Filtration patterns for representations of algebraic groups and their Frobenius kernels (Q1080952)

Let T be a maximal torus contained in a Borel subgroup B of a simply connected semisimple algebraic group G over an algebraically closed field of characteristic \(p\neq 0\). For a character \(\lambda\) on T, let \({\mathcal L}(\lambda)\) be the corresponding induced line bundle on the homogeneous space G/B, (or on \(TG_ n/TB_ n\), where \(G_ n\) is the kernel of the nth power of the Frobenius map on G). In a recent paper, J.-C. Ye determined the set of composition factors of the \(TG_ 1\) module \(H^ 0(TG_ 1/TB_ 1,{\mathcal L}(\lambda))\) associated to any p-regular weight \(\lambda\). We extend this result to arbitrary weight \(\lambda\). We then determine the set of composition factors for \(H^ 0(TG_ n/TB_ n,{\mathcal L}(\lambda))\), for \(n>1\), and show that their multiplicities in a composition series for \(H^ 0(TG_ n/TB_ n,{\mathcal L}(\lambda))\) would be determined if those multiplicities for \(n=1\) were known. We also relate the multiplicities of the composition factors of \(H^ 0(G/B,{\mathcal L}(\lambda))\) to those of \(H^ 0(TG_ n/TB_ n,{\mathcal L}(\lambda))\), extending results of Jantzen and Chastkofsky.