1,464 research outputs found

### Weyl group action and semicanonical bases

Let U be the enveloping algebra of a symmetric Kac-Moody algebra. The Weyl
group acts on U, up to a sign. In addition, the positive subalgebra U^+
contains a so-called semicanonical basis, with remarkable properties. The aim
of this paper is to show that these two structures are as compatible as
possible

### How to compute the Frobenius-Schur indicator of a unipotent character of a finite Coxeter system

For each finite, irreducible Coxeter system $(W,S)$, Lusztig has associated a
set of "unipotent characters" \Uch(W). There is also a notion of a "Fourier
transform" on the space of functions \Uch(W) \to \RR, due to Lusztig for Weyl
groups and to Brou\'e, Lusztig, and Malle in the remaining cases. This paper
concerns a certain $W$-representation $\varrho_{W}$ in the vector space
generated by the involutions of $W$. Our main result is to show that the
irreducible multiplicities of $\varrho_W$ are given by the Fourier transform of
a unique function \epsilon : \Uch(W) \to \{-1,0,1\}, which for various
reasons serves naturally as a heuristic definition of the Frobenius-Schur
indicator on \Uch(W). The formula we obtain for $\epsilon$ extends prior work
of Casselman, Kottwitz, Lusztig, and Vogan addressing the case in which $W$ is
a Weyl group. We include in addition a succinct description of the irreducible
decomposition of $\varrho_W$ derived by Kottwitz when $(W,S)$ is classical, and
prove that $\varrho_{W}$ defines a Gelfand model if and only if $(W,S)$ has
type $A_n$, $H_3$, or $I_2(m)$ with $m$ odd. We show finally that a conjecture
of Kottwitz connecting the decomposition of $\varrho_W$ to the left cells of
$W$ holds in all non-crystallographic types, and observe that a weaker form of
Kottwitz's conjecture holds in general. In giving these results, we carefully
survey the construction and notable properties of the set \Uch(W) and its
attached Fourier transform.Comment: 38 pages, 4 tables; v2, v3, v4: some corrections and additional
reference

### Higher Laminations and Affine Buildings

We give a Thurston-like definition for laminations on higher Teichmuller
spaces associated to a surface $S$ and a semi-simple group $G$ for $G-SL_m$ and
$PGL_m$. The case $G=SL_2$ or $PGL_2$ corresponds to the classical theory of
laminations. Our construction involves positive configurations of points in the
affine building. We show that these laminations are parameterized by the
tropical points of the spaces \X_{G,S} and \A_{G,S} of Fock and Goncharov.
Finally, we explain how these laminations give a compactification of higher
Teichmuller spaces.Comment: 46 page

### Induced characters of the projective general linear group over a finite field

Using a general result of Lusztig, we find the decomposition into
irreducibles of certain induced characters of the projective general linear
group over a finite field of odd characteristic.Comment: 17 page

### Graded Hecke algebras for disconnected reductive groups

We introduce graded Hecke algebras H based on a (possibly disconnected)
complex reductive group G and a cuspidal local system L on a unipotent orbit of
a Levi subgroup M of G. These generalize the graded Hecke algebras defined and
investigated by Lusztig for connected G.
We develop the representation theory of the algebras H. obtaining complete
and canonical parametrizations of the irreducible, the irreducible tempered and
the discrete series representations. All the modules are constructed in terms
of perverse sheaves and equivariant homology, relying on work of Lusztig. The
parameters come directly from the data (G,M,L) and they are closely related to
Langlands parameters.
Our main motivation for considering these graded Hecke algebras is that the
space of irreducible H-representations is canonically in bijection with a
certain set of "logarithms" of enhanced L-parameters. Therefore we expect these
algebras to play a role in the local Langlands program. We will make their
relation with the local Langlands correspondence, which goes via affine Hecke
algebras, precise in a sequel to this paper.Comment: Theorem 3.4 and Proposition 3.22 in version 1 were not entirely
correct as stated. This is repaired in a new appendi

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