Cheng-Chiang

Webpage of Cheng-Chiang Tsai 蔡政江

Address: Room 629, Astro-Math building, Taipei, Taiwan.
Email: chengchiangtsai@gmail.com

I am a faculty member of the Institute of Mathematics in Academia Sinica in Taipei, Taiwan. I am having a joint appointment with National Taiwan University and National Sun Yat-Sen University¹. I am a student of Benedict Gross, probably the one with the longest name.

I am interested in character sheaves, Langlands correspondence, representation theory and harmonic analysis of p-adic groups, graded Springer theory, endoscopy, and most importantly the interplay among these subjects. Recently I've been interested in application of (some of) these subjects to mod-ℓ representations.

In 2026 Spring, I am teaching a course on characters of p-adic reductive groups.

With Tsao-Hsien Chen and Mark Macerato we are organizing a Summer Workshop on Character Sheaves on Loop Groups on June 8th to 12th, 2026.

Selected works:

(click to expand comments)

(by main author: Wille Liu, with Wei-Hsuan Hsin and myself) Lusztig constants and endoscopy, arxiv:2604.21703

A cuspidal function of g(F_q) (arising from generalized Springer theory) and the Fourier transform of the function differ by a fourth root of unity that we call Lusztig constant following Emmanuel Letellier. In this article, Wille Liu (in discussions with Wei-Hsuan Hsin and myself) finds a uniform formula for Lusztig constant (given by the Weil index) and proves the formula using Waldspurger's works on endoscopy. In short, the problem is reduced via smooth (endoscopic) matching to the base case which is either the trivial group, or one of the mysterious isolated cases of G2, F4 or E8.

On local integrability results for p-adic reductive groups, arxiv:2604.15079

We give a new short proof of Harish-Chandra's celebrated theorem on local integrability, both that a function given by restricting a character (irreducible admissible) to regular semisimple locus is locally-L^1, and that the locally-L^1 function does represent the character on the whole G. The proof ultimately expresses the integral locally as a finite sum of geometric series, and is therefore algebraic and works for any coefficient of characteristic 0. It also works for mod-ℓ representations for exponentially large ℓ. The proof is based on local character expansions at the level of distributions; Harish-Chandra wrote the local character expansions in a simpler language using local integrability, but gave a proof that is actually independent. The proof also shows results similar to (but likely different) from that of Glazer, Gordon and Hendel, that a character is locally L^α for some specified α>1.

Local character expansion for mod-ℓ representations, arxiv:2510.20510

In Local characters of mod-ℓ representations of a p-adic reductive group we show that local character expansions exist for mod-ℓ representations. This paper contains a different proof of the expansion near the identity. Note that the expansion has Q-coefficient despite that the representations are mod-ℓ. We think the proof of the other paper is better (and definitely shorter), and advertise the other paper as our main paper. Nevertheless in this paper we use degenerate Moy-Prasad types to study local character expansions, and that might be useful for other purposes, e.g. to compute wave-front sets.

Local characters of mod-ℓ representations of a p-adic reductive group, arxiv:2510.20509

Howe, Harish-Chandra, Waldspurger and DeBacker have established local character expansion results for irreducible admissible complex representations of p-adic reductive groups, each with some working conditions. Suppose ℓ is a prime different from p. Under the same working conditions, we show that irreducible admissible mod-ℓ representations also have similar local character expansion for its Brauer character, that has been defined on regular semisimple elements by Dat and Vignéras. Near the identity this gives the Brauer character as a Q-linear combination of Fourier transforms of nilpotent orbital integrals. In some sense such a local character expansion is as good as the classical one for complex representations, and knows more than the mod-ℓ characters. For example, we show that this local character expansion generalize classical results such as fixed subspace dimension and the Mœglin-Waldspurger theorem on degenerate Whittaker models from complex representations to mod-ℓ representations.

Warning. The quality of the writing is not good (though results are correct modulo possible typos, as far as I know), and better results are in progess. We hope to update to arXiv a version 2 by ~~late February, 2026~~ (alas, still unfinished ...).

(with Tsao-Hsien Chen and Stephen DeBacker) Langlands parameters for Moy-Prasad types, arxiv:2509.07780

Let G be a tamely ramified connected reductive group over a non-archimedean local field. For irreducible smooth representations Moy-Prasad has defined the notion of its depth and unrefined minimal K-types, whose associate class is unique. For any such associate class (of depth r) we construct a restricted Langlands parameter: a homomorphism from the upper numbering subgroup (of depth r) of the Weil group to the Langlands dual group. The construction is canonical, and works for any p as long as G is tamely ramified.

Motivated by (the existence of) our construction, we conjecture that when G is tamely ramified the depth on the Galois side is always less than or equal to the depth on the representation side, and propose a conjecture that the equality happens iff the non-degenerate Moy-Prasad type is "non-nilpotent" in a precise sense.

(with Wille Liu, Kari Vilonen and Ting Xue) Cuspidal character sheaves on graded Lie algebras, arxiv:2409.04030

Consider a reductive group with a grading on its Lie algebra by a cyclic group, so that the 0th graded group G_0 acts on the 1st graded subspace of the Lie algebra g_1. We study G_0-equivariant sheaves on g_1 whose Fourier transforms are supported on the nilcone. (We call such simple perverse sheaves character sheaves.) We give a structural theory that classifies cuspidal character sheaves, showing that they all arise in a precise manner from nearby cycle constructions using data in the theory of Lusztig and Yun, which eventually boils down to cuspidal character sheaves in the original ungraded generalized Springer theory.

The study of these sheaves is of interest to p-adic groups, loop groups, symmetric spaces, and likely relative Langlands program. For example, I would hope that a well-developed theory of all such sheaves will give an upper bound for wave-front sets for p-adic Lie algebra, that is sharp in the sense that equality is always achieved when p is very very large. However, at this moment it seems that the study of these sheaves is so hard and we're pretty far from such a theory.

In a subsequent work, the other three authors successfully addressed several further questions, including a classification of such graded Lie algebra, the role of the Cartan subspaces in the construction, and that (interesting similar to Lusztig's classical result) that the relative little Weyl group is always a complex reflection group.

Wave-front sets for p-adic Lie algebras, arxiv:2311.08078

We adapt an algorithm of Waldspurger to compute any wave-front set for the p-adic Lie algebras, and hence for all regular (supercuspidal) representations. Quite some effort is spent to analyze the algorithm so that it better reveals the behavior of wave-front sets. For example, the algorithm produces more examples whose the geometric wave-front sets are not singleton for arbitrarily large p. I hope that further analysis of the algorithm can eventually tell us how the wave-front sets for positive-depth representations are different from wave-front sets for depth-0 representations. I also hope that these experience can be useful to the study of characters and of affine Springer fibers.

Moreover, it seems very interesting to compare the algorithm with the seemingly very different arithmetic wave-front set of Dihua Jiang, Dongwen Liu and Lei Zhang as well as their recent advance with Cheng Chen.

On two definitions of wave-front sets for p-adic groups, link, ANT 2025

For an irreducible admissible representation of a p-adic reductive group, let us define its wave-front set to be the set of maximal elements in the local character expansion, where maximality is defined via analytic closure as in Mœglin-Waldspurger. We then have two definitions of the geometric wave-front set: the set of geometric orbits of the elements in the wave-front set, or the set of largest geometric orbits appearing in the local character expansion. The latter is contained in the former. This work shows that they can be unequal already for Sp_4.

(with main author: Oscar Salomon Kivinen) Shalika germs for tamely ramified elements in GL_n, arxiv:2209.02509

In this work, Oscar (partly with me) obtains explicit, combinatorially and geometrically meaningful formulas for all Shalika germs and "basic" orbital integrals for regular semisimple orbits in GL_n, and weight polynomials for all regular semisimple affine Springer fiber for SL_n, using actions of the elliptic Hall algebra. This work relies on results of Waldspurger in 1991 and before.
Some side applications of this work:
(a) The weight polynomial for all regular semisimple affine Springer fiber for SL_n (any parahoric) is a polynomial in q with non-negative integral coefficients, suggesting the folklore conjecture that it is paved by affine. (It seems not well-known, but the fact that it is a polynomial in q with rational coefficients that depends only on the root valuations is a consequence of Waldspurger's 1991 result.)
(b) The number of components of any Iwahori affine Springer fiber for SL_n is the index of an explicit parabolic subgroup of the Weyl group S_n in S_n itself, in particular a divisor of n!. This suggests a possible connection to finite Springer fibers, and providing a hint for situations outside type A.
The "explicit parabolic subgroup" is rather interesting. Let us describe it when the regular semisimple element is elliptic, to which the general case can be reduced. In the elliptic case, the minimal root valuation is m/n for some integer m. Consider the rectangular integral-length path from (0,0) to (n,m) that is below the diagonal segment from (0,0) to (n,m) but otherwise closest to it. The lengths d_i of the horizontal steps are integers above or below n/m nearest to it. The parabolic subgroup of S_n we seek for is the product of S_{d_i}.

Geometric wave-front set may not be a singleton, link, JAMS 2023

I show that the wave-front set of specific half-integral depth representations of ramified U_7 have orbits of type (511) and of type (43) in the wave-front set. At MIT in Sep. 2022 (see note), I talked about the intuition of this work from graded Springer theory. Some questions that I am curious about:
(a) Is the wave-front set of a depth-0 representation of a p-adic reductive group contained in a single geometric orbit?
(b) Can we construct an analogous global example?

Uniform bounds of orbital integrals, in preparation.

Assuming p is huge and some exponential map exists. We give a uniform bound for the orbital integral of the characteristic function of any parahoric subgroup on any regular semisimple orbit. The bound is controlled by the regular Shalika germs, and thus sharp up to multiplying by a constant that depends only on the rank. A downside is that the constant I give grows faster than exponential in the rank.
This work was mostly done in 2016-2018. Embarrassingly, I have never disciplined myself to write it down. The method of this work is a combination of harmonic analysis and geometry of affine Springer fibers. Thanks to the organizers of this workshop, I delivered a talk in which I presented the key ideas, with video available in the link. (That being said, it's fully my problem that it's not yet written down. I am also aware that before I write anything down, other authors have written results on the similar problem with different and apparently original methods. The results seem to be different, as far as I know by 2024.)

Components of affine Springer fibers, link, IMRN 2018

Suppose X is a regular semisimple element in the loop algebra with root valuations \ge 1 (the paper says >1, but the method actually works for \ge 1). Let Fl_X be the Iwahori affine Springer fiber above X. Consider the components of Fl_X modulo the centralizer of X in the loop group. We show that the number of such orbits is equal to the order of the Weyl group.

Erratum. Theorem 4.2 in the above paper is straight wrong. This error does not affect the main theorems in the paper, see Erratum. I wish to thank Oscar Kivinen for pointing out the error.

¹Both are titles without duty nor pay.