# Normality of the dual nilcone in positive characteristic

## TL;DR

This paper proves the normality of the dual nilcone in certain positive characteristics and extends representation theory results for $p$-adic Lie groups beyond very good characteristic.

## Contribution

It establishes the normality of the dual nilcone in characteristics not necessarily very good, broadening geometric understanding in positive characteristic.

## Key findings

- Dual nilcone is normal in certain non-very good characteristics.
- Extends Ardakov and Wadsley's results on $p$-adic Lie group representations.
- Shows canonical dimension is zero or at least half the orbit dimension under specific conditions.

## Abstract

Let $G$ be a connected semisimple algebraic group of adjoint type defined over an algebraically closed field $K$ of positive characteristic. The characteristic $p$ is very good for $G$ when $p$ is suitably large and, if $G$ is of type $A_n$, $p$ does not divide $n+1$. The majority of results concerning the geometric structure of algebraic groups in positive characteristic are valid only in very good characteristic. We demonstrate that the dual nilcone $\mathcal{N}^* \subseteq \mathfrak{g}^*$ is a normal variety in certain characteristics which are not very good for $G$. As an application, we extend the results of Ardakov and Wadsley on representations of $p$-adic Lie groups. Under further restrictions on the characteristic, we show that the canonical dimension of a coadmissible representation of a semisimple $p$-adic Lie group in a $p$-adic Banach space is either zero or at least half the dimension of a nonzero coadjoint orbit.