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Branching problems and \({\mathfrak{sl}}(2,\mathbb{C})\)-actions. (English) Zbl 1363.22006
Let \((\mathfrak g^\prime, \mathfrak p^\prime), (\mathfrak g, \mathfrak p)\) be a compatible pair of semisimple Lie algebras and parabolic subalgebras with \(\mathfrak g^\prime \subset \mathfrak g, \mathfrak p^\prime \subset \mathfrak p.\) Under certain hypotheses each scalar Verma module induced from \(\mathfrak p\) is an algebraic direct sum of Verma modules for \(\mathfrak g^\prime\) induced from \(\mathfrak p^\prime. \) The authors construct an explicit \(sl(2,\mathbb C)\)-module structure on the \(\mathfrak g^\prime-\)singular vectors (the generators of the \(\mathfrak g^\prime-\)modules). They achieve the construction for the pairs \(\mathfrak g=\mathfrak{so}(n + 1,1,\mathbb R), \mathfrak p = (\mathfrak{so}(n,\mathbb R) \times \mathbb R) \ltimes \mathbb R^n ,\mathfrak g^\prime = \mathfrak {so}(n,1,\mathbb R), \mathfrak p^\prime = (\mathfrak {so}(n-1,\mathbb R) \times \mathbb R) \ltimes \mathbb R^{n-1}\); \(\mathfrak g =\mathfrak{sl}(2,\mathbb R) \times \mathfrak{sl}(2,\mathbb R), \mathfrak p = (\mathbb R \times \mathbb R) \ltimes (\mathbb R \times \mathbb R), \mathfrak g^\prime = \text{diag}(\mathfrak{sl}(2,\mathbb R)), \mathfrak p^\prime = \text{diag}((\mathbb R \times \mathbb R) \ltimes (\mathbb R \times \mathbb R))\). The answer is written by using Gegenbauer and Jacobi polynomials. Finally the authors relate the construction of singular vectors to Dirac cohomology.

MSC:
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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