# The Order of the Centralizer of a Regular Element

## Author

**David G. A. Jackson**

## Status

Research Report 98-9

Date: 8 April 1998

## Abstract

It is well-known that, in the group of rational points of a connected
reductive algebraic group with connected centre, the centralizer of a
rational regular semisimple element is just the number of rational points
in a certain (twisted) rational maximal torus. The purpose of this paper
is to prove a formula expressing the order of the centralizer of a rational
regular (not necessarily semisimple) element in terms of the class function
of the Weyl group which associates to an element w the number of rational
points in a w-twisted rational maximal torus. The formula uses the notion of
truncation of class functions, introduced in the author's PhD thesis.

## Key phrases

algebraic groups. Lie algebras. regular elements. truncation.

## AMS Subject Classification (1991)

Primary: 20G40

Secondary: 17B45, 20C15, 20F55

## Content

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