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The goal of this chapter is to study the branching problem for a holomorphic discrete series representation of the conformal group of a simple Euclidean Jordan algebra V  restricted to the subgroup \( \operatorname {\mathrm {PSL}}_2(\mathbb {R})\times \operatorname {Aut}(V)\) where \(\operatorname {Aut}(V)\) denotes the compact group of automorphisms of V . We use a realization of the holomorphic discrete series on a space of vector-values \(L^2\) -functions as well as the stratified model developed by the second author to relate the branching problem to the decomposition of certain representations of the compact group \(\operatorname {Aut}(V)\) and to vector-valued orthogonal polynomials.

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Restricting Holomorphic Discrete Series Representations to a Compact Dual Pair

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  • 发表日期：

    2025年

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  • Jan Frahm,
  • Quentin Labriet

摘要

The goal of this chapter is to study the branching problem for a holomorphic discrete series representation of the conformal group of a simple Euclidean Jordan algebra V  restricted to the subgroup \( \operatorname {\mathrm {PSL}}_2(\mathbb {R})\times \operatorname {Aut}(V)\) where \(\operatorname {Aut}(V)\) denotes the compact group of automorphisms of V . We use a realization of the holomorphic discrete series on a space of vector-values \(L^2\) -functions as well as the stratified model developed by the second author to relate the branching problem to the decomposition of certain representations of the compact group \(\operatorname {Aut}(V)\) and to vector-valued orthogonal polynomials.