Simple linear compactifications of odd orthogonal groups

Abstract

We classify the simple linear compactifications of SO(2r + 1), namely those compactifications with a unique closed orbit which are obtained by taking the closure of the SO(2r + 1) x SO(2r + 1)-orbit of the identity in a projective space P(End(V)), where V is a finite dimensional rational SO(2r + 1)-module

Log homogeneous compactifications of some classical groups

“…Recently, Timashev [28], Gandini and Ruzzi [11], found combinatorial criterions for certain linear compactifications to be normal, or smooth. In [10], Gandini classifies the linear compactifications of the odd special orthogonal group having one closed orbit. By a very new and elegant approach, Martens and Thaddeus [22] recently discovered a general construction of the toroidal compactifications of a connected reductive group G as the coarse moduli spaces of certain algebraic stacks parametrizing objects called "framed principal G-bundles over chain of lines".…”

Normality and smoothness of simple linear group compactifications

“…In the case of an odd orthogonal group, a combinatorial classification of its simple linear compactifications was given by the first author in [10] by means of a partial order tightly related to λ , which makes use only of the positive long roots which are non-orthogonal to the dominant weight λ. A similar classification should be expectable in the case of any (non simply-laced) semisimple group by using similar partial orders.…”

“…The paper is organized as follows. In Section 1 we introduce the variety X Π and we give some preliminary results: almost all of these results come from [3] and [10] and although some of them are claimed in a more general form than the original ones, the proofs are substantially the same. In Section 2 we introduce the partial orders λ and λ Q .…”

“…Following lemma and proposition were given in [10] in the case of a simple linear adjoint compactification, however their proofs generalize straightforwardly to the non-adjoint case.…”