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Restrictions of generalized Verma modules to symmetric pairs. (English) Zbl 1257.22014
The branching problem for generalized Verma modules with respect to reductive symmetric pairs $$(g,g')$$ is studied. This problem in representation theory means to determine how irreducible modules decompose when restricted to subalgebras. A necessary and sufficient condition on the triple $$(g,g',p)$$ such that the restriction $$X/{}_g{}_{\prime}$$ always contains $$g'$$-modules for any $$g$$-module $$X$$ in a parabolic Bernstein-Gelfand-Gelfand category $$O^p$$ is given. The results are obtained for the Gelfand-Kirillov dimension of any simple module occurring in a simple generalized Verma module. The cases of parabolic subalgebras with $$p$$ or Heisenberg nilpotent radicals are presented as illustrative examples. A complete classification of the triples $$(g,p,g^\tau)$$, with $$\tau$$ an involutive automorphism of the Lie algebra $$g$$, is given. It is shown that the restriction $$X/{}_g{}_{\prime}$$ is generically multiplicity free for any $$p$$ and $$X\in$$ $$O^p$$ if and only if the pair $$(g,g')$$ is isomorphic to $$(A_n,A{}_n{}_-{}_1)$$, $$(B_n,D_n)$$ or $$(D_n{}_+{}_1,B{}_n)$$. Explicit branching laws for such pairs are also presented.

##### MSC:
 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 22F30 Homogeneous spaces 53C35 Differential geometry of symmetric spaces
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