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

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

2001-12-31

arxiv.org/abs/math/0112314v2

J. London Math. Soc. 69, Part 2 (2004), 273-290

Mathematics

Algebraic Geometry

19 pages, Latex2e; connection with equivariant cohomology is added

Scientific paper

$\frak g$-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.

Affiliated with

Also associated with

No associations

Rating

Weight multiplicity free representations, $\frak g$-endomorphism algebras, and Dynkin polynomials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Weight multiplicity free representations, $\frak g$-endomorphism algebras, and Dynkin polynomials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Weight multiplicity free representations, $\frak g$-endomorphism algebras, and Dynkin polynomials will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-457692