Non-Local Equivariant Star Product on the Minimal Nilpotent Orbit

Cohomology of the minimal nilpotent orbit

We compute the integral cohomology of the minimal non-trivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the sub-root system generated by the long simple roots. The modulo l reduction of the Springer correspondent representation involves the sign representation exactl...

Dimension of a minimal nilpotent orbit

We show that the dimension of the minimal nilpotent coadjoint orbit for a complex simple Lie algebra is equal to twice the dual Coxeter number minus two. Let g be a finite dimensional complex simple Lie algebra. We fix a Cartan subalgebra h, a root system ∆ ⊂ h and a set of positive roots ∆+ ⊂ ∆. Let ρ be half the sum of all positive roots. Denote by θ the highest root and normalize the Killing...

On the Nilpotent Multiplier of a Free Product

From Dixmier Algebras to Star Products

Let M be a Galois cover of a nilpotent coadjoint orbit of a complex semisimple Lie group. We define the notion of a perfect Dixmier algebra for M and show how this produces a graded (non-local) equivariant star product on M with several very nice properties. This is part of a larger program we have been developing for working out the orbit method for nilpotent orbits.

Non-locality of Equivariant Star Products on T∗(rp)

Lecomte and Ovsienko constructed SLn+1(R)-equivariant quantization maps Qλ for symbols of differential operators on λ-densities on RP . We derive some formulas for the associated graded equivariant star products ⋆λ on the symbol algebra Pol(T ∗RP). These give some measure of the failure of locality. Our main result expresses (for n odd) the coefficients Cp(·, ·) of ⋆λ when λ = 1 2 in terms of s...