induction. We begin by recalling some preliminary facts. Let Π_{M} be a subset of the simple roots Π and let P = MN ⊃ B be the corresponding parabolic subgroup and I_{M} = I ∩ M. Denote by W_{M} the subgroup of W corresponding to M. Then the Iwahori Hecke algebra H(M//I_{M}) of M is a subalgebra of H(G//I). The algebra H(G//I) is obtained from the generic Hecke algebra H by specializing the variable z to q. The subalgebra H(M//I_{M}) ⊂ H(G//I) is obtained from a subalgebra of H by the same specialization. Let H^{M} denote the generic Hecke algebra of M, and let H_{W_{M}} be the subalgebra of H_{W} generated by T_{s_{α}}, (α ∈ Π_{M}). Then,

(6.1)  H^{M} ≡ H_{W_{M}} ⊗_{ℂ[z,z^{−1}]} A[z, z^{−1}].

A representation V of M is I_{M}–spherical if it is admissible, of finite length and if each submodule is generated by its I_{M}–fixed vectors.

Theorem 6.1. (Bernstein)

(Ind_{P}^{G}(V))^{I} = H(G//I) ⊗_{H(M//I_{M})} V^{I_{M}} .

The analogue of (6.1) in ℍ_{O} is the subalgebra

(6.2)  ℍ_{O}^{M} = ℂW_{M} ⊗ (E ⊗_{ℂ} S[r]).

If V is a finite dimensional ℍ_{O}^{M} module, we define the induced module Ind(V) to be

(6.3)  Ind(V) = ℍ_{O} ⊗_{ℍ_{O}^{M}} V.

Theorem 6.2. View H and ℍ_{O} as analytic families and recall Φ = Φ_{ν} from Theorem 4.3. Then the diagram

(6.4)  \begin{matrix}H_{χ}^{M} & ⟶^{Φ} & ℍM ; \overline{χ} \\ ^{Ind}↓ & ^{Ind}↓ \\ H_{χ} & ⟶^{Φ} & ℍ\overline{χ} \end{matrix}

commutes.

Proof. This follows from Theorem 4.3 and the discussion above. □

We recall some results on tempered representations. Let V be a tempered representation (of H_{χ} or equivalently of ℍ\overline{χ} via the isomorphism Φ). Since the subalgebra S[r] ⊂ ℍ_{O} is abelian, there is a filtration

(6.5) {0} = V_{0} ⊂ V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{d} = V (dim_{ℂ}V_{i} = i)

such that each V_{i} is S invariant and S acts by a character on V_{i}/V_{i−1} (1 ≤ i ≤ d). This character is completely determined by its restriction to L = ℂ ⊗_{ℤ} X ⊂ S. Moreover, L can be viewed as the dual to the Lie algebra L = ℂ ⊗_{ℤ} Y of ^{L}A. Thus it acts via a linear functional ϕ_{i} ∈ Ľ = Hom_{ℂ}(L, ℂ) on V_{i}/V_{i−1} (1 ≤ i ≤ d). The ϕ_{i}’s are called the generalized coweights of the action of S on V. If we write the infinitesimal character as χ = (e^{r}, e^{t}), then e^{t} and e^{ϕ_{i}} are in the same Weyl group orbit. Let 𝔽 denote either the real ℝ or the complex ℂ numbers, and set

(6.6)

Ľ_{R}(𝔽) = 𝔽–span of simple roots α̌ in Ř
Ľ_{R}^{⊥}(𝔽) = {v ∈ L(𝔽) | < α, v >= 0 for all α ∈ R}.

Then Ľ = Ľ_{R}(ℂ) ⊕ Ľ_{R}^{⊥}(ℂ). With the obvious notation, we write iĽ_{R}^{⊥}(ℝ) for the imaginary elements in Ľ_{R}^{⊥}(ℂ). Set

(6.7)

C = {y_{1} + y_{2} | y_{1} ∈ Ľ_{R}(ℂ), y_{2} ∈ iĽ_{R}^{⊥}(ℝ),
and Re(< y_{1}, ω >) < 0 for ω a fundamental weight}
\overline{C} = \{y_{1} + y_{2} | y_{1} ∈ Ľ_{R}(ℂ), y_{2} ∈ iĽ_{R}^{⊥}(ℝ),
and Re(< y_{1}, ω >) ≤ 0 for ω a fundamental weight}

A restatement of the condition on y_{1} is that

(6.8)  y_{1} ∈ ∑_{α̌∈Ř^{+}} c_{α} ⊗ α̌ , where Re(c_{α}) < 0 for y_{1} ∈ C, (Re(c_{α}) ≤ 0 for y_{1} ∈ \overline{C}).

Suppose (π, V) is an irreducible I–spherical representation of G, such that V^{I} has infinitesimal character χ.

Theorem 6.3. (Casselman) The representation (π, V) is tempered (resp. discrete series) if and only if all the coweights ϕ_{i} of V^{I} lie in \overline{C} (resp. C).

Theorem 6.4.

(1) Suppose V is a module for ℍ_{O}^{M} and {ϕ_{i}} are the coweights of V. Let {w_{1} = 1, w_{2}, …, w_{n}} be the coset representatives of W/W_{M} of minimal length. Then, {w_{j}ϕ_{i}} are the coweights of Ind(V).

(2) Suppose σ ∈ O, V is a module for Γ_{σ}ℍ_{σ} and {ϕ_{i}} are the coweights of V. Let {w_{1} = 1, w_{2}, …, w_{n}} be the coset representatives of W/Stab(σ) of minimal length. Then, {w_{j}ϕ_{i}} are the coweights of F(V).

Proof. The proof of case (1) and (2) are essentially the same. We give the details only in the case of parabolic induction. Let {0} = V_{0} ⊂ V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{d} = V be as in (6.5). For each i > 0 choose a vector v_{i} ∈ V_{i} − V_{i−1}. The set {t_{w_{k}} ⊗ v_{i}} is a basis for Ind(V) (over ℂ).

Arrange the coset representatives w_{j}’s so that ℓ(w_{j}) ≤ ℓ(w_{j+1}). Let ≺ be the lexicographical ordering on the pairs (j, i) (1 ≤ j ≤ n, 1 ≤ i ≤ d), ie.

(j, i) ≺ (k, l) if j ≤ k or j = k and i ≤ l

and set

V_{(i,j)} = span of all w_{k} ⊗ v_{l} with (k, l) ⪯ (j, i).

If (1, 0) ≺ (j, i), let

(j, i)′ = \{ \begin{matrix}(j, i − 1) & if i > 0 \\ (j − 1, d) & if i = 0. \end{matrix}

Then V_{(j,i)}/V_{(j,i)′} is one dimensional generated by the image of w_{j} ⊗v_{i}. Relation (3.8) shows that in general

ωt_{w} = w(ω)t_{w} + ∑_{x≺w} t_{x}ω_{x}.

where ≺ is the Bruhat order. In particular l(x) < l(w), so by (6.5),

ω(t_{w_{j}} ⊗ v_{i}) = w_{j}(ϕ_{i})(ω)w_{j} ⊗ v_{i} (mod V_{(j,i)′}).

for ω ∈ L. □

Corollary 6.5. If V is a tempered module for ℍ_{O}^{M} (resp. Γ_{σ}ℍ_{σ}), then Ind(V) (resp. F(V)) is a tempered module of ℍ_{O}.

Proof. In the case of parabolic induction, let R_{M}^{+} be the positive roots of M. For W′ = W_{M} (resp. W_{σ}), let R′^{+} be R_{M}^{+} (resp. R_{σ}^{+}). The representatives {w_{i}} of W/W′ of minimal length are the elements

(6.9)  {w_{i}} = {w ∈ W | w^{−1}(R′^{+}) ⊂ R^{+}}.

The Corollary follows from this characterization of the minimal length elements. □

We recast the Langlands classification in terms of the graded algebra ℍ_{O}. We assume r has been specialized to log(q^{\frac{1}{2}} ). For Π_{M} ⊂ Π, let L = L_{R_{M}} (ℂ) ⊕ L_{R_{M}}^{⊥} (ℂ) where L_{R_{M}} (ℂ) and L_{R_{M}}^{⊥} (ℂ) are defined, with the obvious change in notation, as in (6.6). Denote by S_{R_{M}} (resp. S_{R_{M}}^{⊥}) the symmetric algebra of L_{R_{M}} (ℂ) (resp. L_{R_{M}}^{⊥} (ℂ)). The symmetric algebra S is then the algebra tensor product S = S_{R_{M}} ⊗_{ℂ} S_{R_{M}}^{⊥}. Then S_{R_{M}}^{⊥} commutes with the subalgebra

B = ℂW_{M} ⊗ S_{R_{M}} [r]

and ℍ_{O}^{M} decomposes as

(6.10)  ℍ_{O}^{M} = B ⊗_{ℂ} S_{R_{M}}^{⊥} .

Each ν ∈ Ľ_{R_{M}} (ℂ) defines an algebra homomorphism ϕ_{ν} : S_{R_{M}}^{⊥} ⟶ ℂ. If (π, V) is a ℍ_{O}^{M}–module, define (π_{n}u, V_{ν}) to be the ℍ_{O}^{M}–module whose space is V with the action

π_{ν}(h · λ) v = ϕ_{ν}(λ)π(h · λ) v,

where h ∈ B and λ ∈ S_{R_{M}}^{⊥}. This construction is the analogue of twisting a representation of M ⊂ G by a one dimensional character of M.

Given data (M, V, ν) such that V is tempered irreducible, define the standard module X(V, ν) to be Ind(π_{ν}, V_{ν}).

Theorem 6.6. (Langlands classification)

(1) Suppose V is an irreducible tempered module for ℍ_{O}^{M} and ν ∈ Ľ_{R_{M}}^{⊥} (ℝ) with ⟨α, ν⟩ > 0 for all α ∈ R^{+} − R_{M}^{+}. Then, X(V, ν) = Ind(V ⊗ ν) has a unique irreducible quotient \overline{X}(V, ν).

(2) Any irreducible module of ℍ_{O} occurs as some \overline{X}(V, ν). The data (M, V, ν) are unique (up to conjugation).

7. COMPARISON

We need to relate the functor F with parabolic induction. Let s = s_{e}s_{h} be the polar decomposition of an element in ^{L}A such that O = W · s_{e} is finite. Write s_{h} as e^{ν}, where ν ∈ X̌ ⊗_{ℤ} ℝ. The centralizer ^{L}M = C_{^{L}G}(s_{h}) of s_{h} in ^{L}G is obviously a Levi subgroup. The Weyl group W_{M} of ^{L}M is generated by the roots

Π_{M} = {α ∈ Π|s_{α}(s_{h}) = s_{h}}.

For σ ∈ O, define R_{σ}^{+} ⊂ R^{+} and R_{M,σ}^{+} ⊂ R_{M}^{+} as in (3.12). Obviously,

(7.1)  if ⟨ν, α⟩ > 0 ∀ α ∈ R^{+} − R_{M}^{+}, then ⟨ν, α⟩ > 0 ∀ α ∈ R_{σ}^{+} − R_{M,σ}^{+}.

The orbit O, decomposes under the action of W_{M}. These W_{M}–orbits in fact lead to a decomposition of ℍ_{O}^{M} into two sided ideals. In order to formulate this decomposition we need to parametrize the W_{M}–orbits. This is accomplished by considering the double cosets W_{M}\W/Stab(s_{e}). Suppose w_{1}, w_{2}, …, w_{k} are representatives for the double cosets. The desired parameterization is given by

W_{M}w_{i}Stab(s_{e}) ⟷ O_{i} the W_{M} orbit of w_{i}(s_{e}).

Let τ_{i} = w_{i}(s_{e}). The stabilizer Stab(τ_{i}) of τ_{i} in W is w_{i}Stab(s_{e})w_{i}^{−1}. Likewise, the stablizer W_{M,τ_{i}} of τ_{i} in W_{M} is W_{M} ∩ Stab(τ_{i}). It is the Weyl group of

LM_{i} = ^{L}M ∩C_{^{L}G}(τ_{i}) = Cent(τ_{i},^{L}M).

In particular,

|O_{i}| = [W_{M} : W_{M,τ_{i}}] = [W_{M}w_{i}Stab(s_{e}) : Stab(s_{e})].

Let H^{M_{i}} be the Hecke algebra associated to ^{L}M_{i} . Then we can form the graded algebras ℍ_{O_{i}}^{M} and ℍ_{i}^{M} ≅ ℍ_{τ_{i}}^{M} and (ℍ_{τ_{i}}^{M})′ as in section 3.

Proposition 7.1.

(1) For each orbit O_{i},

ℍ_{O_{i}}^{M} = ℂW_{M} ⊗_{ℂ} ( ⊕_{τ∈O_{i}}ℂE_{τ}) ⊗_{ℂ} S[r]

is a two sided ideal of ℍ_{O}^{M}. It is isomorphic to (ℍ_{τ_{i}}^{M})′ ⊗ M_{n_{i}}(ℂ).

(2) the ideals ℍ_{O_{i}}^{M} commute and

ℍ_{O}^{M} = ⊕_{i}ℍ_{O_{i}}^{M}.

Proof. The fact that ℍ_{O_{i}}^{M} are commuting ideals and the decomposition in (2) follow from the commutation relations (3.3) between the E_{τ} ’s. The isomorphism

ℍ_{O_{i}}^{M} ≅ (ℍ_{τ_{i}}^{M})′ ⊗ M_{n_{i}}(ℂ)

is Theorem (3.3) applied to this setting. □

We now apply these results to compare Langlands standard modules for the Hecke algebras ℍ_{O} and ℍ_{s_{e}}.

Write ^{L}G_{i} for the group w_{i} ^{L}G(s_{e})w_{i}^{−1}, where ^{L}G(s_{e}) is the stabilizer of s_{e}. It is important to note that ^{L}M_{i} is a Levi subgroup of ^{L}G_{i}; indeed, it is the centralizer of w_{i}^{−1}s_{h}w_{i}. Thus, given a module V′ of (ℍ_{τ_{i}}^{M})′ we have two ways to form a module of ℍ_{O}.

One way is to induce V′ from (ℍ_{τ_{i}}^{M})′ to (ℍ_{τ_{i}})′, and then apply Theorem 3.3. This gives

V_{1} = [Ind_{(ℍ_{τ_{i}}^{M})′}^{ℍ′_{τ_{i}}}V′] ⊗_{ℂ} C_{n}.

The other way is first obtain a module V′ ⊗_{ℂ} C_{n_{i}} of ℍ_{O_{i}}^{M} via Proposition 7.1. Since ℍ_{O_{i}}^{M} is a two sided ideal of ℍ_{O}^{M}, we can view this module as a module for ℍ_{O}^{M} and therefore induce it to ℍ_{O}. This gives

V_{2} = Ind_{ℍ_{O}^{M}}^{ℍ_{O}} (V′ ⊗_{ℂ} C_{n_{i}}).

Theorem 7.2. The modules V_{1} and V_{2} of ℍ_{O} obtained in this fashion are isomorphic. The isomorphism is compatible with *.

Proof. The modules V_{1} and V_{2} are spanned (over ℍ_{O}) by vectors of the form

E_{1,1} ⊗_{ℍ_{τ_{i}}} (h_{τ_{i}} ⊗ v) (h_{τ_{i}} ∈ ℍ_{τ_{i}})

and

1 ⊗_{ℍ_{O}^{M}} (E_{1,1} ⊗ v) (h ∈ ℍ_{O}, h′ ∈ ℍ_{O}^{M})

respectively. Then the correspondence

E_{1,1} ⊗_{ℍ_{τ_{i}}} (h_{τ_{i}} ⊗ v) ⟷ h_{τ_{i}} ⊗_{ℍ_{O}^{M}} (E_{1,1} ⊗_{ℍ′_{τ_{i}}}^{M} v).

defines an ℍ_{O} isomorphism between V_{1} and V_{2}. This isomorphism is in fact a ∗ isomorphism. □

We are now ready to compare standard modules.

Corollary 7.3. For ν ∈ Ľ_{R_{M}}^{⊥} satisfying ⟨ν, α⟩ > 0 for all α ∈ R^{+} − R_{M}^{+}, and V tempered irreducible,

X(V, ν) = X(V′, ν) ⊗_{ℂ} C_{n_{i}}.

Proof. Suppose V is an irreducible tempered module of ℍ_{O}^{M}. By Proposition 7.1 (2), there is a W_{M}–orbit O_{i} ⊂ O such that V is a representation of the ideal ℍ_{O_{i}}^{M}. As a module of ℍ_{O_{i}}^{M}, V is tempered; so there is a tempered module V′ of (ℍ_{τ_{i}}^{M})′ so that V = V′ ⊗_{ℂ} C_{n_{i}}. If ν ∈ Ľ_{R_{M}}^{⊥} satisfies the positivity conditions needed to form a standard module X(V, ν), then by (7.1), X(V′, ν) is a standard module of (ℍ_{τ_{i}})′. □

8. UNITARITY

Let G be a split reductive p–adic group. Our goal is Theorem 8.1. It shows that the unitarity of an Iwahori spherical representation of G can be detected on the Iwahori fixed vectors. An easy reduction is that we need only consider the case when G has connected center. This means the centralizer of any semisimple element s ∈ ^{L}G is a connected reductive group.

Recall

\overline{X}(V, ν) = unique irreducible quotient of X(V, ν),

and that \overline{X} admits a hermitian form if and only if there is a w ∈ W eyl(G, Z(^{L}M)) satisfying

w(ν) = −ν and V ≈ w(V).

There are two ways to place a hermitian form on the family of standard modules

X_{t} = Iwahori fixed vectors of Ind_{P}^{G}(V ⊗ e^{tν}) (t ∈ ℝ).

They are (1) <, >_{t} : the form obtained from the standard intertwining operator

𝕃 : I(P, σ, ν) = Ind_{P}^{G}(V ⊗ ν) ⟶ I(\overline{P} , σ, ν) = IndG ; \overline{P}(V ⊗ ν)

(2) <, >′_{t} : the form gotten, via Corollary 7.3, from the form on X′_{t} = X(V′, tν) and the

isomorphism

X_{t} = X′_{t} ⊗ C_{n}.

Both forms have the property that

\overline{X}_{t} = X_{t}/\{radical of form\} (t > 0)
radical of form = unique subrepresentation of X_{t} (t < 0)

We conclude <, >_{t} and <, >′_{t} differ by a memomorphic function which has no zeros or poles along the real axis.

Theorem 8.1.

(1) \overline{X}_{1} is unitary if and only if \overline{X}′_{1} is unitary.

(2) the unitarity of \overline{X}_{1} is detected on the I–fixed vectors.

Proof. It is obvious that the unitarity of the Iwahori fixed vectors of \overline{X}_{1} is equivalent to the unitarity of the Iwahori fixed vectors of \overline{X}′_{1}. For G′, the group corresponding to ^{L}G(s_{e}), we are in the situation of real infinitesimal character as in [BM]. Therefore, the unitarity of \overline{X}′_{1} is equivalent to the unitarity of its Iwahori fixed vectors. We need only show that \overline{X}′_{1} unitary implies \overline{X}_{1} unitary. Let T (s_{e}) (resp. T′(s_{e})) be the set of tempered representations of G (resp. G′) whose infinitesimal character has compact part s_{e}. Corollary 6.5 establishes a natural one–one map

π ⟷ π′

between these two sets so that

\overline{X}_{1}^{±} = ∑_{π∈T (s_{e})}a_{π}^{±}θ_{π}^{K}

\overline{X}′_{1}^{±} = ∑_{π∈T ′(s_{e})}a_{π′}^{±}θ_{π′}^{K′} (a_{π′}^{±} = a_{π}^{±}).

The unitarity of \overline{X}′_{1} means either a_{π′}^{+} or a_{π′}^{−} is identically zero for all π′. This means a_{π}^{+} or a_{π}^{−} is identically zero for all π and thus \overline{X}_{1} is unitary. □

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