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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.

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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