Toroidal Schubert Varieties
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Toroidal Schubert Varieties Mahir Bilen Can1
· Reuven Hodges2 · Venkatramani Lakshmibai3
Received: 3 August 2019 / Accepted: 12 August 2019 / © Springer Nature B.V. 2019
Abstract Levi subgroup actions on Schubert varieties are studied. In the case of partial flag varieties, the horospherical actions are determined. This leads to a characterization of the toroidal and horospherical partial flag varieties with Picard number 1. In the more general case, we provide a set of necessary conditions for the action of a Levi subgroup on a Schubert variety to be toroidal. The singular locus of a (co)minuscule Schubert variety is shown to contain all the Lmax -stable Schubert subvarieties, where Lmax is the standard Levi subgroup of the maximal parabolic which acts on the Schubert variety by left multiplication. In type A, the effect of the Billey-Postnikov decomposition on toroidal Schubert varieties is obtained. Keywords Toroidal Schubert varieties · Horospherical actions · Billey-Postnikov decomposition Mathematics Subject Classification (2010) 14M15 · 14M27
1 Introduction This paper was first motivated by the following two questions. • •
Which Schubert varieties are toric varieties? More generally, which Schubert varieties are spherical varieties?
On the one hand, it has been known for some time that, except for a few trivial cases, such as the projective space, in general, Schubert varieties are not toric varieties. Nevertheless, one knows that every Schubert variety has a degeneration to a toric variety, see [10]. On the other hand, less is known with regard to the second question. Let us briefly describe our Presented by: Peter Littelmann Mahir Bilen Can
[email protected] Reuven Hodges [email protected] Venkatramani Lakshmibai [email protected]
1
Tulane University, New Orleans, LA USA
2
The University of Illinois at Urbana-Champaign, Urbana, IL, USA
3
Northeastern University, Boston, MA, USA
M.B. Can et al.
current state of knowledge. Let G be a connected reductive complex algebraic group and let P be a parabolic subgroup of G. We fix a Borel subgroup B of G that is contained in P . Clearly, a Schubert variety X in G/P is not G-stable unless the equality X = G/P holds true. Therefore, we look for the reductive subgroups of G which act on X. Some natural candidates are given by the Levi subgroups of the stabilizer subgroup Q := StabG (X). At one extreme, in the Grassmann variety of k dimensional subspaces of Cn , denoted by Gr(k, n), we know the complete classification of the Schubert varieties which are spherical with respect to the action of a Levi subgroup of GLn = GLn (C), see [14]. At the other extreme, if we assume that X is a smooth Schubert variety in a partial flag variety G/P , where G does not have any G2 -factors, then we have by [11] that X is a spherical Lmax variety, Lmax being the Levi factor of the stabilizer subgroup in G of X. By definition, a spherical G-variety X is called toroidal if in X every B-stable irreducible divisor which contains a G-orbit is G-stable. It turns o
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