Webb's conjecture and generalised Harish-Chandra theory
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911984065183744 |
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| author | Rossi, Damiano |
| author_facet | Rossi, Damiano |
| contents | Webb's conjecture states that the orbit space of the Brown complex of a finite group at any given prime $\ell$ is contractible. This conjecture was proved by Symonds in 1998. In this paper, we suggest a generalisation of Webb's conjecture for finite reductive groups. This is done by associating to each irreducible character a new simplicial complex defined in terms of Deligne--Lusztig theory. We then show that our conjecture follows from a condition, called ($e$-HC-conj) below, related to generalised Harish-Chandra theory. In particular, using earlier results of the author, we prove our conjecture and recover Symonds result for finite reductive groups under mild restrictions on the prime $\ell$. Finally, we show that the condition ($e$-HC-conj) is implied by the contractibility of the orbit spaces associated to our newly defined complex offering an unexplored topological approach to proving the uniqueness of $e$-cuspidal pairs up to conjugation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06129 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Webb's conjecture and generalised Harish-Chandra theory Rossi, Damiano Representation Theory Algebraic Topology Group Theory 20G40, 20C20, 55P10, 55P91 Webb's conjecture states that the orbit space of the Brown complex of a finite group at any given prime $\ell$ is contractible. This conjecture was proved by Symonds in 1998. In this paper, we suggest a generalisation of Webb's conjecture for finite reductive groups. This is done by associating to each irreducible character a new simplicial complex defined in terms of Deligne--Lusztig theory. We then show that our conjecture follows from a condition, called ($e$-HC-conj) below, related to generalised Harish-Chandra theory. In particular, using earlier results of the author, we prove our conjecture and recover Symonds result for finite reductive groups under mild restrictions on the prime $\ell$. Finally, we show that the condition ($e$-HC-conj) is implied by the contractibility of the orbit spaces associated to our newly defined complex offering an unexplored topological approach to proving the uniqueness of $e$-cuspidal pairs up to conjugation. |
| title | Webb's conjecture and generalised Harish-Chandra theory |
| topic | Representation Theory Algebraic Topology Group Theory 20G40, 20C20, 55P10, 55P91 |
| url | https://arxiv.org/abs/2408.06129 |