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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.13088 |
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| _version_ | 1866912589776158720 |
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| author | Berry, Rose |
| author_facet | Berry, Rose |
| contents | For a non-Archimedean local field $F$ of residue cardinality $q=p^r$, we give an explicit classical generator $V$ for the bounded derived category $D_{fg}^b(\mathsf{H}_1(G))$ of finitely generated unipotent representations of $G=\mathrm{GL}_n(F)$ over an algebraically closed field of characteristic $l\neq p$. The generator $V$ has an explicit description that is much simpler than any known progenerator in the underived setting. This generalises a previous result of the author in the case where $n=2$ and $l$ is odd dividing $q+1$, and provides a triangulated equivalence between $D_{fg}^b(\mathsf{H}_1(G))$ and the category of perfect complexes over the dg algebra of dg endomorphisms of a projective resolution of $V$. This dg algebra can be thought of as a dg-enhanced Schur algebra. As an intermediate step, we also prove the analogous result for the case where $F$ is a finite field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13088 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Derived $l$-Modular Unipotent Block of $p$-adic $\mathrm{GL}_n$ Berry, Rose Representation Theory Number Theory For a non-Archimedean local field $F$ of residue cardinality $q=p^r$, we give an explicit classical generator $V$ for the bounded derived category $D_{fg}^b(\mathsf{H}_1(G))$ of finitely generated unipotent representations of $G=\mathrm{GL}_n(F)$ over an algebraically closed field of characteristic $l\neq p$. The generator $V$ has an explicit description that is much simpler than any known progenerator in the underived setting. This generalises a previous result of the author in the case where $n=2$ and $l$ is odd dividing $q+1$, and provides a triangulated equivalence between $D_{fg}^b(\mathsf{H}_1(G))$ and the category of perfect complexes over the dg algebra of dg endomorphisms of a projective resolution of $V$. This dg algebra can be thought of as a dg-enhanced Schur algebra. As an intermediate step, we also prove the analogous result for the case where $F$ is a finite field. |
| title | The Derived $l$-Modular Unipotent Block of $p$-adic $\mathrm{GL}_n$ |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2509.13088 |