Cycle spaces of infinite dimensional flag domains
Cycle spaces of infinite dimensional flag domains
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无限维标志域的循环空间
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发表时间：
2015
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通讯作者：
J. Wolf
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作者：
J. Wolf
设G为复简单直接极限群，具体为$$SL(infty ;mathbb {C})$$ SL（∞；C）、$$SO(infty ;mathbb {C})$$ SO（∞；C）或$$Sp(infty ;mathbb {C})$$ Sp（∞；C）。设$$mathcal {F}$$ F是$$mathbb {C}^infty $$ C∞上的一个（广义）标志。若G为$$SO(infty ;mathbb {C})$$ SO（∞；C）或$$Sp(infty ;mathbb {C})$$ Sp(∞；C)，则进一步假设$$mathcal {F}$$ F是各向同性的。设$$mathcal {Z}$$ Z表示对应的标志流形；因此$$mathcal {Z}= G/Q$$ Z=G/Q，其中Q是G的一个抛物子群。在最近的一篇论文（$$SL_{infty } ({mathbb {C}})$$ SL∞(C)的flag -varieties上的Real群轨道中的Penkov和Wolf，将发表在《数学与统计学学报》上）中，我们研究了G的实形式$$G_0$$ G0及其在$$mathcal {Z}$$ Z上的轨道的性质。这里我们集中讨论开放的$$G_0$$ G0-轨道$$D subset mathcal {Z}$$ D∧Z。当$$G_0$$ G0为厄米特型时，给出了$$G_0$$ G0对偶于有界对称区域的标志流形的完整的$$G_0$$ G0-轨道结构。然后给出了相应的循环空间$$mathcal {M}_D$$ MD的结构，最后研究了这些理论的实数和四元数类似物。所有这些都扩展了有限维情况下关于赫米对称空间和循环空间结构的结果(按时间顺序排列：Wolf in Bull Am Math Soc 75:1121-1237, 1969; Wolf et al. in Ann Math 105:397-448, 1977; Wolf in Ann Math 136:541-555, 1992； Wolf in flag域的紧子集，1994;Wolf and Zierau in Math Ann 316:529-545, 2000; Huckleberry et al. in Journal r die reine und angewandte Mathematik 2001:171 - 208,2001；《循环空间中的哈克贝利与狼》（$$SL_n(C)$$ SLn， C），纽约，第111-133页，2002；Wolf and Zierau，《J Lie Theory》，2003；《中国师范大学学报》(5):573-580,2010)。
Let G be a complex simple direct limit group, specifically $$SL(infty ;mathbb {C})$$SL(∞;C), $$SO(infty ;mathbb {C})$$SO(∞;C) or $$Sp(infty ;mathbb {C})$$Sp(∞;C). Let $$mathcal {F}$$F be a (generalized) flag in $$mathbb {C}^infty $$C∞. If G is $$SO(infty ;mathbb {C})$$SO(∞;C) or $$Sp(infty ;mathbb {C})$$Sp(∞;C) we suppose further that $$mathcal {F}$$F is isotropic. Let $$mathcal {Z}$$Z denote the corresponding flag manifold; thus $$mathcal {Z}= G/Q$$Z=G/Q where Q is a parabolic subgroup of G. In a recent paper (Penkov and Wolf in Real group orbits on flag ind-varieties of $$SL_{infty } ({mathbb {C}})$$SL∞(C), to appear in Proceedings in Mathematics and Statistics) we studied real forms $$G_0$$G0 of G and properties of their orbits on $$mathcal {Z}$$Z. Here we concentrate on open $$G_0$$G0-orbits $$D subset mathcal {Z}$$D⊂Z. When $$G_0$$G0 is of hermitian type we work out the complete $$G_0$$G0-orbit structure of flag manifolds dual to the bounded symmetric domain for $$G_0$$G0. Then we develop the structure of the corresponding cycle spaces $$mathcal {M}_D$$MD. Finally we study the real and quaternionic analogs of these theories. All this extends results from the finite-dimensional cases on the structure of hermitian symmetric spaces and cycle spaces (in chronological order: Wolf in Bull Am Math Soc 75:1121–1237, 1969; Wolf et al. in Ann Math 105:397–448, 1977; Wolf in Ann Math 136:541–555, 1992; Wolf in Compact subvarieties in flag domains, 1994; Wolf and Zierau in Math Ann 316:529–545, 2000; Huckleberry et al. in Journal für die reine und angewandte Mathematik 2001:171–208, 2001; Huckleberry and Wolf in Cycle spaces of real forms of $$SL_n(C)$$SLn(C), Springer, New York, pp 111–133, 2002; Wolf and Zierau in J Lie Theory 13:189–191, 2003; Huckleberry and Wolf in Ann Scuola Norm Sup Pisa Cl Sci (5) 9:573-580, 2010).