# Kempf Vanishing theorem and Representation of Lie algebras.

Let $G$ be a reductive connected algebraic group and let $B$ a Borel subgroup. One of central themes of the representation theory of $G$ is the study of the induction functor $H^0$ from $B$ representations to $G$ representations. Many of the features of $H^0$ in the characteristic zero case also hold in the modular case. On the other hand the Borel-Weil-Bott theorem fails in general in the modular case and hence the simplicity of $H^0(λ)$ also breaks down in general. Still, we consider the $H^0(λ)$’s to be the fundamental objects of study, the reason being tha their characters, like in the characteristic zero case, are given by the Weyl character formula. This fact in turn relies on the Kempf Vanishing theorem, i.e. $$H^i(λ)=0 \text{ for } i>0 \text{ and } λ∈P^+.$$

Beside this introduction, in the realm of representation theory of Lie algebras, I've heard few times that Kempf's Vanishing theorem is used to prove some universal properties of Weyl modules in the family of finite-dimensional highest-weight modules.

QUESTION: How is used this Kempf theorem to get this conclusions about universality?

Thanks,

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As usual the history is a little complicated to track, but it should be understood first that the term Weyl module and notation $V(\lambda)$ were first used to describe the module obtained by a natural reduction modulo $p$ process from the usual finite dimensional simple module (of dominant highest weight $\lambda$). Here one uses integral bases derived by Steinberg from a Chevalley basis of the relevant complex Lie algebra. From the construction one knows that the formal character of $V(\lambda)$ is given by Weyl's formula.

On the other hand, there are intrinsic finite dimensional modules $H^i(G/B, \mathcal{L}(\lambda))$ for the algebraic group $G$ in characteristic $p$ given in terms of a line bundle for the weight $\lambda$ (or its negative depending on conventions about $B$ and positive roots). For dominant weights Kempf's theorem now implies that only $H^0$ survives; its Euler character is then the Weyl character for that weight, so the dimension of the module is the same as that of the corresponding Weyl module.

The key point to establish is that the Weyl module is the Serre dual of such a module of global sections, by naturally identifying it with the top cohomology relative to the weight "linked" to $\lambda$ by the longest element of the Weyl group. This follows by dimension comparison because there is a natural embedding of $V(\lambda)$ (or any other "highest weight module") into this highest cohomology. So the universal property of $V(\lambda)$ follows indirectly from Kempf's vanishing theorem.

This was first written down as Satz 1 in a 1977 paper by J.C. Jantzen in J. Reine angew. Math.. His book Representations of Algebraic Groups (AMS 2003 edition) has a more systematic treatment, with Kempf's theorem proved in Chapter II.4 and with "Weyl modules" redefined as the top cohomology groups. So the universal property is a bit hidden in the background.

One other comment on the question: the word "hence" is inappropriate, since the failure of Borel-Weil-Bott is not directly the reason for Weyl modules to be in most cases non-simple. Here you have to get into the linkage principle and other parts of the development, to see the analogy with (infinite dimensional) Verma modules in characteristic 0 and the Kazhdan-Lusztig conjecture, etc.

P.S. I didn't want to reproduce the short argument in Jantzen's paper because of the special notation needed, but the embedding idea (which goes back at least to Chevalley) involves realizing global sections of a line bundle inside the infinite dimensional algebra of regular functions on the algebraic group. Then the classical ideas about "representative functions" on a group come into play for finite dimensional representations.

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