##### Department of Mathematics,

University of California San Diego

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### Final Defense

## Mark Colarusso

#### UCSD, Graduate Student

## The Gelfand-Zeitlin algebra and polarizations of generic adjoint orbits for classical groups

##### Abstract:

We will discuss our current research which describes and constructs polarizations of regular adjoint orbits for certain classical groups. This research generalizes recent work of Bertram Kostant and Nolan Wallach. Kostant and Wallach construct polarizations of regular adjoint orbits in the space of $n\times n$ complex matrices $M(n)$. They accomplish this by defining an $\frac{n(n-1)}{2}$ dimensional abelian complex Lie group $A$ that acts on $M(n)$ and stabilizes adjoint orbits. Note that the dimension of this group is exactly half the dimension of a regular adjoint orbit in $M(n)$. This fact allows $A$ orbits of dimension $\frac{n(n-1)}{2}$ contained in a given regular adjoint orbit to form the leaves of a polarization of an open submanifold of that orbit. We study the $A$ orbit structure on $M(n)$ and generalize the construction to complex orthogonal Lie algebras $\mathfrak{so}(n)$. In the case of $M(n)$, we obtain complete descriptions of $A$ orbits of dimension $\frac{n(n-1)}{2}$ and thus of leaves of polarizations of all regular adjoint orbits. For $\mathfrak{so}(n)$, we construct polarizations of certain regular semi-simple adjoint orbits.

Advisor: Nolan Wallach

### May 30, 2007

### 3:00 PM

### AP&M 5829

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