Minimal depth $K$-types for wild double covers and Shimura correspondences
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arXiv
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| Natura: | Preprint |
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2025
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| author | Karasiewicz, Edmund Takeda, Shuichiro |
| author_facet | Karasiewicz, Edmund Takeda, Shuichiro |
| contents | We construct some Iwahori types, in the sense of Bushnell-Kutzko, for the double cover of an almost simple simply-laced simply-connected Chevalley group $\widetilde{G}$ over any $2$-adic field. These types capture the covering group analog of the Bernstein block of unramified principal series.
We also prove that the associated Hecke algebra essentially admits an Iwahori-Matsumoto (IM) presentation. The complete presentation is obtained for types $A_{r}$, $D_{2r+1}$, $E_{6}$, $E_{7}$; for the other types, some technical obstacles remain. Those Hecke algebras with the complete IM presentation are isomorphic to Iwahori-Hecke algebras of explicit linear Chevalley groups, giving rise to Shimura correspondences.
Along the way, we show that the Iwahori type extends to a hyperspecial maximal compact subgroup $\widetilde{K}\subseteq \widetilde{G}$. This extension has minimal depth among the genuine $\widetilde{K}$-representations and allows us to construct a finite Shimura correspondence, generalizing a result of Savin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23265 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal depth $K$-types for wild double covers and Shimura correspondences Karasiewicz, Edmund Takeda, Shuichiro Representation Theory Number Theory 11F70, 22E50 We construct some Iwahori types, in the sense of Bushnell-Kutzko, for the double cover of an almost simple simply-laced simply-connected Chevalley group $\widetilde{G}$ over any $2$-adic field. These types capture the covering group analog of the Bernstein block of unramified principal series. We also prove that the associated Hecke algebra essentially admits an Iwahori-Matsumoto (IM) presentation. The complete presentation is obtained for types $A_{r}$, $D_{2r+1}$, $E_{6}$, $E_{7}$; for the other types, some technical obstacles remain. Those Hecke algebras with the complete IM presentation are isomorphic to Iwahori-Hecke algebras of explicit linear Chevalley groups, giving rise to Shimura correspondences. Along the way, we show that the Iwahori type extends to a hyperspecial maximal compact subgroup $\widetilde{K}\subseteq \widetilde{G}$. This extension has minimal depth among the genuine $\widetilde{K}$-representations and allows us to construct a finite Shimura correspondence, generalizing a result of Savin. |
| title | Minimal depth $K$-types for wild double covers and Shimura correspondences |
| topic | Representation Theory Number Theory 11F70, 22E50 |
| url | https://arxiv.org/abs/2510.23265 |