On irreducible modules of twisted groups of Lie type (Q790254)

The author extends results on Chevalley groups [Math. Z. 175, 143-159 (1980; Zbl 0432.20008)] to twisted groups. Namely, let G be a finite group of Lie type of characteristic p. Let k be a field of characteristic p and M a simple kG-module. Theorem. If M is not a projective module then the Sylow p-subgroups S of G are vertices of M. Moreover, if k is a splitting field of M then the restriction of M to any S is a source of M. Recall that a vertex of M is a minimal subgroup H (defined up to conjugacy) such that every short sequence \(0\to R\to S\to M\to 0\) of G- modules which splits as a sequence of H-modules splits also as a sequence of G-modules. And, for a vertex S of a kG-module M, a k-module V is a source of M if M is a component of the kG-module induced from V to G. This result is somewhat stronger, and implies the older results of \textit{S. W. Dagger} [J. Lond. Math. Soc., II. Ser. 3, 21-29 (1971; Zbl 0213.305)] and \textit{J. E. Humphreys} [Math. Z. 119, 149-152 (1971; Zbl 0198.045)] on the structure of blocks of groups of Lie type.