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Orbit duality in ind-varieties of maximal generalized flags

Authors: Lucas Fresse and Ivan Penkov
Translated by:
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 78 (2017), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2017, 131-160
MSC (2010): Primary 14L30, 14M15, 22E65, 22F30
Published electronically: December 1, 2017
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Abstract: We extend Matsuki duality to arbitrary ind-varieties of maximal generalized flags, in other words, to any homogeneous ind-variety $ \mathbf {G}/\mathbf {B}$ for a classical ind-group $ \mathbf {G}$ and a splitting Borel ind-subgroup $ \mathbf {B}\subset \mathbf {G}$. As a first step, we present an explicit combinatorial version of Matsuki duality in the finite-dimensional case, involving an explicit parametrization of $ K$- and $ G^0$-orbits on $ G/B$. After proving Matsuki duality in the infinite-dimensional case, we give necessary and sufficient conditions on a Borel ind-subgroup $ \mathbf {B}\subset \mathbf {G}$ for the existence of open and closed $ \mathbf {K}$- and $ \mathbf {G}^0$-orbits on $ \mathbf {G}/\mathbf {B}$, where $ \left (\mathbf {K},\mathbf {G}^0\right )$ is an aligned pair of a symmetric ind-subgroup $ \mathbf {K}$ and a real form $ \mathbf {G}^0$ of $ \mathbf {G}$.

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Additional Information

Lucas Fresse
Affiliation: Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506 France

Ivan Penkov
Affiliation: Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany

Keywords: Classical ind-groups, generalized flags, symmetric pairs, rest forms, Matsuki duality.
Published electronically: December 1, 2017
Additional Notes: The first author was supported in part by ISF Grant Nr. 797/14 and by ANR project GeoLie (ANR-15-CE40-0012).
The second author was supported in part by DFG Grant PE 980/6-1.
Dedicated: To Ernest Borisovich Vinberg on the occasion of his 80th birthday
Article copyright: © Copyright 2017 American Mathematical Society

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