Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups
Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups
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作为经典 ind 群的同质空间的广义标志的 ind 变种
DOI：
10.1155/s1073792804140828
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发表时间：
2004
影响因子：
1
通讯作者：
I. Penkov
中科院分区：
文献类型：
--
作者：
I. Dimitrov;I. Penkov
本文的目的是双重的，在无限维向量空间V中引入广义标志的概念（推广了向量空间中子空间的标志的概念），并根据广义标志给出ind-group SL（∞），SO（∞）和Sp（∞）的齐性空间的几何实现。V中的广义标志是子空间的链，一般不能用整数枚举。给定V的一个基E，我们定义了广义标志的E-可扩性概念，并证明了在V中具有固定广义标志的广义标志E-可扩性的广义标志集E-可扩性具有一个ind-簇的自然结构.当V是G = SL（∞）的标准表示时，所有包含G的固定分裂Cartan子群的抛物子群P的齐次内空间G/P都具有形式φ l（φ，E）.我们还考虑了各向同性广义旗。相应的ind-空间是SO（∞）和Sp（∞）的齐性空间。作为该构造的应用，我们计算了Picard群，证明了Picard群是一个投射的ind-簇当且仅当Picard群是V中子空间的一个通常的、可能无限的标志。
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(∞), SO(∞), and Sp(∞) 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 ℱl(ℱ,E) of generalized flags E-commensurable with a fixed generalized flag ℱ in V has a natural structure of an ind-variety. In the case when V is the standard representation of G = SL(∞), all homogeneous ind-spaces G/P for parabolic subgroups P containing a fixed splitting Cartan subgroup of G are of the form ℱl(ℱ,E). We also consider isotropic generalized flags. The corresponding ind-spaces are homogeneous spaces for SO(∞) and Sp(∞). As an application of the construction, we compute the Picard group of ℱl(ℱ,E) (and of its isotropic analogs) and show that ℱl(ℱ,E) is a projective ind-variety if and only if ℱ is a usual, possibly infinite flag of subspaces in V.