Orders of finite algebraic groups

K. F. Lai

Vol. 97 (1981), No. 2, 425–435

DOI: 10.2140/pjm.1981.97.425

Abstract

Let Ḡ be a simply connected simple algebraic group over a finite field F_q of q elements. The order of the group Ḡ(F_q) of F_q-rational points of Ḡ is well-known (cf: Steinberg, Carter). The proof makes use of the Bruhat decomposition and the study of polynomials invariant under the action the Weyl group. In this paper we deduce the order of Ḡ(F_q) from an explicit formula for the integral M(s,Λ) which occurs in Langlands’ theory of Eisenstein series.

Mathematical Subject Classification 2000

Primary: 20G40

Secondary: 10D40

Milestones

Received: 2 May 1979

Published: 1 December 1981