Colloquium: Tuesday Dec. 16, 2025 (14:00, room 614). Speaker: Yotam Hendel (Ben-Gurion University). Title: “Integrability of Harish-Chandra characters and singularities of representations”.
Abstract: Given a finite group G, and a finite dimensional complex representation π: G \to GL(V), the character χπ(g):=tr(π(g)) of π is a function on G which completely determines π. Let G = GL_n(F), where F is the field of real or p-adic numbers, or more generally let G be the group of F-points of a reductive algebraic group defined over a local field F of characteristic 0. In this setting, irreducible representations π of G may be infinite dimensional. Nevertheless, one may define the Harish-Chandra character Θπ of π, a distribution on G serving as the analogue of a finite-group character. A classical result due to Harish-Chandra asserts that Θπ is regular, namely it is given by integration against a locally integrable function fπ on G. In fact, f_π has stronger integrability properties: it is locally L^{1+r}-integrable for some r>0.
In recent joint work together with Itay Glazer and Julia Gordon, we define a new singularity invariant ϵπ of a representation π, by considering the largest r such that fπ is locally L^{1+r}-integrable. We explore ϵπ, show it is bounded from below only in terms of the group G, and calculate it in the case of p-adic GL_n. To do so, we relate ϵπ to the integrability of certain natural measures on the Lie algebra of G arising from nilpotent matrices, and use the theory of hyperplane arrangements. We plan to discuss some of the above, as well as applications, as time permits.
