Noncommutative Harmonic Analysis pp 375-393 | Cite as

# A localization argument for characters of reductive Lie groups: an introduction and examples

## Abstract

In this article I describe my recent geometric localization argument dealing with actions of *noncompact* groups which provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed in [Sch].

A corresponding problem in the compact group setting was solved by N. Berline, E. Getzler and M. Vergne in [BGV] by an application of the theory of equivariant forms and, particularly, the fixed point integral localization formula.

This localization argument seems to be the first successful attempt in the direction of building a similar theory for integrals of differential forms, equivariant with respect to actions of noncompact groups.

I will explain how the argument works in the *SL*(2, ℝ) case, where the key ideas are not obstructed by technical details and where it becomes clear how it extends to the general case. The general argument appears in [L].

I have made every effort to present this article so that it is widely accessible. Also, although characteristic cycles of sheaves is mentioned, I do not assume that the reader is familiar with this notion.

## Keywords

Irreducible Character South Pole Characteristic Cycle Character Formula Distinguished Triangle## Preview

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