Descriptions of strongly multiplicity free representations for simple Lie algebras

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra and $Z(\mathfrak{g})$ be the center of the universal enveloping algebra $U(\mathfrak{g})$. Denote by $V_λ$ the finite-dimensional irreducible $\mathfrak{g}$-module with highest weight $λ$. Lehrer and Zhang defined the notion of strongly multiplicity free representations for simple Lie algebras motivated by studying the structure of the endomorphism algebra $\text{End}_{U(\mathfrak{g})}(V_λ^{\otimes r})$ in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the $\mathfrak{g}$-invariant endomorphism algebras $R_λ(\mathfrak{g})= (\text{End} V_λ\otimes U(\mathfrak{g}))^\mathfrak{g}$ and $R_{λ,π}(\mathfrak{g})=(\text{End} V_λ\otimes π(U(\mathfrak{g})))^\mathfrak{g}.$ In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs $(\mathfrak{g}, V_λ)$, which are multiplicity free and for such pairs, $R_λ(\mathfrak{g})$ and $R_{λ,π}(\mathfrak{g})$ are generated by generalizations of the quadratic Casimir elements of $Z(\mathfrak{g})$.

Theorems & Definitions (56)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4
• Proposition 2.5
• proof
• Proposition 2.6
• Lemma 2.7
• Lemma 2.8
• Theorem 3.3