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.


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