Speaker:  Zhiwei Yun (MIT)

Where:      In-Person at MIT Room 2-135 and on

Zoom         https://mit.zoom.us/j/94469771032?pwd=d3JaSVhVV0xDOGpkUDdveXdOYmNXQT09

Title: Counting indecomposable G-bundles over a curve

Abstract: In an influential 1980 paper of Victor Kac, he proved (among many other things) that the number of absolutely indecomposable representations of a quiver over a finite field behaves like point-counting on a variety over F_q. This variety has been made precise by Crawley-Boevey and Van den Bergh using deformed preprojective algebras. 

A parallel problem is to count absolutely indecomposable vector bundles on a curve over a finite field. About 10 years ago, Schiffmann proved that the number of such (with degree coprime to the rank) is equal to the number of stable Higgs bundles of the same rank and degree (up to a power of q). Dobrovolska, Ginzburg and Travkin gave a slightly different formulation of this result and a very different proof. Neither argument obviously generalizes to G-bundles for other reductive groups G.

In joint work with Konstantin Jakob, we generalize the above-mentioned results to G-bundles. Namely, we show that the number of absolutely indecomposable G-bundles (suitably defined) on a curve over a finite field can be expressed using the number of stable (parabolic) G-Higgs bundles on the same curve.