Gerbes on complex reductive Lie groups

TL;DR

This paper constructs a geometric gerbe over complex reductive Lie groups using invariant bilinear forms, exploring their properties and restrictions to subgroups, with implications for understanding conjugation actions and orbit triviality.

Contribution

It introduces a new construction of gerbes on complex reductive Lie groups based on geometric methods and invariant forms, extending previous work by Deligne and others.

Findings

Gerbes are constructed using the Grothendieck manifold.

The gerbe's restriction to semisimple orbits may be non-trivial.

The paper clarifies the relation between different gerbe formalisms.

Abstract

We construct a gerbe over a complex reductive Lie group G attached to an invariant bilinear form on a maximal diagonalizable subalgebra which is Weyl group invariant and satisfies a parity condition. By restriction to a maximal compact subgroup K, one then gets a gerbe over K. For a simply-connected group, the parity condition is the same used by Pressley and Segal; in general, it was introduced by Deligne and the author. The gerbe is defined by geometric methods, using the so-called Grothendieck manifold. It is equivariant under the conjugation action of G; its restriction to a semisimple orbit is not always trivial. The paper starts with a discussion of gerbe data (in the sense of Chatterjee and Hitchin) and of gerbes as geometric objects (sheaves of groupoids); the relation between the two approaches is presented. There is an Appendix on equivariant gerbes, discussed from both points of view.