Finite multiplicity theorems for induced representations of semisimple Lie groups and their applications to generalized Gelfand-Graev representations (Q1097978)

The author announces results whose proofs are to appear elsewhere. They concern finite multiplicity results for induced representations in the following setting. Let G be a real semisimple Lie group with Cartan involution \(\theta\), let \(Q=LN\) be a parabolic subgroup with \(\theta\)- stable Levi component L, suppose that L carries an involution \(\sigma\) which commutes with \(\theta| L\), and let H be an open subgroup of its group of fixed points; thus L/H is a reductive symmetric space. One of the results announced is: if \(\zeta\) is a representation of HN with finite multiplicities, then \(C^{\infty}IND^ G_{HN}(\zeta)\) also has the finite multiplicity property. In the second half of the paper these results are applied to generalized Gelfand-Graev representations associated with irreducible Hermitian symmetric spaces.