Nilpotent varieties in symmetric spaces and twisted affine Schubert varieties

Hong, Jiuzu; Korkeathikhun, Korkeat

doi:10.1090/ert/613

Nilpotent varieties in symmetric spaces and twisted affine Schubert varieties
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by Jiuzu Hong and Korkeat Korkeathikhun

Represent. Theory 26 (2022), 585-615

DOI: https://doi.org/10.1090/ert/613

Published electronically: June 2, 2022

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Abstract:

We relate the geometry of Schubert varieties in twisted affine Grassmannian and the nilpotent varieties in symmetric spaces. This extends some results of Achar–Henderson in the twisted setting. We also get some applications to the geometry of the order 2 nilpotent varieties in certain classical symmetric spaces.

References

Pramod N. Achar and Anthony Henderson, Geometric Satake, Springer correspondence and small representations, Selecta Math. (N.S.) 19 (2013), no. 4, 949–986. MR 3131493, DOI 10.1007/s00029-013-0125-7
Pramod N. Achar, Anthony Henderson, and Simon Riche, Geometric Satake, Springer correspondence, and small representations II, Represent. Theory 19 (2015), 94–166. MR 3347990, DOI 10.1090/ert/465
Marc Besson, Twisted Affine Schubert Varieties and Twisted Affine Demazure Modules, ProQuest LLC, Ann Arbor, MI, 2021. Thesis (Ph.D.)–The University of North Carolina at Chapel Hill. MR 4285356
Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
Abraham Broer, The sum of generalized exponents and Chevalley’s restriction theorem for modules of covariants, Indag. Math. (N.S.) 6 (1995), no. 4, 385–396. MR 1365182, DOI 10.1016/0019-3577(96)81754-X
Ranee Kathryn Brylinski, Limits of weight spaces, Lusztig’s $q$-analogs, and fiberings of adjoint orbits, J. Amer. Math. Soc. 2 (1989), no. 3, 517–533. MR 984511, DOI 10.1090/S0894-0347-1989-0984511-X
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
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
Tsao-Hsien Chen, Kari Vilonen, and Ting Xue, Springer correspondence, hyperelliptic curves, and cohomology of Fano varieties, Math. Res. Lett. 27 (2020), no. 5, 1281–1323. MR 4216588, DOI 10.4310/MRL.2020.v27.n5.a2
A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras, Lie groups and invariant theory, Amer. Math. Soc. Transl. Ser. 2, vol. 213, Amer. Math. Soc., Providence, RI, 2005, pp. 85–104. MR 2140715, DOI 10.1090/trans2/213/05
T. Haines, J. Lourenço, and T. Richarz, On the normality of Schubert varieties: remaining cases in positive characteristic, arXiv:1806.11001, 2020.
Thomas J. Haines and Timo Richarz, Smoothness of Schubert varieties in twisted affine Grassmannians, Duke Math. J. 169 (2020), no. 17, 3223–3260. MR 4173154, DOI 10.1215/00127094-2020-0025
Jiuzu Hong and Linhui Shen, Tensor invariants, saturation problems, and Dynkin automorphisms, Adv. Math. 285 (2015), 629–657. MR 3406511, DOI 10.1016/j.aim.2015.08.015
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842, DOI 10.1007/978-1-4612-6398-2
Jens Carsten Jantzen, Nilpotent orbits in representation theory, Lie theory, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1–211. MR 2042689
K. Korkeathikhun, Nullcones of symmetric spaces and twisted affine Grassmannian, Oral exam at UNC, March 4, 2020.
Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255–339. MR 1485921, DOI 10.1023/A:1000102604688
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
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
David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203. MR 573434
Yiqiang Li, Quiver varieties and symmetric pairs, Represent. Theory 23 (2019), 1–56. MR 3900699, DOI 10.1090/ert/522
G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), no. 2, 169–178. MR 641425, DOI 10.1016/0001-8708(81)90038-4
Ivan Mirković and Maxim Vybornov, On quiver varieties and affine Grassmannians of type $A$, C. R. Math. Acad. Sci. Paris 336 (2003), no. 3, 207–212 (English, with English and French summaries). MR 1968260, DOI 10.1016/S1631-073X(03)00022-0
G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), no. 1, 118–198. With an appendix by T. Haines and Rapoport. MR 2435422, DOI 10.1016/j.aim.2008.04.006
Takuya Ohta, The singularities of the closures of nilpotent orbits in certain symmetric pairs, Tohoku Math. J. (2) 38 (1986), no. 3, 441–468. MR 854462, DOI 10.2748/tmj/1178228456
Mark Reeder, Zero weight spaces and the Springer correspondence, Indag. Math. (N.S.) 9 (1998), no. 3, 431–441. MR 1692153, DOI 10.1016/S0019-3577(98)80010-4
Timo Richarz, Schubert varieties in twisted affine flag varieties and local models, J. Algebra 375 (2013), 121–147. MR 2998951, DOI 10.1016/j.jalgebra.2012.11.013
Jir\B{o} Sekiguchi, The nilpotent subvariety of the vector space associated to a symmetric pair, Publ. Res. Inst. Math. Sci. 20 (1984), no. 1, 155–212. MR 736100, DOI 10.2977/prims/1195181836
T. Shoji, Springer correspondence for symmetric spaces, arXiv:1909.06744, 2019.
Xinwen Zhu, The geometric Satake correspondence for ramified groups, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 2, 409–451 (English, with English and French summaries). MR 3346175, DOI 10.24033/asens.2248