Schubert decompositions for ind-varieties of generalized flags

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

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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