##### Department of Mathematics,

University of California San Diego

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### Math 256 - Lie Groups

## Gail Letzer

#### UCSD Visitor/Virginia Tech

## Quantum zonal spherical functions and Macdonald polynomials

##### Abstract:

Given a semisimple Lie algebra and one of its involutions, it is possible to construct a coideal subalgebra B in the Êquantized enveloping algebra U which is a quantum analog of the classical enveloping algebra of the fixed Lie subalgebra. We study the space of B bi-invariants inside the associated quantized function algebra. Under the obvious restriction map, the space of bi-invariants is proved isomorphic to the Weyl group invariants of the character group ring associated to the restricted roots. The quantum Peter-Weyl decomposition and the classification of finite-dimensional spherical modules associated to U,B implies that this space of bi-invariants is a direct sum of one-dimensional eigenspaces for the action of the center of U. When the restricted root system is reduced, we show that the zonal spherical functions, i.e. representations of each eigenspace, correspond to Macdonald polynomials under a standard

Host: Nolan Wallach

### November 19, 2002

### 2:00 PM

### AP&M 7321

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