Normality and smoothness of simple linear group compactifications

TL;DR

This paper characterizes when simple linear compactifications of complex semisimple algebraic groups are normal or smooth, using combinatorial criteria based on highest weights, and identifies Sp(2r) as uniquely admitting a smooth compactification among non-adjoint groups.

Contribution

It provides the first combinatorial characterizations of normality and smoothness for simple linear compactifications of complex semisimple groups.

Findings

Normality and smoothness are characterized by combinatorial conditions on highest weights.

Sp(2r) is the only non-adjoint simple group with a smooth compactification.

Abstract

If G is a complex semisimple algebraic group, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant GxG-compactifications which possess a unique closed orbit and which arise in a projective space of the shape P(End(V)), where V is finite dimensional rational G-module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of V. In particular, we show that Sp(2r) (with r > 0) is the unique non-adjoint simple group which admits a simple smooth compactification.