\(K\)-invariants in the algebra \(U(\mathfrak{g})\otimes C(\mathfrak{p})\) for the group \(SU(2,1)\) (Q2786995)

Let \(G=\mathrm{SU}(2,1)\) and \(K=S(U(2)\times U(1))\) be a maximal compact subgroup of \(G\). Consider the complexified Lie algebra \(\mathfrak{g}\cong \mathfrak{sl}_3(\mathbb{C})\) of \(G\) and let \(\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}\) be a Cartan decomposition of \(\mathfrak{g}\). Let \(C(\mathfrak{p})\) be the Clifford algebra with respect to the trace form on \(\mathfrak{p}\). The main result in the paper under review asserts that the algebra of \(K\)-invariants in \(U(\mathfrak{g})\otimes C(\mathfrak{p})\) is generated by five explicitly given elements. Two of them are in the center of \(U(\mathfrak{k})\), one is in \(C(\mathfrak{p})^K\), the forth generator is the Dirac operator and the last generator is a certain \(\mathfrak{k}\)-analogue of the Dirac operator. All this is applied to study algebraic Dirac induction for \((\mathfrak{g},K)\)-modules. The classical results on the structure of \(U(\mathfrak{g})^K\) are recovered along he way.