Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.21703

Lusztig constants and endoscopy

Wille Liu·Wei-Hsuan Hsin·Cheng-Chiang Tsai
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 $\hat f = \gamma^{-1}f$ for an explicit quadratic Gauss sum $\gamma$. 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.

Explore related subjects

math.RTmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Wille Liu, Wei-Hsuan Hsin, Cheng-Chiang Tsai. 2026-04-23. Lusztig constants and endoscopy. https://arxiv.org/abs/2604.21703

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Whittaker supermodules over Lie superalgebras

In this paper, we develop a general theory of quasi-Whittaker supermodules over Lie superalgebras induced from an arbitrary ideal. We determine the quasi-Whittaker vectors in universal supermodules, establish an irreducibility criterion, and classify several families of irreducible supermodules. The odd part produces a new irreducibility phenomenon absent from the Lie algebra setting. As applications, we determine all irreducible quasi-Whittaker supermodules over the $N=1$ super Schr\"odinger algebra and the $N=1$ $\frac{3}{2}$-conformal Galilei superalgebra, and over the complete spectrum-generating superalgebra in a special case.

math.RT↗
arXiv2026

Rankin--Selberg integrals of opposite conductor--one newforms

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni--Zhang's conjecture on the nonvanishing of local relative characters.

math.RT↗
arXiv2026

Obstructions to Jacobi-Finiteness of Quivers with Potentials

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod--Shafarevich--Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

math.RT↗