×

Graham’s variety and perverse sheaves on the nilpotent cone. (English) Zbl 1481.20172

Summary: Graham has constructed a variety with a map to the nilpotent cone which is similar in some ways to the Springer resolution. One aspect in which Graham’s map differs is that it is not in general an isomorphism over the principal orbit, but rather the universal covering map. This map gives rise to a certain semisimple perverse sheaf on the nilpotent cone, and we discuss here the problem of describing the summands of this perverse sheaf. For type \(A_n\), a key tool is a known description of an affine paving of Springer fibers.

MSC:

20G05 Representation theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
17B08 Coadjoint orbits; nilpotent varieties
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alexeev, V., Complete moduli in the presence of semiabelian group action, Ann. Math., 155, 3, 611-708 (2002) · Zbl 1052.14017
[2] Alexeevski, A., Component groups of the centralizers of unipotent elements in semisimple algebraic groups, Am. Math. Soc. Transl. (2), 213, 15-31 (2005) · Zbl 1093.20502
[3] Beilinson, A.; Bernstein, J.; Deligne, P., Faisceaux pervers, Astérisque, 100, 5-171 (1982) · Zbl 0536.14011
[4] Borho, W.; MacPherson, R., Partial resolutions of nilpotent varieties, Analysis and Topology on Singular Spaces, II, III. Analysis and Topology on Singular Spaces, II, III, Luminy, 1981. Analysis and Topology on Singular Spaces, II, III. Analysis and Topology on Singular Spaces, II, III, Luminy, 1981, Astérisque, 101-102, 23-74 (1983) · Zbl 0576.14046
[5] Brion, M., Rational smoothness and fixed points of torus actions, Transform. Groups, 4, 2-3, 127-156 (1999) · Zbl 0953.14004
[6] Collingwood, D. H.; McGovern, W. M., Nilpotent Orbits in Semisimple Lie Algebras (1993), Van Nostrand Reinhold: Van Nostrand Reinhold New York · Zbl 0972.17008
[7] Fulton, W., Introduction to Toric Varieties (1993), Princeton University Press: Princeton University Press Princeton, New Jersey · Zbl 0813.14039
[8] Graham, W., Toric varieties and a generalization of the Springer resolution (2019)
[9] Humphreys, J. E., Introduction to Lie Algebras and Representation Theory (1972), Springer Verlag: Springer Verlag New York · Zbl 0254.17004
[10] Jantzen, J. C., Nilpotent Orbits in Representation Theory, 1-211 (2004), Birkhäuser Boston: Birkhäuser Boston Boston, MA · Zbl 1169.14319
[11] Kostant, B., Lie group representations on polynomial rings, Am. J. Math., 85, 3, 327-404 (1963) · Zbl 0124.26802
[12] Lusztig, G., Intersection cohomology complexes on a reductive group, Invent. Math., 75, 2, 205-272 (1984) · Zbl 0547.20032
[13] Mizuno, K., The conjugate classes of unipotent elements of the Chevalley groups \(E_7\) and \(E_8\), Tokyo J. Math., 3, 2, 391-459 (1980) · Zbl 0454.20046
[14] Russell, A., Graham’s variety and perverse sheaves on the nilpotent cone (2012), Ph.D. thesis, Louisiana State University · Zbl 1481.20172
[15] Shimomura, N., A theorem on the fixed point set of a unipotent transformation on the flag manifold, J. Math. Soc. Jpn., 32, 1, 55-64 (1980) · Zbl 0413.20037
[16] Springer, T., Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math., 36, 1, 173-207 (1976) · Zbl 0374.20054
[17] Springer, T., A construction of representations of Weyl groups, Invent. Math., 44, 3, 279-293 (1978) · Zbl 0376.17002
[18] Tymoczko, J. S., Linear conditions imposed on flag varieties, Am. J. Math., 128, 6, 1587-1604 (2006) · Zbl 1106.14038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.