Real Group Orbits on Flag Ind-Varieties of SL(∞,C)

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Real Group Orbits on Flag Ind-Varieties of SL(∞,C)

Ignatyev Mikhail V., Penkov I., Wolf J.

We consider the complex ind-group G = SL(∞,C) and its real forms G0 = SU(∞,∞), SU(p,∞), SL(∞,R), SL(∞,H). Our main object of study are the G0-orbits on an ind-variety G/P for an arbitrary splitting parabolic ind-subgroup P ⊂ G, under the assumption that the subgroups G0 ⊂ G and P ⊂ G are aligned in a natural way. We prove that the intersection of any G0-orbit on G/P with a finite-dimensional flag variety Gn/Pn from a given exhaustion of G / P via Gn/Pn for n → ∞, is a single (G0∩Gn)-orbit. We also characterize all ind-varieties G/P on which there are finitely many G0-orbits, and provide criteria for the existence of open and closed G0-orbits on G/P in the case of infinitely many G0-orbits.

Language: English

DOI

In book

Lie Theory and Its Applications in Physics. Varna, Bulgaria, June 2015

Vol. 191. , Springer, 2016.

An algebraic–geometric construction of ind-varieties of generalized flags

Penkov I., Tikhomirov A. S., Annali di Matematica Pura ed Applicata 2022 Vol. 201 No. 5 P. 2287–2314

We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra. This result enables us to show that any direct limit of admissible ...

Added: August 19, 2022