On conjugating representations and adjoint representations of semisimple groups (Q1100278)

Let G be a reductive algebraic group over an algebraically closed field k and let G act (as k-algebra automorphisms) on a finitely generated k- algebra A. Suppose the algebra of invariants, \(C=A\) G, is a free polynomial k-algebra and that A is flat as a C-module. Then the author proves that a good filtration of A in a natural way gives rise to an ascending (C,G)-filtration of A whose quotients have the form \(E_{\lambda}\otimes C\), \(\lambda\) dominant weight (w.r.t. appropriate maximal torus and Borel subgroup). Here \(E_{\lambda}\) is a direct sum of (dual) Weyl modules with highest weight \(\lambda\) and C acts trivially on \(E_{\lambda}\). In particular, this result implies that A is a free C-module. Taking A to be the coordinate ring of G with the conjugation action (resp. the polynomial algebra on the Lie algebra of G with the adjoint action), the author obtains in this way generalizations to prime characteristics of \textit{R. W. Richardson}'s theorem [Invent. Math. 54, 229-245 (1979; Zbl 0424.20035)] (resp. \textit{B. Kostant}'s theorem [Am. J. Math. 85, 327-404 (1963; Zbl 0124.268)]). Key ingredients in the proof are the author's deep study of good filtrations [``Rational representations of algebraic groups: Tensor products and filtrations'', Lect. Notes Math. 1140 (1985; Zbl 0586.20017)] and the use of his ``weight bounding functors''.