##### Department of Mathematics,

University of California San Diego

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### Math 256 - Lie Groups and Lie Algebras

## Mark Colarusso

#### University of Wisconsin, Milwaukee

## K-orbits on the flag variety and the Gelfand-Zeitlin integrable system

##### Abstract:

In 2006, Kostant and Wallach constructed an integrable system on the n x n complex matrices $M_{n}( C)$ using Gelfand-Zeitlin theory. This system can be viewed as a complexified version of the one studied by Guillemin and Sternberg on the n x n Hermitian matrices, which is related to the classical Gelfand-Zeitlin basis for irreducible representations of the unitary group via geometric quantization. In this talk, we discuss joint work with Sam Evens in which we develop a geometric description of the fibres of the moment map for the complexified Gelfand-Zeitlin system. Our approach uses the theory of orbits of a symmetric subgroup K of the group G of all invertible n x n complex matrices on the flag variety of $M_{n}( C)$ . These orbits play a central role in the geometric construction of Harish-Chandra modules for the pair $(M_{n} (C ), K)$ using the Beilinson-Bernstein correspondence. We indicate how our work provides the foundation for the geometric construction of a category of generalized Harish-Chandra modules studied by Drozd, Futorny, and Ovsienko.

Host: Nolan Wallach

### July 2, 2014

### 1:00 PM

### AP&M 7218

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