2000/10/27 by Alexander Astashkevich, Astashkevich, Alexander, Ranee Brylinski +1
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Symplectic Geometry (math.SG) #math.QA #math.RT #math.SG
paper · pdf · doi:10.48550/arxiv.math/0010257
latex file, 13 pages. In this new version we use the star product to construct a unitary representation attached to the orbit
arxiv created 2001/03/20 · arxiv updated 2009/11/30
We construct a unique G-equivariant graded star product on the algebra S(g)/I of polynomial functions on the minimal nilpotent coadjoint orbit \Omin of G where G is a complex simple Lie group and g≠\sl2(C). 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 μx with any function f. We find μx⋆ f=μxf+\half\μx,f\t+Λx(f)t2. For \g different from spn(\C), Λx is not a differential operator. Instead \Lamdax is the left quotient of an explicit order 4 algebraic differential operator Dx by an order 2 invertible diagonalizable operator. Precisely, Λx=-1/4(1)/(E'(E'+1))Dx where E' is a positive shift of the Euler vector field. Thus μx⋆ f is not local in f. Using ⋆ we construct a positive definite hermitian inner product on Sg/I. The Hilbert space completion of Sg/I is then a unitary representation of G. This quantizes \Omin in the sense of geometric quantization and the orbit method.