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How does the restriction of representations change under translations? A story for the general linear groups and the unitary groups

Toshiyuki Kobayashi, Birgit Speh

Abstract

We present a new approach to symmetry breaking for pairs of real forms of $(GL(n, \mathbb{C}), GL(n-1, \mathbb{C}))$. Translation functors are powerful tools for studying families of representations of a single reductive group $G$. However, when applied to a pair of groups $G \supset G'$, they can significantly alter the nature of symmetry breaking between the representations of $G$ and $G'$, even within the same Weyl chamber of the direct product group $G \times G'$. We introduce the concept of "fences for the interleaving pattern", which provides a refinement of the usual notion of walls of Weyl chambers. We then establish a theorem stating that the multiplicity remains constant unless these "fences" are crossed, together with a new general vanishing theorem for symmetry breaking. These general results are illustrated with examples involving both tempered and non-tempered representations. In addition, we present a new non-vanishing theorem for period integrals for pairs of reductive symmetric spaces, which is further strengthened by this approach.

How does the restriction of representations change under translations? A story for the general linear groups and the unitary groups

Abstract

We present a new approach to symmetry breaking for pairs of real forms of . Translation functors are powerful tools for studying families of representations of a single reductive group . However, when applied to a pair of groups , they can significantly alter the nature of symmetry breaking between the representations of and , even within the same Weyl chamber of the direct product group . We introduce the concept of "fences for the interleaving pattern", which provides a refinement of the usual notion of walls of Weyl chambers. We then establish a theorem stating that the multiplicity remains constant unless these "fences" are crossed, together with a new general vanishing theorem for symmetry breaking. These general results are illustrated with examples involving both tempered and non-tempered representations. In addition, we present a new non-vanishing theorem for period integrals for pairs of reductive symmetric spaces, which is further strengthened by this approach.

Paper Structure

This paper contains 41 sections, 20 theorems, 162 equations, 1 table.

Table of Contents

  1. Introduction
  2. Symmetry Breaking Under Translations
  3. Harish-Chandra Isomorphism and Translation Functor.
  4. Non-Vanishing Condition for Translating SBOs.
  5. Interleaving Pattern
  6. Stability Theorem for Multiplicities in Symmetry Breaking inside Fences
  7. A General Vanishing Theorem for Symmetry Breaking
  8. Known Examples for $(G, G')=(U(p,q), U(p-1,q))$
  9. Weyl's Branching Law for $U(n) \downarrow U(n-1)$
  10. Gan--Gross--Prasad conjecture for $U(p,q)\downarrow U(p-1,q)$
  11. Branching of Some Special Representations for $GL(2m,{\mathbb{R}}) \downarrow GL(2m-1,{\mathbb{R}})$
  12. A Family of Representations of $GL(2m, {\mathbb{R}})$
  13. A Family of Representations of $GL(2m-1,{\mathbb{R}})$
  14. Branching for $GL(2m,{\mathbb{R}})\downarrow GL(2m-1,{\mathbb{R}})$
  15. A Non-Vanishing Theorem for Period Integrals
  16. ...and 26 more sections

Key Result

Theorem 2.1

Let $\Pi \in {\mathcal{M}}(G)$ and $\pi \in {\mathcal{M}}(G')$. Suppose that any generalized eigenspaces of ${\mathfrak{Z}}({\mathfrak{g}}_{\mathbb{C}})$ in $\Pi \otimes {\mathbb{C}}^{n}$ are eigenspaces. (1) If $\operatorname{Hom}_{G'}(\Pi|_{G'},\pi)\ne \{0\}$, then $\operatorname{Hom}_{G'}(\phi_{

Theorems & Definitions (42)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3: Interleaving Pattern and Fence
  • Example 2.4
  • Theorem 2.5: Stability Theorem in Symmetry Breaking
  • Remark 2.6
  • Lemma 2.7
  • proof
  • proof : Proof of Theorem \ref{['thm:24120703']}
  • Definition 2.8: $\tau$-invariant
  • ...and 32 more