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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:
 2.2e+48 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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