Borel subgroups adapted to nilpotent elements of standard Levi type

Fresse, Lucas

doi:10.1090/S1088-4165-2014-00458-8

Borel subgroups adapted to nilpotent elements of standard Levi type
HTML articles powered by AMS MathViewer

by Lucas Fresse

Represent. Theory 18 (2014), 341-360

DOI: https://doi.org/10.1090/S1088-4165-2014-00458-8

Published electronically: October 27, 2014

PDF | Request permission

Previous version: Original version posted October 27, 2014
Corrected version: Current version corrects changes to the support information.

Abstract:

Let a reductive algebraic group over an algebraically closed field of good characteristic be given. Attached to a nilpotent element of its Lie algebra, we consider a family of algebraic varieties, which incorporates classical objects such as Springer fiber, Spaltenstein varieties, and Hessenberg varieties. When the nilpotent element is of standard Levi type, we show that the varieties of this family admit affine pavings that can be obtained by intersecting with the Schubert cells corresponding to a suitable Borel subgroup.

References

A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497. MR 366940, DOI 10.2307/1970915
Walter Borho and Hanspeter Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), no. 1, 61–104 (German, with English summary). MR 522032, DOI 10.1007/BF02566256
Walter Borho and Robert MacPherson, Partial resolutions of nilpotent varieties, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 23–74. MR 737927
Jonathan Brundan and Victor Ostrik, Cohomology of Spaltenstein varieties, Transform. Groups 16 (2011), no. 3, 619–648. MR 2827037, DOI 10.1007/s00031-011-9149-2
Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
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
C. De Concini, G. Lusztig, and C. Procesi, Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc. 1 (1988), no. 1, 15–34. MR 924700, DOI 10.1090/S0894-0347-1988-0924700-2
F. De Mari, C. Procesi, and M. A. Shayman, Hessenberg varieties, Trans. Amer. Math. Soc. 332 (1992), no. 2, 529–534. MR 1043857, DOI 10.1090/S0002-9947-1992-1043857-6
Lucas Fresse and Anna Melnikov, Some characterizations of singular components of Springer fibers in the two-column case, Algebr. Represent. Theory 14 (2011), no. 6, 1063–1086. MR 2844756, DOI 10.1007/s10468-010-9227-5
Lucas Fresse and Anna Melnikov, Smooth orbital varieties and orbital varieties with a dense $B$-orbit, Int. Math. Res. Not. IMRN 5 (2013), 1122–1203. MR 3031828, DOI 10.1093/imrn/rns018
Jens Carsten Jantzen, Nilpotent orbits in representation theory, Lie theory, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1–211. MR 2042689
Daniel Juteau, Carl Mautner, and Geordie Williamson, Parity sheaves, J. Amer. Math. Soc. 27 (2014), no. 4, 1169–1212. MR 3230821, DOI 10.1090/S0894-0347-2014-00804-3
Hiraku Nakajima, Homology of moduli spaces of instantons on ALE spaces. I, J. Differential Geom. 40 (1994), no. 1, 105–127. MR 1285530
D. I. Panyushev, Rationality of singularities and the Gorenstein property of nilpotent orbits, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 76–78 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 3, 225–226 (1992). MR 1139878, DOI 10.1007/BF01085494
Martha Precup, Affine pavings of Hessenberg varieties for semisimple groups, Selecta Math. (N.S.) 19 (2013), no. 4, 903–922. MR 3131491, DOI 10.1007/s00029-012-0109-z
Naohisa Shimomura, The fixed point subvarieties of unipotent transformations on the flag varieties, J. Math. Soc. Japan 37 (1985), no. 3, 537–556. MR 792991, DOI 10.2969/jmsj/03730537
N. Spaltenstein, On unipotent and nilpotent elements of groups of type $E_{6}$, J. London Math. Soc. (2) 27 (1983), no. 3, 413–420. MR 697134, DOI 10.1112/jlms/s2-27.3.413
Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610, DOI 10.1007/BFb0096302
T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
Robert Steinberg, On the desingularization of the unipotent variety, Invent. Math. 36 (1976), 209–224. MR 430094, DOI 10.1007/BF01390010
Julianna S. Tymoczko, Linear conditions imposed on flag varieties, Amer. J. Math. 128 (2006), no. 6, 1587–1604. MR 2275912, DOI 10.1353/ajm.2006.0050
Nanhua Xi, A partition of the Springer fibers $\scr B_N$ for type $A_{n-1},B_2,G_2$ and some applications, Indag. Math. (N.S.) 10 (1999), no. 2, 307–320. MR 1816223, DOI 10.1016/S0019-3577(99)80024-X