Abstract: We extend the classification of irreducible finite dimensional representations of almost simple algebraic groups over an algebraically closed field of characteristic zero to certain non-connected groups $G$ where the component group is cyclic. We also extend some of Steinberg's results on the adjoint quotient $G\rightarrow T/W$ to these non-connected groups. These results are used to describe the geometry of $\theta$-conjugacy classes of $G^o$, where $\theta$ is an automorphism of the connected group $G^o$. As an application, we show 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 $G$.
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