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.


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