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Irreducible components of the nilpotent commuting variety of a symmetric semisimple Lie algebra. (Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple.) (French. English summary) Zbl 1189.17008

Given an involution \(\theta \) of a finite dimensional semisimple Lie algebra \(\mathfrak g\) and the associated Cartan decomposition \(\mathfrak g=\mathfrak k\oplus \mathfrak p\), the nilpotent commuting variety of \((\mathfrak g,\theta )\) consists of pairs of nilpotent elements \((x,y)\) of \(\mathfrak p\) satisfying the condition \([x,y]=0\). An interesting structural problem concerns the questions whether this variety is equidimensional and its irreducible components are indexed by the orbits of \(\mathfrak{p}\) distinguished elements (i.e., those elements whose centralizer in \(\mathfrak{p}\) is formed by nilpotent elements).
The validity of this assertion was already established by A. Premet in the case \((\mathfrak g\times \mathfrak g,\theta )\) where \(\theta (x,y)=(y,x)\) [Invent. Math. 154, No. 3, 653–683 (2003; Zbl 1068.17006)]. The main purpose of this to prove the conjecture for other types. In all, 18 cases are covered and proven to satisfy the assertion. The proof is based largely on two technical lemmas on the orbits and a reduction principle.

MSC:

17B08 Coadjoint orbits; nilpotent varieties
17B20 Simple, semisimple, reductive (super)algebras
14L30 Group actions on varieties or schemes (quotients)

Citations:

Zbl 1068.17006
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References:

[1] Baranovski, V., The variety of pairs of commuting nilpotent matrices is irreducible, Transform. Groups, 6, 3-8 (2001) · Zbl 0980.15012 · doi:10.1007/BF01236059
[2] Bourbaki, N., Groupes et algèbres de Lie. Chapitres 4, 5 et 6 (1968) · Zbl 0483.22001
[3] Djokovic, D. Z., Classification of nilpotent elements in simple exceptional real algebras of inner type and description of their centralizers, J. Algebra, 112, 503-524 (1988) · Zbl 0639.17005 · doi:10.1016/0021-8693(88)90104-4
[4] Djokovic, D. Z., Classification of nilpotent elements in simple real lie algebras \(E_{6(6)}\) and \(E_{6(-26)}\) and description of their centralizers, J. Algebra, 116, 196-207 (1988) · Zbl 0653.17004 · doi:10.1016/0021-8693(88)90201-3
[5] Djokovic, D. Z., Explicit Cayley triples in real forms of \(F_2, G_4\) and \(E_6\), Pacific J. Math., 184, 231-255 (1998) · Zbl 1040.17004 · doi:10.2140/pjm.1998.184.231
[6] Djokovic, D. Z., Explicit Cayley triples in real forms of \(E_8\), Pacific J. of Math., 194, 57-82 (2000) · Zbl 1013.22003 · doi:10.2140/pjm.2000.194.57
[7] Djokovic, D. Z., The closure diagram for nilpotent orbits of the split real form of \(E_7\), Represent. Theory, 5, 284-316 (2001) · Zbl 1050.17007 · doi:10.1090/S1088-4165-01-00124-8
[8] Djokovic, D. Z., The closure diagrams for nilpotent orbits of real forms of \(E_6\), J. Lie Theory, 11, 381-413 (2001) · Zbl 1049.17006
[9] Djokovic, D. Z., The closure diagram for nilpotent orbits of the split real form of \(E_8\), Centr. Europ. J. Math., 4, 573-643 (2003) · Zbl 1050.17006 · doi:10.2478/BF02475183
[10] Elashvili, E. B., The centralisers of nilpotent elements in semisimple Lie algebras, Trudy Tbiliss. Inst. Mat. Nats. Nauk Gruzin., 46, 109-132 (1975) · Zbl 0323.17004
[11] Goodman, R.; Wallach, N. R., An algebraic group approach to compact symetric spaces (1997)
[12] Helgason, S., Differential geometry, Lie groups, and symmetric spaces (1978) · Zbl 0451.53038
[13] Jackson, S. G.; Noel, A. G., Prehomogeneous spaces associated with nilpotent orbits (2005) · Zbl 1147.17300
[14] Jantzen, J. C., Lie Theory, 228, 1-211 (2004) · Zbl 1169.14319
[15] Kawanaka, N., Orbits and stabilizers of nilpotent elements of a graded semisimple Lie algebra, J. Fac. Sci. Univ. Tokyo, 34, 573-597 (1987) · Zbl 0651.20046
[16] King, D. R., The component groups of nilpotents in exceptionnal simple real Lie algebras, Comm. Algebra, 20, 219-284 (1992) · Zbl 0758.17006 · doi:10.1080/00927879208824339
[17] Kostant, B.; Rallis, S., Orbits and representations associated with symmetric spaces, Amer. J. Math., 93, 753-809 (1971) · Zbl 0224.22013 · doi:10.2307/2373470
[18] Otha, T., The singularities of the closure of nilpotent orbits in certain symmetric pairs, Tôhoku Math. J., 38, 441-468 (1986) · Zbl 0654.22004 · doi:10.2748/tmj/1178228456
[19] Otha, T., The closure of nilpotent orbits in the classical symmetric pairs and their singularities, Tôhoku Math. J., 43, 161-211 (1991) · Zbl 0738.22007 · doi:10.2748/tmj/1178227492
[20] Panyushev, D. I., The Jacobian modules of a representation of a Lie algebra and geometry of commuting varieties, Compositio Math., 94, 181-199 (1994) · Zbl 0834.17003
[21] Panyushev, D. I., On the conormal bundle of a \(G\)-stable subvariety, Manuscripta Math., 99, 185-202 (1999) · Zbl 0961.14030 · doi:10.1007/s002290050169
[22] Panyushev, D. I., On the irreducibility of commuting varieties associated with involutions of simple Lie algebras, Func. Anal. Appl., 38, 38-44 (2004) · Zbl 1125.17001 · doi:10.1023/B:FAIA.0000024866.28468.c2
[23] Panyushev, D. I., Two results on centralisers of nilpotent elements, J. Pure Appl. Algebra, 212, 774-779 (2008) · Zbl 1137.17017 · doi:10.1016/j.jpaa.2007.07.003
[24] Panyushev, D. I.; Yakimova, O., Symmetric pairs and associated commuting varieties, Math. Proc., 143, 307-321 (2007) · Zbl 1126.17010
[25] Popov, V. L.; Tevelev, E. A., Self-dual projective algebraic varieties associated with symmetric spaces, Algebraic transformation groups and algebraic varieties, 132, 131-167 (2004) · Zbl 1093.14072
[26] Premet, A., Nilpotent commuting varieties of reductive Lie algebras, Invent. Math., 154, 653-683 (2003) · Zbl 1068.17006 · doi:10.1007/s00222-003-0315-6
[27] Richardson, R. W., Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math., 38, 311-327 (1979) · Zbl 0409.17006
[28] Sabourin, H.; Yu, R. W. T., Sur l’irréductibilité de la variété commutante d’une paire symétrique réductive de rang 1, Bull. Sci. Math., 126, 143-150 (2002) · Zbl 1017.17010 · doi:10.1016/S0007-4497(01)01091-0
[29] Sabourin, H.; Yu, R. W. T., On the irreducibility of the commuting variety of the symmetric pair \((\mathfrak{so}_{p+2},\mathfrak{so}_p\times \mathfrak{so}_2)\), J. Lie Theory, 16, 57-65 (2006) · Zbl 1128.17009
[30] Sekiguchi, J., The nilpotent subvariety of the vector space associated to a symmetric pair, Publ. RIMS Kyoto Univ., 20, 155-212 (1984) · Zbl 0556.14022 · doi:10.2977/prims/1195181836
[31] Springer, T. A.; Steinberg, R., Seminar on algebraic groups and related finite groups, 131, 167-266 (1970) · Zbl 0249.20024
[32] Tauvel, P.; Yu, R. W. T., Lie algebras and algebraic groups (2005) · Zbl 1068.17001
[33] Vinberg, E. B., Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra, Selecta Math. Sovietica, 6, 15-35 (1987) · Zbl 0612.17010
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