---
abstract: 'If $G$ is a complex simply connected semisimple algebraic group and if ${\lambda}$ is a dominant weight, we consider the compactification $X_{\lambda}\subset {\mathbb P}\big(\operatorname{End}({V({{\lambda}})})\big)$ obtained as the closure of the $G\times G$-orbit of the identity and we give necessary and sufficient conditions on the support of ${\lambda}$ so that $X_{\lambda}$ is normal; as well, we give necessary and sufficient conditions on the support of ${\lambda}$ so that $X_{\lambda}$ is smooth.'
author:
- 'Paolo Bravi, Jacopo Gandini, Andrea Maffei, Alessandro Ruzzi'
title: 'Normality and non-normality of group compactifications in simple projective spaces'
---

Introduction {#introduction .unnumbered}
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Consider a semisimple simply connected algebraic group $G$ over an algebraically closed field ${\Bbbk}$ of characteristic zero. If ${\lambda}$ is a dominant weight (with respect to a fixed maximal torus $T$ and a fixed Borel subgroup $B\supset T$) and if ${V({{\lambda}})}$ is the simple $G$-module of highest weight ${\lambda}$, then $\operatorname{End}\big({V({{\lambda}})}\big)$ is a simple $G\times G$-module. Let $I_{\lambda}\in \operatorname{End}\big({V({{\lambda}})}\big)$ be the identity map and consider the variety $X_{\lambda}\subset {\mathbb P}\big(\operatorname{End}({V({{\lambda}})})\big)$ given by the closure of the $G\times G$-orbit of $[I_{\lambda}]$. In [@Ka], S. Kannan studied for which ${\lambda}$ this variety is projectively normal, and this happens precisely when ${\lambda}$ is minuscule. In 