K-invariants in the algebra U(g) $\otimes$ C(p) for the group SU(2,1)

Abstract

Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the Cartan decomposition of the complexified Lie algebra $\mathfrak{g}=\mathfrak{sl}(3,\mathbb{C})$ of the group $G=SU(2,1)$. Let $K=S(U(2) \times U(1))$; so $K$ is a maximal compact subgroup of $G$. Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$, and let $C(\mathfrak{p})$ be the Clifford algebra with respect to the trace form $B(X,Y)=\text{tr}(XY)$ on $\mathfrak{p}$. We are going to prove that the algebra of K-invariants in $U(\mathfrak{g}) \otimes C(\mathfrak{p})$ is generated by five explicitly given elements. This is useful for studying algebraic Dirac induction for $(\mathfrak{g},K)$-modules. Along the way we will also recover the (well known) structure of the algebra $U(\mathfrak{g})^K$.