  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.
