Quasi-split symmetric pairs of 𝑈(𝔰𝔩_{𝔫}) and Steinberg varieties of classical type

Li, Yiqiang

doi:10.1090/ert/570

Quasi-split symmetric pairs of $\mathrm {U}(\mathfrak {sl}_n)$ and Steinberg varieties of classical type
HTML articles powered by AMS MathViewer

by Yiqiang Li

Represent. Theory 25 (2021), 903-934

DOI: https://doi.org/10.1090/ert/570

Published electronically: October 21, 2021

PDF | Request permission

Abstract:

We provide a Lagrangian construction for the fixed-point subalgebra, together with its idempotent form, in a quasi-split symmetric pair of type $A_{n-1}$. This is obtained inside the limit of a projective system of Borel-Moore homologies of the Steinberg varieties of $n$-step isotropic flag varieties. Arising from the construction are a basis of homological origin for the idempotent form and a geometric realization of rational modules.

References

Tomoyuki Arakawa and Anne Moreau, On the irreducibility of associated varieties of W-algebras, J. Algebra 500 (2018), 542–568. MR 3765468, DOI 10.1016/j.jalgebra.2017.06.007
Huanchen Bao, Jonathan Kujawa, Yiqiang Li, and Weiqiang Wang, Geometric Schur duality of classical type, Transform. Groups 23 (2018), no. 2, 329–389. MR 3805209, DOI 10.1007/s00031-017-9447-4
A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, DOI 10.1215/S0012-7094-90-06124-1
Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, and Cornelius Pillen, Cohomology for quantum groups via the geometry of the nullcone, Mem. Amer. Math. Soc. 229 (2014), no. 1077, x+93. MR 3204911
Alexander Braverman and Dennis Gaitsgory, On Ginzburg’s Lagrangian construction of representations of $\textrm {GL}(n)$, Math. Res. Lett. 6 (1999), no. 2, 195–201. MR 1689209, DOI 10.4310/MRL.1999.v6.n2.a7
Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
Zhaobing Fan, Chun-Ju Lai, Yiqiang Li, Li Luo, and Weiqiang Wang, Affine flag varieties and quantum symmetric pairs, Mem. Amer. Math. Soc. 265 (2020), no. 1285, v+123. MR 4080913, DOI 10.1090/memo/1285
Zhaobing Fan and Yiqiang Li, Positivity of canonical bases under comultiplication, Int. Math. Res. Not. IMRN 9 (2021), 6871–6931. MR 4251292, DOI 10.1093/imrn/rnz047
Baohua Fu, Daniel Juteau, Paul Levy, and Eric Sommers, Generic singularities of nilpotent orbit closures, Adv. Math. 305 (2017), 1–77. MR 3570131, DOI 10.1016/j.aim.2016.09.010
Victor Ginzburg, Lagrangian construction of the enveloping algebra $U(\textrm {sl}_n)$, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 12, 907–912 (English, with French summary). MR 1111326
Victor Ginzburg, Geometric methods in the representation theory of Hecke algebras and quantum groups, Representation theories and algebraic geometry (Montreal, PQ, 1997) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 127–183. Notes by Vladimir Baranovsky [V. Yu. Baranovskiĭ]. MR 1649626
R. M. Green, Hyperoctahedral Schur algebras, J. Algebra 192 (1997), no. 1, 418–438. MR 1449968, DOI 10.1006/jabr.1996.6935
Hanspeter Kraft and Claudio Procesi, On the geometry of conjugacy classes in classical groups, Comment. Math. Helv. 57 (1982), no. 4, 539–602. MR 694606, DOI 10.1007/BF02565876
Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1923198, DOI 10.1007/978-1-4612-0105-2
Yiqiang Li, Quiver varieties and symmetric pairs, Represent. Theory 23 (2019), 1–56. MR 3900699, DOI 10.1090/ert/522
Yiqiang Li, Spaltenstein varieties of pure dimension, Proc. Amer. Math. Soc. 148 (2020), no. 1, 133–144. MR 4042837, DOI 10.1090/proc/14726
Yiqiang Li and Weiqiang Wang, Positivity vs negativity of canonical bases, Bull. Inst. Math. Acad. Sin. (N.S.) 13 (2018), no. 2, 143–198. MR 3792711
Y. Li and J. Zhu, Quasi-split symmetric pairs of $\mathrm {U}(\mathfrak {gl}_n)$ and their Schur algebras, Nagoya Math. J., 1-27, DOI 10.1017/nmj.2020.16.
George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145–202. MR 972345, DOI 10.1007/BF02699129
G. Lusztig, A class of perverse sheaves on a partial flag manifold, Represent. Theory 11 (2007), 122–171. MR 2336607, DOI 10.1090/S1088-4165-07-00320-2
Hiraku Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515–560. MR 1604167, DOI 10.1215/S0012-7094-98-09120-7
Masahiro Sakamoto and Toshiaki Shoji, Schur-Weyl reciprocity for Ariki-Koike algebras, J. Algebra 221 (1999), no. 1, 293–314. MR 1722914, DOI 10.1006/jabr.1999.7973
Éric Vasserot, Représentations de groupes quantiques et permutations, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 6, 747–773 (French, with English summary). MR 1251151, DOI 10.24033/asens.1686
M. Varagnolo and E. Vasserot, Perverse sheaves and quantum Grothendieck rings, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 345–365. MR 1985732
Hideya Watanabe, Crystal basis theory for a quantum symmetric pair $(\mathrm {U}, \mathrm {U}^{\jmath })$, Int. Math. Res. Not. IMRN 22 (2020), 8292–8352. MR 4216690, DOI 10.1093/imrn/rny227