Dedicated to Claudio Procesi on the occasion of his 60th birthday
In this article we discuss an expansion formula for the character of the cohomology of an induced bundle on G/B, where G is a semisimple, simply connected algebraic group over an algebraically closed field k of characteristic p > 0 and B is a Borel subgroup. In its basic form this is an easy consequence of the degeneration of the two spectral sequences in Section 12.2 of part II of Jantzen's book [12]. However, in this form, the expansion involves choosing G-modules which are projective on restriction to the first infinitesimal subgroup G 1 of G. It therefore seems worthwhile to recast the formula in a way that is independent of choice and this we do by expressing the result in terms of certain operations on a quotient group rep(B) of the representation ring of B. At this stage we have an expansion formula which involves, for Ξ» a restricted weight, the characters of the irreducible modules L(Ξ») and injective indecomposable modules Q 1 (Ξ») for G 1 T , where T is a maximal torus of G. Since these characters are not usually explicitly known it also seems worthwhile to derive a more general version of the formula in which the characters of L(Ξ») and Q 1 (Ξ») are replaced by any pair of "p-dual bases" Ο Ξ» , Ο Ξ» . The most general form of the expansion is then the formula