B-cohomology

Abstract

Let B be a Borel subgroup in a reductive algebraic group G over a field k. We study the cohomology H • (B, λ) of 1-dimensional B-modules λ. When char k = 0 there is an easy and well-known description of this cohomology whereas the corresponding problem in characteristic p > 0 is wide open. We develop some new techniques which enable us to calculate all such cohomology in degrees at most 3 when p is larger than the Coxeter number for G. Our methods also apply to the corresponding question for quantum groups at roots of unity.

Second cohomology groups for Frobenius kernels and related structures

“…First one can verify the statement directly for root systems of rank less than or equal to two (i.e., Φ = A 1 , A 1 × A 1 , A 2 , B 2 , G 2 ) by using the restriction on the allowable weights of H 1 (u, u * ) given in part (b) of Proposition 4.4.1. Alternatively, we can also appeal to [AR,Prop. 4.3] since our assumptions on p imply p > h for these small rank cases.…”

On liftings of projective indecomposable G(1)-modules

“…The subgroup scheme G (1) H ≤ G is isomorphic to the quotient of the external semi-direct product G (1) ⋊ H by G (1) ∩ H [8, I.6.2]. The conjugation action of G ⋄ on j G (1) (G (1) ) factors via π since U A acts trivally, therefore j H (H) acts on j G (1) (G (1) ) as H acts on G (1) . We then get a morphism…”

“…Second cohomology of B. We require the following computations for Bcohomology given by Bendel, Nakano, and Pillen [3], and also obtained by Andersen and Rian [1] when p > h. As we will often apply these results to Borel subgroups of Levi subgroups of G, we note that these formulas hold for arbitrary reductive groups for which the assumptions on the prime hold.…”

“…Let G be a simple simply connected algebraic group over an algebraically closed field k of characteristic p, with Frobenius kernel G (1) . A conjecture by Humphreys and Verma, at this point more than 40 years old, states that the projective indecomposable G (1) -modules (PIMs) should lift to G. Donkin later added to this by conjecturing that each PIM should come via restriction of a certain indecomposable tilting module for G. These conjectures are known to be true if p ≥ 2h − 2, where h is the Coxeter number of G, but remain open for general p (see Section 2.2).…”

Bounding the Dimensions of Rational Cohomology Groups

“…In this context we are able to exploit the existence of a Borel subgroup B in G (i.e., a maximal closed connected solvable subgroup in G) and a maximal torus T in B. In Section 2 we apply intricate calculations of Bendel, Nakano, and Pillen [5], Wright [27], and Andersen and Rian [2], summarized in Theorem 2.3.1, to prove for a finite-dimensional rational B-module M that…”