Categorification of the function-sheaf correspondence

I will describe a categorification of Grothendieck's function-sheaf correspondence which associates a constructible function to an l-adic on a variety over a finite field. Moreover, I will describe some applications of this to geometric representation theory, focusing on Deligne-Lusztig theory using higher categorical traces. The first part of the talk will describe the general theory of higher categorical traces in this context and the second will focus on representation theoretic applications. This is joint work with Gaitsgory and Varshavsky.