Frobenius Splittings and B-modules (Tata Institute Lectures on Mathematics and Physics)

The course given by the author at the Tata Institute in 1992 explains the solution by O. Mathieu of some conjectures in the representation theory of arbitrary semisimple algebraic groups. The conjectures concern filtrations of (tensor products of) "standard" representations; while Demazure studied the representations by means of a particular resolutionof singularities of Schubert varieties, the method of Mathieu involves sheafcohomology and "Frobenius splittings" on a Demazure resolution in finite characteristic p. The notes explain all this, but assume some familiarity with algebraic varieties and semisimple algebraic groups (or Liegroups).