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Generalized Deligne-Lusztig characters. (English) Zbl 0812.20024
Let $${\mathbf G}^ 0$$ be a connected reductive algebraic group defined over the algebraic closure of a finite field $${\mathbf F}_ q$$ and let $$\sigma$$ be an outer automorphism of $$\mathbf G^ 0$$. Form the semi-direct product $${\mathbf G} = {\mathbf G}^ 0 \cdot \langle \sigma\rangle$$ and consider the finite group $${\mathbf G}^ F$$ where $$F$$ is a Frobenius endomorphism. As the reductive group $$\mathbf G$$ is not connected, one can not immediately apply Deligne-Lusztig theory to obtain characters of $${\mathbf G}^ F$$. Therefore the author tries to extend the theory to this disconnected case. He also studies several small rank cases in more detail. He finds similarities to and differences from the connected case.

##### MSC:
 20G05 Representation theory for linear algebraic groups 20G40 Linear algebraic groups over finite fields
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