The Euler character and cancellation theorems for Weyl modules (Q1115974)

Let G be a simple, simply connected algebraic group over an algebraically closed field of characteristic p and B be a Borel subgroup of G, \(G_ n\) be the kernel of the n-th power of the Frobenius map on G. Let \(H^ 0(G/B,\lambda)\) be the G-module induced from a character \(\lambda\) on B, \(H^ 0(G_ nB/B,\lambda)\) the \(G_ nB\)-module induced from \(\lambda\) and \(H^ 0(G/G_ nB,M)\) the G-module induced from a \(G_ nB\)-module M. The author asks whether the socle levels of \(H^ 0(G/B,\lambda)\) for \(\lambda\) in singular position can be computed in terms of the socle levels of \(H^ 0(G/B,\mu)\) for \(\mu\) in general position. Qualitatively, the author shows that the socle levels of \(H^ 0(GB/B,\lambda)\) are induced from the \(G_ n\)-socle levels of \(H^ 0(G_ nB/B,\lambda)\) assuming only that Bott's Theorem applies to the composition factors of \(H^ 0(G_ nB/B,\lambda)\) and shows that the multiplicities of the composition factors of the G-module induced from a \(G_ n\)-socle layer M of \(H^ 0(G_ nB/B,\lambda)\) are determined by the \(G_ nB\)-structure of the \(G_ n\)-semisimple module M.