A generalization of Kostant theorem to integral cohomology

TL;DR

This paper extends Kostant's theorem to integral cohomology, analyzing weight decompositions and ranks in the (co)homology of semi-simple Lie algebra systems, with implications for characteristic p fields.

Contribution

It generalizes Kostant's theorem to integral cohomology, providing new insights into weight decomposition and rank behavior over various fields.

Findings

Integral (co)homology weight decomposition established.

Cohomology vanishes over fields of characteristic p if weight rank isn't divisible by p.

Generalization of Kostant's theorem to integral cohomology achieved.

Abstract

In this paper, we find weight decomposition and rank of a weight in the integral (co)homology of the positive system of a semi-simple Lie algebra over $\Bbb C$ and prove that the (co)homology of the weight subcomplex over a field of characteristic p is 0 if the rank of the weight is not divisible by p. This generalizes Kostant theorem to the integral cohomology of the positive system.