I have been reading through several papers on Deformation Quantization and the terms "symbol" and "symbol calculus" keep cropping up. I am somewhat well acquainted with most of the basic notions in differential geometry, but I have yet to run into this in any textbook and I'm having a hard time finding a definition for this.
Here's an excerpt of the paper "On representations of star product algebras over cotangent spaces on Hermitian line bundles" by Bordemann et al. where the term is used:
... Under an operator representation we understand a $\mathbb{C}$-linear assignment of a pseudo-differential operator $\mathrm{Op}_{\hbar}(a)$ on $\Gamma^\infty(L)$ [smooth sections of a Hermitian line bundle $L$ over $Q$] for every $\hbar\in \mathbb{R}^+$ ... and every symbol $a\in S^\infty(Q)\subset C^\infty(T^*Q)$...
I understand that $S^\infty(Q)$ is the set of smooth symbols on $Q$, which somehow is a subset of smooth functions on $T^*Q$, but I am still somewhat confused. What is a "symbol"?