Upper triangular parts of conjugacy classes of nilpotent matrices with finite number of \(B\)-orbits. (English) Zbl 1350.17006

Fresse, Lucas

J. Math. Soc. Japan 65, No. 3, 967-992 (2013).

Summary: We consider the intersection of the conjugacy class of a nilpotent matrix with the space of upper triangular matrices. We give necessary and sufficient conditions for this intersection to be a union of finitely many orbits for the action by conjugation of the group of invertible upper triangular matrices.

MSC:

17B08	Coadjoint orbits; nilpotent varieties
20G15	Linear algebraic groups over arbitrary fields
05E10	Combinatorial aspects of representation theory

Keywords:

nilpotent matrices; nilpotent orbits; spherical varieties; orbital varieties

References:

[1]	M. Brion, Quelques propriétés des espaces homogènes sphériques, Manuscripta Math., 55 (1986), 191-198. · Zbl 0604.14048
[2]	L. Fresse, On the singularity of some special components of Springer fibers, J. Lie Theory, 21 (2011), 205-242. · Zbl 1222.14106
[3]	L. Fresse and A. Melnikov, On the singularity of the irreducible components of a Springer fiber in \(\mathfrak{sl}_n\), Selecta Math. (N.S.), 16 (2010), 393-418. · Zbl 1209.14037
[4]	L. Fresse and A. Melnikov, Smooth orbital varieties and orbital varieties with a dense \(B\)-orbit, Int. Math. Res. Not., 2013 (2013), 1122-1203. · Zbl 1361.17011
[5]	F. Y. C. Fung, On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory, Adv. Math., 178 (2003), 244-276. · Zbl 1035.20004
[6]	M. Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. (2), 73 (1961), 324-348. · Zbl 0168.28201
[7]	W. H. Hesselink, A classification of the nilpotent triangular matrices, Compositio Math., 55 (1985), 89-133. · Zbl 0579.15011
[8]	L. Hille and G. Röhrle, A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, Transform. Groups, 4 (1999), 35-52. · Zbl 0924.20035
[9]	A. Joseph, On the variety of a highest weight module, J. Algebra, 88 (1984), 238-278. · Zbl 0539.17006
[10]	A. Melnikov, Description of B-orbit closures of order 2 in upper-triangular matrices, Transform. Groups, 11 (2006), 217-247. · Zbl 1196.14040
[11]	A. Melnikov, B-orbits of nilpotent order 2 and link patterns, · Zbl 1276.14068
[12]	D. Mertens, Über einen Satz von Spaltenstein, University of Wuppertal, · JFM 38.0253.02
[13]	D. I. Panyushev, Complexity and nilpotent orbits, Manuscripta Math., 83 (1994), 223-237. · Zbl 0822.14024
[14]	R. W. Richardson, Jr., Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bull. London Math. Soc., 6 (1974), 21-24. · Zbl 0287.20036
[15]	N. Spaltenstein, On the fixed point set of a unipotent element on the variety of Borel subgroups, Topology, 16 (1977), 203-204. · Zbl 0445.20021
[16]	T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math., 36 (1976), 173-207. · Zbl 0374.20054
[17]	T. A. Springer, A construction of representations of Weyl groups, Invent. Math., 44 (1978), 279-293. · Zbl 0376.17002
[18]	R. Steinberg, On the desingularization of the unipotent variety, Invent. Math., 36 (1976), 209-224. · Zbl 0352.20035
[19]	J. A. Vargas, Fixed points under the action of unipotent elements of \(\mathrm{SL}_n\) in the flag variety, Bol. Soc. Mat. Mexicana (2), 24 (1979), 1-14. · Zbl 0458.14019
[20]	È. B. Vinberg, Complexity of actions of reductive groups, Funktsional. Anal. i Prilozhen., 20 (1986), 1-13. In Russian; English translation in Funct. Anal. Appl., 20 (1986), 1-11. · Zbl 0601.14038