August 23, 2007

Graded Hecke algebras

Dan Ciubotaru

Based on joint work with D. Barbasch, P. Trapa.

11Unequal parameters graded Hecke algebra

TypeBncc(1 − 2) = 1c(n) = c > 0geometricc ∈ Z (SO(m, C))c ∈ 1/2 + Z (Sp(m, C))c ∈ ±1/4 + Z (Spin(m, C))c∈/ 1/2Z (Kato “exotic nilpotentcone” of Sp(2n, C))G2c(αs ) = 1, c(αl ) = c

c = 3 (E6)F4c(αs ) = 1, c(αl ) = cc = 2 (E7)

There is also an independent “harmonic analysis” classification due to Opdam for the affine Hecke H in all (?)
cases.

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Classification of G0,χ\g1,χ

Kawanaka 1987 classified the G0,χ orbits on g1,χ.
There are finitely many orbits, so there is a unique
open orbit.
The simplest example is when χ = ρ̌. ThenM
g1,ρ̌ =C · Xα ,α∈Π

where Xα are root vectors, and G0,ρ̌ is the Cartan H.
SoG0,ρ̌\g1,χ ←→ 2Π,XC∗ · Xα ←− ΠP ⊂ Π.
OP =α∈ΠPIn this case, all orbits have smooth closure. Moreover,the closure ordering is just the inclusion of subsets ofsimple roots.
In general, Lusztig 2006 gives a reformulation of Kawanaka’sclassification by “good” parabolic subalgebras, and an algorithm for computing KL polynomials.

33Zelevinski’s matching

G = GL(n, C), W = Sn . The geometric parametersfor H can be viewed as multisegments. The geometricparameters for category O correspond to elements of Sn .
There matching of geometric parameters so that the KL
polynomials are the same.
G0,χ\g1,χ ,→ Sn
assigns to every multisegment a product of cycle permutations.
For example, at χ = ρ̌ for GL(3):Multisegments Sn• • •()•–• •(12)(23)• •–••–•–•
(123)

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Matching for GL(n, R) (with P. Trapa)
Example of parametrization

Take χ = (5, 5, 4, 4, 4, 4, 3, 3, 3, 2, 2, 1). So
G0,χ ' GL(2) × GL(4) × GL(3) × GL(2) × GL(1)For instance, one multisegment is5
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Defining Φ : G0,χ\g1,χ −→ Σ±(n)5 4 3 2• • − ++ − + •− +−1•

Flatten:•t + − − − •u − + + )• + *•55Reducibility of standard modules

We consider the Hecke algebra with equal parameters.
Let ΠP ⊂ Π be maximal, {α} = Π − ΠP . Let ω̌ denotethe fundamental coweight for α.
If p = m + n is the corresponding Lie algebra, σ isattached to a map sl(2) ,→ m.
Then n is an sl(2)-module, via the adjoint action of m.
The coweight ω̌ commutes with the sl(2). Decompose
n as n = ⊕ki=1ni, where ni is the i-eigenspace of ω̌.
Decompose each ni into simple sl(2)-modules ni =⊕j (dij ), where (d) is the simple sl(2)-module of dimension d.
The following result is known from Muić-Shahidi; theHecke algebra proof follows from the geometric classification.
Theorem 5.1 (M-S, Barbasch-C). Let σ be a generictempered module. Then the reducibility points ν > 0of X(P, σ, ν) are
dij + 1.
2ii,jProof. Let χ be the central character.
Let Oχ(P, σ) be the orbit corresponding to X(P, σ, ν).
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The generic module is parametrized by the trivial localχ
. So X(P, σ, ν) is irreducible
system on the open orbit Oop
if and only ifOχ(P, σ) = Oop .
From Lusztig’s algorithm 2006:
dim Oχ(P, σ) = dim g0,χ − dim(g0,χ ∩ p̄) + dim(g1,χ ∩ p̄).
But dim Oop = dim g1,χ.
Write the roots ∆(p̄) = ∆(m) ∪ ∆(n̄). The dimension
argument says that Oχ (P, σ) = Oop if and only if#{α ∈ ∆(n) : hα, χi = 1} = #{α ∈ ∆(n) : hα, χi = 0}.
Consider the rational function of ν:Y 1 − hα, χi.
hα, χiα∈∆(n)

The reducibility points are given by the zeros of this function.
One partial result for non-generic data:
Corollary 5.2. Let σ 0 be a tempered module, in thesame L-packet with the generic tempered σ. Then the
reducibility points of X(P, σ 0, ν) are a subset of the
reducibility points of X(P, σ, ν).
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6Generic unitary dual

The spherical principal series X(χ) is reducible if andonly if hα, χi = 1, for some positive root α.
Theorem 6.1 (Barbasch-C.). The set of unitary spherical generic parameters CS(0) is a union of k simplices (alcoves) in the dominant Weyl chamber, where:
G2 : k = 2
F4 : k = 2
E6: k = 2
E7: k = 8
E8: k = 16.

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Figure 1: Generic unitary dual of G2
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One partitions the spherical unitary dual into pieces
CS(O) parametrized by nilpotent orbits.
Let Exc denote the following set of nilpotent orbits:
e1 , A23A1, A4A2A1, A4 A2, D4(a1)A2, A32A1, A23A1, 4A1}.
Exc = {A1 A}| {z| {z } |{z}F4E7E8(The notation is as in the Bala-Carter classification.)

Then the generic unitary dual of H can be describedas follows.
Theorem 6.2 (Barbasch-C.).
1. If O ∈/ Exc, thenCS(O) = CSz(O) (0).
2. If O ∈ Exc, and O 6= (4A1 ⊂ E8), then CS(O) (CSz(O)(0).
3. If O = (4A1 ⊂ E8), then CS(O) ) CSz(O)(0).
Note that by the Iwahori-Matsumoto involution, thisgives the spherical unitary dual as well.

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