arXiv subjects

Wille Liu

Publications and source records attributed to Wille Liu.

8 recordsLinked to original sources

arXiv2026

Lusztig constants and endoscopy

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.

math.RT↗

arXiv2025

Cuspidal character sheaves on graded Lie algebras II

In this paper we give a complete classification of cyclically graded semisimple Lie algebras that afford cuspidal character sheaves and determine the support of the cuspidal character sheaves. This constitutes a major step towards the explicit classification of cuspidal character sheaves for graded Lie algebras.

math.RT↗

arXiv2024

Cuspidal character sheaves on graded Lie algebras

We show in this paper that in the context of graded Lie algebras, all cuspidal character sheaves arise from a nearby-cycle construction followed by a Fourier--Sato transform in a very specific manner. Combined with results of the last two named authors, this completes the classification of cuspidal character sheaves for Vinberg's type I graded classical Lie algebras.

math.RT↗

arXiv2024

Bi-orbital sheaves and affine Hecke algebras at roots of unity

Continuing the study of perverse sheaves on the nilpotent cone of a $\mathbb{Z}/m$-graded Lie algebra initiated by Lusztig--Yun, we study in this work the parabolic induction and introduce the notion of supercuspidal sheaves on the nilpotent cone. One of our main results shows that simple perverse sheaves with nilpotent singular support (called bi-orbital sheaves) are produced by parabolic induction from supercuspidal sheaves. As application, we provide a proof for a theorem announced by I. Grojnowski on the parametrisation of simple modules of affine Hecke algebras at roots of unity via Deligne--Langlands--Lusztig parameters with nilpotent singular support.

math.RT↗

arXiv2022

Derived equivalences for trigonometric double affine Hecke algebras

The trigonometric double affine Hecke algebra $\mathbf{H}_c$ for an irreducible root system depends on a family of complex parameters $c$ Given two families of parameters $c$ and $c'$ which differ by integers, we construct the translation functor from $\mathbf{H}_{c}\operatorname{-Mod}$ to $\mathbf{H}_{c'}\operatorname{-Mod}$ and prove that it induces equivalence of derived categories. This is a trigonometric counterpart of a theorem of Losev on the derived equivalences for rational Cherednik algebras.

math.RT↗

arXiv2022

Knizhnik--Zamolodchikov functor for degenerate double affine Hecke algebras : algebraic theory

In this article, we define an algebraic version of the Knizhnik--Zamolodchikov functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo--Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases.

math.RT↗

arXiv2021

Generalised Springer correspondence for Z/m-graded Lie algebras

Let $G$ be a simple simply connected complex algebraic group and let $\mathfrak{g}_*$ be a $\mathbf{Z}/m$-grading on its Lie algebra $\mathfrak{g}$. In a recent series of articles, G. Lusztig and Z. Yun, studied the classification of simple $G_0$-equivariant perverse sheaves on the nilpotent cone of $\mathfrak{g}_i$ for $i\in \mathbf{Z}/m$, where $G_0$ is the exponentiation of the degree zero piece $\mathfrak{g}_0$. They proved a decomposition of the equivariant derived category of $\ell$-adic sheaves on the nilpotent cone of $\mathfrak{g}_i$ into blocks, each generated by a certain cuspidal local system via {\itshape spiral inductions}. We prove a conjecture of them, which predicts the bijectivity of a map from 1) the set of simple perverse sheaves in a fixed block to 2) the set of simple modules of a block of a (trigonometric) degenerate double affine Hecke algebra (dDAHA). This is a dDAHA analogue of the Deligne--Langlands correspondence for affine Hecke algebras proven by Kazhdan--Lusztig. Our results generalise a previous work of E. Vasserot, where the perverse sheaves in the principal block were considered.

math.RT↗

arXiv2016

Motivic equivalence under semismall flops

We prove that under semismall smooth flops, smooth projective varieties have (non-canonically) isomorphic Chow motives with coefficients in any noetherian local ring $Λ$ by comparing the pushforward of the constant intersection complexes through flopping contractions.

math.AG↗