Non-Local Equivariant Star Product on the Minimal Nilpotent Orbit

✍ Scribed by Alexander Astashkevich; Ranee Brylinski

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 204 KB
• Volume: 171
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2002.2073

✦ Synopsis

We construct a unique G-equivariant graded star product on the algebra SðgÞ=I of polynomial functions on the minimal nilpotent coadjoint orbit O min of G where G is a complex simple Lie group and gaslð2; CÞ: This strengthens the result of Arnal, Benamor and Cahen.

Our main result is to compute, for G classical, the star product of a momentum function m x with any function f :

For g different from spð2n; CÞ; L x is not a differential operator. Instead L x is the left quotient of an explicit order 4 algebraic differential operator D x by an order 2 invertible diagonalizable operator. Precisely, L x ¼ À 1 4 1 E 0 ðE 0 þ1Þ D x where E 0 is a positive (half-form) shift of the Euler vector field. Thus m x $ f is not local in f :

Using $ we construct a positive definite hermitian inner product on SðgÞ=I: The Hilbert space completion of SðgÞ=I is then a unitary representation of G: This quantizes O min in the sense of geometric quantization and the orbit method. # 2002 Elsevier Science (USA)