Distinguished Orbits of Reductive Groups

Stable Reductive Varieties I: Affine Varieties

0. Introduction 1 1. Main definitions and results 3 2. General criteria 6 2.1. Seminormality and connectedness of isotropy groups 6 2.2. Finiteness of number of orbits and group–like condition 9 3. Orbits in stable reductive varieties 11 3.1. Isotropy groups 11 3.2. Algebras of regular functions 14 4. Reductive varieties 18 4.1. Classification 18 4.2. Associated stable toric varieties 20 5. Sta...

ADMISSIBLE NILPOTENT COADJOINT ORBITS OF p-ADIC REDUCTIVE LIE GROUPS

The orbit method conjectures a close relationship between the set of irreducible unitary representations of a Lie group G, and admissible coadjoint orbits in the dual of the Lie algebra. We define admissibility for nilpotent coadjoint orbits of p-adic reductive Lie groups, and compute the set of admissible orbits for a range of examples. We find that for unitary, symplectic, orthogonal, general...

Branching Rules for the Weyl Group Orbits of the Lie Algebra

The orbits of Weyl groups W (An) of simple An type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of An. Matrices transforming points of the orbits of W (An) into points of subalgebra orbits are listed for all cases n ≤ 8 and for the infinite series of algebra-subalgebra pairs An ⊃ An−k−1 × Ak × U1, A2n ⊃ Bn, A2n−1 ⊃ Cn, A2n−1 ⊃ Dn. Numerous ...

The method of coadjoint orbits for real reductive groups

Self–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 ...