##### Department of Mathematics,

University of California San Diego

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### University of California Lie Theory Workshop

## Edward Frenkel

#### University of California, Berkeley

## Opers with irregular singularity and spectra of the shift of argument subalgebra

##### Abstract:

The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras. Recently B. Feigin, L. Rybnikov and myself have proved that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with differential geometric objects on the projective line called "opers". They have regular singularity at one point, irregular singularity at another point and are monodromy free. Interestingly, they are associated not to G, but to the Langlands dual group of G. In addition, we have shown that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of g has a cyclic vector in any irreducible finite-dimensional g-module. As a byproduct, we obtain the structure of a Gorenstein ring on any such module. I will talk about these results and explain the connection to the geometric Langlands correspondence.

Host: Efim Zelmanov

### February 16, 2008

### 9:00 AM

### NSB 1205

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