Koszul duality for generalised Steinberg representations of $p$-adic groups
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908392018149376 |
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| author | Cunningham, Clifton Steele, James |
| author_facet | Cunningham, Clifton Steele, James |
| contents | Let $G$ be a semisimple group, split over a non-Archimedean field $F$. We prove that the category of modules over the extension algebra of generalised Steinberg representations of $G(F)$ is equivalent to a full subcategory of equivariant perverse sheaves on the variety of Langlands parameters for these representations. Specifically, we establish an equivalence \[ \textbf{Mod}(\text{Ext}_G^\bullet(Σ_λ, Σ_λ)) \simeq \textbf{Per}_{\widehat{G}}^\circ(X_λ), \] where $Σ_λ$ is the direct sum of generalised Steinberg representations and $\textbf{Per}_{\widehat{G}}^\circ(X_λ)$ is the subcategory of perverse sheaves on the variety of Langlands parameters $X_λ$ corresponding to these representations under Vogan's geometrisation of the Langlands correspondence. Furthermore, we demonstrate that this equivalence is a true Koszul duality by showing that the extension algebra of generalised Steinberg representations is Koszul dual to the endomorphism algebra of the direct sum of corresponding equivariant perverse sheaves, taken in the equivariant derived category $D_{\widehat{G}}^b(X_λ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Koszul duality for generalised Steinberg representations of $p$-adic groups Cunningham, Clifton Steele, James Representation Theory 11F70, 22E50, 35A27, 32S30 Let $G$ be a semisimple group, split over a non-Archimedean field $F$. We prove that the category of modules over the extension algebra of generalised Steinberg representations of $G(F)$ is equivalent to a full subcategory of equivariant perverse sheaves on the variety of Langlands parameters for these representations. Specifically, we establish an equivalence \[ \textbf{Mod}(\text{Ext}_G^\bullet(Σ_λ, Σ_λ)) \simeq \textbf{Per}_{\widehat{G}}^\circ(X_λ), \] where $Σ_λ$ is the direct sum of generalised Steinberg representations and $\textbf{Per}_{\widehat{G}}^\circ(X_λ)$ is the subcategory of perverse sheaves on the variety of Langlands parameters $X_λ$ corresponding to these representations under Vogan's geometrisation of the Langlands correspondence. Furthermore, we demonstrate that this equivalence is a true Koszul duality by showing that the extension algebra of generalised Steinberg representations is Koszul dual to the endomorphism algebra of the direct sum of corresponding equivariant perverse sheaves, taken in the equivariant derived category $D_{\widehat{G}}^b(X_λ)$. |
| title | Koszul duality for generalised Steinberg representations of $p$-adic groups |
| topic | Representation Theory 11F70, 22E50, 35A27, 32S30 |
| url | https://arxiv.org/abs/2408.05103 |