2009 Algebraic Geometry Seminar - 04

We prove a conjecture of Kottwitz and Rapoport formulated in terms of root systems and use it to imply the vanishing of higher cohomology groups for certain line bundles on toric varieties associated with root systems. During the proof, we will also discuss the so-called numbers game with a cutoff. These results are also relevant to a converse to Mazur's inequality, which will be discussed on Friday at the Algebraic Groups, Galois Cohomology and Related Topics Seminar. (The part of the talk about the numbers game is joint work with T. Schedler).