Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of $G$.

Let $C$ be the Casimir of $\mathfrak{g}$. It is in the center of the universal enveloping algebra $U(\mathfrak{g})$. Hence, it acts as a scalar $C_\pi\in \mathbf{C}$ on $\pi\in \widehat{G}_u$.

Is it true that for a given $c\in \mathbf{C}$, there exits only finite $\pi\in \widehat{G}_u$ such that $C_\pi=c$?

complexifiedLie algebra and its universal enveloping algebra. For a Casimir-type operator to act by a scalar may require an algebraically closed field. $\endgroup$ – Jim Humphreys Feb 28 '15 at 20:44