  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 $Z(\mathfrak{g})$.
