Glasnik Matematicki, Vol. 50, No. 2 (2015), 397-414.

K-INVARIANTS IN THE ALGEBRA U(𝔤) ⊗ C(𝔭) FOR THE GROUP SU(2,1)

Ana Prlić

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: anaprlic@math.hr

Abstract.   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.

2010 Mathematics Subject Classification.   22E47, 22E46.

Key words and phrases.   Lie group, Lie algebra, representation, special unipotent representation, Dirac operator, Dirac cohomology.

Full text (PDF) (free access)

DOI: 10.3336/gm.50.2.09

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