Article

Inventiones mathematicae

, Volume 117, Issue 1, pp 181-205

Discrete decomposability of the restriction ofA q(λ) with respect to reductive subgroups and its applications

  • Toshiyuki KobayashiAffiliated withDepartment of Mathematical Sciences, University of Tokyo

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Summary

LetG′⊂G be real reductive Lie groups and q a θ-stable parabolic subalgebra of Lie (G) ⊗ ℂ. This paper offers a sufficient condition on (G, G′, q) that the irreducible unitary representation\(\mathop {A_q }\limits^--- \) ofG with non-zero continuous cohomology splits into a discrete sum of irreducible unitary representations of a subgroupG′, each of finite multiplicity. As an application to purely analytic problems, new results on discrete series are also obtained for some pseudo-Riemannian (non-symmetric) spherical homogeneous spaces, which fit nicely into this framework. Some explicit examples of a decomposition formula are also found in the cases whereA q is not necessarily a highest weight module.