On Varieties in an Orbital Variety Closure in Semisimple Lie Algebras
نویسنده
چکیده
In this note we discuss the closure of an orbital variety as a union of varieties. We show that if semisimple Lie algebra g contains factors not of type An then there are orbital varieties whose closure contains components which are not Lagrangian. We show that the argument does not work if all the factors are of type An and provide the facts supporting the conjecture claiming that if all the factors of g are of type An then the closure of an orbital variety is a union of orbital varieties.
منابع مشابه
ON ORBITAL VARIETY CLOSURES IN sln II. DESCENDANTS OF A RICHARDSON ORBITAL VARIETY
For a semisimple Lie algebra g the orbit method attempts to assign representations of g to (coadjoint) orbits in g∗. Orbital varieties are particular Lagrangian subvarieties of such orbits leading to highest weight representations of g. In sln orbital varieties are described by Young tableaux. In this paper we consider so called Richardson orbital varieties in sln. A Richardson orbital variety ...
متن کاملON ORBITAL VARIETY CLOSURES IN sln I. INDUCED DUFLO ORDER
For a semisimple Lie algebra g the orbit method attempts to assign representations of g to (coadjoint) orbits in g∗. Orbital varieties are particular Lagrangian subvarieties of such orbits leading to highest weight representations of g. In sln orbital varieties are described by Young tableaux. Inclusion relation on orbital variety closures defines a partial order on Young tableaux. Our aim is t...
متن کاملSelf–Dual Algebraic Varieties and Nilpotent Orbits
We give a construction of nonsmooth self-dual projective algebraic varieties. They appear as certain projectivized orbit closures for some linear actions of reductive algebraic groups. Applying this construction to adjoint representations, we obtain geometric characterization of distinguished nilpotent elements of semisimple Lie algebras [BC1], [BC2] (i.e., nilpotent elements whose centralizer ...
متن کاملA characterization of nilpotent varieties of complex semisimple Lie algebras
A normal complex algebraic variety X is called a symplectic variety (cf. [Be]) if its regular locus Xreg admits a holomorphic symplectic 2-form ω such that it extends to a holomorphic 2-form on a resolution f : X̃ → X. Affine symplectic varieties are constructed in various ways such as nilpotent orbit closures of a semisimple complex Lie algebra (cf. [CM]), Slodowy slices to nilpotent orbits (cf...
متن کاملOn Primitive Ideals
We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the Irreducibility theorem and Duflo theorem, to much wider classes of algebras. Our general version of Irreducibility theorem says that if A is a positively filtered associative algebra such that grA is a commutative Poisson algebra with finitely many symplectic leaves, then the associated v...
متن کامل