On the cohomology of integral p-adic unipotent radicals

Abstract

Let G be a reductive split p-adic group and let 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.

Keywords:

• p-adic groups
• cohomology

MATHEMATICS SUBJECT CLASSIFICATION:

• 20G25
• 20J06

Acknowledgments

This paper owes a debt of gratitude to my advisor Akshay Venkatesh, who encouraged me to pursue this project to extend my thesis’s results to a larger generality and suggested the correct strategy to approach the problem. I also want to thank Nivedita Bhaskhar, Rita Fioresi and Mihalis Savvas for helpful conversations, and an anonymous referee for suggesting some improvements.

Notes

1 In other words, $V_{{\mathcal{O}}_F}\left(\lambda \right)\cong {\mathcal{O}}_F$ with $\mathrm{T}\left({\mathcal{O}}_F\right)$ acting via λ.

2 Our notion for Weil restriction of Lie algebras follows that of Oesterle in [21], proposition A.3.3: of the ${\mathcal{O}}_F$-module $\mathfrak{g}$ we only remember its structure as a ${\mathbb{Z}}_p$-module, and the bracket operation is also seen as a ${\mathbb{Z}}_p$-bilinear map.

3 In other words, $V_{{\mathcal{O}}_F}\left(\lambda \right)\cong {\mathcal{O}}_F$ with $\mathrm{T}\left({\mathcal{O}}_F\right)$ acting via λ.

4 For a filtered A-module M, the shift $M^{\left(h\right)}$ is the filtered A-module defined by $F^n{M}^{\left(h\right)}=F^{n-h}M.$

5 For a graded B-module M, the shift $M^{\left(h\right)}$ is the graded B-module defined by ${\left(M^{\left(h\right)}\right)}_n=M_{n-h}.$

6 Equivalently, a (filtered) $\Lambda ♯A$-module is a (filtered) A-module M with a Λ-action (preserving the filtration) such that $\lambda (a.m)=\lambda \left(a\right).\lambda \left(m\right)$ for all $a\in A,\lambda \in \Lambda$ and $m\in M.$