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Cheng-Chiang Tsai

Publications and source records attributed to Cheng-Chiang Tsai.

At least 19 recordsLinked to original sources
arXiv2026

Admissibility of Bernstein centers

We provide a criterion for determining when elements of the Bernstein center of a totally disconnected locally compact group are admissible invariant distributions in the sense of Harish-Chandra \cite{HC99}. As a consequence, we deduce the local integrability results for elements of bounded depth or Bernstein supports in the Bernstein centers of reductive $p$-adic groups. Our methods apply uniformly to both complex and mod-$\ell$ coefficients, generalizing results of Moy and Tadić \cite{MT02} in the complex case.

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arXiv2026

Shalika germs for tamely ramified elements in $GL_n$

We prove explicit combinatorial formulas for various germ expansions of orbital integrals of tamely ramified elements in $GL_n(F)$, where $F$ is a nonarchimedean local field. The relevant combinatorics arises from the theory of the elliptic Hall algebra and the representation-theoretic knot superpolynomials defined by Cherednik--Danilenko and Morton--Samuelson. As a byproduct, we give explicit formulas for the weight polynomials of affine Springer fibers in type A and standard orbital integrals of tamely ramified regular semisimple elements. Our formulas subsume most earlier facts about the structure of Shalika germs of $GL_n$ in the literature. As a further corollary, we show that point-counts of compactified Jacobians of locally planar curves are given by non-negative integral polynomials. Our results also provide further evidence for the Oblomkov-Rasmussen-Shende conjecture relating compactified Jacobians and HOMFLY-PT invariants of algebraic knots.

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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 $\hat f = γ^{-1}f$ 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.

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arXiv2026

On local integrability results for $p$-adic reductive groups

We present a short proof, based on local character expansions, of the celebrated theorem of Harish-Chandra about local integrability of complex characters of $p$-adic reductive groups. The proof gives an algebraic incarnation of the local integrability that works for some coefficients different from $\mathbb{C}$, verifies local integrability in cases that appear not covered in the literature, and shows that a character is locally-$L^α$ for some specified $α>1$ as in [GGH23].

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arXiv2026

Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems

We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.

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arXiv2026

Jordan decompositions in Lie algebras and their duals

We provide a discussion of Jordan decompositions in the Lie algebra, and the dual Lie algebra, of a reductive group in as uniform a way as possible. We give a counterexample to the claim that Jordan decompositions on the dual Lie algebra are unique, and state an upper bound on how non-unique they can be. We also prove some Chevalley-restriction-type claims about GIT quotients for the adjoint and co-adjoint actions of $G$.

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arXiv2025

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.

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arXiv2025

Local characters of mod-$\ell$ representations of a $p$-adic reductive group

We define the ``lifted character'' of mod-$\ell$ representations of $p$-adic reductive groups where $\ell\not=p$, on compact elements with pro-orders not divisible by $\ell$. We generalize the local character expansion results of Howe, Harish-Chandra and DeBacker to such lifted characters. We show that the result of Moeglin-Waldspurger and Varma on degenerate Whittaker models is valid for the character expansion.

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arXiv2025

Local character expansion for mod-$\ell$ representations

Let $G$ be a $p$-adic reductive group with $p$ ``very large.'' For any irreducible admissible representation $π$ of $G$ over an algebraically closed field $C$ of characteristic $\not=p$, we define a ``local character expansion'' of $π$ with coefficients $c_{\mathcal{O}}(π)\in\mathbb{Q}$, that does not use the character of $π$ directly but instead use the multiplicities of degenerate Moy-Prasad types. Note that the existence of local character expansion for mod-$\ell$ representations is shown by another paper of the author using a different and quicker method.

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arXiv2025

Normalized Indexing for Ramification Subgroups

This expository note introduces a normalization of the indexing of the lower and upper numbering ramification subgroups of local class field theory. We then look at how this normalization interacts with base change for Langlands parameters.

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arXiv2025

Langlands parameters for Moy-Prasad types

Suppose $G$ is a tamely ramified $p$-adic reductive group. We construct a partial local Langlands correspondence between the set of irreducible smooth representations of $G$ having depth $r$ and a certain set of $G^\vee$-conjugacy classes of continuous homomorphisms $φ:I_F^r\rightarrow G^{\vee}$. Here $G^\vee$ is the dual group of $G$, and $I_F^r$ is the $r^{\text{th}}$ upper-numbering filtration subgroup of the inertia subgroup $I_F$.

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arXiv2025

On two definitions of wave-front sets for $p$-adic groups

The wave-front set for an irreducible admissible representation of a $p$-adic reductive group is the set of maximal nilpotent orbits which appear in the local character expansion. By Mœglin-Waldspurger, they are also the maximal nilpotent orbits whose associated degenerate Whittaker models are non-zero. However, in the literature there are two versions commonly used, one defining maximality using analytic closure and the other using Zariski closure. We show that these two definitions are non-equivalent for $G=Sp_4$.

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arXiv2023

Wave-front sets for $p$-adic Lie algebras

We study the wave-front set of an element in a $p$-adic reductive Lie algebra (for $p\gg\operatorname{rank}$), namely the set of maximal nilpotent orbits appearing in its Shalika germ expansion. By adapting an algorithm of Waldspurger that computes orbital integrals, we obtain an inductive algorithm to compute an invariant that determines the wave-front set. This gives an algorithm to compute the wave-front sets for regular supercuspidal representations and reveals examples whose wave-front sets are not contained in a single geometric orbit, for arbitrarily large $p$ within a fixed rank.

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arXiv2022

Computations of orbital integrals and Shalika germs

For a reductive group $G$ over a non-archimedean local field, with some assumptions on (residue) characteristic we give an method to compute certain orbital integrals using a method close to that of Goresky-Kottiwitz-MacPherson but in a different language. These orbital integrals allow us to compute the Shalika germs at some ``very elliptic'' elements in terms of number of rational points on some quasi-finite covers of the Hessenberg varieties of GKM, which are subvarieties of (partial) flag varieties. Such values of Shalika germs determine the Harish-Chandra local character expansions of the so-called very supercuspidal representations.

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arXiv2020

Epipelagic Langlands parameters and L-packets for unitary groups

Reeder and Yu have recently given a new construction of a class of supercuspidal representations called epipelagic representations. We explicitly calculate the Local Langlands Correspondence for certain families of epipelagic representations of unitary groups, following the general construction of Kaletha [Kal15]. The interesting feature of our computation is that we find simplifications within L-packets of the two novel invariants introduced in [Kal15], the toral invariant and the admissible L-embedding.

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arXiv2017

Components of affine Springer fibers

Let $\mathbf{G}$ be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, $γ_0\in(\operatorname{Lie}\mathbf{G})((t))$ be any topologically nilpotent regular semisimple element, and $γ=tγ_0$. Using methods from $p$-adic orbital integrals, we show that the number of components of the Iwahori affine Springer fiber over $γ$ modulo $Z_{\mathbf{G}((t))}(γ)$ is equal to the order of the Weyl group.

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arXiv2015

Inductive structure of Shalika germs and affine Springer fibers

This article has two parallel perspectives: to demonstrate an inductive structure of Shalika germs, and to show an analogous inductive structure for affine Springer fibers. More precisely, we give an algorithm to compute arbitrary Shalika germs (resp. affine Springer fibers up to stratification) in terms of three ingredients: Shalika germs (resp. affine Springer fibers) for twisted Levi subgroups, a finite list of combinatorial objects, and the numbers of rational points on varieties over the residue field (resp. varieties themselves) among an explicit finite list of such. We also discuss some formal applications of the algorithm to Shalika germs and orbital integrals.

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