How does the restriction of representations change under translations? A story for the general linear groups and the unitary groups
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866918204642689024 |
|---|---|
| author | Kobayashi, Toshiyuki Speh, Birgit |
| author_facet | Kobayashi, Toshiyuki Speh, Birgit |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08479 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | How does the restriction of representations change under translations? A story for the general linear groups and the unitary groups Kobayashi, Toshiyuki Speh, Birgit Representation Theory 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. |
| title | How does the restriction of representations change under translations? A story for the general linear groups and the unitary groups |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2502.08479 |