On the Endomorphism Algebra of the Steinberg Module

Abstract

Let $\bar{\mathfrak{g}} = {\mathfrak{g}_{0}} \otimes k((z)) \otimes k.c$ be an affine Kac–Moody Lie algebra over a field k of finite characteristic p and let $St = \Delta ((p - 1)\rho )$ be the Steinberg module. By contrast with the finite dimensional case, the Steinberg module is not irreducible [M3]. Indeed its endomorphism algebra E is a very large commutative algebra on which the group Г= Autk((Z)) acts. It is shown that E is Г-equivariantly isomorphic with the algebra of regular functions on the space of all $\mathfrak{h}_0^*$-valued rational one-forms Ω over Spec k((Z)) satisfying the following conditions:

• has a simple pole with residue-

• where is the Cartier operator