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

DOI:
https://doi.org/10.1090/mosc/266

Published electronically:
December 1, 2017

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We extend Matsuki duality to arbitrary ind-varieties of maximal generalized flags, in other words, to any homogeneous ind-variety for a classical ind-group and a splitting Borel ind-subgroup . As a first step, we present an explicit combinatorial version of Matsuki duality in the finite-dimensional case, involving an explicit parametrization of - and -orbits on . After proving Matsuki duality in the infinite-dimensional case, we give necessary and sufficient conditions on a Borel ind-subgroup for the existence of open and closed - and -orbits on , where is an aligned pair of a symmetric ind-subgroup and a real form of .

- [1]
A. A. Baranov: Finitary simple Lie algebras. J. Algebra
**219**(1999), 299-329. MR**1707673** - [2]
M. Berger: Les espaces symétriques non compacts. Ann. Sci. École Norm. Sup.
**74**(1957) 85-177. MR**0104763** - [3]
M. Brion and G. Helminck: On orbit closures of symmetric subgroups in flag varities. Canad. J. Math.
**52**(2000), 265-292. MR**1755778** - [4]
I. Dimitrov and I. Penkov: Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups. Int. Math. Res. Not.
**2004**(2004), 2935-2953. MR**2099177** - [5] L. Fresse and I. Penkov: Schubert decompositions for ind-varieties of generalized flags. Asian J. Math., to appear.
- [6]
S. Gindikin and T. Matsuki: Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds. Transform. Groups
**8**(2003), 333-376. MR**2015255** - [7]
G. Fels, A. Huckleberry, and J. A. Wolf: Cycle spaces of flag domains. A complex geometric viewpoint. Progr. Math., vol. 245, Birkhäuser, Boston, MA, 2006. MR
**2188135** - [8] M. V. Ignatyev and I. Penkov: Ind-varieties of generalized flags: a survey of results. Preprint (2016).
- [9]
M. V. Ignatyev, I. Penkov, and J. A. Wolf: Real group orbits on flag ind-varieties of . In: Lie Theory and Its Applications in Physics, pp. 111-135. Springer Proc. Math. Stat., vol. 191, Springer, Singapore, 2016. MR
**3613922** - [10]
J. C. Jantzen: Nilpotent orbits in representation theory. Lie theory, 1-211, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004. MR
**2042689** - [11]
T. Matsuki: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Japan.
**31**(1979) 331-357. MR**527548** - [12]
T. Matsuki: Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups. Intersections of associated orbits. Hiroshima Math. J.
**18**(1988) 59-67. MR**935882** - [13]
T. Matsuki and T. Oshima: Embeddings of discrete series into principal series. In: The orbit method in representation theory (Copenhagen, 1988), pp. 147-175. Progr. Math., vol. 82, Birkhäuser, Boston, MA, 1990. MR
**1095345** - [14]
K. Nishiyama: Enhanced orbit embedding. Comment. Math. Univ. St. Pauli
**63**(2014), 223-232. MR**3328431** - [15]
A. L. Onishchik and È. B. Vinberg: Lie groups and algebraic groups. Translated from the Russian and with a preface by D. A. Leites. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990. MR
**1064110** - [16]
T. Ohta: The closures of nilpotent orbits in the classical symmetric pairs and their singularities. Tohoku Math. J.
**43**(1991), 161-211. MR**1104427** - [17]
T. Ohta: An inclusion between sets of orbits and surjectivity of the restriction map of rings of invariants. Hokkaido Math. J.
**37**(2008), 437-454. MR**2441911** - [18]
R. W. Richardson and T. A. Springer: The Bruhat order on symmetric varieties. Geom. Dedicata
**35**(1990), 389-436. MR**1066573** - [19]
J. A. Wolf: The action of a real semisimple Lie group on a complex flag manifold. I: Orbit structure and holomorphic arc components. Bull. Amer. Math. Soc.
**75**(1969), 1121-1237. MR**0251246** - [20]
J. A. Wolf: Cycle spaces of infinite dimensional flag domains. Annals of Global Analysis and Geometry
**50**(2016), 315-346. MR**3573989** - [21]
A. Yamamoto: Orbits in the flag variety and images of the moment map for classical groups I. Represent. Theory
**1**(1997), 329-404. MR**1479152**

Retrieve articles in *Transactions of the Moscow Mathematical Society*
with MSC (2010):
14L30,
14M15,
22E65,
22F30

Retrieve articles in all journals with MSC (2010): 14L30, 14M15, 22E65, 22F30

Additional Information

**Lucas Fresse**

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

Email:
lucas.fresse@univ-lorraine.fr

**Ivan Penkov**

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

Email:
i.penkov@jacobs-university.de

DOI:
https://doi.org/10.1090/mosc/266

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