Descriptions of strongly multiplicity free representations for simple Lie algebras

Let g be a complex simple Lie algebra and Z(g) be the center of the universal enveloping algebra U(g). Denote by V lambda the finite -dimensional irreducible g -module with highest weight lambda. Lehrer and Zhang defined the notion of strongly multiplicity free representations for simple Lie algebras motivated by studying the structure of the endomorphism algebra EndU(g)(V lambda circle times r) in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the g -invariant endomorphism algebras R lambda (g) = (End V lambda circle times U(g))g and R lambda,pi(g) = (End V lambda circle times pi(U(g)))g. In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs (g, V lambda), which are multiplicity free and for such pairs, R lambda (g) and R lambda,pi(g) are generated by generalizations of the quadratic Casimir elements of Z(g). (c) 2024 Elsevier Inc. All rights reserved.