BACKGROUND ON THE SPHERICAL UNITARY DUAL

D. BARBASCH

1. Preliminaries

{1}

                                                                       {1.1}

1.1. Spherical representations. Let 𝔽 be a p-adic field. Let G be the 𝔽rational points of a connected linear algebraic reductive group defined over 𝔽. We assume that G is split. Denote by A a maximally split torus and fix a Borel subgroup with decomposition B = AN.

Let ν ∈ Â be an unramified character and I(ν) be the unramified principal series. Then it is important to study the composition series and irreducible subquotients of I(ν). This is done via the Hecke algebra attached to the Iwahori subgroup. Let

R = {x ∈ 𝔽 : |x| ≤ 1}, P = {x ∈ 𝔽 : |x| < 1}, Ω = {x ∈ 𝔽 : |x| = 1}.

(1.1.1) {1.1.1}

Then G has a maximal compact subgroup K = G(R), and there is an exact sequence

1 ⟶ K_{1} ⟶ K ⟶ G(R/P) ≅ G(𝔽_{q}) ⟶ 1.  (1.1.2) {1.1.2}

Fix a Borel subgroup in G(𝔽_{q}). Then its inverse image is an open compact subgroup called the Iwahori subgroup and denoted I. The algebra of compact supported biinvariant functions H := H(I\G/I) is called the Hecke algebra. Let

C(I) = the category of admissible finite length representations so that all their

subquotients are generated by their Iwahori fixed vectors.

(1.1.3) {1.1.3}

Theorem. ([B]) The functor V ↦ V^{I} is an equivalence of categories from C(I) to C(H), the category of finite dimensional representations of H. The inverse is given by W ↦ H(G/I) ⊗ W.

An irreducible representation π is in C(I) iff it is a subquotient of an I(ν).

H also has a star operation given by

f^{∗}(x) = \overline{f(x^{−1})},  (1.1.4) \{1.1.4\}

so we can talk about hermitian and unitary modules. It is more or less clear that an admissible representation V ∈ C(I) of G is hermitian iff V^{I} is hermitian. It is also clear that if V is unitary, then so is V^{I}. The converse is the subject of two papers [BM1] and [BM2] . The result is summarized in the next theorem.

Theorem. An irreducible representation in C(I) is unitary if and only if V^{I} is unitary.

1.2. The Graded Hecke Algebra. The Hecke algebra H can be described by generators and relations. Let z be an indeterminate (which can then be specialized to q^{1/2}). Let Π ⊂ R^{+} ⊂ R be the simple roots, positive roots and roots corresponding to A ⊂ B, and S be the simple root reflections. Let G_{m} = GL(1, 𝔽) and X = Y̌ = Hom(G_{m}, A) be the (algebraic) lattice of 1-parameter subgroups and Y = X̌ = Hom(A, G_{m}) the lattice of algebraic characters. Then H can be characterized as the Hecke algebra over ℂ[z, z^{−1}] attached to the root datum R = (Y, X , Ř, R, Π̌). The set of generators we will use is the one first introduced by Bernstein. Denote by^{L}G the (complex) dual group, with maximal torus ^{L}T and Borel subgroup ^{L}B containing ^{L}T . Let

A = rational functions on ℂ^{∗} ×^{L}T .

(1.2.1) {1.2.1}

Then H is generated by {T_{w}}_{w∈W} and {θ_{x}}_{x∈X} where X is the lattice of algebraic characters of ^{L}T , subject to the relations

\begin{matrix}T_{w}T_{w′} = & T_{ww′} (l(w) + l(w′) = l(ww′)), \\ θ_{x}θ_{y} & = θ_{x+y}, \\ T_{s}^{2} = & (z^{2} − 1)T_{s} + z^{2}, \\ θ_{x}T_{s} = & T_{s}θ_{sx} + (z^{2} − 1)\frac{θ_{x} − θ_{sx}}{1 − θ_{α}} . \end{matrix} {(1.2.2) \atop \{1.2.2\}}

This realization is very convenient for detemining the center of H and thus computing infinitesimal characters of representations.

Theorem. (Bernstein-Lusztig) The center of H is given by the Weyl group invariants in A.

In particular, infinitesimal characters are parametrized by W -orbits χ = (q, t) ∈ ℂ^{∗} ×^{L}T . We always assume that q is real or at least not a root of unity. In particular, such an infinitesimal character is called real if t has no elliptic part.

The study can be simplified significantly by using the graded Hecke algebra introduced by Lusztig. Let

J = \{f ∈ A : f(1, 1) = 0\} {(1.2.3) \atop \{1.2.3\}}

This ideal satisfies HJ = J H, so we can introduce a filtration

H = H^{0} ⊃ ⋯ ⊃ H^{i} ⊃ H^{i+1} ⊃ …, {(1.2.4) \atop \{1.2.4\}}

and form the graded object ℍ. It can be written as

ℍ = ℂ[r]W × 𝔸, {(1.2.5) \atop \{1.2.5\}}

where r ≡ z − 1(modJ), and 𝔸 is the symmetric algebra over ^{L}𝔱 = X ⊗_{ℤ} ℂ. The previous relations become

\begin{matrix}t_{w}t_{w′} = & t_{ww′}, \\ t_{s}^{2} = & 1, \\ t_{s}ω = & s(ω)t_{s} + 2r < ω, α̌ >, (s = s_{α}). \end{matrix} {(1.2.6) \atop \{1.2.6\}}

The center of ℍ is 𝔸^{W} . In particular, infinitesimal characters are parametrized by W -orbits of elements \overline{χ} = (r, t) ∈ ℂ ×^{L}𝔱.

                                                                       {t:1.2}

Theorem. ([L2]) There is a matching χ ⟷ \overline{χ} between real infinitesimal characters χ of H and infinitesimal characters \overline{χ} of ℍ so that if H_{χ} and ℍ\overline{χ} are the quotients by the corresponding ideals, then

H_{χ} ≅ ℍ\overline{χ}.

As a consequence, we fix r = 1 and transfer the study of the representation theory of H to ℍ. The classification of irreducible representations of H is as according to [KL] and [L5]; irreducible representations are parametrized by ^{L}G conjugacy classes (t, e, ψ), where t ∈ ^{L}G is semisimple, e ∈ ^{L}𝔤 is nilpotent, and such that Ad(t)e = qe and ψ ∈ { \atop A(x, e)} is an irreducible representation of the component group of the centralizer of x and e.

Write t = t_{0}t_{H} where t_{H} = e^{\frac{1}{2} log qH} and t_{0} semisimple centralizes the Lie triple. The results in [BM2] allow us to specialize to the case when t_{0} is hyperbolic. In this case we may consider representations of ℍ only. The classification of irreducible representations is then given ([L5 ]) by ^{L}G conjugacy classes {s, e, ψ} such that s ∈ ^{L}𝔤 is semisimple, e ∈ ^{L}𝔤 is nilpotent such that [s, e] = re with r = \frac{1}{2} log q and ψ ∈ { \atop A(s, e)}. The results in [BM2] imply that for unitarity it is sufficient to consider the same situation.

The results in [KL] attach to each (^{L}G conjugacy class) (e, s) a standard module X(e, s) which decomposes under the action of A(s, e) as

X(e, s) := ⨁_{ψ∈{ \atop A(s,e)}} X(e, s, ψ) ⊗ V_{ψ}.

As a ℂ[W] module,

X(e, s) ≅ H^{∗}(B_{e}).  (1.2.7) {1.2.7}

The component group A(s, e) is naturally a subgroup of A(e) because in a connected algebraic group, the centralizer of a torus is connected. The group A(e) acts on the right hand side of (1.2.7), and the action of A(s, e) on the left hand side is compatible with the inclusion and the isomorphism. According to the Springer correspondence, the ϕ ∈ Â(e) isotypic component of H^{top}(B_{e}) is an irreducible representation of W denoted σ(O, ϕ). The correspondence is normalized so that if e is the principal nilpotent, then σ(O, ϕ) = sgn. Furthermore, as a W representation,

X(e, s, ψ) = H^{∗}(B_{e})^{ψ}.  (1.2.8) {1.2.8}

We recall furthermore that the σ(O′, ψ) occuring in H^{∗}(B_{e}) all correspond to O′ such that O ⊂ \overline{O′}.

Write s = H/2 + ν such that ν centralizes e, H, f. The analogous formula to (1.2.8) holds whenever the data (e, s) factor through a Levi component LM :

\begin{matrix}X(e, s) = ℍ ⊗_{ℍ_{M}} X_{M}(e, s), \\ ℍ ⊗_{ℍ_{M}} X_{M}(e, s, τ) = ⨁ [ψ|_{A_{M} (s,e)} : τ] X(e, s, ψ) \end{matrix} {(1.2.9) \atop \{1.2.9\}}

We write Ind_{M}^{G}[π] for the module ℍ ⊗_{ℍ_{M}} π.

Suppose^{L}M is the centralizer of ν, and^{L}P = ^{L}M^{L}N is such that ⟨ν, α⟩ > 0 for all roots α ∈ Δ(^{L}𝔫). Write ^{L}M = ^{L}M_{0}^{L}A, where ^{L}A is the center. Then A_{M}(s, e) = A_{G}(s, e). This is because the centralizer of e is of the form LU with U connected unipotent, and L the centralizer of e and H. It follows that every component of A_{G}(s, e) meets L, and therefore A_{G}(s, e) =

A_{G}(ν, H, e) = A_{M}(H.e) = A_{M}(s, e). For τ ∈ { \atop A_{M}(s, e)} = { \atop A_{M_{0}}(e)},

X_{M}(e, s, τ) = X_{M_{0}}(e, H/2, τ) ⊗ ℂ_{ν}. {(1.2.10) \atop \{1.2.10\}}

The representation X_{M_{0}}(e, H/2, τ) is a tempered irreducible module, and if ψ is the representation τ of A_{M}(s, e) viewed as a representation of A_{G}(s, e), then X(e, s, ψ) = ℍ ⊗_{ℍ_{M}} X_{M}(e, s, τ). The usual Langlands classification implies that X(e, s, ψ) has a unique irreducible quotient.

We apply these ideas to the case when O is the trivial orbit. In this case, X(e, s) is the full induced module, and we abbreviate it as X(s). Fix s. According to [BM1], the spherical irreducible subquotient \overline{X}(s) is obtained as follows. There is a unique orbit which we denote O(s) of maximal dimension such that [s, e(s)] = e(s) (recall that we set r = 1 earlier) for some e(s) ∈ O(s). Then the Iwahori-Matsumoto involution IM takes \overline{X}(s) to \overline{X}(e(s), s, triv). In turn this equals X(e(s), s, triv) because any factor would have to have parameter an orbit of strictly larger dimension. Write again s = H/2 + ν where H is the middle element of the triple containing e(s). By the above discussion, X(e(s), s, triv) is also induced irreducible from \overline{X}_{M}(e(s), s, triv) the M determined by ν as defined right above formula (1.2.10). Applying IM, we get

\overline{X}(s) = Ind_{M}^{G}[\overline{X}_{M}(s)].

We call a face a set in the dominant chamber determined by the positive roots being = 0, = 1, < 1, or > 1. We want to show that the multiplicities of the W-types in the spherical irreducible module \overline{X}(s) are constant as s ranges over a face. The set of roots equal to zero and equal to one on s do not change as s varies over a face. Thus the orbit O(s) does not change, because it is the unique orbit whose intersection with 𝔤_{1}(s) is open. This intersection is also a single orbit under G(s) which also doesn’t change over a face. Up to conjugation, we can write the parameter as s = H/2 + ν, and \overline{X}(s) = Ind_{M}^{G}(σ ⊗ ℂ_{ν}). Here σ is anti-tempered, i.e. it is IM of a tempered module. In fact IM(σ) has parameter (e, H/2, triv). Up to conjugation there are only finitely many such (M, σ). Suppose ν ranges over the center of ^{L}M. The only way that the W-structure of \overline{X}(s) can change is if A(s, e) differs from A_{M}(s, e). Consider the centralizer C(e) The one parameter group exp(tH) acts on it, and by [BV], the components coincide with the components of C(H, e). But s commutes with H and stabilizes C(e). The fixed points of s in C(e) also have an action of exp(tH), and the argument in [BV] shows that all the components meet the centralizer of H in C(s, e) which is also the centralizer of ν in C(s, e). Thus every component of C(s, e) meets C_{M}(s, e), so the only way the multiplicities of \overline{X}(s) can change is if the centralizer of ν becomes larger. Let ν_{0} be such a point, and assume there are ν_{i} → ν_{0} such that H/2 + ν_{i} are conjugate by a single w ∈ W to points in the same face for all i ≥ 0. Assume that the centralizer of all the ν_{i} for i > 0 is 𝔪′ Then 𝔪′ ⊊ 𝔪(ν_{0}), is a Levi component. Since the zero subspace of s cannot change in a face, H cannot have any kernel on 𝔪(ν_{0}). There is a decomposition 𝔪(ν_{0}) = 𝔫′ + 𝔪′ + \overline{𝔫′}. The Lie triple is in 𝔪′, so the corresponding sl(2) acts on both 𝔫′, \overline{𝔫′}. Since it has no eigenvalues of zero, all the eigenvalues of H/2 are half-integers. But the component group A(e) in any semisimple Lie algebra 𝔤 is the same as the component group in A(H, e), so it is the same as A(exp(iπH), e). The Lie algebra of the centralizer of exp(iπH) is formed of even eigenspaces of H only. Applied to 𝔤 = 𝔪(ν_{0}), this implies that the component group with respect to M′ and M(ν_{0}) are the same. The argument implies that a face is a finite union of closed (in the face) sets on which the W -structure of \overline{X}(s) does not change. Since the face is connected, these sets cannot have empty intersection, and this implies that the W multiplicities in \overline{X}(s) are constant for s over a face.

1.3. Hermitian Modules. The ^{∗} operation also transfers to the graded version. We refer to §5 of [BM2] for the details. Here is a summary of what we need. Let w_{0} ∈ W be the longest element, t_{0} be the corresponding element in ℂW. Since ^{L}𝔱 = X ⊗_{ℤ} ℂ, it has a conjugation coming from the complex conjugation on ℂ. We denote it by . Let ι(ω) = (−1)^{degω}\overline{ω} and ω̃ = w_{0}ι(ω).

                                                                       {t:1.3}

Theorem. (§5 in [BM2]) Let ω ∈ 𝔸. Then

\begin{matrix}t_{w}^{∗} = & t_{w^{−1}}, \\ ω^{∗} = & t_{0} · ω̃ · t_{0}. \end{matrix}

In particular, if ω ∈ ^{L}𝔱, then

ω^{∗} = −\overline{ω} + 2r ∑_{β∈R^{+}} < \overline{ω}, β̌ >t(β),  (1.3.1) \{1.3.1\}

where t(β) ∈ ℂ[W] is the reflection about β.

The results in [BM2] imply the following. Let (e, H, f) be a Lie triple corresponding to e. Write s = s_{0} + s_{H}, where s_{0} is a semisimple element centralizing the triple, and s_{H} = 1/2 log qH. Let s_{0} = s_{e} + s_{h} be the decomposition of s_{0} into elliptic and hyperbolic parts. Denote by \overline{s} := s_{e}−s_{h}. An irreducible representation admits a hermitian form if and only if (s, e, ψ) is conjugate to (s′ = \overline{s_{0}} + s_{H}, e, ψ). A parameter will be called real if s has trivial elliptic part.

Corollary. The classification of the unitary dual of a split p-adic group is the same as the classification of the unitary dual of the corresponding graded Hecke algebra. Furthermore, using certain isomorphisms of these algebras it is sufficient to consider the case of real parameter.

{1.4}

1.4. Hermitian Duals. Fix a (standard) parabolic subgroup P with Levi decomposition P = MN. Let ℍ_{M} be the corresponding (graded) Hecke subalgebra. Every element a ∈ ℍ can be written as a = ∑_{w∈W/W (M)} t_{w}m_{w} with m_{w} ∈ ℍ_{M}. There is a well defined map

ϵ_{M} : ℍ ⟶ ℍ_{M}, ϵ_{M}(a) := m_{1}, the component of a in ℍ_{M}.

(1.4.1) {1.4.1}

{p:1.4}

Proposition. Denote by ∗_{M} the map corresponding to the star operation ∗_{M} : a ↦ a^{∗} in ℍ_{M} and denote by ∗_{G} the corresponding map in ℍ. Then

ϵ_{M}(∗_{G}a) = ∗_{M}ϵ_{M}(a) for all a ∈ ℍ.

Proof. First observe that

ϵ_{M}(t_{m_{1}}at_{m_{2}}) = t_{m_{1}}ϵ(a)t_{m_{2}}, for m_{i} ∈ W (M). {(1.4.2) \atop \{1.4.2\}}

Let ω ∈ 𝔸. The long element w_{0} decomposes as w_{0} = w_{l}w_{0}^{M}, where w_{0}^{M} is the long element in W (M). Let t_{0}, t_{l} and t_{0}^{M} be the corresponding elements in ℍ. Since t_{0}^{−1} = t_{0} and (t_{0}^{M})^{−1} = t_{0}^{M}, the relation

t_{0} = t_{l}t_{0}^{M} = t_{0}^{M}t_{l}^{−1} {(1.4.3) \atop \{1.4.3\}}

holds. Then we get

∗_{G}(ω) = t_{0} · ω̃ · t_{0} = t_{l}t_{0}^{M} · w_{0}^{M}w_{l}^{−1}(ι(ω)) · t_{l}t_{0}^{M}. {(1.4.4) \atop \{1.4.4\}}

On the other hand,

w_{0}^{M}w_{l}^{−1}ι(ω) · t_{l} = t_{l} · w_{l}^{−1}w_{0}^{M}w_{l}^{−1}ι(ω) + ∑_{x<w_{l}} t_{x}ω_{x} = t_{l} · w_{0}^{M}ι(ω) + ∑_{x<w_{l}} t_{x}ω_{x},

(1.4.5) {1.4.5} where the sum ranges over x ∈ W which are smaller than w_{l} in the Bruhat order. Combining this with 1.4.4 and the formula for ∗_{M} in theorem 1.3, we get

∗_{G}(ω) = ∗_{M}(ω) + ∑_{x<w_{l}} t_{l}t_{0}^{M}t_{x}ω_{x}t_{0}^{M}. {(1.4.6) \atop \{1.4.6\}}

Therefore, using 1.4.2,

ϵ(∗_{G}(ω)) = ∗_{M}(ω) + t_{0}^{M}ϵ(t_{l}^{−1}t_{x})ω_{x}t_{0}^{M} = ∗_{M}(ω) + t_{0}^{M}ϵ(t_{w_{l}^{−1}x})ω_{x}t_{0}^{M}. {(1.4.7) \atop \{1.4.7\}}

To prove the formula in the proposition for a = ω, we must show that ϵ(t_{w_{l}^{−1}x}) = 0 for x < w_{l}. But if w_{l}^{−1}x = m ∈ W (M), then w_{l} = xm^{−1} with x < w_{l}, contradicting the minimality property of w_{l}. Now consider an

element t_{w}ω such that w ∉ W (M). Then ϵ_{M}(t_{w}ω) = 0. We need to show that ϵ_{M}(∗_{G}(ω)t_{w^{−1}}) = 0 as well. For this, write ∗_{G}(ω) = t_{0} · ω̃ · t_{0} and decompose

t_{0} · ω̃ · t_{0} · t_{w^{−1}} = t_{0} · ω̃ · t_{w_{0}w^{−1}} = ∑_{x<w_{0}w^{−1}} t_{w_{0}x}ω_{x},  ω_{x} ∈ 𝔸. (1.4.8) {1.4.8}

To prove the claim we need to show w_{0}x ∉ W (M). Suppose w_{0}x = m. Then x = w_{0}m, and so w_{0}m < w_{0}w^{−1}. It follows that w_{0}w^{−1} ∈ w_{0}W (M), because w_{0} is the long element. But this implies w^{−1} ∈ W (M), a contradiction. Thus ϵ_{M}(∗_{G}(ω)t_{w^{−1}}) = 0 as claimed. □

If W is a module for ℍ_{M}, then we can form the induced module

I(W) := ℍ ⊗_{ℍ_{M}} W.  (1.4.9) {1.4.9}

This has a basis {t_{x} ⊗ v} where v ∈ W and x ∈ W/W (M).

Corollary. The hermitian dual of I(W) is I(W^{h}) = ℍ ⊗_{ℍ_{M}} W^{h}. More precisely, let t_{x} ⊗ v_{x} ∈ I(W), t_{y} ⊗ v_{y} ∈ I(W^{h}), and ⟨ , ⟩_{M} be the pairing of W with W^{h}. Then the pairing between I(W) and I(W^{h}) is given by

⟨t_{x} ⊗ v_{x}, t_{y} ⊗ v_{y}⟩ := ⟨ϵ_{M}(t_{y}^{∗}t_{x})v_{x}, v_{y}⟩_{M}.

Proof. First observe that these spaces are finite dimensional, so it is sufficient to construct an injection

ℍ ⊗_{ℍ_{M}} W^{h} ⟶ [ℍ ⊗_{ℍ_{M}} W]^{h}.  (1.4.10) {1.4.10}

The pairing ⟨ , ⟩ gives such an injection with the required properties by Proposition 1.4. □

                                                                       {1.5}

1.5. Langlands Classification. Suppose a is an automorphism between two root data R_{1} and R_{2}. This induces an automorphism of the corresponding Hecke algebras,

a : ℍ_{2} ⟶ ℍ_{1}.  (1.5.1) {1.5.1}

In particular, let M_{1}, M_{2} be two Levi components of standard parabolic subgroups. Suppose w ∈ W satisfies wM_{1}w^{−1} = M_{2} and is minimal in its double coset W (M_{2})wW (M_{1}). Then the above discussion gives an isomorphism

a_{w} : ℍ_{M_{2}} ⟶ ℍ_{M_{1}}  (1.5.2) {1.5.2}

which we use to transfer representations V of ℍ_{M_{1}} to wV of ℍ_{M_{2}}. Recall

that according to the (classical version of the) Langlands classification, every irreducible module can be realized as the unique irreducible quotient L(M, W, ν) of a standard module

X(M, W, ν) := ℍ ⊗_{ℍ_{M}} [W ⊗ 11_{ν}],  (1.5.3) {1.5.3}

where W is tempered irreducible and

⟨Re ν, α̌⟩ > 0, for all α ∈ Δ(𝔫), 𝔫 := Lie(N).  (1.5.4) {1.5.4}

The module X(M, W, ν) of course coincides with I(M, W ⊗ 11_{ν}), but we use the notation (1.5.3) to emphasize that it satisfies (1.5.4). According to theorem 2.1 in [E], two Langlands quotients L(M, W, ν) and L(M′, W′, ν′) are isomorphic if and only if

(M, ν) = (M′, ν′), and W ≅ W′. {(1.5.5) \atop \{1.5.5\}}

If on the other hand ⟨Re ν, α̌⟩ < 0, then the module I(M, W ⊗ 11_{ν}) has a unique irreducible submodule. We can determine the parameter of this module as follows. The long element w_{0} has the property that w_{0}ν is dominant. Let M′ := w_{0}(M). Write w_{m} for the shortest element in the double coset W (M′)w_{0}W (M). This element has the property that w_{m}(Δ^{+}(M′)) = Δ^{+}(M). Thus it determines an automorphism between ℍ_{M} and ℍ_{M′}. Let σ′ be the module of ℍ_{M′} corresponding to σ. The data (M′, σ′, w_{0}ν) are the Langlands data of the unique irreducible submodule of I(M, σ, ν). Of course w_{0}ν = w_{m}ν.

{p:1.5}

Proposition. The hermitian dual of L(M, W, ν) is L(w_{m}M, w_{m}W, −w_{m}\overline{ν}). In particular, L is hermitian if and only if w_{0}(M, ν) = (M, −ν), and w_{m} · W ≅ W.

w · (M, W, ν) = (M, W, −\overline{ν}).

Proof. The hermitian dual of X(M, W, ν) is I(M, W, −\overline{ν}). This is not a standard module because −\overline{ν} fails to satisfy (1.5.4). However ⟨−Re \overline{ν}, α̌⟩ < 0, so I(M, W, −\overline{ν}) has L(w_{m}M, w_{m}W, −w_{m}\overline{ν}) as unique irreducible submodule. Thus

L(M, W, ν)^{h} ≅ L(w_{m}M, w_{m}W, −w_{m}\overline{ν}) {(1.5.6) \atop \{1.5.6\}}

as claimed. □

{1.6}

1.6. Intertwining operators. We use the notation of 1.5. Suppose that wM_{1}w^{−1} = M_{2} is minimal in its double coset W (M_{2})wW (M_{1}) (or rather the ^{L}M’s). Given a simple reflection s_{α}, we can form the element r_{α} ∈ ℍ,

r_{α} := t_{α}α − 2r. {(1.6.1) \atop \{1.6.1\}}

Lemma. The elements r_{α} satisfy ωr_{α} = r_{α}s(ω). Let w = s_{1} · · · · · s_{k} be a reduced decomposition. Then r_{w} = ∏ r_{i} is well defined.

Proof. The first relation is a simple application of the defining relations for ℍ :

\begin{matrix}ω(t_{α}α − 2r) = & (t_{α}s(ω) + 2r⟨ω, α̌⟩)α − 2rω = \\ = & (t_{α}α − 2r)s(ω) − 2r(s(ω) − ω) + 2r⟨ω, α̌⟩α = (t_{α}α − 2r)s(ω). \end{matrix}

(1.6.2) {1.6.2} More generally if f is a rational function in one variable, then f(ω)r_{α} = r_{α}f(s_{α}ω). Recall from [L1] that

\overline{τ}_{α} = t_{α} \frac{α}{α + 2r} − \frac{2r}{α + 2r} = \frac{1}{−α + 2r} r_{α}, {(1.6.3) \atop \{1.6.3\}}

and that \overline{τ}_{w} = ∏ \overline{τ}_{i} is well defined (i.e. independent of the reduced decomposition). Write

R_{w} = {α_{1}, s_{1}α_{2}, …, s_{1}s_{2} … s_{k−1}α_{k}} = {β > 0 : w^{−1}β < 0}. (1.6.4) {1.6.4}

Then substituting (1.6.3) for each r_{i} and applying (1.6.2) repeatedly we get

∏ r_{i} = ∏(−α_{i} + 2r)\overline{τ}_{i} = ( ∏_{β∈R_{w}}(−β + 2r)) ∏ \overline{τ}_{i} = ( ∏_{β∈R_{w}} (−β + 2r))\overline{τ}_{w}.

(1.6.5) {1.6.5}

The claim follows. □

                                                                       {p:1.6}

Proposition. Let V be a representation of ℍ_{M_{1}}. Then the operator A_{w} : I(M_{1}, V) ⟶ I(M_{2}, wV) defined by

A_{w}(t_{x} ⊗_{ℍ_{M_{1}}} v) := t_{x}r_{w} ⊗_{ℍ_{M_{2}}} v

is an intertwining operator.

Proof. We need to check that A_{w} is well defined, i.e. it satisfies

ωr_{w} = r_{w}w^{−1}(ω), t_{α}r_{w} = r_{w}t_{w^{−1}α}, α ∈ Δ(M_{1}) a simple root.
(1.6.6) {1.6.6}

The first relation follows from a repeated application of Lemma 1.6. For the second relation, r_{α}r_{w} = r_{w}r_{w^{−1}α} holds. Substituting r_{α} = t_{α}α − 2r and using αr_{w} = r_{w}w^{−1}α, we get the second relation. The fact that the map is an intertwining operator follows from the nature of the action of ℍ which is by multiplication on the left. □

                                                                       {1.7}

1.7. We apply the results in 1.6 to the Langlands classification. Recall w_{m} the minimal element in the double coset W (M)w_{0}W (w_{0}M). Then multiplication on the right by r_{m} := r_{w_{m}} is an intertwining operator

A_{m} : X(M, W, ν) ⟶ I(w_{0}M, w_{m}W, w_{0}ν).  (1.7.1) {1.7.1}

This is identically zero precisely when r_{m} ⊗ [v11_{ν}] = 0 for all v ∈ W. When it is not identically zero, the image is precisely L(M, W, ν) because this submodule is generated by any vector whose generalized eigenvalue under 𝔸 projects onto ν. On the other hand, observe that the leading term of r_{m} is t_{m} ∏_{(α,ν)>0} α. But

( ∏_{(α,ν)>0} α) · v11_{ν} = ∏_{(α,ν)>0}(α, ν + χ) v11_{ν},  (1.7.2) {1.7.2}

where χ is the infinitesimal character of W. This is because ∏_{(α,ν)>0} α is invariant under W (M) therefore in the center of ℍ_{M}. Thus A_{m} is not identically zero as a function of ν. For a full principal series 1.7.2 implies that A_{m} is not identically zero for any ν satisfying 1.5.4. But for more general induced representations A_{m} could be zero for certain values of ν. On the other hand, it follows from the theory of intertwining operators on the p-adic group, that there exists a family of intertwining operators B_{m}(ν)

B_{m} : X(M, W, ν) ⟶ I(w_{m}M, w_{m}W, w_{m}ν) {(1.7.3) \atop \{1.7.3\}}

which is analytic as a function of ν and whose image is L(M, W, ν) for all ν satisfying 1.5.4. We claim that there is a meromorphic function f(ν) such that f(ν)A_{m} = B_{m}(ν). This goes as follows. Recall that X has a lowest K-type μ occuring with multiplicity 1. Then there are analytic functions g and h such that for any vector v ∈ X transforming according to μ, we have A_{m}(v) = g(ν)v and B_{m}v = h(ν)v. Then f = hg^{−1}.

We will not need this refinement.

{1.8}

1.8. Suppose L(M, W, ν) is hermitian with ν real. Let w be such that w(M, W, ν) = (M, W, −ν), minimal in its double coset W (M)wW (M). This double coset is also W (M)w_{0}W (M). This is because both wν and w_{0}ν are antidominant (so w_{0}ν = wν = −ν) and Δ(M) = {α ∈ Δ : (α, ν) = 0}. Denote by a_{w} the isomorphism 1.5.2 and by τ : W ⟶ W an isomorphism (unique up to a scalar) satisfying τ(h · v) = a_{w}(h) · τ(v). Recall that we can decompose M = ^{0}M · A, where A is the (split part of the) center of M. Then we can write any element m ∈ ℍ_{M} as m = ∑ m_{i}a_{i} with m_{i} ∈ ℍ _{^{0}M} and a_{i} ∈ A. Thus it makes sense to evaluate m ∈ ℍ_{M} at ν. We write this as m(ν). With this notation

∑ m_{i}a_{i}(v11_{ν}) = ∑ m_{i}a_{i}(ν)v11_{v} = (m(ν)v)11_{ν}.  (1.8.1)

{c:1.8}

Corollary. Suppose X(M, W, ν) is hermitian irreducible and ν is real. Then up to a nonzero scalar the inner product is given by the formula

⟨t_{x} ⊗ v_{x}11_{ν}, t_{y} ⊗ v_{y}11_{ν}⟩ = ⟨ϵ(t_{y}^{∗}t_{x}r_{w})(ν)τ(v_{x}), v_{y}⟩_{W}.

Proof. This follows from 1.4, 1.6 and 1.7. □

2. Character theory

{2}

We review some material from [BM1]. Recall the existence of the Iwahori-Matsumoto involution IM on the Hecke algebra,

IM(t_{w}) = (−1)^{l(w)}t_{w},  IM(ω) = −ω,  ω ∈ ^{L}𝔱. {(2.0.2) \atop \{2.1.1\}}

Given the infinitesimal character χ, there is a unique irreducible L(χ; sgn) (containing sgn with multiplicity 1); and IM takes it to a spherical module. This is due to the form of IM on W. The classification theorem of representations of ℍ then gives the following result.

Theorem. ([BM1]) Let χ be an infinitesimal character corresponding to the semisimple element h ∈ ^{L}𝔤 Then there is a unique orbit of maximal dimension (with representative e_{max}) such that [h, e_{max}] = 2e_{max}. Then L(χ, sgn) equals the standard module representation X(χ, e_{max}, ϕ = triv). In particular if the orbit of e meets a Levi coponent of a proper parabolic subalgebra, then L(χ, sgn) is induced irreducible from the corresponding L_{M}(χ, sgn).

Modulo the characters of tempered representations (for which there is a formula in [KL]) this theorem computes the character of L(χ, sgn). Applying IM, we get a corresponding result for spherical representations.

2.1. Unitary representations. One of the main features of theorem 1.1 is that it implies that IM preserves unitarity. A special case of the representations X(χ, sgn) is when in fact (h, e) can be made into a Lie triple, i.e. there is f such that [h, e] = 2e, [h, f] = −2f, [e, f] = h. Then X(χ, sgn) consists of the I-fixed vectors of a tempered representation, therefore unitary. These are special cases of Arthur parameters. The Main Theorem says that they are essentially the only unitary representations containing sgn.

References

[B] A. Borel Admissible representations of a semisimple group over a local field with fixed vectors under an Iwahori subgroup, Invent. Math., vol 35, 1976, 233-259

[BM1] D. Barbasch, A. Moy A unitarity criterion for p-adic groups Invent. Math. , vol 98, 1989, 19-38

[BM2] Reduction to real infinitesimal character in affine Hecke algebras, Journal of the AMS, vol 6, Number 3, 1993, 611-635

[BV] D. Barbasch, D. Vogan Unipotent representations of complex semisimple groups Ann. of Math., 121, (1985), 41-110

[C] W. Casselman A new nonunitarity argument for p-adic representations, J. Fac. Sci. Tokyo Univ. , sect IA Math. 28, no. 3, 1981, 907-928

[Ca] R. Carter, Finite groups of Lie type , Wiley-Interscience, New York, 1985

[E] The Langlands classification for graded Hecke algebras, S. Evens, Proceedings of AMS, vol. 124, no. 4, April 1996

[KL] D. Kazhdan, G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Inv Math., vol 87, 1987 , 153-215

[L1] G. Lusztig Characters of reductive groups over a finite field, Annals of Math. Studies, Princeton University Press, vol 107

[L2] Affine Hecke algebras and their graded version, Jour. AMS, vol 2, 1989, 599-635

[L3 ] Intersection cohomology complexes on a reductive group, Invent. Math., vol

75, 1984, 205-272 [L4 ] Cuspidal local systems and graded algebras I, Publ. Math de l’IHES, vol 67,

1988, 145-202 [L5 ] Cuspidal local systems and graded algebras II, Proceedings of conference in

Banff, 1994,

[T] M. Tadic Classification of unitary representations in irreducible representations of general linear group (non-archimedean case), Ann. Scient. Ec. Norm. Sup., vol 19 , 1986, 335-382