Gelfand-Zeitlin theory from the perspective of classical mechanics. I

Abstract

A commutative Poisson subalgebra of the Poisson algebra of polynomials on the Lie algebra of n x n matrices over $\mathbb{C}$ is introduced which is the Poisson analogue of the Gelfand-Zeitlin subalgebra of the universal enveloping algebra. As a commutative algebra it is a polynomial ring in $n(n+1)/2$ generators, $n$ of which can be taken to be basic generators of the polynomial invariants. Any choice of the next $n(n-1)/2$ generators yields a Lie algebra of vector fields that generates a global holomorphic action of the additive group ${\mathbb{C}}^{n(n-1)/2}$. This paper proves several remarkable properties of this group action and relates it to the theory of orthogonal polynomials.

plain tex, 54 pages