Completed Iwahori-Hecke algebra for Kac-Moody groups over local fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909858439102464 |
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| author | Hébert, Auguste Muthiah, Dinakar |
| author_facet | Hébert, Auguste Muthiah, Dinakar |
| contents | Let $G$ be a split Kac-Moody group over a non-Archimedean local field, and let $\mathcal{H}$ be the Iwahori-Hecke algebra of $G$. In this paper, we construct a completed Iwahori-Hecke algebra $\widehat{\mathcal{H}}$ and prove that it contains a large center isomorphic to Looijenga's invariant ring. By the Kac-Moody Satake isomorphism, Looijenga's invariant ring is isomorphic to the spherical Hecke algebra. Our completion is constructed by considering Iwahori biinvariant functions on $G$ satisfying a support condition that we call Weyl almost finite support. We contrast our construction with another completion $\widetilde{\mathcal{H}}$, defined early by Abdellatif and Hébert, which is defined algebraically via the Bernstein-Lusztig presentation and not in terms of functions on $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17559 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Completed Iwahori-Hecke algebra for Kac-Moody groups over local fields Hébert, Auguste Muthiah, Dinakar Representation Theory Let $G$ be a split Kac-Moody group over a non-Archimedean local field, and let $\mathcal{H}$ be the Iwahori-Hecke algebra of $G$. In this paper, we construct a completed Iwahori-Hecke algebra $\widehat{\mathcal{H}}$ and prove that it contains a large center isomorphic to Looijenga's invariant ring. By the Kac-Moody Satake isomorphism, Looijenga's invariant ring is isomorphic to the spherical Hecke algebra. Our completion is constructed by considering Iwahori biinvariant functions on $G$ satisfying a support condition that we call Weyl almost finite support. We contrast our construction with another completion $\widetilde{\mathcal{H}}$, defined early by Abdellatif and Hébert, which is defined algebraically via the Bernstein-Lusztig presentation and not in terms of functions on $G$. |
| title | Completed Iwahori-Hecke algebra for Kac-Moody groups over local fields |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2510.17559 |