On the cohomology of integral $p$-adic unipotent radicals

Abstract

Let $\mathrm{G}/F$ be a reductive split $p$-adic group and let $\mathrm{U}$ be the unipotent radical of a Borel subgroup. We study the cohomology with trivial ${\mathbb{Z}}_p$-coefficients of the profinite nilpotent group $N=\mathrm{U}({\mathcal{O}}_F)$ and its Lie algebra $\mathfrak{n}$, by extending a classical result of Kostant to our integral $p$-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.

24 pages. referee's suggestions addressed, theorem 4 slightly weakened