Asian Journal of Mathematics

Volume 21 (2017)

Number 4

Schubert decompositions for ind-varieties of generalized flags

Pages: 599 – 630

DOI: http://dx.doi.org/10.4310/AJM.2017.v21.n4.a1

Authors

Lucas Fresse (CNRS, Institut Élie Cartan de Lorraine, Université de Lorraine, Vandoeuvre-lès-Nancy, France)

Ivan Penkov (Jacobs University, Bremen, Germany)

Abstract

Let $\mathbf{G}$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$ and $\mathbf{P} \subset \mathbf{G}$ be a splitting parabolic ind-subgroup. The ind-variety $\mathbf{G/P}$ has been identified with an ind-variety of generalized flags in [4]. In the present paper we define a Schubert cell on $\mathbf{G/P}$ as a $\mathbf{B}$-orbit on $\mathbf{G/P}$, where $\mathbf{B}$ is any Borel ind-subgroup of $\mathbf{G}$ which intersects $\mathbf{P}$ in a maximal ind-torus. A significant difference with the finite-dimensional case is that in general $\mathbf{B}$ is not conjugate to an ind-subgroup of $\mathbf{P}$, whence $\mathbf{G/P}$ admits many non-conjugate Schubert decompositions. We study the basic properties of the Schubert cells, proving in particular that they are usual finite-dimensional cells or are isomorphic to affine ind-spaces.

We then define Schubert ind-varieties as closures of Schubert cells and study the smoothness of Schubert ind-varieties. Our approach to Schubert ind-varieties differs from an earlier approach by H. Salmasian [12].

Keywords

classical ind-group, Bruhat decomposition, Schubert decomposition, generalized flag, homogeneous ind-variety

2010 Mathematics Subject Classification

14M15, 14M17, 20G99

Full Text (PDF format)

This project was supported in part by the Priority Program “Representation Theory” of the DFG (SPP 1388) and by DFG Grant PE980/6-1. L. Fresse acknowledges partial support through ISF Grant Nr. 882/10 and ANR Grants Nr. ANR-12-PDOC-0031 and ANR-15-CE40-0012. I. Penkov thanks the Mittag-Leffler Institute in Djursholm for its hospitality.

Received 2 July 2015

Published 25 August 2017