Department of Mathematics
Colloquium
by
Prof. Victor Ginzburg

Some new unexpected relations between Hilbert schemes and Lie theory will be discussed. These relations are similar to the classically known relations between the combinatorics of symmetric functions, the geometry of conjugacy classes of nxn-matrices, and representation theory of the general linear group. We 'double' the standard setup by replacing polynomials in n variables invariant under the action of the symmetric group S_n by polynomials in two n-tuples of variables invariant under the simultaneous S_n-action. This corresponds in geometry to replacing the set of conjugacy classes of matrices by the punctual Hilbert scheme of C^2.

We propose an interpretation and a generalisation of this in the framework of general complex reductive groups. Our goal is to extend to the 'doubled' setup some of the fundamental results of Harish-Chandra and Kostant about regular functions and invariant eigen-distributions on a semisimple Lie algebra.

Additional Reference: H. Nakajima "Lectures on Hilbert schemes" homepage at nakajima@kusm.kyoto-u.ac.jp.