Hilbert modules associated to parabolically induced representations of semisimple Lie groups

Details Hilbert modules associated to parabolically induced representations of semisimple Lie groups Hilbert modules associated to parabolically induced representations of semisimple Lie groups

2009-12-21

arxiv.org/abs/0912.4190v2

Mathematics

Operator Algebras

21 pages, to appear in J. Operator Theory

Scientific paper

Given a measured space X with commuting actions of two groups G and H satisfying certain conditions, we construct a Hilbert C*(H)-module E(X) equipped with a left action of C*(G), which generalises Rieffel's construction of inducing modules. Considering G to be a semisimple Lie group and H to be the Levi component L of a parabolic subgroup P=LN, the Hilbert module associated to X=G/N encodes the P-series representations of G coming from parabolic subgroups associated to P. We provide several descriptions of this Hilbert module, corresponding to the classical pictures of P-series. We then characterise the bounded operators on E(G/N) that commute to the left action of C*(G) as central multipliers of C*(L) and interpret this result as a globalised generic irreducibility theorem. Finally, we establish the convergence of intertwining integrals on a dense subset of E(G/N).

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