ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: 17, 22 Barberis, María Laura, FaMAF - Universidad Nacional de Córdoba, Argentina Quaternionic representations Let G be a simply connected, simple Lie group with maximal compact subgroup K such that the symmetric space G/K has a G-invariant quaternionic structure, that is, K contains a normal subgroup S isomorphic to SU(2) which acts on 𝔭_{0} (the tangent space of G/K at eK) by left quaternionic multiplication. In this case K decomposes as a product K = S × M where M is a compact, connected subgroup of G. Let U(1) ⊂ S and set L = U(1) × M ⊂ K. Consider the generalized Verma modules N(λ) = U(𝔤) ⊗_{U(𝔮)} F (λ) where F (λ) is an arbitrary irreducible finite dimensional representation of L = U(1) × M, 𝔤 is the complexification of the Lie algebra of G and 𝔮 is a canonically defined parabolic subalgebra of 𝔤. Let L(λ) denote the unique irreducible quotient of N(λ). By applying the Zuckerman functor (Γ_{L}^{K})^{1} to L(λ) we obtain a nonzero (𝔤, K)-module Γ^{1}L(λ) which admits a nondegenerate (𝔤, K)-invariant Hermitian form. We study the parameters λ for which the corresponding Hermitian form is positive definite and obtain some generalizations of the results in [G-W]. Reference: [G-W] B. H. Gross and N. R. Wallach, On quaternionic discrete series representations and their continuations, J. reine angew. Math. 481 (1996), 73–123.