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Lower semicontinuity of bounded property in the branching problem and sphericity of flag variety

Masatoshi Kitagawa

TL;DR

This work connects bounded multiplicities in branching problems for reductive Lie groups to the sphericity of partial flag varieties. By exploiting the lower semicontinuity of the PI-degree on the primitive spectrum and establishing a tight link between bounded multiplicities and PI-deg finiteness, the authors derive criteria ensuring $G/P$ is $G'$-spherical whenever cohomologically induced modules have uniformly bounded restrictions. The results extend to almost irreducible restrictions and transitivity, showing parallel criteria between sphericity and bounded lengths of restricted modules. Overall, the paper provides a ring-theoretic and geometric framework to detect spherical actions via invariants of universal enveloping algebras and their primitive ideals, with implications for degenerate principal series and visible/ multiplicity-free phenomena.

Abstract

Vinberg--Kimel'fel'd [Funct. Anal. Appl., 1978] established that a quasi-projective normal $G$-variety $X$ is spherical if and only if $G$-modules on the spaces $Γ(X, \mathcal{L})$ of global sections of $G$-equivariant line bundles are multiplicity-free. This result was generalized by Kobayashi--Oshima [Adv. Math., 2013] and several researchers to (degenerate) principal series representations of reductive Lie groups. The purpose of this short article is to show that the boundedness of the multiplicities in the restrictions of cohomologically induced modules implies the sphericity of some partial flag variety. In our previous paper, we reduce the boundedness of the multiplicities to the finiteness of a ring-theoretic invariant $\mathrm{PIdeg}$. To show the main result, we discuss the lower semicontinuity of $\mathrm{PIdeg}$ on the space $\mathrm{Prim}(\mathcal{U}(\mathfrak{g}))$ of primitive ideals. We also treat the finiteness of the lengths of the restrictions of cohomologically induced modules.

Lower semicontinuity of bounded property in the branching problem and sphericity of flag variety

TL;DR

This work connects bounded multiplicities in branching problems for reductive Lie groups to the sphericity of partial flag varieties. By exploiting the lower semicontinuity of the PI-degree on the primitive spectrum and establishing a tight link between bounded multiplicities and PI-deg finiteness, the authors derive criteria ensuring is -spherical whenever cohomologically induced modules have uniformly bounded restrictions. The results extend to almost irreducible restrictions and transitivity, showing parallel criteria between sphericity and bounded lengths of restricted modules. Overall, the paper provides a ring-theoretic and geometric framework to detect spherical actions via invariants of universal enveloping algebras and their primitive ideals, with implications for degenerate principal series and visible/ multiplicity-free phenomena.

Abstract

Vinberg--Kimel'fel'd [Funct. Anal. Appl., 1978] established that a quasi-projective normal -variety is spherical if and only if -modules on the spaces of global sections of -equivariant line bundles are multiplicity-free. This result was generalized by Kobayashi--Oshima [Adv. Math., 2013] and several researchers to (degenerate) principal series representations of reductive Lie groups. The purpose of this short article is to show that the boundedness of the multiplicities in the restrictions of cohomologically induced modules implies the sphericity of some partial flag variety. In our previous paper, we reduce the boundedness of the multiplicities to the finiteness of a ring-theoretic invariant . To show the main result, we discuss the lower semicontinuity of on the space of primitive ideals. We also treat the finiteness of the lengths of the restrictions of cohomologically induced modules.

Paper Structure

This paper contains 12 sections, 29 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Primitive spectrum
  3. Family of irreducible highest weight modules
  4. Primitive spectrum
  5. Uniformly bounded multiplicities
  6. PI degree
  7. Uniformly bounded multiplicities
  8. Lower semicontinuity of PIdeg
  9. Spherical v.s. uniformly bounded multiplicities
  10. Almost irreducible restriction
  11. Almost irreducibility
  12. Transitivity v.s. almost irreducibility

Key Result

Theorem 1.3

The map $\mathop{\mathrm{Prim}}\nolimits(\mathcal{U}(\mathfrak{g})) \ni I \mapsto \mathrm{PI.deg}((\mathcal{U}(\mathfrak{g})/I)^{G'}) \in \mathbb{N} \cup \left\{\infty \right\}$ is lower semicontinuous, where $\mathop{\mathrm{Prim}}\nolimits(\mathcal{U}(\mathfrak{g}))$ is the set of primitive ideals

Theorems & Definitions (54)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • ...and 44 more