X (and hence a polynomial function on the fibers) is an order d symbol. The order d symbol map sends Vd(X) into Rd(T* X) where Vd(X) is the space of differential operators of order at most d and Rd(T* X) is the space of order d symbols. We say that ¢ E Rd(T* X) quantizes into D if D E Vd(X) has order d and the principal symbol of D is ¢.
4. 1. Coordinate representations of vector fields. 3, is to quantize the symbol S = foro into an operator S which satisfies condition (ii) in Prop. 4. To construct S and compute S(fj), we will simply work out everything in terms of our local coordinates fo, f~, Ii, fI on 0 from (43). 1 in [A-B2J, we first found a formula for the highest weight vector ft/J E R1(O) ~ g. 1. [A-B2] The unique expression for the function f,p E R1(O) in terms of our local coordinates fo, f~, fi, f:, i = 1, ... , m, on 0 is (63) The unique expressions for the vector fields "lXi, "Ix; ,TJxo in terms of our local coordinates fi, fI, fo, f~ are, where i = 1, ...
Advances in Geometry by Alexander Astashkevich (auth.), Jean-Luc Brylinski, Ranee Brylinski, Victor Nistor, Boris Tsygan, Ping Xu (eds.)