**Webpage of Cheng-Chiang Tsai 蔡政江**

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.

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.

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}.

(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?

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.)

**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.