Activities This Week

Nuclear dimension, introduced by Winter and Zacharias, is an invariant for C-algebras which generalizes covering dimension for compact Hausdorff spaces, and plays an important role in structure theory for amenable C-algebras. It is usually mainly interesting to show that it is finite, as opposed to computing its actual value. Given an action of a group G on a locally compact Hausdorff space X, one forms the crossed product C*-algebra C_0(X) \rtimes G; this construction has been heavily studied in the field.

I will discuss joint work with Jianchao Wu, in which we find bounds on the nuclear dimension of nuclear dimension of the crossed product for a large class of group actions, including arbitrary actions of finitely generated virtually nilpotent groups on finite dimensional spaces and certain boundary actions of hyperbolic groups. This involves introducing a notion of “long and thin covers” which serves as the appropriate generalization of Rokhlin-type towers for non-free actions. As another application of the result, we generalize a result of Joseph and construct a family of profinite actions of wreath products of finite abelian groups by Z^d which are allosteric (that is, are minimal and topologically free, but not essentially free, meaning that fixed points sets are meager but have non-zero measure with respect to the unique invariant measure), and show that the resulting crossed product are well behaved from the perspective of structure and classification of C*-algebras.

As the paper is rather long, in the talk I will just give an overview of some of the definitions and techniques, intended for people from dynamical systems who are not experts in C*-algebras.

The theory of characters is central to the representation theory of finite groups. 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 p-adic reductive groups, Harish-Chandra obtained a beautiful formula for the characters of cuspidal representations in terms of orbital integrals.

One of Harish-Chandra’s 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 the characters of irreducible representations of such groups are represented by functions that are locally in L1(G). He achieved this by obtaining a bound on the character in terms of a power of a rather elementary function, namely the discriminant function on the group (which measures the distances between eigenvalues). In the positive-characteristic case, this approach to proving local integrability fails, and the problem remains open.

In this talk, we will describe our approach and results concerning the regularity of characters of representations of p-adic groups over fields of positive characteristic. In the case of GLn, this leads to a proof of the Harish-Chandra regularity theorem for cuspidal representations. Our key tool is the so-called Chevalley map (for G=GLn, this is the map sending a matrix to the coefficients of its characteristic polynomial): we show that it sends compactly supported smooth measures to essentially bounded measures, and from this we deduce the regularity theorem.

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

This talk is based on a series of joint works with Aizenbud, Gourevitch, and Kazhdan (see arXiv:2602.16389), as well as on very recent joint work by the same authors with Avni.