Geodesic
Orbit duality in ind-varieties of maximal generalized flags
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 1, pp. 155-194.

Voir la notice de l'article provenant de la source Math-Net.Ru

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}$. 
@article{MMO_2017_78_1_a7,
     author = {Ivan Penkov and Lucas Fresse},
     title = {Orbit duality in ind-varieties of maximal generalized flags},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {155--194},
     publisher = {mathdoc},
     volume = {78},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MMO_2017_78_1_a7/}
}
TY  - JOUR
AU  - Ivan Penkov
AU  - Lucas Fresse
TI  - Orbit duality in ind-varieties of maximal generalized flags
JO  - Trudy Moskovskogo matematičeskogo obŝestva
PY  - 2017
SP  - 155
EP  - 194
VL  - 78
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MMO_2017_78_1_a7/
LA  - ru
ID  - MMO_2017_78_1_a7
ER  - 
%0 Journal Article
%A Ivan Penkov
%A Lucas Fresse
%T Orbit duality in ind-varieties of maximal generalized flags
%J Trudy Moskovskogo matematičeskogo obŝestva
%D 2017
%P 155-194
%V 78
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MMO_2017_78_1_a7/
%G ru
%F MMO_2017_78_1_a7
Ivan Penkov; Lucas Fresse. Orbit duality in ind-varieties of maximal generalized flags. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 1, pp. 155-194. http://geodesic.mathdoc.fr/item/MMO_2017_78_1_a7/

[1] Baranov A. A., “Finitary simple Lie algebras”, J. Algebra, 219 (1999), 299–329 | DOI | MR | Zbl

[2] Berger M., “Les espaces symétriques non compacts”, Ann. Sci. École Norm. Sup., 74 (1957), 85–177 | DOI | MR | Zbl

[3] Brion M., Helminck A. G., “On orbit closures of symmetric subgroups in flag varieties”, Canad. J. Math., 52 (2000), 265–292 | DOI | MR | Zbl

[4] Dimitrov I., Penkov I., “Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups”, Int. Math. Res. Not., 2004 (2004), 2935–2953 | DOI | MR | Zbl

[5] Fresse L., Penkov I., “Schubert decompositions for ind-varieties of generalized flags”, Asian J. Math. (to appear) | MR

[6] Gindikin S., Matsuki T., “Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds”, Transform. Groups, 8 (2003), 333–376 | DOI | MR | Zbl

[7] Fels G., Huckleberry A., Wolf J. A., Cycle spaces of flag domains. A complex geometric viewpoint, Progress in Math., 245, Birkhäuser, Boston, MA, 2006 | MR | Zbl

[8] Ignatyev M. V., Penkov I., Ind-varieties of generalized flags: a survey of results, 2016, arXiv: 1701.08478

[9] Ignatyev M. V., Penkov I., Wolf J. A., “Real group orbits on flag ind-varieties of $SL(\infty,\mathbb{C})$”, Lie Theory and Its Applications in Physics, Springer Proc. Math. Stat., 191, Springer, Singapore, 2016, 111–135 | DOI | MR | Zbl

[10] Jantzen J. C., “Nilpotent orbits in representation theory”, Lie theory, Progress in Math., 228, Birkhäuser, Boston, MA, 2004, 1–211 | MR | Zbl

[11] Matsuki T., “The orbits of affine symmetric spaces under the action of minimal parabolic subgroups”, J. Math. Soc. Japan, 31 (1979), 331–357 | DOI | MR | Zbl

[12] Matsuki T., “Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups. Intersections of associated orbits”, Hiroshima Math. J., 18 (1988 pages 59–67) | MR | Zbl

[13] Matsuki T., Oshima T., “Embeddings of discrete series into principal series”, The orbit method in representation theory (Copenhagen, 1988), Progress in Math., 82, Birkhäuser, Boston, MA, 1990, 147–175 | MR

[14] Nishiyama K., “Enhanced orbit embedding”, Comment. Math. Univ. St. Pauli, 63 (2014), 223–232 | MR | Zbl

[15] Vinberg E. B., Onischik A. L., Seminar po gruppam Li i algebraicheskim gruppam, URSS, M., 1995 ; Наука, М., 1988 | MR

[16] Ohta T., “The closures of nilpotent orbits in the classical symmetric pairs and their singularities”, Tohoku Math. J., 43 (1991), 161–211 | DOI | MR | Zbl

[17] Ohta T., “An inclusion between sets of orbits and surjectivity of the restriction map of rings of invariants”, Hokkaido Math. J., 37 (2008), 437–454 | DOI | MR | Zbl

[18] Richardson R. W., Springer T. A., “The Bruhat order on symmetric varieties”, Geom. Dedicata, 35 (1990), 389–436 | DOI | MR | Zbl

[19] Wolf J. A., “The action of a real semisimple group on a complex flag manifold. I: Orbit structure and holomorphic arc components”, Bull. AMS, 75 (1969), 1121–1237 | DOI | MR | Zbl

[20] Wolf J. A., “Cycle spaces of infinite dimensional flag domains”, Ann. Global Anal. Geom., 50 (2016), 315–346 | DOI | MR | Zbl

[21] Yamamoto A., “Orbits in the flag variety and images of the moment map for classical groups. I”, Represent. Theory, 1 (1997), 329–404 | DOI | MR | Zbl