Character formulas and Frobenius subgroups of algebraic groups

Abstract

Let G be a connected semisimple algebraic group over a field of non-zero characteristic p, with a maximal torus T. On G we have the Frobenius endomorphism F induced by the pth power map on the underlying field. The (scheme-theoretic) kernel of F is the first Frobenius kernel G,, and the inverse image of T under F is the “Frobenius subgroup” G, T. The purpose of this paper is to study certain questions in the (rational) representation theory of G, T. Let B be a Bore1 subgroup of G which contains T. In [6], Cline, Parshall, and Scott obtained a result which describes the functor “induction from B to G” as the composite of functors of the form “induction from B to a minimal parabolic subgroup P” over a sequence of minimal parabolics coming from a given reduced expression of the long word in the Weyl group. In Section 5, we obtain the analogous result for G, T. In Section 6, we develop a theory of intertwines for a certain class of G, T-modules. This gives a direct proof of an extended strong linkage principle for G, T, independent of the strong linkage principle for G. (Originally, Jantzen [17] proved that strong linkage for G, T, and more generally for G, T, n > 1, follows from the strong linkage principle for G. 1 Recently we have applied this method to give a new proof of an extended strong linkage principle for G; see [ 121. From the results of Section 5 we can write down a formula for the image of one of the intertwines in our class of modules. (This also follows from results of Jantzen [16, 171.) Composing the intertwines in question, relative to the class of modules corresponding to a given character of T, gives a map whose image is the irreducible G, T-module of that highest weight. In case the character is a restricted dominant character, the image coincides with the irreducible G-module of that highest weight, by a result