Cohomology of induced representations for algebraic groups

Abstract

1.2. Lemma. Let M be a reduced algebraic group over K with Lie algebra m. Then grM is naturally isomorphic to m. Suppose charK=p+0 and denote the r th Frobenius kernel of M (respectively m) by M, (respectively mr). Then gr(Mr) is naturally isomorphic to mr for any r > O. Proof. In order to prove the first statement we have to show that the natural surjection from S(I/1 ~) onto grK[M] is an isomorphism. This however follows from [7], tI, Sect. 5, 2.1 or more explicitly from [26], Lemma t 1.4 and Remark on p. 108. In the second case observe that K[M,]--K[M]/K[M]I (q~ where q=pr and I (q) = {xqJx ~ I}. Similarly K[mr] ~-S(I/t2)/(I/I2) ~q~. It is now easily checked that the isomorphism K[m]~grK[M] already established induces an isomorphism K[m,]--*grK[M,], proving our claim.

The second cohomology of small irreducible modules for simple algebraic groups

“…For h < p, Kumar, Lauritzen and Thomsen [KLT99, Theorem 8] have extended a result of Andersen and Jantzen [AJ84,3.7]; this result implies in particular that the minimal degree for which H * (G 1 , L) is non-0 is (w 0 w i ), and that…”

The Second Cohomology of SimpleSL3-Modules

“…From now on we will drop the L and refer simply to r as the irreducible G-module of high weight r; we write L(0) = K to avoid confusion. We will often need to refer to specific parts of this tensor product, so we write r = r 0 ⊗r [1] 1 ⊗· · ·⊗r…”

“…The second main ingredient is the following proposition, which is a specialisation of the main theorem from [1] to the case G = SL 2 .…”

“…We use the main result of [1]: Let w ∈ W be an element of length l(w) of the Weyl group W = N G (T )/T of the reductive algebraic group G with Borel subgroup B and maximal torus T ≤ B. Let u denote the Lie algebra of the unipotent radical U of B. Then…”

“…Provided [1] and E 01 2 = K as required. If p = 2, recall by the linkage principle that r 0 = s 0 so that E 01 2 = Hom G ((s 1 ± 1) ⊗ r ′′ [1] , s ′ ), as observed already, giving r ′ = (s 1 ± 1) ⊗ s ′′ [1] and E 01 2 = K.…”