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

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

Lie algebra, K-invariants

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