Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over $\Z_p$ for representations with p-small weights

TL;DR

This paper develops integral versions of Bernstein-Gelfand-Gelfand complexes and Kostant formulas for representations with p-small weights over p, providing foundational tools for understanding cohomology of nilpotent groups in this setting.

Contribution

It introduces p-integral and mod p versions of BGG complexes and Kostant formulas for p-small weights, extending classical results to integral and modular contexts.

Findings

Existence of p-integral BGG complexes as direct summands

Validity of p-integral and mod p Kostant formulas

Application to cohomology of nilpotent groups

Abstract

We show that for $p$small highest weight $\lambda$, 1) there is a $\Z_p$-integral version of the Bernstein-Gelfand-Gelfand complex, still a direct summand subcomplex of the standard complex for $V(\lambda)$ 2) Similarly, a $\Z_p$-integral (as well as a mod. p) version of Kostant formula holds true. This paper is a companion paper to the one by Mokrane-Tilouine (AG, subm. 12/12/00), where these results are requested.