Resolutions for Locally Analytic Representations
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arXiv
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| Formato: | Preprint |
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2024
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| _version_ | 1866913493897183232 |
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| author | Agrawal, Shishir Strauch, Matthias |
| author_facet | Agrawal, Shishir Strauch, Matthias |
| contents | The purpose of this paper is to study resolutions of locally analytic representations of a $p$-adic reductive group $G$. Given a locally analytic representation $V$ of $G$, we modify the Schneider-Stuhler complex (originally defined for smooth representations) so as to give an `analytic' variant ${\mathcal S}^A_\bullet(V)$. The representations in this complex are built out of spaces of analytic vectors $A_σ(V)$ for compact open subgroups $U_σ$, indexed by facets $σ$ of the Bruhat-Tits building of $G$. These analytic representations (of compact open subgroups of $G$) are then resolved using the Chevalley-Eilenberg complex from the theory of Lie algebras. This gives rise to a resolution ${\mathcal S}^{\rm CE}_{q,\bullet}(V) \rightarrow {\mathcal S}^A_q(V)$ for each representation ${\mathcal S}^A_q(V)$ in the analytic Schneider-Stuhler complex. In a last step we show that the family of representations ${\mathcal S}^{\rm CE}_{q,j}(V)$ can be given the structure of a Wall complex. The associated total complex ${\mathcal S}^{\rm CE}_\bullet(V)$ has then the same homology as that of ${\mathcal S}^A_\bullet(V)$. If the latter is a resolution of $V$, then one can use ${\mathcal S}^{\rm CE}_\bullet(V)$ to find a complex which computes the extension group $\underline{Ext}^n_G(V,W)$, provided $V$ and $W$ satisfy certain conditions which are satisfied when both are admissible locally analytic representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Resolutions for Locally Analytic Representations Agrawal, Shishir Strauch, Matthias Representation Theory Number Theory 22E50 (Primary), 20G05, 11F85 (Secondary) The purpose of this paper is to study resolutions of locally analytic representations of a $p$-adic reductive group $G$. Given a locally analytic representation $V$ of $G$, we modify the Schneider-Stuhler complex (originally defined for smooth representations) so as to give an `analytic' variant ${\mathcal S}^A_\bullet(V)$. The representations in this complex are built out of spaces of analytic vectors $A_σ(V)$ for compact open subgroups $U_σ$, indexed by facets $σ$ of the Bruhat-Tits building of $G$. These analytic representations (of compact open subgroups of $G$) are then resolved using the Chevalley-Eilenberg complex from the theory of Lie algebras. This gives rise to a resolution ${\mathcal S}^{\rm CE}_{q,\bullet}(V) \rightarrow {\mathcal S}^A_q(V)$ for each representation ${\mathcal S}^A_q(V)$ in the analytic Schneider-Stuhler complex. In a last step we show that the family of representations ${\mathcal S}^{\rm CE}_{q,j}(V)$ can be given the structure of a Wall complex. The associated total complex ${\mathcal S}^{\rm CE}_\bullet(V)$ has then the same homology as that of ${\mathcal S}^A_\bullet(V)$. If the latter is a resolution of $V$, then one can use ${\mathcal S}^{\rm CE}_\bullet(V)$ to find a complex which computes the extension group $\underline{Ext}^n_G(V,W)$, provided $V$ and $W$ satisfy certain conditions which are satisfied when both are admissible locally analytic representations. |
| title | Resolutions for Locally Analytic Representations |
| topic | Representation Theory Number Theory 22E50 (Primary), 20G05, 11F85 (Secondary) |
| url | https://arxiv.org/abs/2409.05079 |