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On generalized Steinberg theory for type AIII
Fresse, Lucas1; Nishiyama, Kyo2
1 Université de Lorraine CNRS Institut Élie Cartan de Lorraine UMR 7502 Vandoeuvre-lès-Nancy F-54506 France
2 Department of Mathematics Aoyama Gakuin University Fuchinobe 5-10-1 Chuo-ku Sagamihara 252-5258 Japan
Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 165-195.

The multiple flag variety 𝔛=Gr(ℂ p+q ,r)×(Fl(ℂ p )×Fl(ℂ q )) can be considered as a double flag variety associated to the symmetric pair (G,K)=(GL p+q (ℂ), GL p (ℂ)×GL q (ℂ)) of type AIII. We consider the diagonal action of K on 𝔛. There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization (by a certain set of graphs), dimensions, closure relations and cover relations.

In [5], we defined two generalized Steinberg maps from the K-orbits of 𝔛 to the nilpotent K-orbits in 𝔨 and those in the Cartan complement of 𝔨, respectively. The main result in the present paper is a complete, explicit description of these two Steinberg maps by means of a combinatorial algorithm which extends the classical Robinson–Schensted correspondence.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.245
Classification: 14M15, 17B08, 53C35, 05A15
Keywords: double flag variety, conormal bundle, exotic moment map, nilpotent orbits, partial permutations, Robinson–Schensted correspondence, Steinberg variety

Fresse, Lucas 1; Nishiyama, Kyo 2

1 Université de Lorraine CNRS Institut Élie Cartan de Lorraine UMR 7502 Vandoeuvre-lès-Nancy F-54506 France
2 Department of Mathematics Aoyama Gakuin University Fuchinobe 5-10-1 Chuo-ku Sagamihara 252-5258 Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Fresse, Lucas; Nishiyama, Kyo. On generalized Steinberg theory for type AIII. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 165-195. doi : 10.5802/alco.245. https://alco.centre-mersenne.org/articles/10.5802/alco.245/

[1] Chriss, Neil; Ginzburg, Victor Representation theory and complex geometry, Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2010, x+495 pages (Reprint of the 1997 edition) | DOI | MR

[2] Collingwood, David H.; McGovern, William M. Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993, xiv+186 pages | MR

[3] Fresse, Lucas; Nishiyama, Kyo On the exotic Grassmannian and its nilpotent variety, Represent. Theory, Volume 20 (2016), pp. 451-481 ([Paging previously given as 1–31]) | DOI | MR | Zbl

[4] Fresse, Lucas; Nishiyama, Kyo Orbit embedding for double flag varieties and Steinberg maps, Lie groups, number theory, and vertex algebras (Contemp. Math.), Volume 768, Amer. Math. Soc., [Providence], RI, [2021] ©2021, pp. 21-42 | DOI | MR | Zbl

[5] Fresse, Lucas; Nishiyama, Kyo A generalization of Steinberg theory and an exotic moment map, Int. Math. Res. Not. IMRN (2022) no. 1, pp. 1-62 | DOI | MR | Zbl

[6] Fulton, William Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J., Volume 65 (1992) no. 3, pp. 381-420 | DOI | MR | Zbl

[7] Fulton, William Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997, x+260 pages | MR

[8] He, Xuhua; Nishiyama, Kyo; Ochiai, Hiroyuki; Oshima, Yoshiki On orbits in double flag varieties for symmetric pairs, Transform. Groups, Volume 18 (2013) no. 4, pp. 1091-1136 | DOI | MR | Zbl

[9] Matsuki, Toshihiko; Ōshima, Toshio Embeddings of discrete series into principal series, The orbit method in representation theory (Copenhagen, 1988) (Progr. Math.), Volume 82, Birkhäuser Boston, Boston, MA, 1990, pp. 147-175 | DOI | MR | Zbl

[10] Nishiyama, Kyo; Ochiai, Hiroyuki Double flag varieties for a symmetric pair and finiteness of orbits, J. Lie Theory, Volume 21 (2011) no. 1, pp. 79-99 | MR | Zbl

[11] Rosso, Daniele Classic and mirabolic Robinson-Schensted-Knuth correspondence for partial flags, Canad. J. Math., Volume 64 (2012) no. 5, pp. 1090-1121 | DOI | MR | Zbl

[12] Singh, Rahul A Robinson-Schensted correspondence for partial permutations (2020) | arXiv

[13] Steinberg, Robert An occurrence of the Robinson-Schensted correspondence, J. Algebra, Volume 113 (1988) no. 2, pp. 523-528 | DOI | MR | Zbl

[14] Timashev, Dmitry A. A generalization of the Bruhat decomposition, Izv. Ross. Akad. Nauk Ser. Mat., Volume 58 (1994) no. 5, pp. 110-123 | DOI | MR | Zbl

[15] Trapa, Peter E. Generalized Robinson-Schensted algorithms for real groups, Internat. Math. Res. Notices (1999) no. 15, pp. 803-834 | DOI | MR | Zbl

[16] Travkin, Roman Mirabolic Robinson-Schensted-Knuth correspondence, Selecta Math. (N.S.), Volume 14 (2009) no. 3-4, pp. 727-758 | DOI | MR | Zbl

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