Cheng-Chiang
Webpage of Cheng-Chiang Tsai 蔡政江
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 Sun Yat-Sen University in Kaohsiung. I am a student of Benedict Gross, in fact the one with the longest name.
I am interested in character sheaves (especially on loop groups), Langlands correspondence, representation theory of p-adic groups, graded Springer theory, endoscopy, and most importantly the interplay among these subjects. Recently I have paid more attention to the topic of wave-front sets for p-adic groups.
Selected works:
(click for comments)
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, arxiv:2306.09536
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 Moeglin-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 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, there are authors who have written results on the similar problem with different and apparently original methods. The results seem to be different, as far as I know.)
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.
I plan to go to the Summer School and Workshop on Relative Langlands Duality in June 2024.