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 o
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