Lectures on Frobenius splittings and B-modules

Preface These notes are based on a course given at the Tata Institute of Fundamental Research in the beginning of 1990. The aim of the course was to describe the solution by O. Mathieu of some conjectures in the representation theory of semi-simple algebraic groups. These conjectures concern the inner structure of dual Weyl modules and some of their analogues. Recall that if G is a (connected, simply connected) semi-simple complex Lie group and B a Borel subgroup, the Borel–Weil–Bott Theorem tells that one may construct the finite dimensional irreducible G-modules in the following way. Take a line bundle L on the generalized flag variety G/B, such that H 0 (G/B, L) does not vanish. Then H 0 (G/B, L) is an irreducible G-module, called a dual Weyl module or an " induced module " , and by varying L one gets all finite dimensional irreducibles. More generally one may, after Demazure, consider the B-modules H 0 (BwB/B, L) with L as above. (So one still requires that H 0 (G/B, L) does not vanish.) The " Demazure character formula " determines the character of H 0 (BwB/B, L). It was shown by P. Polo that the B-module H 0 (BwB/B, L) has a nice homological characterization in terms of its highest weight λ (see 3.1.10). We therefore use the notation P (λ) for this module. The P (λ) are generalizations of dual Weyl modules. Indeed recall that nothing is lost when restricting a rational module from G to B; inducing back up from B to G one recovers the original module (see 2.1.7). Now the conjectures are about filtrations of the dual Weyl modules H 0 (G/B, L) or their generalizations P (λ), for semi-simple algebraic groups in arbitrary characteristic. (Over the integers, actually.) The strongest conjecture of the series is Polo's conjecture, which says that if one twists a P (λ) by an anti-dominant character the resulting B-module can be filtered with subsequent quotients P (µ i). In Polo's terminology—which we will follow— the twisted module has an excellent filtration. (In Mathieu's terminology the i ii Preface twisted module is strong.) This conjecture, now a theorem of Mathieu, has many nice consequences. For instance, suppose one takes a semi-simple subgroup L of G corresponding with a subset of the set of simple roots. Then if one restricts the representation P (λ) from B to B ∩ L, that …