The \((\mathfrak g,K)\)-module structures of principal series of \(\text{SU}(2,2)\). (English) Zbl 1231.22014

Principal series representations are building blocks in representation theory of real reductive groups. Moreover, Harish-Chandra proved that the study of irreducible unitary representations of such groups can be deduced from the study of irreducible \((g,K)\)-modules. In the paper under review, the author describes explicitly the \((g,K)\)-modules of principal series representations, induced from maximal parabolic subgroups, of the group \(G=SU(2,2)\). Here \(g\) is the complexification of the Lie algebra of \(SU(2,2)\) and \(K=S(U(2)\times U(2))\) is a maximal compact subgroup of \(G\). The strategy is to write in terms of elementary functions the marked basis for each \(K\)-isotopic component of the principal series representations, compute the Clebsch-Gordan coefficients of finite dimensional representations of \(K=S(U(2)\times U(2))\) and deduce the explicit form of some matrix of intertwining constants. The main reason for the author to have such explicit formulas is motivated by the arithmetic of automorphic forms, e.g. the Whittaker models that he had studied before. It should be noted that the Hermitian group \(SU(2,2)\) plays an important role in physics as it is isomorphic to the conformal group \(Spin(4,2)\) via twistors. Therefore the formulas computed by the author will certainly be of some interest to physicists.
Reviewer: Salah Mehdi (Metz)


22E46 Semisimple Lie groups and their representations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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