2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72279A commutative Poisson subalgebra of the Poisson algebra of polynomials on the Lie algebra of n x n matrices over ${\Bbb C}$ 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 ${\Bbb C}^{n(n -1)/2}$. This paper proves several remarkable properties of this group action and relates it to the theory of orthogonal polynomials.plain tex, 54 pagesSymplectic GeometryGroup Theory14L30, 14R20, 33C45, 53D17text