Lusztig constants and endoscopy
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910160555868160 |
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| author | Liu, Wille Hsin, Wei-Hsuan Tsai, Cheng-Chiang |
| author_facet | Liu, Wille Hsin, Wei-Hsuan Tsai, Cheng-Chiang |
| contents | We prove that on a semisimple Lie algebra $\mathfrak{g}$ over a finite field of large characteristic, if a complex-valued invariant function $f$ and its Fourier transform $\hat f$ are both supported in the nilpotent cone of $\mathfrak{g}$, then $\hat f = γ^{-1}f$ for an explicit quadratic Gauss sum $γ$. Consequently, we determine a fourth root of unity appearing in various formulae of generalised Gel'fand--Graev characters, known as Lusztig constant, previously known in special cases due to works of Kawanaka, Digne--Lehrer--Michel, Waldspurger and Geck. As consequence, we show the validity of a conjecture of Letellier on the compatibility of Fourier transform with Deligne--Lusztig induction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21703 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lusztig constants and endoscopy Liu, Wille Hsin, Wei-Hsuan Tsai, Cheng-Chiang Representation Theory Number Theory We prove that on a semisimple Lie algebra $\mathfrak{g}$ over a finite field of large characteristic, if a complex-valued invariant function $f$ and its Fourier transform $\hat f$ are both supported in the nilpotent cone of $\mathfrak{g}$, then $\hat f = γ^{-1}f$ for an explicit quadratic Gauss sum $γ$. Consequently, we determine a fourth root of unity appearing in various formulae of generalised Gel'fand--Graev characters, known as Lusztig constant, previously known in special cases due to works of Kawanaka, Digne--Lehrer--Michel, Waldspurger and Geck. As consequence, we show the validity of a conjecture of Letellier on the compatibility of Fourier transform with Deligne--Lusztig induction. |
| title | Lusztig constants and endoscopy |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2604.21703 |