A Description of Discrete Series for Semisimple Symmetric Spaces II

Abstract

IntroductionIn [F], Flensted-Jensen constructed countably many discrete series for a semisimple symmetric space G/ H whenConversely, [OMI] proved that (1.1) holds if there exist discrete series for G/H.Moreover [OMI] constructed Harish-Chandra modules Bi which parametrize all the discrete series for G/H, where j runs through finite indices and J. runs through lattice points contained in a positive Wey! chamber.In this paper, we give a necessary condition for j and J. so that the module Bj is nontrivial.In the subsequent paper [OM2], we will prove that the condition also assures Bf =i=={O}.We remark that our results also covers "limits of discrete series" for G / H.In the appendix, we give a certain simplification of the proof of a main result in [OMI].To state the precise result in this paper, we prepare some notations.Let B be a semisimple Lie algebra and a an involution (automorphism of order 2) of B• Fix a Cartan involution e of B such that a0=0a.Let B=9+q (resp.B=f+P) be the decomposition of B into the +I and -1 eigenspaces for a (resp.0).Let Be denote the complexification of g and put fd=f n fj+J=T (P n fj),Ba= fa+Pa= 9 a+ qa.pa =-l=I (t n q)+ p n q, qa=-l=I (P n fj)+P n q, Let Ge be a connected complex Lie group with Lie algebra Be, and let G, K, H, ca, Ka, Ha, He and Kc be the analytic subgroups of Ge corresponding to B, f, fj, B\ fd, fja, 9c and fc, respectively.In [OMI], we studied the discrete series for G/H and proved that

Locations

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