Normality and smoothness of simple linear group compactifications

Abstract

Given a semisimple algebraic group G, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant G x G-compactifications possessing a unique closed orbit which arise in a projective space of the shape P(End(V)), where V is a 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 >= 1) is the unique non-adjoint simple group which admits a simple smooth compactification

Kähler–Einstein metrics on group compactifications

“…It is a semisimple group of type B 2 , which means that its associated root system is the root system B 2 . While it is not an adjoint group, it also admits a wonderful compactification (see for example [GR13]). Let X 2 be the blow up of this wonderful compactification along the closed orbit.…”

Simple linear compactifications of odd orthogonal groups

“…Notice that λ depends only on Supp(λ) and that it coincides with the usual dominance order if λ is a regular weight. The partial order λ was studied in the general semisimple case by Gandini and Ruzzi in [6], where it is used to characterize the normality of a simple linear compactification of a semisimple group. In particular, there are proved the following properties.…”

“…The variety X Π was studied by Timashev in [12]: there are studied the local structure and the G × G-orbit structure, and normality and smoothness are characterized as well. The conditions of normality in particular rely on some properties of the tensor product, and together with the conditions of smoothness they were remarkably simplified by Bravi, Gandini, Maffei, Ruzzi in [3] in case X Π is simple and adjoint, and by Gandini, Ruzzi in [6] in case X Π is simple. In particular, in [3] it was shown that every simple adjoint linear compactification is normal if G is simply laced, whereas several examples of non-normal simple adjoint linear compactifications arise in the non-simply laced case.…”

“…In case λ is regular, then λ coincides with the usual dominance order , while if λ = 0 then λ is the trivial order. In the general case of a (possibly non-adjoint) simple subset Π, the partial order λ was used in [6] to characterize combinatorially the normality of the variety X Π .…”

“…Since we will deal only with adjoint subsets, for simplicity we will refer to adjoint simple subsets just as simple subsets. For a general treatment on the case of a possibly non-adjoint simple linear group compactification see [6].…”