##### Department of Mathematics,

University of California San Diego

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### Representation Theory

## Bertram Kostant

#### Massachusetts Institute of Technology

## Cent($U(n)$), the cascade of orthogonal roots and the generalized exponents.

##### Abstract:

Let $\frak{b}$ be a Borel subalgebra of a complex semisimple Lie algebra $\frak{g}$. Let $\frak{h}\subset\frak{b}$ \ be a Cartan subalgebra and let $\frak{n}$ be the nilradical of $\frak{b}$. Let $\Delta_{+}\subset\frak{h}^{\ast}$ be the set of positive roots corresponding to $\frak{b}$. Then there is a distinguished maximal set $B\subset\Delta_{+}$ of strongly orthogonal roots called the cascade of orthogonal roots. The center, Cent($U(\frak{n})$), of the enveloping algebra of $\frak{n}$ is a module for $H=\exp\frak{h}$. \noindent{\bf Theorem.} Cent($U(\frak{n})$) {\it is a polynomial ring in }$m$ {\it generators} $u_{1},...,u_{m}$ {\it where} $m=$ card$B $. {\it Furthermore, all} $H$-{\it weights in} Cent($U(\frak{n})$) {\it are of multiplicity} $1$ {\it and the} $u_{i}${\it can be chosen so that the weight vectors are all the monomials} $u_{1}^{k_{1}}\cdots u_{m}^{k_{m}}$. {\it The} $u_{i}$ {\it are characterized up to scalar multiple as the weight vectors which are also irreducible polynomials}. We also have \noindent{\bf Theorem.} {\it The set of weights in} Theorem 1 {\it is exactly the set} $D_{cas}$ {\it of elements in the semigroup generated by the linearly independent set} $B$ {\it which are also dominant.} The construction of the weight vectors can be given in terms of matrix units for $U(\frak{g})$. Applications of the results are given to the determination if minimal generalized exponents and the proof that a Borel subgroup of $G$ has open coadjoint orbits when $m=rank{\frak g}$.

Host: Nolan Wallach

### March 14, 2006

### 1:30 PM

### AP&M 7218

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