##### Department of Mathematics,

University of California San Diego

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### Symplectic Geometry Seminar

## Gerald Schwarz

#### Brandeis University

## Symplectic Quotients and Orbifolds

##### Abstract:

Let $K$ be a compact Lie group and $V$ a unitary $K$-module. Let $\mu\colon V\to \frak k^*$ be the associated moment mapping and let $M_0$ denote the quotient of $\mu^{-1}(0)$ by $K$. This is the (symplectic) quotient associated to the $K$-action. Now $K$ is a real algebraic subgroup of the unitary group of $V$ and its complex points are a complex reductive subgroup $G$ of $\rm{GL}(V)$. We recall the invariant theory quotient $V{/\!\!/} G$ associated to the $G$-action, and the fact that $V{/\!\!/} G$ is homeomorphic to $M_0$. This fact is enormously useful. The simplest kinds of symplectic quotients are those of the form $W/H$ where $W$ is a unitary $H$-module and $H$ is finite. Let $\dim K>0$. For $K$-modules $V$ which are ``small,'' there are examples of isomorphisms of $M_0$ with some $W/H$. We show that for most $K$-modules, there can be no such isomorphism. We give necessary and sufficient conditions for such isomorphisms for $K=S^1$ and $K=\rm{SU}(2,\Bbb C)$. This is joint work with H.-C. Herbig and C. Seaton.

Host: Alvaro Pelayo

### April 8, 2016

### 4:00 PM

### AP&M 6402

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