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

الأصل · EN

Let G= SL₃(k) where k is a field of characteristic p > 0 and let λ∈ X(T) be any weight with corresponding line bundle L(λ) on G/B. In this paper we compute the support varieties for all modules of the form Hⁱ(λ):= Hⁱ(G/B, L(λ)) over the first Frobenius kernel G₁. The calculation involves certain recursive character formulas given by Donkin 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)≠ 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 to show that the module Hⁱ(λ) is not projective over G₁ whenever it is nonzero and λ lies outside of the Steinberg block.