June 8, 2006

Spherical unitary representations for split groups

Dan Ciubotaru

1. Basic examples.

1.1 Graded Hecke algebra of type A_{1} 1.2 SL(2, ℝ)

2. Generalization.

2.1 Graded Hecke algebra 2.2 Split real groups

3. Spherical unitary dual. 3.1 Relevant W -types 3.2 Explicit description

1 Basic examples

1.1 Graded Hecke algebra of type A_{1}

Let ℍ = ℍ(A_{1}) be the algebra generated over ℂ by s and α subject to the relations

s^{2} = 1
s · α + α · s = 2.

Denote 𝔸 = Sym(ℂα). As a ℂ-vector space ℍ(A_{1}) = ℂℤ/2ℤ ⊗ 𝔸, where ℤ/2ℤ = {1, s}.

The algebra ℍ has a ∗-operation defined on generators by

s^{∗} = s
α^{∗} = −α + 2s.

We say that an ℍ-module U is hermitian (unitary) if it admits a hermitian form (positive definite) ⟨ , ⟩ such that

⟨x · u_{1}, u_{2}⟩ + ⟨u_{1}, x^{∗} · u_{2}⟩ = 0, x ∈ ℍ, u_{1}, u_{2} ∈ U.

(The characters of 𝔸 are determined by the action of α.) Let ℂ_{ν} denote the character of 𝔸 on which α acts by ν.

Define the principal series

X(ν) = ℍ ⊗_{𝔸} ℂ_{ν}, ν ≥ 0.

Consider the element

r_{α} = s · α − 1.

Lemma 1.1.1. The element r_{α} satisfies the following relations

α · r_{α} = r_{α} · (−α) and s · r_{α} = r_{α} · (−s).

Then we immediately have the following result.

Proposition 1.1.2. The map A(ν) : X(ν) → X(−ν), given by

A(ν)(x ⊗ 1_{ν}) = x · r_{α} ⊗ 1_{−ν},

is an intertwining operator.

It is a general fact that an invariant hermitian form on a module is equivalent with an intertwining operator between the module and its hermitian dual.

As a ℤ/2ℤ-representation,

X(ν) = triv ⊕ sgn = span{(1 + s) ⊗ 1_{ν}, (1 − s) ⊗ 1_{ν}}.

Note that (1 + s) · r_{α} = (1 + s)(α − 1) and (1 − s) · r_{α} = (1 − s)(−α − 1). So the hermitian form corresponding to A(ν) has matrix

(\begin{matrix}a_{triv}(ν) & 0 \\ 0 & a_{sgn}(ν) \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & \frac{1−ν}{1+ν} \end{matrix} ) ,

where a_{τ}(ν) denote the normalized operators on ℤ/2ℤ-types. (The normalization is such that on the trivial ℤ/2ℤ-type, the operator is identically 1.)

In conclusion, X(ν), ν ≥ 0, has a unique quotient L(ν), which is unitary for 0 ≤ ν ≤ 1. (At ν = 1, L(1) = triv.)

1.2 SL(2, ℝ)

Let G be the group SL(2, ℝ), B = AN the Borel subgroup (A is the maximal split torus) and K = SO(2) the maximal compact subgroup. Then K̂ ≅ ℤ.

Consider the spherical principal series

X_{B}(ν) = Ind_{B}^{G}(e^{ν} ⊗ 1), ν ≥ 0.

(In Prof. Trapa’s table, this is denoted by P_{+}(ν).)

The Langlands quotient L(ν) is unitary for 0 ≤ ν ≤ 1. (L(1) is the trivial representation.) Recall that as a K-representation,

X_{B}(ν)|_{K} = ∑_{m∈ℤ}(2m).

There is an (integral) intertwining operator

A(ν) : X_{B}(ν) → X_{B}(−ν),

which is normalized so that it is identically 1 on the trivial K-type. One can compute the restriction of A(ν) on each K-type. Since the K-types are onedimensional, these restrictions are scalars. A classical computation shows that these scalars are

A_{(2m)}(ν) = \frac{1 − ν}{1 + ν} · \frac{3 − ν}{3 + ν} · · · · · \frac{2|m| − 1 − ν}{2|m| − 1 + ν} .

Remark. Note that

A_{(2)}(ν) = a_{sgn}(ν) = \frac{1 − ν}{1 + ν} ,

and the (unitary) complementary series is the same in the two cases.

2 Generalization

2.1 Graded Hecke algebra

Let (X , Π, X̌, Π̌) be a based root datum, with Δ the roots and Δ̌ the coroots, W the Weyl group. Set 𝔞 = X̌ ⊗_{ℤ}ℂ and 𝔞̌ = X ⊗_{ℤ}ℂ. Similarly, define 𝔞_{ℝ}, 𝔞̌_{ℝ}.

Definition 2.1.1. (Lusztig) The graded Hecke algebra is the vector space ℍ = ℂW ⊗ 𝔸, where 𝔸 = Sym(𝔞̌), subject to the commutation relation

s_{α} · ω = s_{α}(ω) · s_{α} + ω(α̌), for all α ∈ Π, ω ∈ 𝔞̌.

As in the A_{1} case, ℍ has a ∗-operation, so it makes sense to define hermitian and unitary modules.

Remark. The problem of classifying the unitary representations with Iwahori fixed vectors of split p-adic groups can be reduced to the problem of identifying the unitary dual of graded Hecke algebras ℍ.

Some facts about ℍ:

1. (Bernstein,Lusztig) The center of ℍ is 𝔸^{W} .

2. All simple ℍ-modules are finite dimensional, and the central characters are parametrized by W -orbits in 𝔞.

3. The ℍ-modules have a Kazhdan-Lusztig classification.

4. (Barbasch-Moy) For every w ∈ W, with reduced expression w = s_{α_{1}} … s_{α_{m}}, one can define the element r_{w} = r_{α_{1}} … r_{α_{m}}, which does not depend on the reduced decomposition.

Let X(ν) = ℍ ⊗_{𝔸}ℂ_{ν} be the principal series. Assume ν ∈ 𝔞_{ℝ} is dominant, i.e., ⟨α, ν⟩ ≥ 0, for all α ∈ Π.

Definition 2.1.2. The ℍ-module U is called spherical if Hom_{W} [triv, U] ≠ 0.

The spherical modules (with real central character) are precisely the (unique) Langlands quotients L(ν) of X(ν) with ν dominant.

Let w_{0} be the longest Weyl group element. Define the (Barbasch-Moy) intertwining operator

A(ν) : X(ν) → X(w_{0}ν), x ⊗ 1_{ν} ↦ x · r_{w_{0}} ⊗ 1_{w_{0}ν}.

Then L(ν) is hermitian if and only if w_{0}ν = −ν. Assume this is the case.

If (τ, V_{τ}) is a W -type, A(ν) defines hermitian operators

a_{τ}(ν) : Hom_{W} [V_{τ}, X(ν)] → Hom_{W} [V_{τ}, X(−ν)]
a_{τ}(ν) : (V_{τ})^{∗} → (V_{τ})^{∗},

by the Frobenius reciprocity. Normalize them so that a_{triv}(ν) = Id. The normalization factor is (−1)^{|Δ^{+}|} ∏_{α∈Δ^{+}}(1 + ⟨α, ν⟩).

Proposition 2.1.3. A spherical parameter ν is unitary if and only if w_{0}ν = −ν and a_{τ}(ν) is positive semidefinite for all τ ∈ Ŵ.

If w_{0} has a reduced decomposition w_{0} = s_{1}s_{2} ⋯ s_{n}, then the operators a_{τ}(ν) have a decomposition

a_{τ}(ν) = a_{τ,1}(w_{1}ν) · a_{τ,2}(w_{2}ν) ⋯ a_{τ,n}(w_{n}ν),

where w_{i} = s_{n−i+1} … s_{n}. Each simple operator a_{τ,i}(ν) is induced from an ℍ(A_{1})-operator and corresponds to a simple root α_{i}. Explicitly,

a_{τ,i}(ν) = \begin{cases}1 & on the (+1)-eigenspace of s_{α_{i}} of V_{τ}^{∗} \\ \frac{1−⟨α_{i},ν⟩}{1+⟨α_{i},ν⟩} & on the (+1)-eigenspace of s_{α_{i}} of V_{τ}^{∗} \end{cases} .

2.2 Split real groups

Let B = AN be a Borel subgroup, A maximal split torus, K maximal compact. Set M = A ∩ K. As before, let X_{B}(ν) denote the spherical principal series X_{B}(ν) = Ind_{B}^{G}(e^{ν} ⊗ 1), where ν ∈ 𝔞_{ℝ}^{∗}, and ν is dominant.

There is a (Knapp-Zuckerman) normalized intertwining operator

A(ν) : X_{B}(ν) → X_{B}(−ν).

The Langlands quotient L(ν), which is spherical, is hermitian if and only if w_{0}ν = −ν. If this is the case, for every K-type (μ, V_{μ}), A(ν) induces operators:

A_{μ}(ν) : Hom_{K}[V_{μ}, X_{B}(ν)] → Hom_{K}(V_{μ}, X_{B}(−ν))
A_{μ}(ν) : (V_{μ}^{∗})^{M} → (V_{μ}^{∗})^{M},

by Frobenius reciprocity. The normalization is such that A_{triv}(ν) = Id.

The Weyl group W = N_{G}(A)/A ≅ N_{K}(A)/M, so for every K-type (μ, V_{μ}), the space (V_{μ}^{∗})^{M} is naturally a W -type. Denote it by τ(μ).

The Barbasch-Vogan idea of petite K-types is to identify a class of Ktypes μ such that the operators

A_{μ}(ν) = a_{τ(μ)}(ν).

(As it will follow from the calculation, the Weyl group operators are for the Hecke algebra of the dual root datum.)

The operator A(ν), and consequently A_{μ}(ν), have a (Gindikin-Karpelević) decomposition into operators A(s_{α}, ν) relative to a reduced decomposition of w_{0}.

For each simple root of A in G, consider the root homomorphism Ψ_{α} : SL(2, ℝ) → G. Via Ψ_{α}, the compact group SO(2) embeds into K. Therefore, the K-type (μ, V_{μ}) has a decomposition into Ψ_{α}(SO(2)) isotypic components:

V_{μ} = ⊗_{j∈ℤ}V_{μ}(j).

The action of M preserves V_{μ}(j)+V_{μ}(−j) and it has fixed vectors if and only if j is even. On the spaced of M-fixed vectors of V_{μ}(2m) + V_{μ}(−2m), as in the SL(2, ℝ) case, the operator A_{μ}(s_{α}, ν) is

A_{μ}(s_{α}, ν) = ∏_{1≤j≤|m|} \frac{2j − 1 − ⟨α̌, ν⟩}{2j − 1 + ⟨α̌, ν⟩}.

Definition 2.2.1. A K-type (μ, V_{μ}) is called petite if for every simple root α, the decomposition of V_{μ} into Ψ_{α}(SO(2))-types contains only the representations (j), |j| ≤ 3.

The following result is an immediate consequence.

Proposition 2.2.2 (Barbasch,Vogan). If (μ, V_{μ}) is a petite K-type, then A_{μ}(ν) = a_{τ(μ)}(ν), where the second operator is the Hecke algebra of Ǧ.

The condition of being petite is very restrictive. For example, for a group G, few W -types occur in τ(μ) for μ petite K-types.

Barbasch identified all the petite K-types (and their corresponding W - types) for split real groups. There are also extensions of this idea: nonspherical principal series (Barbasch-Pantano), nonlinear covers of split real groups (Adams-Barbasch-Paul-Trapa-Vogan), U(p, q) (Barbasch).

Example. If G = SL(n, ℝ), K = SO(n), W = S_{n}, examples of petite

K-types are μ = (\underbrace{2, 2, …, 2}_{k}, 0, …, 0), k ≤ [\frac{n}{2}], which has τ(μ) = (n − k, k).

3 The spherical unitary dual

3.1 Relevant W -types

Let us return to the setting of the Hecke algebra ℍ. We need to determine the spherical unitary dual of ℍ. In addition, in order to be able to use the calculations for real split groups, one must find a set of relevant W -types which detect unitarity and come from petite K-types.

Let 𝔤 denote the complex Lie algebra attached to ℍ. Recall that by the Springer correspondence, to every nilpotent orbit O in 𝔤, one attaches a subset of Ŵ. If e ∈ O, define the height of O to be

ht(O) = max{ℓ ≥ 0 : ad(e)^{ℓ} ≠ 0}.

Definition 3.1.1. A W -type τ is called relevant if the nilpotent orbit O corresponding to τ in the Springer’s correspondence has height ht(O) ≤ 4.

Then we have the first form of the answer for the spherical unitary dual problem.

Theorem 3.1.2. A spherical parameter ν for the Hecke algebra ℍ is unitary if and only if a_{τ}(ν) is positive semidefinite for all relevant W -types τ.

This result was proved in the classical cases by Barbasch, in the exceptional cases by Barbasch-C.

Theorem 3.1.3 (Barbasch). Every relevant W -type comes from a petite Ktype of the split real group.

Corollary 3.1.4. A spherical parameter ν for the real split group G is unitary only if it is unitary for the Hecke algebra associated to Ǧ.

For classical real split groups this condition is also sufficient, as proved by Barbasch.

3.2 Explicit description

We are still in the setting of the graded Hecke algebra ℍ.

Definition 3.2.1. A spherical parameter ν is called generic if the principal series X(ν) is irreducible.

The module X(ν) is reducible if and only if ⟨α, ν⟩ = 1, for some positive root α.

Let us denote by SU_{0} the set of unitary spherical generic parameters. This set can be described explicitly (combinatorially).

Theorem 3.2.2. The set of unitary spherical generic parameters SU_{0} is a union of k simplices (alcoves) in the dominant Weyl chamber, where:

\begin{matrix}A_{n}: & k = 1 \\ B_{n}: & k = 2^{[(n−1)/2]} \\ C_{n}: & k = 1 \\ D_{n}: & k = 2^{[(n−2)/2]} \\ G_{2}: & k = 2 \\ F_{4}: & k = 2 \\ E_{6}: & k = 2 \\ E_{7}: & k = 8 \\ E_{8}: & k = 16. \end{matrix}

Note that the root systems above refer to the Hecke algebra, so they are the dual root systems of the split real group.

Let O be a nilpotent orbit in 𝔤. Any e ∈ O can be embedded into a Lie triple {e, h, f}. The centralizer of the Lie triple in 𝔤 is a reductive Lie subalgebra. Denote it by 𝔷(O).

To every dominant spherical parameter ν ∈ 𝔞_{ℝ}, one can attach uniquely a nilpotent orbit O in 𝔤. The orbit O is the unique G-orbit meeting the 1eigenspace of ad(ν) in a dense orbit. (It is the orbit attached in the Kazhdan-Lusztig classification to the Iwahori-Matsumoto dual of the spherical module parametrized by ν.)

One partitions the spherical unitary dual into pieces CS(O) parametrized by nilpotent orbits. Note that by definition CS(0) = SU_{0}.

Let Exc denote the following set of nilpotent orbits:

Exc = \{\underbrace{A_{1}Ã_{1}}_{F_{4}}, \underbrace{A_{2}3A_{1}}_{E_{7}}, \underbrace{A_{4}A_{2}A_{1}, A_{4}A_{2}, D_{4}(a_{1})A_{2}, A_{3}2A_{1}, A_{2}3A_{1}, 4A_{1}}_{E_{8}}\}.

(The notation is as in the Bala-Carter classification.)

Then the spherical unitary dual of ℍ can be described as follows.

Theorem 3.2.3 (Barbasch,Barbasch-C.).

1. If O ∉ Exc, then

CS(O) = SU_{0}(𝔷(O)).

2. If O ∈ Exc, and O ≠ (4A_{1} ⊂ E_{8}), then CS(O) ⊊ SU_{0}(𝔷(O)).

3. If O = (4A_{1} ⊂ E_{8}), then CS(O) ⊋ SU_{0}(𝔷(O)).