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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 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 proof 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,

1

# 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 proof 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,