Descriptions of strongly multiplicity free representations for simple Lie algebras

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 $V_\lambda$ the finite-dimensional irreducible $\mathfrak{g}$-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 $\text{End}_{U(\mathfrak{g})}(V_{\lambda}^{\otimes r})$ in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the $\mathfrak{g}$-invariant endomorphism algebras $R_\lambda(\mathfrak{g})= (\text{End} V_\lambda\otimes U(\mathfrak{g}))^\mathfrak{g}$ and $R_{\lambda,\pi}(\mathfrak{g})=(\text{End} V_\lambda\otimes \pi (U(\mathfrak{g})))^\mathfrak{g}.$ In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs $(\mathfrak{g}, V_\lambda)$, which are multiplicity free and for such pairs, $R_\lambda(\mathfrak{g})$ and $R_{\lambda,\pi}(\mathfrak{g})$ are generated by generalizations of the quadratic Casimir elements of $Z(\mathfrak{g})$.