I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion.

### Background

I think I understand the basic idea on ℝ^{n}, so for readers who know as little as I do, I will provide some ideas. Any differential operator on ℝ^{n} is (uniquely) of the form Σ p_{i1,...,ik}(x) ∂^{k}/(∂x_{i1}...∂x_{ik}), where x_{1},...,x_{n} are the canonical coordinate functions on ℝ^{n}, the p_{i1,...,ik}(x) are smooth functions, and the sum ranges over (finitely many) possible indexes (of varying length). Then the **symbol** of such an operator is Σ p_{i1,...,ik}(x) ξ^{i1}...ξ^{ik}, where ξ^{1},...,ξ^{n} are new variables; the symbol is a polynomial in the variables {ξ^{1},...,ξ^{n}} with coefficients in the algebra of smooth functions on ℝ^{n}.

Ok, great. So symbols are well-defined for ℝ^{n}. But most spaces are not ℝ^{n} — most spaces are formed by gluing together copies of (open sets in) ℝ^{n} along smooth maps. So what happens to symbols under changes of coordinates? An **affine change of coordinates** is a map y_{j}(x) = a_{j} + Σ_{i} Y_{j}^{i}x_{i} for some vector (a_{1},...,a_{n}) and some invertible matrix Y. It's straightforward to describe how the differential operators change under such a transformation, and thus how their symbols transform. In fact, you can forget about the fact that indices range 1,...,n, and think of them as keeping track of tensor contraction; then everything transforms as tensors under affine coordinate changes, e.g. the variables ξ^{i} transform as coordinates on the cotangent bundle.

On the other hand, consider the operator D = ∂^{2}/∂x^{2} on ℝ, with symbol ξ^{2}; and consider the change of coordinates y = f(x). By the chain rule, the operator D transforms to (f'(y))^{2} ∂^{2}/∂y^{2} + f''(y) ∂/∂y, with symbol (f'(y))^{2} ψ^{2} + f''(y) ψ. In particular, the symbol did not transform as a function on the cotangent space. Which is to say that I don't actually understand where the symbol of a differential operator lives in a coordinate-free way.

### Why I care

One reason I care is because I'm interested in quantum mechanics. If the symbol of a differential operator on a space X were canonically a function on the cotangent space T*X, then the inverse of this Symbol map would determine a "quantization" of the functions on T*X, corresponding to the QP quantization of ℝ^{n}.

But the main reason I was thinking about this is from Lie algebras. I'd like to understand the following proof of the PBW theorem:

Let

Lbe a Lie algebra over ℝ or ℂ,Ga group integrating the Lie algebra, ULthe universal enveloping algebra ofLand SLthe symmetric algebra of the vector spaceL. Then ULis naturally the space of left-invariant differential operators onG, and SLis naturally the space of symbols of left-invariant differential operators onG. Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism UL→ SL.