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