##### Department of Mathematics,

University of California San Diego

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### Representation Theory

## Peter Trapa

#### University of Utah

## Unipotent representations and the theta correspondence.

##### Abstract:

Fix a real reductive group $G$. Suppose $mathcal{O}'$ is a nilpotent orbit in $mathfrak{g}'$, the dual of the complexified Lie algebra of $G$. To each $X' in mathcal{O}'$, one may associate an sl(2) triple, say $X', Y'$, and $H'$. Since $(1/2)H'$ lives in a Cartan subalgebra of $mathfrak{g}'$, it defines an infinitesimal character for $G$. One piece of the Arthur conjectures predicts that the smallest representations of $G$ with infinitesimal character $(1/2)H'$ should appear as local components of automorphic forms; in particular, they should be unitary. (Two good examples to keep in mind are the trivial representation and limits of discrete series with zero infinitesimal character; the former corresponds to the principal orbit $mathcal{O'}$ and the latter to the zero orbit.) In this talk, we explain how to prove a large part of this conjecture for certain classical groups using the theta correspondence.

Host: Wee Teck Gan

### February 24, 2004

### 1:30 PM

### AP&M 7321

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