### Ben-Zvi on Geometric Function Theory

#### Posted by Urs Schreiber

*guest post by David Ben-Zvi*

Dear Café Patrons,

In this guest post I want to briefly discuss correspondences, integral transforms and their categorification as they apply to representation theory, a topic that might be called geometric function theory.

This has been one of the fundamental paradigms of geometric representation theory (together with localization of representations and, on a much grander scale, the Langlands program) for at least the past twenty years (some key names to mention in this context prior to the last decade are Kazhdan, Lusztig, Beilinson, Bernstein, Drinfeld, Ginzburg, I. Frenkel, Nakajima and Grojnowski).

These ideas are closely related to topics often discussed in this Café (in particular groupoidification and extended topological field theories), and my hope is to facilitate communication between the schools by focussing on some toy examples of geometric function theory and suppressing techincal details.

I will conclude self-centeredly by describing my recent work *Integral transforms and Drinfeld centers in derived algebraic geometry* (arXiv:0805.0157) with John Francis and David Nadler, in which we prove some basics of categorified function theory using tools from higher category theory and derived algebraic geometry.
(Of course this is out of all proportion to its role relative to the seminal works I glancingly mention or omit, but it provides my excuse to be writing here, so please indulge me.)

We were motivated by a desire to understand aspects of one of the more exciting developments of the past five years, namely the convergence of categorified representation theory (in particular the geometric Langlands program), derived algebraic geometry and topological field theory, which was at the center of last year’s special program at the IAS.

I will not attempt to describe these developments here but refer readers to collected lecture notes on my webpage.

I would also like to apologize in advance for the highly informal and imprecise style, inaccuracies and mis- or un-attributions. Finally I want to thank Urs Schreiber and Bruce Bartlett for their encouragement towards the writing of this post.

[*The guest post continues in this pdf file*:]

David Ben-Zvi,

Geometric function theory(pdf, 8 pages).

*See also the following related entries at the $n$Lab*

## Re: Ben-Zvi on Geometric Function Theory

Enjoyable post, but you skirted a question I’ve had unresolved for awhile now. In reading Edward Frenkel’s expository paper ‘Lectures on the Langland’s Program and CFT’, he mentions Grothendieck’s functions-sheaves dictionary, and then states that the right analog of functions isn’t the category of sheaves, but that of perverse sheaves. However, he doesn’t really go into

whythis is the right category, aside from it being stable under Verdier duality.Since I still don’t feel like I understand the significance of perverse sheaves, this claim has always bothered me. In your mind, what propert(y/ies) of perverse sheaves seems to make them the right analog of functions?