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Construction of Irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-Modules and Discretely Decomposable Restrictions

Masatoshi Kitagawa

Abstract

In this paper, we study the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-modules on the spaces of intertwining operators in the branching problem of reductive Lie algebras, and construct a family of finite-dimensional irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-modules using the Zuckerman derived functors. We provide criteria for the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-modules in the cases of generalized Verma modules, cohomologically induced modules, and discrete series representations. We treat only discrete decomposable restrictions with certain dominance conditions (quasi-abelian and in the good range). To describe the $\mathcal{U}(\mathfrak{g})^{G'}$-modules, we give branching laws of cohomologically induced modules using ones of generalized Verma modules when $K'$ acts on $K/L_K$ transitively.

Construction of Irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-Modules and Discretely Decomposable Restrictions

Abstract

In this paper, we study the irreducibility of -modules on the spaces of intertwining operators in the branching problem of reductive Lie algebras, and construct a family of finite-dimensional irreducible -modules using the Zuckerman derived functors. We provide criteria for the irreducibility of -modules in the cases of generalized Verma modules, cohomologically induced modules, and discrete series representations. We treat only discrete decomposable restrictions with certain dominance conditions (quasi-abelian and in the good range). To describe the -modules, we give branching laws of cohomologically induced modules using ones of generalized Verma modules when acts on transitively.

Paper Structure

This paper contains 22 sections, 52 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. $\boldsymbol{(\mathfrak{g}, K)}$-module
  4. Translation functor
  5. Discrete decomposability
  6. Generalized Verma module
  7. Zuckerman derived functor
  8. Cohomological induction
  9. Vanishing theorem
  10. Cyclic subspace
  11. Embedding of categories
  12. Setting and main theorem
  13. Proof of Theorem \ref{['thm:Exactness']}
  14. Equivalence of categories in special case
  15. Embedding 2
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $F$ be a finite-dimensional irreducible $\mathfrak{l}$-module with infinitesimal character $\lambda \in \mathfrak{t}^*$ in the good range, i.e., Then

Theorems & Definitions (77)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.6
  • Definition 2.8
  • ...and 67 more