## On finite dimensional representations of non-connected reductive groups.(English)Zbl 0986.20043

The article is devoted to finite-dimensional representations of classical Lie groups over an algebraically closed field of characteristic zero.
A twisted analog of the adjoint quotient $$G\to T/W$$ is described, where $$T$$ is a Cartan subgroup of $$G$$ and $$W$$ is its Weyl group. It is shown that there is a functorial correspondence between virtual (finite-dimensional) characters of $$\theta$$-invariant representations of $$G$$ and virtual characters of an endoscopic group $$H$$ of $$G$$. Moreover, some of Steinberg’s results on the adjoint quotient $$G\to T/W$$ are extended to these groups. Dynkin diagrams are used for a description of irreducible representations.

### MSC:

 20G05 Representation theory for linear algebraic groups 22E46 Semisimple Lie groups and their representations
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