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

Strictly small representations and a reduction theorem for the unitary dual

First published in Representation Theory in Vol 5, 2001. Published by the American Mathematical Society.To any irreducible unitary representation X of a real reductive Lie group we associate in a canonical way, a Levi subgroup Gsu and a representation of this subgroup. Assuming a conjecture of the authors on the infinitesimal character of X, we show that X is cohomologically induced from a unitary representation of the subgroup Gsu. This subgroup is in some cases smaller than the subgroup Gu that the authors attached to X in earlier work. In those cases this provides a further reduction to the classification problem

Laplacians on spheres

Spheres can be written as homogeneous spaces $G/H$ for compact Lie groups in a small number of ways. In each case, the decomposition of $L^2(G/H)$ into irreducible representations of $G$ contains interesting information. We recall these decompositions, and see what they can reveal about the analogous problem for noncompact real forms of $G$ and $H$

Mathematicians’ Central Role in Educating the STEM Workforce

In the recent report Engage to Excel,1 President Obama’s Council of Advisors on Science and Technology (PCAST) identifies mathematics as a bottleneck in undergraduate Science, Technology, Engineering, and Mathematics (STEM) education. Among PCAST’s recommendations are ones calling for the development and teaching of college-level mathematics courses “by faculty from mathematics-intensive disciplines other than mathematics” and for “a new pathway for producing K–12 mathematics teachers…in programs in mathematics-intensive fields other than mathematics.”2 While we are in sharp disagreement with these specific recommendations, we do share PCAST’s concern for the state of STEM education. We encourage the mathematics community to focus constructively on the broad view the report sketches. We appeal to the community to amplify its communications with other STEM disciplines, to publicize its teaching innovations, and to redouble its efforts to meet the challenges discussed by PCAST