Harish-Chandra regularity, flatness of Chevalley map and Resolutions of Singularities

Harish-Chandra introduced the theory of characters as a fundamental tool for studying infinite-dimensional representations of real and p-adic groups G. 

In this setting, the character is a conjugation-invariant distribution on the group. For real reductive groups, Harish-Chandra used characters to show the existence of discrete series representations. For p-adic reductive groups he obtained a  beautiful formula for the characters of cuspidal representations. One of his most remarkable results—valid for real reductive groups and for reductive p-adic groups over fields of characteristic zero —is the regularity theorem, which establishes that characters are represented by functions that are locally in L^1(G).  This he achieved by obtaining a bound on the character in terms of a power of the discriminant function. 

In this talk, we will sketch our approach to regularity of characters of representations of p-adic groups over fields of positive characteristics. 

We will introduce the Chevalley map (for G=GL_n, this is the map sending a matrix to the coefficients of its characteristic polynomial), state some results on the jets of this map and explain how one can deduce from 

(i) Certain additional properties of the Chevalley map (such as flatness of the jets),  

(ii) The formula of Harish-Chandra mentioned above

(iii) a substitute for the discriminant function that we introduce and 

(iv) Existence of resolution of singularities in positive characteristics  - 

The regularity of characters of cuspidal representations of the p-adic groups GL_n(F_p((t))).

We will discuss what are the limits of our technique and what unconditional results we can obtain at the moment.

No prior familiarity with p-adic groups or representation theory will be assumed.

This talk is based on joint work with Aizenbud, Gourevitch, and Kazhdan (see arXiv:2602.16389).