CU Algebraic Lie Theory Seminar

CU Algebraic Lie Theory Seminar
Germs of matrix coefficients and their geometric interpretation Mark Reeder, Boston College
April 9, 2013 Math 350 2-3pm	Abstract The character ring R(G) of a complex Lie group G has many interesting bases, each of whose elements is parametrized by dominant weights. The Weyl characters and the Macdonald-Hall polynomials are two examples of such bases. Each basis is given by a certain function F(t,d), defined on T x D, where T is a maximal torus of G and D is the lattice-cone of dominant weights of T. Normally one fixes a weight d in D and studies the analytic function t --> F(t,d). For example if F is the Weyl character then F(t,d) is the trace of t on the representation of G with highest weight d. It is also interesting to fix t and study the asymptotic behavior of the function d --> F(t,d). Such functions arise as matrix coefficients of representations of the p-adic group G* dual to G. Now the functions blow up, according to the singularity of t, and the blowing up is measured, at least in the two examples above, by the homology classes of certain subvarieties of the flag variety G/B of G. I will show this explicitly in the model of the homology of G/B given by harmonic polynomials on the Lie algebra of T, and will discuss implications for the representation theory of G*.

Germs of matrix coefficients and their geometric interpretation

Mark Reeder, Boston College

April 9, 2013

Math 350

2-3pm

Abstract

The character ring R(G) of a complex Lie group G has many interesting bases, each of whose elements is parametrized by dominant weights. The Weyl characters and the Macdonald-Hall polynomials are two examples of such bases.

Each basis is given by a certain function F(t,d), defined on T x D, where T is a maximal torus of G and D is the lattice-cone of dominant weights of T. Normally one fixes a weight d in D and studies the analytic function t --> F(t,d). For example if F is the Weyl character then F(t,d) is the trace of t on the representation of G with highest weight d.

It is also interesting to fix t and study the asymptotic behavior of the function d --> F(t,d). Such functions arise as matrix coefficients of representations of the p-adic group G* dual to G. Now the functions blow up, according to the singularity of t, and the blowing up is measured, at least in the two examples above, by the homology classes of certain subvarieties of the flag variety G/B of G.

I will show this explicitly in the model of the homology of G/B given by harmonic polynomials on the Lie algebra of T, and will discuss implications for the representation theory of G*.