On the Iwahori-Weyl group

2013/10/17 by Richarz, Timo
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1310.4635

Abstract

Let F be a discretely valued complete field with valuation ring O_F and perfect residue field k of cohomological dimension ≤ 1. In this paper, we generalize the Bruhat decomposition in Bruhat and Tits from the case of simply connected F-groups to the case of arbitrary connected reductive F-groups. If k is algebraically closed, Haines and Rapoport define the Iwahori-Weyl group, and use it to solve this problem. Here we define the Iwahori-Weyl group in general, and relate our definition of the Iwahori-Weyl group to that of Haines and Rapoport. Furthermore, we study the length function on the Iwahori-Weyl group, and use it to determine the number of points in a Bruhat cell, when k is a finite field.

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