Spectral mod p Satake isomorphism for GL_n
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909115367817216 |
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| author | Lee, Heejong |
| author_facet | Lee, Heejong |
| contents | Let $K/\mathbb{Q}_p$ be a finite extension with residue field $k$. By a work of Emerton--Gee, irreducible components inside the reduced special fiber of the moduli stack of rank $n$ étale $(φ,Γ)$-modules are labeled by Serre weights of $\mathrm{GL}_n(k)$. Let $σ$ be a non-Steinberg Serre weight and $\mathcal{C}_σ$ be the corresponding irreducible component. Motivated by the categorical $p$-adic local Langlands program, we construct a natural injective map $\mathcal{O}(\mathcal{C}_σ) \hookrightarrow \mathcal{H}(σ)$ from the ring of global functions on $\mathcal{C}_σ$ to the Hecke algebra of $σ$ compatible with the mod $p$ Satake isomorphism by Herzig and Henniart--Vignéras in a suitable sense. For sufficiently generic $σ$, we prove that it is an isomorphism. As an application, we obtain a natural stratification of the irreducible component whose strata are equipped with a parabolic structure. Our main input is a construction of a morphism from an integral Hecke algebra of a generic tame type to the ring of global functions on a tamely potentially crystalline Emerton--Gee stack. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14011 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral mod p Satake isomorphism for GL_n Lee, Heejong Number Theory Representation Theory Let $K/\mathbb{Q}_p$ be a finite extension with residue field $k$. By a work of Emerton--Gee, irreducible components inside the reduced special fiber of the moduli stack of rank $n$ étale $(φ,Γ)$-modules are labeled by Serre weights of $\mathrm{GL}_n(k)$. Let $σ$ be a non-Steinberg Serre weight and $\mathcal{C}_σ$ be the corresponding irreducible component. Motivated by the categorical $p$-adic local Langlands program, we construct a natural injective map $\mathcal{O}(\mathcal{C}_σ) \hookrightarrow \mathcal{H}(σ)$ from the ring of global functions on $\mathcal{C}_σ$ to the Hecke algebra of $σ$ compatible with the mod $p$ Satake isomorphism by Herzig and Henniart--Vignéras in a suitable sense. For sufficiently generic $σ$, we prove that it is an isomorphism. As an application, we obtain a natural stratification of the irreducible component whose strata are equipped with a parabolic structure. Our main input is a construction of a morphism from an integral Hecke algebra of a generic tame type to the ring of global functions on a tamely potentially crystalline Emerton--Gee stack. |
| title | Spectral mod p Satake isomorphism for GL_n |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2402.14011 |