[Paper Review] Ind--varieties of generalized flags as homogeneous spaces for classical ind--groups

Ivan Dimitrov, Ivan Penkov|ArXiv.org|Mar 26, 2004

Advanced Algebra and Geometry4 references4 citations

TL;DR

This paper introduces generalized flags in infinite-dimensional vector spaces as chains that need not be indexed by integers, and constructs ind-varieties of $E$-commensurable generalized flags as geometric realizations of homogeneous spaces for classical ind-groups $SL(\infty)$, $SO(\infty)$, and $Sp(\infty)$. The key contribution is a geometric classification of $G/P$ spaces for these ind-groups via generalized flags, with a complete computation of the Picard group and a criterion for projectivity: the ind-variety is projective if and only if the generalized flag is a standard flag.

ABSTRACT

The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space $V$ (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of homogeneous spaces of the ind--groups $SL(\infty)$, $SO(\infty)$ and $Sp(\infty)$ in terms of generalized flags. Generalized flags in $V$ are chains of subspaces which in general cannot be enumerated by integers. Given a basis $E$ of $V$, we define a notion of $E$--commensurability for generalized flags, and prove that the set $\cFl (\cF, E)$ of generalized flags E$--commensurable with a fixed generalized flag $\cF$ in $V$ has a natural structure of an ind--variety. In the case when $V$ is the standard representation of $G = SL(\infty)$, all homogeneous ind--spaces $G/P$ for parabolic subgroups $P$ containing a fixed splitting Cartan subgroup of $G$, are of the form $\cFl (\cF, E)$. We also consider isotropic generalized flags. The corresponding ind--spaces are homogeneous spaces for $SO(\infty)$ and $Sp(\infty)$. As an application of the construction, we compute the Picard group of $\cFl (\cF, E)$ (and of its isotropic analogs) and show that $\cFl (\cF, E)$ is a projective ind--variety if and only if $\cF$ is a usual, possibly infinite, flag of subspaces in $V$.

Motivation & Objective

• To extend the classical notion of a flag in a finite-dimensional vector space to infinite-dimensional settings by introducing generalized flags that are not necessarily indexed by integers.
• To provide a geometric construction of homogeneous ind-spaces $G/P$ for classical ind-groups $SL(\infty)$, $SO(\infty)$, and $Sp(\infty)$ using generalized flags.
• To define $E$-commensurability between generalized flags and endow the set of $E$-commensurable flags with a natural ind-variety structure.
• To compute the Picard group of the ind-variety of generalized flags and establish a criterion for its projectivity.
• To show that the ind-variety of generalized flags is projective if and only if the flag is a standard (possibly infinite) flag.

Proposed method

• Introduce generalized flags as chains of subspaces in an infinite-dimensional vector space $V$ that are not necessarily indexed by integers, satisfying specific density and adjacency conditions.
• Define $E$-commensurability between generalized flags using a fixed basis $E$ of $V$, where two flags differ only in a finite-dimensional subspace.
• Construct the ind-variety $\mathcal{F}\ell(\mathcal{F},E)$ as the set of all generalized flags $E$-commensurable with a fixed generalized flag $\mathcal{F}$, equipped with an ind-variety structure via direct limits of finite-dimensional flag varieties.
• Prove that all homogeneous ind-spaces $G/P$ for $G = SL(\infty)$, $SO(\infty)$, or $Sp(\infty)$ and parabolic subgroups $P$ containing a fixed splitting Cartan subgroup are isomorphic to $\mathcal{F}\ell(\mathcal{F},E)$ for some generalized flag $\mathcal{F}$.
• Define isotropic generalized flags in the presence of a non-degenerate symmetric or skew-symmetric form, and construct corresponding ind-variety models for $SO(\infty)$ and $Sp(\infty)$.
• Use homomorphisms $\varphi_n$ and $\varphi$ from a product of $\mathbb{Z}$-copies to the Picard groups of finite-dimensional approximations to compute $\operatorname{Pic}(\mathcal{F}\ell(\mathcal{F},E))$ and $\operatorname{Pic}(\mathcal{F}\ell(\mathcal{F},w,E))$ via kernel analysis.

Experimental results

Research questions

• RQ1 Can homogeneous spaces for classical ind-groups $SL(\infty)$, $SO(\infty)$, and $Sp(\infty)$ be geometrically realized as ind-varieties of generalized flags?
• RQ2 What is the structure of the Picard group of the ind-variety of $E$-commensurable generalized flags?
• RQ3 Under what conditions is the ind-variety of generalized flags projective?
• RQ4 How does the notion of $E$-commensurability allow for a classification of these ind-variety models?
• RQ5 What is the relationship between the geometric structure of the generalized flag and the projectivity of the associated ind-variety?

Key findings

• The ind-variety $\mathcal{F}\ell(\mathcal{F},E)$ of $E$-commensurable generalized flags has a natural ind-variety structure, and every homogeneous ind-space $G/P$ for $G = SL(\infty)$, $SO(\infty)$, or $Sp(\infty)$ arises as such a space.
• The Picard group of $\mathcal{F}\ell(\mathcal{F},E)$ is isomorphic to $\left(\prod_{F' \in \mathcal{F}'} \mathbb{Z} \gamma_{F'} \right) / \left( \mathbb{Z} \prod_{F' \in \mathcal{F}'} \gamma_{F'} \right)$, reflecting a structure analogous to the classical case.
• The Picard group of the isotropic ind-variety $\mathcal{F}\ell(\mathcal{F},w,E)$ is isomorphic to $\prod_{F' \in \mathcal{F}', F' \subset \tau(F')} \mathbb{Z} \gamma_{F'}$, where $\tau$ is the duality involution.
• The ind-variety $\mathcal{F}\ell(\mathcal{F},E)$ is projective if and only if $\mathcal{F}$ is a standard flag, i.e., a chain indexed by a well-ordered set with consecutive successors and predecessors.
• If $\mathcal{F}$ is a flag of finite length, then $\mathcal{F}\ell(\mathcal{F},E)$ is not isomorphic to $\mathcal{F}\ell(\mathcal{G},L)$ for any generalized flag $\mathcal{G}$ of different length, due to non-isomorphic Picard groups.
• The ind-variety $\mathcal{F}\ell(\mathcal{F},E)$ is not isomorphic to any Grassmannian $Gr(l;V)$ if $\mathcal{F} = \{0 \subset F \subset V\}$ with $F$ infinite-dimensional and of infinite codimension in $V$.