The Derived Unipotent Block of $p$-Adic $\mathrm{GL}_2$ as Perfect Complexes over a dg Schur Algebra
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915065037324288 |
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| author | Berry, Rose |
| author_facet | Berry, Rose |
| contents | For a $p$-adic field $F$ of residual cardinality $q$, we provide a triangulated equivalence between the bounded derived category $D^b(\mathcal{B}_{1}(G)_{fg})$ of finitely generated unipotent representations of $G=\mathrm{GL}_2(F)$ and perfect complexes over a dg enriched Schur algebra, in the non-banal case of odd characteristic $l$ dividing $q+1$. The dg Schur algebra is the dg endomorphism algebra of a projective resolution of a direct sum $V$ of the parahoric inductions of the trivial representations of the reductive quotients of $G$, and $V$ is shown to be a classical generator of $D^b(\mathcal{B}_{1}(G)_{fg})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17469 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Derived Unipotent Block of $p$-Adic $\mathrm{GL}_2$ as Perfect Complexes over a dg Schur Algebra Berry, Rose Representation Theory Number Theory For a $p$-adic field $F$ of residual cardinality $q$, we provide a triangulated equivalence between the bounded derived category $D^b(\mathcal{B}_{1}(G)_{fg})$ of finitely generated unipotent representations of $G=\mathrm{GL}_2(F)$ and perfect complexes over a dg enriched Schur algebra, in the non-banal case of odd characteristic $l$ dividing $q+1$. The dg Schur algebra is the dg endomorphism algebra of a projective resolution of a direct sum $V$ of the parahoric inductions of the trivial representations of the reductive quotients of $G$, and $V$ is shown to be a classical generator of $D^b(\mathcal{B}_{1}(G)_{fg})$. |
| title | The Derived Unipotent Block of $p$-Adic $\mathrm{GL}_2$ as Perfect Complexes over a dg Schur Algebra |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2411.17469 |