On the existence of smooth orbital varieties in simple Lie algebras

Autor: Lucas Fresse, Anna Melnikov
Přispěvatelé: Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), University of Haifa [Haifa], L. Fresse is supported in part by the ISF Grant Nr. 797/14 and by the ANR project GeoLie ANR-15-CE40-0012., ANR-15-CE40-0012,GéoLie,Méthodes géométriques en théorie de Lie(2015)
Rok vydání: 2019
Předmět:
Pure mathematics
General Mathematics
domino tableaux
Nilpotent orbit
classical groups
Type (model theory)
01 natural sciences
[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]
Springer fibers
Simple (abstract algebra)
induced nilpotent orbits
0103 physical sciences
Lie algebra
FOS: Mathematics
2010 Mathematics Subject Classification. 14L30
14M15
17B08
05E15
Representation Theory (math.RT)
0101 mathematics
Mathematics::Representation Theory
Mathematics
14L30
14M15
17B08
05E15
[MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT]
010102 general mathematics
Subalgebra
Reductive group
Nilpotent
[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
Astrophysics::Earth and Planetary Astrophysics
010307 mathematical physics
Variety (universal algebra)
orbital varieties
Mathematics - Representation Theory
Zdroj: Journal of the London Mathematical Society
Journal of the London Mathematical Society, London Mathematical Society, 2020, 101 (3), pp.960-983. ⟨10.1112/jlms.12293⟩
ISSN: 1469-7750
0024-6107
Popis: International audience; Orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible components of Springer fibers. In type A, a construction due to Richardson implies that every nilpotent orbit contains at least one smooth orbital variety and every Springer fiber contains at least one smooth component. In this paper, we show that this property is also true for the other classical cases. Our proof uses the interpretation of Springer fibers as varieties of isotropic flags and van Leeuwen's parametrization of their components in terms of domino tableaux. In the exceptional cases, smooth orbital varieties do not arise in every nilpotent orbit, as already noted by Spaltenstein. We however give a (nonexhaustive) list of nilpotent orbits which have this property. Our treatment of exceptional cases relies on an induction procedure for orbital varieties, similar to the induction procedure for nilpotent orbits.
Databáze: OpenAIRE
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