Weight multiplicity free representations, $\frak g$-endomorphism algebras, and Dynkin polynomials

Abstract

$\frak g$-endomorphism algebras form an interesting class of associative algebras related to the adjoint representation of a semisimple Lie algebra $\frak g$. These algebras were recently introduced by A.Kirillov, who used the term `family algebras'. Let $C_\lambda$ denote the $\frak g$-endomorphism algebra associated with a simple $\frak g$-module $V_\lambda$. Most of our results concern the case in which $C_\lambda$ is commutative, i.e., $V_\lambda$ is a weight multiplicity free $\frak g$-module. It is proved that $C_\lambda$ is a polynomial algebra if and only if $\lambda$ is minuscule. We also characterise in general the number of the irreducible components of the corresponding affine variety. The main result is that the commutative $\frak g$-endomorphism algebra is always Gorenstein. We explicitly compute the Poincare series of $C_\lambda$ for any $\lambda$, and show that in the commutative case the numerator coincides with the polynomial that was introduced by E.B.Dynkin in 1950. We also discuss a connection between commutative $\frak g$-endomorphism algebras and equivariant cohomology.