Support varieties of line bundle cohomology groups for SL3(k)SL3(k)

Let G=SL3(k)G=SL3(k) where k   is a field of characteristic p>0p>0 and let λ∈X(T)λ∈X(T) be any weight with corresponding line bundle L(λ)L(λ) on G/BG/B. In this paper we compute the support varieties for all modules of the form Hi(λ):=Hi(G/B,L(λ))Hi(λ):=Hi(G/B,L(λ)) over the first Frobenius kernel G1G1. The calculation involves certain recursive character formulas given by Donkin (cf. [5]) which can be used to compute the characters of the line bundle cohomology groups. In the case where λ is a p  -regular weight and M=Hi(λ)≠0M=Hi(λ)≠0 for some i, these formulas are used to show that any pth root of unity ζ is not a root of the generic dimension of M. To handle the case where λ is not p-regular, we employ techniques similar to those used by Drupieski, Nakano and Parshall (cf. [7]) to show that the module Hi(λ)Hi(λ) is not projective over G1G1 whenever it is nonzero and λ lies outside of the Steinberg block.