On sharpness in Local Converse Theorems for classical groups and $G_2$
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911178513448960 |
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| author | Adrian, Moshe Stevens, Shaun |
| author_facet | Adrian, Moshe Stevens, Shaun |
| contents | We prove various results about the Local Converse Problem for split reductive groups $G$ over a non-archimedean local field~$F$ of characteristic $0$ and residual characteristic $p$. In particular, we prove that when $G$ is a symplectic or special orthogonal group, or the exceptional group $G_2$, and $p$ is large enough, then the optimal standard Local Converse Theorem for $G(F)$ requires twisting by representations of $GL_r(F)$ with $r$ up to half the dimension of the standard representation of the dual group of $G$. However, if we restrict to generic supercuspidal representations of $G(F)$ then it can be improved when $G=SO_{2N}$; we conjecture that the same is true for symplectic and odd special orthogonal groups. We also consider the possibility of using non-standard representations of the dual group to distinguish representations, giving counterexamples to possible improvements for general linear groups, $G_2$ and $SO_{2N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22390 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On sharpness in Local Converse Theorems for classical groups and $G_2$ Adrian, Moshe Stevens, Shaun Representation Theory 11S70, 22E50 We prove various results about the Local Converse Problem for split reductive groups $G$ over a non-archimedean local field~$F$ of characteristic $0$ and residual characteristic $p$. In particular, we prove that when $G$ is a symplectic or special orthogonal group, or the exceptional group $G_2$, and $p$ is large enough, then the optimal standard Local Converse Theorem for $G(F)$ requires twisting by representations of $GL_r(F)$ with $r$ up to half the dimension of the standard representation of the dual group of $G$. However, if we restrict to generic supercuspidal representations of $G(F)$ then it can be improved when $G=SO_{2N}$; we conjecture that the same is true for symplectic and odd special orthogonal groups. We also consider the possibility of using non-standard representations of the dual group to distinguish representations, giving counterexamples to possible improvements for general linear groups, $G_2$ and $SO_{2N}$. |
| title | On sharpness in Local Converse Theorems for classical groups and $G_2$ |
| topic | Representation Theory 11S70, 22E50 |
| url | https://arxiv.org/abs/2509.22390 |