2001/12/31 by Dmitri I. Panyushev · 1 citation
Mathematics · #math.AG #math.RT
published as J. London Math. Soc. 69, Part 2 (2004), 273-290 · 19 pages, Latex2e; connection with equivariant cohomology is added
arxiv created 2002/04/02 · arxiv updated 2009/11/30
\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 Cλ denote the \frak g-endomorphism algebra associated with a simple \frak g-module Vλ. Most of our results concern the case in which Cλ is commutative, i.e., Vλ is a weight multiplicity free \frak g-module. It is proved that Cλ is a polynomial algebra if and only if λ 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 Cλ for any λ, 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.