The Real Chevalley Involution

We consider the Chevalley involution in the context of real reductive groups. We show that if G(R) is the real points of a connected reductive group, there is an involution, unique up to conjugacy by G(R), taking any semisimple element to a conjugate of its inverse. As applications we give a condition for every irreducible representation of G(R) to be self-dual, and to the Frobenius Schur indicator for such groups

Buffalo Sewer Authority

The Buffalo Sewer Authority is a public benefit corporation created by the New York State legislature in 1935 to clean wastewater before it is released into the environment. The BSA also maintains the storm drains for the City of Buffalo. The BSA serves the residents and businesses of the Buffalo area as well as some neighboring communities. Currently, around 98,000 Buffalo residents and nearly 400 businesses in the City of Buffalo are served by the BSA

Duality for nonlinear simply laced groups

Let G be a nonlinear double cover of the real points of a connected reductive complex algebraic group with simply laced root system. We establish a uniform character multiplicity duality theory for the category of Harish-Chandra modules for G.Comment: 51 pages, 1 figur

Analysis on the minimal representation of O(p,q) -- II. Branching laws

This is a second paper in a series devoted to the minimal unitary representation of O(p,q). By explicit methods from conformal geometry of pseudo-Riemannian manifolds, we find the branching law corresponding to restricting the minimal unitary representation to natural symmetric subgroups. In the case of purely discrete spectrum we obtain the full spectrum and give an explicit Parseval-Plancherel formula, and in the general case we construct an infinite discrete spectrum.Comment: 27 page

Quasisplit Hecke algebras and symmetric spaces

Let (G,K) be a symmetric pair over an algebraically closed field of characteristic different of 2 and let sigma be an automorphism with square 1 of G preserving K. In this paper we consider the set of pairs (O,L) where O is a sigma-stable K-orbit on the flag manifold of G and L is an irreducible K-equivariant local system on O which is "fixed" by sigma. Given two such pairs (O,L), (O',L'), with O' in the closure \bar O of O, the multiplicity space of L' in the a cohomology sheaf of the intersection cohomology of \bar O with coefficients in L (restricted to O') carries an involution induced by sigma and we are interested in computing the dimensions of its +1 and -1 eigenspaces. We show that this computation can be done in terms of a certain module structure over a quasisplit Hecke algebra on a space spanned by the pairs (O,L) as above.Comment: 46 pages. Version 2 reorganizes the explicit calculation of the Hecke module, includes details about computing \bar, and corrects small misprints. Version 3 adds two pages relating this paper to unitary representation theory, corrects misprints, and displays more equations. Version 4 corrects misprints, and adds two cases previously neglected at the end of 7.

Contragredient representations and characterizing the local Langlands correspondence

We consider the question: what is the contragredient in terms of L-homomorphisms? We conjecture that it corresponds to the Chevalley automorphism of the L-group, and prove this in the case of real groups. The proof uses a characterization of the local Langlands correspondence over R. We also consider the related notion of Hermitian dual, in the case of GL(n,R)

Hecke algebras and involutions in Weyl groups

For any two involutions y,w in a Weyl group (y\le w), let P_{y,w} be the polynomial defined in [KL]. In this paper we define a new polynomial P^\sigma_{y,w} whose i-th coefficient is a_i-b_i where the i-th coefficient of P_{y,w} is a_i+b_i (a_i,b_i are natural numbers). These new polynomials are of interest for the theory of unitary representations of complex reductive groups. We present an algorithm for computing these polynomials.Comment: 25 page