CLASSIFICATION OF ONE K–TYPE REPRESENTATIONS

Dan Barbasch and Allen Moy

1. Introduction

Let G be a split reductive p-adic group. In [BM1,BM2], the authors reduce the classification of the irreducible unitary representations of G which have a nonzero Iwahori fixed vector, to similar questions regarding unitary representations of the Iwahori and graded Hecke algebras of G. It is therefore natural to investigate the unitary representations of the Iwahori and graded Hecke algebras.

The relation between the Iwahori Hecke algebra and the graded algebra is described in [Ls1]. There, Lusztig begins with a Weyl group W and a reduced simple root system (X , Y, R, Ř, Π) whose reflection group is W. The graded Hecke algebra ℍ associated to the root system is an algebra containing the group algebra ℂW and the algebra S of polynomials on ℂ ⊗_{ℤ} Y. As a vector space, ℍ = ℂW ⊗_{ℂ} S. The algebra ℍ comes equipped with a ∗ operation. A finite dimensional irreducible representation (π, V_{π}) of ℍ is said to be hermitian (resp. unitary) if there is a nondegenerate (resp. definite) hermitian form ⟨ , ⟩ on V_{π} such that ⟨π(h)v_{1}, v_{2}⟩ = ⟨v_{1}, π(h^{∗})v_{2}⟩ for all h ∈ ℍ and v_{1}, v_{2} ∈ V_{π}.

The purpose of this paper is to investigate the following problem.

Problem. Determine the irreducible representations of ℂW which can be extended as hermitian representations to ℍ.

We call such a representation a one K–type representation. Such representations are clearly unitary.

The relation between the unitary dual of the Hecke algebra and the corresponding graded Hecke algebra is studied in [BM2]. In particular, the most important case is when the elements x ∈ X ⊂ S act with real eigenvalues. This situation is known as the real infinitesimal character case. In [BM2], the description of the unitary dual is reduced to the description in the case of real infinitesimal character. Thus we shall deal only with real infinitesimal character.

The one K–type representations play a role in the residual spectrum of a reductive algebraic group. For GL(n) they appear as local factors of the automorphic forms constructed in [S] later generalized in [J]. In the work of [T], certain representations a(n, d)

Typeset by AMS-TEX

are the building blocks for the unitary dual of GL(n). These are cases of one K–type representations.

The main result of this paper is a classification of one K–type representations for simple root systems. This is obtained as follows. Given σ ∈ Ŵ and a linear functional ω ∈ X set

σ(ω) = \frac{1}{2} ∑_{β∈R^{+}} h_{β}⟨ω, β̌⟩σ(t_{β})

where the sum runs over the positive roots and h_{β} is a scalar as in [BM2]. After a review of the ∗ operation in section 2, it is shown that assuming a multiplicity one result (which holds for all σ ∈ Ŵ except certain representations of E_{8}) a necessary and sufficient condition for σ to extend to a hermitian representation of ℍ is that the commutator [σ(ω), σ(ω′)] vanish for all ω, ω′ ∈ X. In sections 3 and 4, the individual cases of the classical and the exceptional Lie types are treated. The results are tabulated in Theorems 3.15 and 3.28 and in the tables of section 4. In section 5, we deal with the remaining cases in E_{8} which are not amenable to the aforementioned techniques (c.f. section 2).

Acknowledgements. The authors were supported in part by NSF grants DMS–9401176 and DMS–9500973.

2. The graded Hecke algebra

We recall the definition [Ls1] of the graded Hecke algebra. Let (X , Y, R, Ř, Π) be a reduced simple root system. Let R^{+} be the positive system corresponding to Π. For α ∈ R, let s_{α} ∈ GL(X) be the reflection

s_{α}(x) = x − ⟨x, α̌⟩α.

Let W ⊂ GL(X) be the Weyl group generated by the s_{α}’s, and ℂW the group algebra of W. When we view s_{α} as an element in W, we shall find it convenient to also denote it as t_{α}. To define a graded Hecke algebra, recall that we need a function h defined on the roots R and taking values in the positive rationals ℚ^{+} so that h(α) = h(α′) if α and α′ are in the same Weyl orbit. We write h_{α} for h(α).

Let S be the symmetric algebra of ℂ ⊗_{ℤ} X. The commutative algebra S is the algebra of polynomials on the vector space ^{L}𝔞 = ℂ ⊗_{ℤ} Y. Let ℍ be the graded Hecke algebra associated to h as in [Ls1]. It has the algebras ℂW and S as subalgebras. In fact as a ℂ vector space, ℍ = ℂW ⊗_{ℂ} S. Given α ∈ Π and x ∈ X ⊂ S, the element t_{α} commutes past the element x according to the relation

(2.1)  t_{α} · x = s_{α}(x) · t_{α} + ⟨x, α̌⟩h_{α}

Observe that if we multiply the function h by a positive rational number c, the graded Hecke algebra obtained from ch is isomorphic to the original graded Hecke algebra. As such, we can and often do assume that the function h takes values in the natural numbers ℕ.

Theorem 2.2. ([BM2]) The algebra ℍ is a ∗ algebra with the ∗ involution given by

\begin{aligned}t^{∗} & = t^{−1}  t ∈ W \\ ω^{∗} & = −ω + ∑_{β∈R^{+}} t_{β}⟨ω, β̌⟩h_{β}  ω ∈ X \end{aligned}

We use Theorem 2.2 to give a necessary and sufficient condition for a σ ∈ Ŵ to extend to a hermitian representation of ℍ. Let ⟨ , ⟩ be a positive definite W –invariant hermitian form on V_{σ}. Up to a positive scalar multiple ⟨ , ⟩ is unique. Given A ∈ End_{ℂ}(V_{σ}), the adjoint of A is the endomorphism A^{∗} ∈ End_{ℂ}(V_{σ}) satisfying ⟨Av_{1}, v_{2}⟩ = ⟨v_{1}, A^{∗}v_{2}⟩ for all v_{1}, v_{2} ∈ V_{σ}. For ω ∈ X, let p_{ω} ∈ ℂW be the element

p_{ω} = \frac{1}{2} ∑_{β∈R^{+}}⟨ω, β̌⟩h_{β}t_{β}

and set

σ(ω) = \frac{1}{2} ∑_{β∈R^{+}}⟨ω, β̌⟩h_{β}σ(t_{β})

so that σ(ω) = σ(p_{ω}).

Proposition 2.3. Suppose σ restricts in a multiplicity free fashion to some parabolic subgroup W_{P} ≠ W. Then σ extends to a one K–type representation of ℍ if and only if

σ([p_{ω}, p_{ω′}]) = 0 ∀ ω, ω′ ∈ X .

Proof. We first establish that

(2.4)  π(ω^{∗}) = π(ω)^{∗} for all ω ∈ X .

Suppose σ extends to a real hermitian representation π of ℍ. Then S is generated by the W –translates of the x ∈ X satisfying w(x) = x for all w ∈ W_{P} . In ℍ, such an x commutes with W_{P}; hence, π(x) must commute with π(W_{P}). This and the hypothesis that π_{|W_{P}} is multiplicity free implies π(x) preserves each W_{P} –subspace of V_{σ}. Hence the action of π(x) on V_{σ} can be diagonalized. Since the eigenvalues of π(x) are assumed to be real, it follows that π(x)^{∗} = π(x). Since s_{α}(x) = t_{α}xt_{α} − ⟨x, β̌⟩h_{β}, it follows that π(s_{α}(x))^{∗} = π(s_{α}(x)) for every α ∈ Π. Thus (2.4) follows. As a consequence,

π(ω) = σ(ω) = \frac{1}{2} ∑_{β∈R^{+}}⟨ω, β̌⟩h_{β} σ(t_{β}) ∀ ω ∈ X .

It is obvious now that a necessary condition for σ to extend to a real hermitian representation of ℍ is [σ(ω), σ(ω′)] = 0. Conversely, if [σ(ω), σ(ω′)] = 0 for all ω, ω′ ∈ X, then σ can be extended to a real hermitian representation by letting ω act as σ(ω). □

The following pairs (W, W_{P}) have the property that any σ ∈ Ŵ restricts to W_{P} as a multiplicity free representation:

\begin{matrix}(W (A_{n}) & , W (A_{n−1})) \\ (W (B_{n}) & , W (B_{n−1})) \\ (W (D_{n}) & , W (D_{n−1})) \\ (W (F_{4}) & , W (B_{3})) \\ (W (G_{2}) & , W (A_{1})) \\ (W (E_{6}) & , W (D_{5})) \\ (W (E_{7}) & , W (E_{6})) \end{matrix}

The exceptional cases follow from a case by case inspection of the restriction tables of [A]. The classical cases follow from the Littlewood–Richardson rule for type A and its modifications for the other types.

The pair (W (E_{8}), W (E_{7})) also has this property except for the representations ϕ_{6075,14}, ϕ_{6075,22}, ϕ_{3240,31}, ϕ_{3240,9}, ϕ_{5600,19}, ϕ_{5600,21}, ϕ_{4536,23}, ϕ_{4536,18} and ϕ_{7168,17}. The notation is as in [C]. We call these the sporadic cases. Except in section 5, we will always assume that σ is not one of the sporadic cases.

Proposition 2.4. Let ω_{1}, ω_{2} ∈ X be linearly independent, and such that they satisfy the following conditions:

(1) the W –translates of ω_{1} span ℂ ⊗_{ℤ} X modulo its W –invariants,

(2) The translates of ω_{2} under Stab(ω_{1}) together with ω_{1} span ℂ ⊗ X .

Then σ ∈ Ŵ extends to a one K–type representation if and only if [σ(ω_{1}), σ(ω_{2})] = 0.

Proof. It is enough to check that if [σ(ω_{1}), σ(ω_{2})] = 0, then [σ(ω), σ(ω′)] = 0 for all ω, ω′ ∈ X . We observe that

(2.5)  [σ(s_{α}(ω)), σ(s_{α}(ω′))] = t_{α}[σ(ω), σ(ω′)]t_{α}.

If [σ(ω_{1}), σ(ω_{2})] = 0, then it is the case that [σ(ω_{1}), σ(t(ω_{2}))] = 0 for every t ∈ Stab(ω_{1}). Therefore

(2.6)  [σ(ω_{1}), σ(x)] = 0  for all x ∈ ℂ ⊗ X

Using (2.5) again, we find

(2.7)  [σ(s_{α}(ω_{1})), σ(s_{α}(ω_{2}))] = t_{α}[σ(ω_{1}), σ(ω_{2})]t_{α} = 0,

so [σ(w(ω_{1})), σ(w(x))] = 0 for all w ∈ W, and the claim follows. □

Example 2.8. We give a simple illustration with G_{2} of how we intend to use Proposition 2.4 to classify the one K–type representations. The G_{2} root system lies in the plane in real 3-space defined by x_{1} + x_{2} + x_{3} = 0. The twelve roots are

R = {±(ϵ_{i} − ϵ_{j}), ±(2ϵ_{i} − ϵ_{j} − ϵ_{k})}_{i,j,k distinct}

As simple root, we choose

α = (1, −1, 0) and β = (−1, 2, −1).

Take

ω_{1} = (1, 0, −1) and ω_{2} = (1, 1, −2).

Then

\begin{aligned}ρ_{ω_{1}} & = h_{α} (t_{(1,−1,0)} + t_{(0,1,−1)} + 2t_{(1,0,−1)}) + h_{β} (t_{(2,−1,−1)} + t_{(1,1,−2)}) \\ ρ_{ω_{2}} & = 3h_{α} (t_{(0,1,−1)} + t_{(1,0,−1)}) + h_{β} (t_{(−1,2,−1)} + t_{(2,−1,−1)} + 2t_{(1,1,−2)}) \end{aligned}

Of course [σ(ω_{1}), σ(ω_{2})] = 0 for any one dimensional representation σ of W (G_{2}) so they are one K–type representations. In the situation when σ is either the reflection representation σ_{ref} or the other irreducible two dimensional representation σ_{2′} we find

σ_{ref}([p_{ω_{1}}, p_{ω_{2}}]) ≠ 0 for any h_{α}, h_{β} ∈ ℕ
σ_{2′}([p_{ω_{1}}, p_{ω_{2}}]) = 0 if and only if h_{β} = h_{α} or h_{β} = 3h_{α}

3. The Classical Cases

The arguments we present for the diagrams of classical type are different from the exceptional types, even though the two strategies apply equally well. For the classical cases we take advantage of the realizations of the representations in terms of harmonic polynomials and the combinatorics for the symmetric group. The main feature is that in lowest degree there are representatives which look like the sgn representations of smaller groups.

We use the standard bases and positive systems for each of the types, A, B, C, D. Denote the system by (R, Π, Ř, Π̌) and the positive root system by R^{+}.

Type A_{n−1}.

(3.1)  \begin{matrix}R = Ř = & \{ϵ_{i} − ϵ_{j}\}_{i≠j≤n}, \\ Π = Π̌ = & \{ϵ_{i} − ϵ_{i+1}\}, \\ R^{+} = & \{ϵ_{i} − ϵ_{j}\}_{i<j≤n}. \end{matrix}

Most of the time it is more convenient to use the lattices corresponding to gl(n) instead of sl(n). We will do so without special mention.

Type B_{n}.

(3.2)  \begin{matrix}R = & \{±ϵ_{i} ± ϵ_{j}, ±ϵ_{i}\}_{i<j≤n}, Ř = \{±ϵ_{i} ± ϵ_{j}\}_{i<j≤n}, 2ϵ_{i}\}_{i<j≤n}, \\ Π = & \{ϵ_{i} − ϵ_{i+1}, ϵ_{n}\}, Π̌ = \{ϵ_{i} − ϵ_{i+1}, 2ϵ_{n}\}, \\ R^{+} = & \{ϵ_{i} ± ϵ_{j}, ϵ_{i}\}_{i<j≤n}. \end{matrix}

Type C_{n}.

(3.3)  \begin{matrix}R = & \{±ϵ_{i} ± ϵ_{j}, 2ϵ_{i}\}_{i<j≤n}, Ř = \{±ϵ_{i} ± ϵ_{j}, ϵ_{i}\}_{i<j≤n}, \\ Π = & \{ϵ_{i} − ϵ_{i+1}, 2ϵ_{n}\}, Π̌ = \{ϵ_{i} − ϵ_{i+1}, ϵ_{n}\}, \\ R^{+} = & \{ϵ_{i} ± ϵ_{j}, 2ϵ_{i}\}_{i<j}. \end{matrix}

Type D_{n}.

(3.4)  \begin{matrix}R = Ř = & \{±ϵ_{i} ± ϵ_{j}\}_{i<j≤n}, \\ Π = Π̌ = & \{ϵ_{i} − ϵ_{i+1}, ϵ_{n−1} + ϵ_{n}\}, \\ R^{+} = & \{ϵ_{i} ± ϵ_{j}\}_{i<j}. \end{matrix}

We now translate Proposition 2.3 and Proposition 2.4 into coordinates. In all cases we take ω_{1} = ϵ_{1} and ω_{2} = ϵ_{2}. Denote by p_{ω} the expression ∑ h_{α}⟨ω, α̌⟩t_{α}. To establish some shorthand notation set

t_{ij} = t_{ϵ_{i}−ϵ_{j}} t′_{ij} = t_{ϵ_{i}+ϵ_{j}} t_{i} = t_{ϵ_{i}} = t_{2ϵ_{i}}

For an element w ∈ W, we write t_{w} for its representative in ℂW and s_{w} for its action (the reflection representation except in type A) on the vector space ℝ^{n} and similarly for its action on S(ℝ^{n}) and S((ℝ^{n})^{∗}).

We may as well assume that h_{ϵ_{i}±ϵ_{j}} = 1. Then

Type A_{n−1}

\begin{aligned}p_{ω_{1}} & = ∑_{i=2}^{n} t_{1i} \\ p_{ω_{2}} & = −t_{12} + ∑_{i=3}^{n} t_{2i} \end{aligned}

Type B_{n}

\begin{aligned}p_{ω_{1}} & = 2h_{ϵ_{1}}t_{1} + ∑_{i=2}^{n}(t_{1i} + t′_{1i}) \\ p_{ω_{2}} & = 2h_{ϵ_{2}}t_{2} − t_{12} + t′_{12} + ∑_{i=3}^{n}(t_{2i} + t′_{2i}) \end{aligned}

Type C_{n}

\begin{aligned}p_{ω_{1}} & = h_{2ϵ_{1}}t_{1} + ∑_{i=2}^{n}(t_{1i} + t′_{1i}) \\ p_{ω_{2}} & = h_{2ϵ_{2}}t_{2} − t_{12} + t′_{12} + ∑_{i=3}^{n}(t_{2i} + t′_{2i}) \end{aligned}

Type D_{n}

\begin{aligned}p_{ω_{1}} & = ∑_{i=2}^{n}(t_{1i} + t′_{1i}) \\ p_{ω_{2}} & = −t_{12} + t′_{12} + ∑_{i=3}^{n}(t_{2i} + t′_{2i}) \end{aligned}

Of course h_{ϵ_{1}} = h_{ϵ_{2}} and similarly h_{2ϵ_{1}} = h_{2ϵ_{2}}.

Theorem 3.6. Write t = t_{12}, and t′ = t′_{12}. The condition [σ(ω_{1}), σ(ω_{2})] = 0 on a K–type σ is equivalent to

[σ(t), σ(p_{ω_{1}})] = 0 for type A,
[σ(t) − σ(t′), σ(p_{ω_{1}})] = 0 for types B, C, D.

Proof. Consider type A. Note that [t_{ij}, p_{1}] = 0 for all i, j ≥ 2 (p_{1} is invariant under the symmetric group permuting {2, …, n}). Thus [p_{1}, p_{2}] = [p_{1}, t_{12}] as claimed. For types B, C, D, the same proof applies, [p_{1}, t_{ij}] = 0 as well as [p_{1}, t′_{ij}] = 0 for all i, j ≥ 2, and in types B, C [p_{1}, t_{i}] = 0 for all i as well. □

We now describe the realizations of the irreducible representations as in [Ls2]. This uses MacDonald’s construction, and is well known in the case of type A when W is the symmetric group. In all cases, given a representation σ, we write down a harmonic polynomial π_{σ} with the property that the canonical action of W on V_{σ} := span{w · π_{σ}}_{w∈W} , provides a realization of σ. With coordinates as before, let I ⊂ {1, …, n}. Then define

(3.7)  \begin{matrix}π_{A,I} = & ∏_{\begin{matrix}i<j \\ i,j∈I \end{matrix}}(ϵ_{i} − ϵ_{j}), \\ π_{B,I} = π_{C,I} = & ∏_{\begin{matrix}i<j \\ i,j∈A \end{matrix}}(ϵ_{i}^{2} − ϵ_{j}^{2}) · ∏ ϵ_{i}, \\ π_{D,I} = & ∏_{\begin{matrix}i<j \\ i,j∈A \end{matrix}}(ϵ_{i}^{2} − ϵ_{j}^{2}). \end{matrix}

When I = {1, …, n}, the polynomial π affords the sgn representation of the corresponding Weyl group.

Type A_{n−1}. The representations of the Weyl group are parametrized by partitions. Let (n_{1}, …, n_{k}) with n_{i} ≤ n_{i+1} be a partition of n. In terms of tableaus, the n_{i} are the sizes of the columns. Fix a partition of {1, … n} into subsets I^{i} (i = 1, …, k) of sizes n_{i}, and set

(3.8)  π_{I_{1},...,I_{k}} = ∏_{i} π_{A,I_{i}}.

This is a harmonic polynomial. The span of these polynomials ranging over all partitions into subsets of sizes (n_{1}, …, n_{k}) generate an irreducible representation V_{σ} which is by definition the irreducible representation attached to the partition σ. The action of W is s_{w}π_{I_{1},...,I_{k}} = π_{wI_{1},...,wI_{k}}.

Types B, C, D. In these cases, σ is parametrized by a pair of partitions (σ_{L}, σ_{R}), adding up to n. In type D, the two partitions (σ_{L}, σ_{R}) and (σ_{R}, σ_{L}) parameterize the same representation which is the restriction of the corresponding representation of type B(n), C(n). When σ_{L} = σ_{R} in which case the restriction splits into two irreducible inequivalent composition factors, (σ_{R}, σ_{L})_{I} and (σ_{R}, σ_{L})_{II}.

Using the same conventions as for type A, partition {1, …, n} into subsets J_{s}, K_{r} of sizes equal to the elements of the partitions σ_{L} and σ_{R} respectively. Then the span of the

(3.9)  π_{I,J} = ∏_{s} π_{D,I_{s}} · ∏_{r} π_{B,J_{r}}

generate an irreducible representation which we call V_{σ}.

The next proposition is the crucial combinatorial tool we will use. Assume we are in type A. Given a polynomial p ∈ V_{σ}, let p_{ij} := s_{ij}p.

Proposition 3.10. Assume σ corresponds to (n_{1}, …, n_{k}) and let I_{i} (i = 1, …, k) be a partition into disjoint sets of sizes n_{i}. Suppose that x ∈ I_{r}, with n_{r} ≤ n_{s} and let p = π_{I_{1},...,I_{k}}. Then

∑_{j∈I_{s}} p_{xj} = p.

Proof. It is enough to consider σ corresponding to (n_{1}, n_{2}) with n_{1} ≤ n_{2}, I_{1} = {1, …, n_{1}}, I_{2} = {n_{1} + 1, …, n_{1} + n_{2}}, and x = 1.

The sum in the Proposition is

(3.11)  ∑_{j∈I_{2}} s_{1j}(p).

Then any s_{kl} with 1 < k, l ≤ n_{1} and k, l ∈ I_{1} commutes with the operator ∑_{j∈I_{2}} s_{1j}. Similarly any s_{kl} with n_{1} < k, l ≤ n_{1} + n_{2} commutes with ∑_{j∈I_{2}} s_{1j}. Thus the vector on the left transforms like the sgn representation of the subgroup S_{n_{1}−1} × S_{n_{2}}. We claim that there is only one such vector (up to a scalar) in the representation σ. We apply Frobenius Reciprocity. By the Littlewood–Richardson rule, the induced module from Sgn of S_{n_{1}−1} × S_{n_{2}} to S_{n_{1}+n_{2}−1} is formed of representations corresponding to partitions with columns (a, b) with b ≥ n_{2}, a ≤ b, and a+b = n_{2} +n_{1} −1. Each occurs with multiplicity 1. The induced module from a representation corresponding to (a, b) of S_{n_{1}+n_{2}−1} to S_{n_{1}+n_{2}} is formed of representations corresponding to partitions with columns (1, a, b), (a, b + 1) (only for a > b) and (a + 1, b), each occuring with multiplicity 1. The claim now follows from the fact that b ≥ n_{2}; the representation with columns (n_{1}, n_{2}) occurs exactly once (this is where the fact that n_{1} ≤ n_{2} is crucial). Therefore

(3.12)  ∑_{j∈I_{2}} p_{1j} = cp,

for some constant c. It remains to show that c = 1. In general, for a partition (n_{1}, …, n_{k}) of n, write

(3.13)  ρ(n_{1}, …, n_{k}) := (n_{1}, …, 1, n_{2}, …, 1, …, n_{k}, …, 1).

We plug ρ(n_{1}, n_{2}) into the two sides of (3.10). On the Left Hand Side, we get

(3.14)  ∑_{j∈I_{2}} p(s_{1j}ρ(n_{1}, n_{2}))

Unless s_{1j} fixes ρ, the term is 0 because p is evaluated at a point that is singular with respect to a root in I_{1} or I_{2}. There is only one such s_{1j} namely the one for j = n_{1} + 1. In this case the value is exactly the Right Hand Side of (3.12). □

Example. Consider the case when n_{1} = 2 and n_{2} = 3. The induced module from the sgn representation of S_{1} × S_{3} is formed of the representations corresponding to partitions (1, 1, 3), (2, 3) and (5) occuring once and (1, 4) occuring twice.

Take the partition I_{1} = {1, 2} and I_{2} = {3, 4, 5}. The polynomial p is

[(ϵ_{1} − ϵ_{2})] · [(ϵ_{3} − ϵ_{4})(ϵ_{4} − ϵ_{5})(ϵ_{3} − ϵ_{5})].

The element ρ = ρ(2, 3) is

(2, 1, 3, 2, 1).

The polynomial p vanishes on any s_{1j}ρ unless j = 4, in which case s_{14}ρ = ρ.

On the other hand if we use the same σ but n_{1} = 3 and n_{2} = 2, the induced module from sgn of S_{2} × S_{2} is formed of the representations corresponding to the partitions (1, 2, 2), ((1, 1, 3), (5) once and (2, 3), (1, 4) twice. Formula (3.12) does not necessarily hold any more.

Theorem 3.15. In type A a representation σ of W (A) extends to a representation of ℍ if and only if n_{1} = ⋯ = n_{k} = d.

Proof. The relation σ([t, ω_{1}]) = 0 is equivalent to σ(ω_{1}) = σ(tω_{1}t), which in turn is equivalent to

(3.16)  ∑_{i≠1} s_{1i}(p) = ∑_{j≠2} s_{2j}(p).

It is also equivalent to

(3.17)  ∑_{i≠1} t_{1i}(p) = ∑_{j≠k} t_{mj}(p) for any m ≠ 1.

Assume first that n_{1} = ⋯ = n_{k} = d. We check (3.16). With the same notational conventions as before, let I_{1}, …, I_{k} be any partition of {1, …, n} into subsets of size d, and let p = π_{I_{1},...,I_{k}} be as in (3.8). Note that s_{1j}p = −p if j is in the same set as 1. Using this and applying Proposition 3.10, we find that

(3.18)  σ(p_{ω_{1}})(p) = [−(d − 1) + (k − 1)]p = (k − d)p.

Then

(3.19)  \begin{matrix}σ(p_{ω_{1}})σ(t_{12})(p) = σ(p_{ω_{1}})(p_{12}) = (k − d)p_{12}, \\ σ(t_{12})σ(p_{ω_{1}})(p) = σ(t_{12})(k − d)p = (k − d)p_{12} \end{matrix}

The claim follows.

Now assume that the n_{i} are not all equal. It is enough to show that (3.17) cannot hold for some choice of p. Write the sizes of the partition as

\underbrace{n_{1}, …, n_{1}}_{d_{1}} < \underbrace{n_{2}, …, n_{2}}_{d_{2}} < ….

Assume that I_{1}, …, I_{k} is the partition where 1, 2, … n_{1} ∈ I_{1}, n_{1} + 1, … 2n_{1} ∈ I_{2} and so on, and let p be the corresponding polynomial. We use (3.17) with m = n_{1}d_{1} + 1. Then by applying Proposition 3.10 to the p corresponding to this partition, we can rewrite (3.17) as

(3.20)  (d_{1} − n_{1} + d_{2} + ⋯ + d_{r})p = (d_{2} − n_{2} + d_{3} + ⋯ + d_{r})p + ∑_{\begin{matrix}i∈I_{l}, \\ 1≤l≤d_{1} \end{matrix}} p_{mi}.

This can be rewritten further as

(3.21)  ∑_{\begin{matrix}i∈I_{l}, \\ 1≤l≤d_{1} \end{matrix}} p_{mi} = (d_{1} + n_{2} − n_{1})p.

To prove that (3.21) cannot hold, we evaluate at a special element. If we substitute in ρ(n_{1}, …, n_{1}, n_{2}, …), we see that the the Right Hand Side is not zero. But the the Left Hand Side is zero. Therefore (3.17) cannot hold. □

We now turn to the cases of type B, C and D. Note first that the relation

(3.22)  [tt′, p_{ω_{1}}] = 0

holds. Decompose the space of the representation σ as V = V_{−} ⊕ V_{+} according to the eigenvalues ±1 of σ(tt′). If p is the polynomial given by (3.9) for a pair of partitions correponding to σ, then p is in V_{+} if both 1, 2 are in sets of the same type, and p is in V_{−} if they are in opposite types.

The relation in Theorem 3.6 is automatically satisfied on V_{+} because σ(t) = σ(t′) on this space. On V_{−} the relation in Theorem 3.6 has the simpler form

(3.23)  [σ(t), σ(p_{ω_{1}})]p = 0, for p ∈ V_{−}.

Proposition 3.24. Assume that σ_{L} = ∅ or σ_{R} = ∅. Then σ extends to a hermitian representation of ℍ.

Proof. In these cases, V_{−} = 0. The claim follows from the above discussion. □

Assume that p is given by (3.9) for a partition {J_{1}, …, J_{s}, …, K_{1}, …, K_{r}, … } of {1, …, n} into subsets, corresponding to σ attached to (σ_{L}, σ_{R}).

Lemma 3.25. Assume that 1 ∈ J_{1} Then

[s_{1j} + s′_{1j}]p = \{ \begin{matrix}0 & if j ∈ K_{s}, \\ 2p_{1j} & if j ∈ J_{r}, r ≠ 1. \end{matrix}

Assume 2 ∈ K_{1}. Then

[s_{2i} + s′_{2i}]p = \{ \begin{matrix}0 & if i ∈ J_{r}, \\ 2p_{2i} & if i ∈ K_{s}, s ≠ 1. \end{matrix}

Proof. This is just the fact that the degree of an ϵ_{i} is even in the case of π_{D,K_{1},...} and odd in the case of π_{B,J_{1},...}. □

Corollary 3.26. Let

δ = \{ \begin{matrix}2h_{ϵ_{1}} & in type B, \\ h_{2ϵ_{1}} & in type C, \\ 0 & in type D. \end{matrix}

Suppose 0 ≠ p ∈ V_{−} is of the form (3.9) with 1 ∈ J_{1} and 2 ∈ K_{1}. Then the relation σ([t, ω_{1}])p = 0 is equivalent to

(3.27)  ∑_{r≠1} ∑_{i∈J_{r}} p_{1i} − ∑_{s≠2} ∑_{j∈K_{s}} p_{2j} = (δ + m_{1} − n_{1})p.

Proof. We rewrite σ(t_{12}ω_{1})p − σ(ω_{1}t_{12})p = 0 as

s_{12}[2δs_{1} + ∑_{i≠1}(s_{1i} + s′_{1i})]p − s_{12}[2δs_{2} + ∑_{j≠2}(s_{2j} + s′_{2j})]p = 0.

Then we use the fact that p is skew invariant with respect to s_{1}, invariant with respect to s_{2} and (3.25) to derive the relation. □

Theorem 3.28. Let δ be as in 3.26. Assume that σ_{L} ≠ ∅ corresponds to the partition m_{1} ≤ ⋯ ≤ m_{k}, and σ_{R} ≠ ∅ corresponds to the partition n_{1} ≤ ⋯ ≤ n_{l}. Then σ extends to a hermitian representation of ℍ if and only all m_{i} = d, all n_{j} = f and k − d = l − f + δ.

Proof. Let p ∈ V_{−} be as in (3.9). We may as well assume 1 ∈ J_{1} and 2 ∈ K_{1}. Suppose p satisfies (3.27). In cases B, C, all the terms in relation (3.27) are divisible by ∏_{r} ∏_{i∈J_{r}} ϵ_{i}. Furthermore, (ϵ_{i} − ϵ_{j})(ϵ_{i} + ϵ_{j}) = ϵ_{i}^{2} − ϵ_{j}^{2}. Divide by the common factor ∏ ϵ_{i} and replace ϵ_{i}^{2} by ϵ_{i}. We get a polynomial p̃ of the form π_{A} which must satisfy

(3.29)  ∑_{r≠1} ∑_{i∈J_{r}} p̃_{1i} − ∑_{s≠2} ∑_{j∈K_{s}} p̃_{2j} = (δ + m_{1} − n_{1})p̃.

Furthermore p̃ factors as p̃_{B} · p̃_{D} where p̃_{B} is the product over the factors for the J_{r} and p̃_{D} is the product over the factors for the K_{s}. Using the skew invariance with respect to the J_{r} and K_{s}, we find that

(3.30)  \begin{matrix}∑_{r≠1} ∑_{i∈B^{r}} p̃_{B,1i} = c_{1}p̃_{B}, \\ ∑_{s≠1} ∑_{i∈D^{s}} p̃_{D,2j} = c_{2}p̃_{D},  c_{1} + c_{2} = δ + m_{1} − n_{1}. \end{matrix}

The terms in the first sum are all skew for permutations of J_{1}, the terms in the second sum are all skew for permutations of K_{1}. Since the difference is skew for both, all the individual terms are skew. By the result in type A, all the m_{r} = d and all the n_{s} = f. Then the first part of the proof of theorem 3.15 allows us to compute c_{1}, c_{2}. The relation c_{1} + c_{2} translates into k − d = l − f + δ.

Now assume that the sizes of the J_{r} satisfy

\underbrace{m_{1}, …, m_{1}}_{d_{1}} < \underbrace{m_{2}, …, m_{2}}_{d_{2}} < ….

Then p̃_{B} is as in the second part of the proof of theorem 3.15. It is enough to show that

(3.31)  ∑_{i≠m} p̃_{B,im} ≠ cp̃_{B}

for m = m_{1}d_{1} + 1. By plugging in ρ(m_{1}, …) this can hold if and only if c = 0. Thus it is enough to show that the Left Hand Side is nonzero. This can be achieved by plugging in wρ(m_{1}, … m_{k}) into the Left Hand Side , where w interchanges the m_{1} in the first entry with the entry m_{2} in the m_{1}d_{1} + 1^{st} coordinate. □

4. The exceptional cases

Proposition 4.1. Let σ ∈ Ŵ and let χ_{σ} be its character. A necessary and sufficient condition that [σ(ω_{1}), σ(ω_{2})] = 0 (ω_{1}, ω_{2} ∈ X) is that

χ_{σ}([p_{ω_{1}}, p_{ω_{2}}]^{2}) = 0

Proof. The representation σ can be taken to be real orthogonal matrices. Each involution σ(t_{α}) in the definition of σ(ω) is therefore a symmetric matrix. The commutator [σ(ω_{1}), σ(ω_{2})] is a real skew symmetric matrix and so \sqrt{−1}[σ(ω_{1}), σ(ω_{2})] is a Hermitian matrix. Hence it must have real eigenvalues. This means

χ_{σ}([p_{ω_{1}}, p_{ω_{2}}]^{2}) ≤ 0,

with equality precisely when σ(ω_{1}) and σ(ω_{2}) commute. □

We use Propositions 3.1 and 4.1 to determine the one K–type representations of the exceptional types. We begin with types G_{2} and F_{4}.

Type G_{2}. Label the Dynkin diagram as

o_{1} ≡ ⟨≡ o_{2}

Let ω_{1} and ω_{2} be as in (2.6) and take h_{α} = 1 and h_{β} = h. To evaluate the character values in Proposition 4.1, we need only compute the terms in the square of the commutator up to conjugacy, i.e. we need to only know the conjugacy classes which appear in the square of the commutator and how many elements of each conjugacy class occurs. One can compute that the element [p_{ω_{1}}, p_{ω_{2}}]^{2} has the same character values as the element

−(4h^{4} + 88h^{2} + 36) − 32(h^{3} + 3h)t_{α}t_{β} + (4h^{4} + 88h^{2} + 36)(t_{α}t_{β})^{2} + 32(h^{3} + 3h)(t_{α}t_{β})^{3}

From this we recover the results of example 2.8.

Type F_{4}. Label the Dynkin diagram as

o_{1}—–o_{2}=⟩=o_{3}—–o_{4}

Let

α_{1} = ϵ_{2} − ϵ_{3}, α_{2} = ϵ_{3} − ϵ_{4} α_{3} = ϵ_{4}, α_{1} = \frac{1}{2}(ϵ_{1} − ϵ_{2} − ϵ_{3} − ϵ_{4})

Let γ = ϵ_{1} basis vector to α̌_{4}. The stablizer of γ is a maximal parabolic subgroup W_{P} and the weights

ω_{1} = γ and ω_{2} = t_{4}(ω_{1}).

obviously satisfy the hypothesis of Proposition 3.1. The expression for p_{ω_{1}} has 15 terms. The calculation of [p_{ω_{1}}, p_{ω_{2}}]^{2}, or equivalent for our purposes an element g which has the same trace on any σ ∈ Ŵ, is best done by machine. For our computations, we used the algebra software GAP [G]. Set

\begin{aligned}w_{3} & = t_{1}t_{2}, w_{3′} = t_{3}t_{4} \\ w_{12} & = t_{1}t_{2}t_{3}t_{4} \\ w_{6} & = w_{12}^{2}, w_{6′} = w_{6}t_{1}t_{2}t_{4}t_{3}, w_{6′′} = w_{12}^{3}t_{12}t_{2} \\ w_{2} & = t_{1}t_{3}, w_{2′} = t_{1}t_{2}t_{4}t_{3}t_{2}t_{1}t_{3}t_{4} \\ w_{4} & = t_{2}t_{3}, w_{4′} = w_{12}^{3} \end{aligned}

Take h_{α_{1}} = h and h_{α_{2}} = 1. A necessary and sufficient condition for a one K–type representation is that χ_{σ} vanish on the group element

\begin{aligned}g = & − 12(h^{4} + 6h^{2} + 4)1 + 480h^{2}(w_{12} − w_{6}) + 24h^{4}(w_{6′} − w_{3′}) − 24(h^{4} + 4)w_{4′} \\ & + 288h(h^{2} + 2)(w_{2} − w_{4}) + 96(w_{6′′} − w_{3}) + 36(h^{4} + 2h^{2} + 4)w_{2′} \end{aligned}

We see that χ_{σ}(g) = 0 for any of the four 1-dimensional and four 2-dimensional representations. It is also the case that χ_{σ}(g) vanishes for the 4-dimensional representation whose character values on the w′s are

\begin{matrix}1 & w_{2} & w_{2′} & w_{3} & w_{3′} & w_{4} & w_{4′} & w_{6} & w_{6′} & w_{6′′′} & w_{12} \\ 4 & 0 & 4 & −2 & −2 & 0 & 4 & 1 & −2 & −2 & 1 \end{matrix}

These nine representations yield one K–type representations for any value of h. When h = 1, there are five additional one K–type representations with dimensions 4, 4, 6, 8, and 8. The two representations of a given dimension differ by a twist of the sgn character. Their values on the w′s are

\begin{matrix}1 & w_{2} & w_{2′} & w_{3} & w_{3′} & w_{4} & w_{4′} & w_{6} & w_{6′} & w_{6′′′} & w_{12} \\ 4 & 0 & 0 & 1 & 1 & −2 & 0 & 2 & −1 & −1 & 0 \\ 6 & 2 & −2 & 0 & 0 & −2 & 2 & 3 & 0 & 0 & −1 \\ 8 & 0 & 0 & −1 & 2 & 0 & 0 & −2 & −2 & 1 & 0 \end{matrix}

When h = 2, the representations of dimension 4 and 6 are again one K–type representations. The two representations of dimension 8 no longer yield one K–type representations. Instead, two other representations of dimension 8, which again differ by sgn, give one K–type representations. Their values on the w′s are

\begin{matrix}1 & w_{2} & w_{2′} & w_{3} & w_{3′} & w_{4} & w_{4′} & w_{6} & w_{6′} & w_{6′′′} & w_{12} \\ 8 & 0 & 0 & 2 & −1 & 0 & 0 & −2 & 1 & −2 & 0 \end{matrix}

Consider now the types E_{6}, E_{7} and E_{8}. Label the Dynkin diagrams of E_{6}, E_{7} and E_{8}

as o_{1}—–o_{3}—–o_{4}—–o_{5}—–o_{6}

|
o_{2}

o_{1}—–o_{3}—–o_{4}—–o_{5}—–o_{6}—–o_{7}
|
o_{2}

o_{1}—–o_{3}—–o_{4}—–o_{5}—–o_{6}—–o_{7}—–o_{8}
|
o_{2}

Let γ ∈ R be the dual basis vector to α̌_{2}, α̌_{1}, α̌_{8} respectively. The stabilizer of γ is a maximal parabolic subgroup W_{P}. In case E_{6}, W_{P} is the subgroup obtained from the Dynkin diagram by removing the node s_{2}. In the cases of E_{7} and E_{8}, W_{P} is the subgroup obtained from the Dynkin diagram by removing the nodes s_{1} and s_{8} respectively. Let s denote the removed node in each of the three cases. Then the weights

ω_{1} = γ and ω_{2} = s(ω_{1})

obviously satisfy the hypothesis of Proposition 3.1. Hence, a necessary and sufficient condition for σ to be a one K–type representation is that [σ(ω_{1}), σ(ω_{2})] = 0. The vanishing of the commutator is equivalent, by Proposition 4.1, to

χ_{σ}([p_{ω_{1}}, p_{ω_{2}}]^{2}) = 0.

In the case of E_{6} the group algebra elements p_{ω_{1}} and p_{ω_{2}} each have 21 terms. For E_{7} and E_{8}, the number of terms increase to 33 and 57 respectively. We again used GAP to perform the tedious calculation of an element g which has the same trace as the element [p_{ω_{1}}, p_{ω_{2}}]^{2} on any σ ∈ Ŵ. We tabulate the results in a table. Set

w_{2} = s_{1}s_{2}, w_{3} = s_{1}s_{3}, w_{4} = s_{1}s_{2}s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}, w_{6} = s_{2}s_{3}s_{4}s_{5}.
conjugacy class order  D_{5}  E_{6}  E_{7}  E_{8}
1  1  −1  −1  −1  −1
w_{3}  3  −4  −8  −14 −26
w_{2}  2  5  9  15  27
w_{4}  4  −4  −12  −30 −90
w_{6}  6  4  12  30  90

Then, σ ∈ Ŵ is a one K–type representation if and only if the character χ_{σ} vanishes on the following elements

type E_{6} − 1 − 8w_{3} + 9w_{2} − 12w_{4} + 12w_{6}
type E_{7} − 1 − 14w_{3} + 15w_{2} − 30w_{4} + 30w_{6}
type E_{8} − 1 − 26w_{3} + 27w_{2} − 90w_{4} + 90w_{6}

Type E_{6}. There are 25 irreducible representations of W and 7 are one K–type representations. The character values of these 7 representations on the elements 1, w_{3}, w_{2}, w_{4}, w_{6} are

\begin{matrix}1 & w_{3} & w_{2} & w_{4} & w_{6} \\ 1′ & 1 & 1 & 1 & 1 \\ 10 & −2 & 2 & 2 & 0 \\ 15′ & 0 & 3 & −1 & −2 \\ 24′ & 0 & 0 & 0 & 2 \end{matrix}

A ′ on the dimension indicates that the representation and its twist by the sgn character are distinct representations.

Type E_{7}. As W (E_{7}) is the direct product of the group of the subgroup W (E_{7})′ of elements of even length and the center {±I}. A representation σ ∈ Ŵ and its twist by the sgn character are distinct. Either both σ and σ ⊗ sgn are one K–type representations or neither is. There are 30 irreducible representations of W (E_{7})′. Nine of the representations of W (E_{7})′ are one K–type representations. The character values of these representations on the elements 1, w_{3}, w_{2}, w_{4}, w_{6} are

\begin{matrix}1 & w_{3} & w_{2} & w_{4} & w_{6} \\ 1 & 1 & 1 & 1 & 1 \\ 15 & 0 & 3 & −1 & −2 \\ 21 & 6 & 5 & 1 & 2 \\ 35 & 5 & 7 & −1 & −1 \\ 70 & −5 & 6 & 2 & −1 \\ 84 & −6 & 4 & 4 & 2 \\ 105 & 0 & −3 & −3 & 2 \\ 105 & 0 & 9 & −3 & −4 \\ 210 & −15 & 10 & 6 & 1 \end{matrix}

Type E_{8}. There are 18 one K–type representations. Their character values on the five conjugacy classes in question are

\begin{matrix}1 & w_{3} & w_{2} & w_{4} & w_{6} \\ 1′ & 1 & 1 & 1 & 1 \\ 50′ & 5 & 10 & −2 & −3 \\ 84′ & 21 & 20 & 4 & 5 \\ 175′ & −5 & 15 & −1 & −5 \\ 525′ & 30 & 5 & −7 & 6 \\ 700′ & −20 & 20 & 0 & −4 \\ 972′ & 0 & 36 & 0 & 0 \\ 168 & −12 & 8 & 8 & 4 \\ 420 & −30 & 20 & 12 & 2 \\ 448′ & 16 & 32 & 0 & 0 \end{matrix}

We note that Proposition 4.1 also applies to the classical types. Let χ_{σ} denote the character of σ ∈ Ŵ. If we combine Proposition 4.1 and Theorem 3.6, we see that in the classical cases, a necessary and sufficient condition for σ to be a one K–type representation is that

χ_{σ}([t, p_{ω_{1}}]^{2}) = 0 for type A,
(4.3)  χ_{σ}([t − t′, p_{ω_{1}})]^{2}) = 0 for types B, C, D.

For type A_{n−1} (n ≥ 3), if we label the Dynkin diagram as

o_{1}—–o_{2}—–o_{3}—– ⋯ —–o_{n}

then this amounts to the character χ_{σ} vanishing on the group element

1 − (n − 3)t_{1}t_{3} + (n − 4)t_{1}t_{2}

As remarked above, [t, p_{ω_{1}}]^{2} ≠ 1 − (n − 3)t_{1}t_{3} + (n − 4)t_{1}t_{2}. Rather it is the case that these two group algebra elements have the same character values.

For type B_{n} (n ≥ 2), label the Dynkin diagram as

o_{1}=⟨=o_{2}—–o_{3}—– ⋯ —–o_{n}

A necessary and sufficient condition for a one K–type representation is that χ_{σ} vanish on the group element

\begin{matrix}(n + 2) & \{1 − t_{1}t_{2}t_{1}t_{2}\} + 8(n − 2)\{t_{1}t_{2} − t_{1}t_{3}\} \\ + 2(n − 2)(n − 3)\{t_{2}t_{3} − t_{2}t_{4} − t_{1}t_{2}t_{1}t_{2}t_{4}t_{3} + t_{1}t_{2}t_{1}t_{2}t_{3}t_{4}t_{2}t_{3}\} \end{matrix}

In type C_{n} (n ≥ 2), with Dynkin diagram

o_{1}=⟩=o_{2}—–o_{3}—– ⋯ —–o_{n}

a necessary and sufficient condition for a one K–type representation is that χ_{σ} vanish on the group element

\begin{matrix}(n − 1) & \{1 − t_{1}t_{2}t_{1}t_{2}\} + 4(n − 2)\{t_{1}t_{2} − t_{1}t_{3}\} \\ + 2(n − 2)(n − 3)\{t_{2}t_{3} − t_{2}t_{4} − t_{1}t_{2}t_{1}t_{2}t_{4}t_{3} + t_{1}t_{2}t_{1}t_{2}t_{3}t_{4}t_{2}t_{3}\} \end{matrix}

In type D_{n} (n ≥ 4), with Dynkin diagram

o_{1}—–o_{3}—–o_{4}—– ⋯ —–o_{n}
|
o_{2}

σ is a one K–type representation if and only if χ_{σ} vanishes on

{1 − t_{1}t_{2}} + 2(n − 3){t_{4}t_{3} − t_{4}t_{1} − t_{4}t_{3}t_{1}t_{2} + t_{4}t_{1}t_{3}t_{1}t_{2}t_{3}}

5. The sporadic cases

It is enough to consider ϕ_{6075,14}, ϕ_{3240,9}, ϕ_{4536,18}, ϕ_{5600,19}, and ϕ_{7168,17} because the other representations are obtained by tensoring these with sgn. We resort to the classification of representations of the Iwahori Hecke algebra in [KL], more precisely the consequences in [BM1]. Let G be the complex reductive group corresponding to the root datum (X , Y, R, Ř, Π). This is dual to the p–adic group in the introduction. We recall that there is a 1–1 correspondence between equivalence classes of irreducible modules of the Iwahori Hecke algebra H and G–conjugacy classes of {(s, u, ϕ)} where s ∈ G is semisimple, u ∈ G is unipotent such that sus^{−1} = u^{q} and ϕ is an irreducible representation of the component group A(s, u) of the centralizer of s and u. More precisely, let A(u) be the component group of the centralizer of u. There is a canonical map π : A(s, u) → A(u). Attached to (s, u) there is a standard module which we call X(s, u) affording an action of H × A(u). Then A(s, u) also acts via the canonical map. Let ϕ ∈ { \atop A(s, u)}. Then the ϕ–isotypic component X(s, u, ϕ) is either zero or else has a unique irreducible submodule L(s, u, ϕ). Let H_{K} ⊂ H be the finite Hecke algebra, which is isomorphic to ℂW. The module X(s, u, ϕ) has the same ℂW structure as the module ∑_{ψ} H^{∗}(B_{u})^{ψ}, where the sum is over the ψ ∈ Â(u) for which ψ ∘ π contains ϕ on A(s, u). The action of W is the Springer action on the cohomology of the incidence variety of Borel subgroups of G containing u. The irreducible module L(s, u, ϕ) is the unique module containing the W –module σ_{u,ψ} which is also called a lowest K–type of L(s, u, ϕ). Note that not all ψ ∈ Â(u) have σ’s attached to them, so X(s, u, ϕ) and L(s, u, ϕ) can be zero. Also an L(s, u, ϕ) will contain all the lowest K–types σ_{u,ψ} such that ψ ∘ π contains ϕ.

Decompose s = s_{0}s_{u} such that s_{0} ∈ Stab(u, s) and Ad(s_{u})u = u^{q}. Then A(s, u) can be identified with the component group of the stabilizer of u in G(s_{0}). In particular, if A(u) is nontrivial and A_{G(s_{0})} = 1, then L(s, u, ϕ) cannot be a one K. We will show that this is the case for the sporadic representations if s_{0} ≠ 1. We use the notation in [C]. In the case when s_{0} = 1, we have A(u) = A(s, u), X(s, u, ϕ) is irreducible, and we use [BS] and [BM1] to exhibit several W –types in H^{∗}(B_{u})^{ϕ}.

ϕ_{5600,19} and ϕ_{7168,17}. These two representations are attached to different ϕ’s of the nilpotent labeled E_{7}(a_{5}). The centralizer (or rather the reductive part of the centralizer which is all that matters for this kind of computation) is a subgroup of type A_{1}. Thus we can conjugate any real s_{0} into a 1–dimensional torus, and if s_{0} ≠ 1, then G(s_{0}) is a Levi component of type A_{5}A_{1}. Then u is the principal nilpotent in this subgroup, so A(s, u) = 1. From [C], A(u) = S_{3}. Thus, there is only one L(s, u, ϕ) and it contains both ϕ_{5600,19} and ϕ_{7168,17}. In particular, neither ϕ_{5600,19} nor ϕ_{7168,17} can be a one K-type representations. It remains to analyze the case s_{0} = 1. Then ϕ_{7168,17} corresponds to the trivial character of A(u) so by [BM1], H^{∗}(B_{u})^{triv} always contains sgn. For ϕ_{5600,19}, [BS] shows that for the corresponding ϕ, H^{∗}(B_{u})^{ϕ} is not irreducible.

ϕ_{6075,14}. This corresponds to the trivial character of A(u) where u is E_{7}(a_{4}). In this case A(u) = S_{2} and the centralizer is of type A_{1}. Arguing as before, if s_{0} ≠ 1 then G(s_{0}) is a Levi subgroup of type A_{6}A_{1}. Again u is the principal nilpotent so A(s, u) = 1. We complete the argument as in the previous case.

ϕ_{4536,13}. This representation corresponds to ϕ = triv for u of type D_{5} + A_{2}. In this case A(u) = S_{2} and the connected component of the centralizer is a torus. if s_{0} ≠ 1 then G(s_{0}) is of type D_{5}A_{1} and u is again the principal nilpotent. The same argument as before applies.

ϕ_{3240,9}. This representation corresponds to ϕ = triv for u of type D_{7}(a_{1}). Then A(u) = S_{2}, and the identity component of the centralizer is a torus. Then if s_{0} ≠ 1, G(s_{0}) is of type D_{7} and u is the principal nilpotent in this subgroup. The argument is as before.

Thus none of the sporadic cases afford a one K-type representation.

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[BM2] , Reduction to real infinitesimal character in affine Hecke algebras, Journal of the AMS 6 (1993), 611–635.

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