Infinitesimal fixed points in modules with good filtration

Abstract

Let g be the Lie algebra of the connected semisimple algebraic group G defined over an algebraically closed field k of characteristic p, p > 0. If M is a G module with good filtration ([D]), we consider the module M g of fixed points under g as a module for the image of G under the Frobenius homomorphism G → G. This module is denoted (M g)[−1], cf. [J], and it has been conjectured that it also has a good filtration ([D]). We will give an example to show that this is too optimistic. In the other direction, we prove something stronger in rank 1. Namely, we show that if B is a Borel subgroup in SL2 or PSL2 and M is a B module with relative Schubert filtration (cf. [vdK]), then so is (Mb)[−1]. Some preliminary work was done at the University of Virginia, where I much enjoyed the hospitality of Brian Parshall and Leonard Scott.