Cycle Intersection for SOp,q-Flag Domains

Faten Abu Shoga

doi:10.1155/2020/1527973

Abstract

A real form G0 of a complex semisimple Lie group G has only finitely many orbits in any given compact G-homogeneous projective algebraic manifold Z=G/Q. A maximal compact subgroup K0 of G0 has special orbits C which are complex submanifolds in the open orbits of G0. These special orbits C are characterized as the closed orbits in Z of the complexification K of K0. These are referred to as cycles. The cycles intersect Schubert varieties S transversely at finitely many points. Describing these points and their multiplicities was carried out for all real forms of SLn,ℂ by Brecan (Brecan, 2014) and (Brecan, 2017) and for the other real forms by Abu-Shoga (Abu-Shoga, 2017) and Huckleberry (Abu-Shoga and Huckleberry). In the present paper, we deal with the real form SOp,q acting on the SO (2n, C)-manifold of maximal isotropic full flags. We give a precise description of the relevant Schubert varieties in terms of certain subsets of the Weyl group and compute their total number. Furthermore, we give an explicit description of the points of intersection in terms of flags and their number. The results in the case of G/Q for all real forms will be given by Abu-Shoga and Huckleberry.

Highlights

Introduction to the Flags in ZRecall that the orthogonal group G SO(p, q) acts on the space of full flags by two orbits: closed orbit and open orbit. e closed orbit is the set of all maximally b-isotropic full flags F with respect to b, where the maximally b-isotropic full flag F is defined as a sequence of (2n + 1)-vector spaces: (5)such that dim Vi i, for all 0 ≤ i ≤ 2n, and V2n− i V⊥i, for 1 ≤ i ≤ n
Fix a Cartan involution θ : G0 ⟶ G0. e fixed point set of θ in G0 is a maximal compact subgroup of G0 denoted by K0, and K is its complexification
Since S ∩ D ≠ ∅ implies that S ∩ C0 ≠ ∅, the first goal of this paper is to determine which Schubert varieties S have nonempty intersection with C0. We describe this intersection in the case where S is of complementary dimension to C0. e Schubert varieties are determined by the elements of the Weyl group WI of a distinguished maximal torus TI in the Borel subgroup BI which fixes a certain base point in ccl. ese Schubert varieties are denoted by Sw where w ∈ WI

Summary

Background

Let G be a complex semisimple Lie group and B a Borel subgroup. e compact algebraic homogeneous space Z. E Schubert varieties are determined by the elements of the Weyl group WI of a distinguished maximal torus TI in the Borel subgroup BI which fixes a certain base point in ccl. We choose a base point FS ∈ Fix(TS) It follows that the set C of closed Korbits can be identified with the Weyl group orbit WG(TS).FS (see [3], Corollary 4.2.4). Recall that the fixed point in the closed orbit ccl is the flag associated with the following ordered basis:. If wj − 2i is the first even number such that − 2i sits to the left of − (2i − 1), the relevant orbit is BI.〈e|wj| + e|2 w􏽥j|− 1〉 which contains points of the form α1􏼐e1 + e2q􏼑 + · · · + αq􏼐eq + eq+1􏼑 + αq+1􏼐eq − eq+1􏼑. For all b ∈ BI, the flag b(Fw) is not TS-fixed, and Sα ∩ Dα ∅. □

Introduction to the Combinatorics

Intersection Points of Schubert Duality