The cohomology of $p$-adic distribution representations

TL;DR

This paper extends Kostant's theorem to certain infinite-dimensional $p$-adic distribution representations, developing new Banach representation theory and constructing eigen orthonormalizable weight completions.

Contribution

It generalizes Lie algebra cohomology results to infinite-dimensional $p$-adic contexts using novel Banach representation techniques.

Findings

Generalized Kostant's theorem to $p$-adic distribution representations

Developed eigen orthonormalizable Banach representations over affinoid algebras

Constructed eigen orthonormalizable weight completions of distribution representations

Abstract

We give a generalization of Kostant's theorem on Lie algebra cohomology of finite dimensional highest weight representations to some infinite dimensional cases over a $p$-adic family of highest weight distribution representations. For proving this, we develop a theory of eigen orthonormalizable Banach representations of $p$-adic torus over an affinoid algebra, and we construct an eigen orthonormalizable weight completion of the distribution representations.