I am currently trying to understand the local expression of a (pseudo)differential operator $ \int_{R^n} e^{(x - y)\cdot \xi} \sigma(x,\xi) \, d \xi $ on a manifold $M$ (compact and boundaryless, say), where $\sigma(x,\xi)$, as usual, is the symbol of the operator. (alternatively one may also write this as $ \int_{R^n} e^{(x - y)\cdot \xi} a(x,y,\xi) \, d \xi $ where $a(x,y,\xi)$ belongs to the same symbol class as $\sigma$.)
now the picture I have in my head is that the variable $(x,\xi)$ stands for a local coordinate in the cotangent bundle $T^*M$. but then, at the same time, the variables $(x,y)$ are local coordinates for some subset $U \times V$ of $M \times M$, so I fear I am mixing something up here?
Also, I was wondering whether the dot product $(x - y) \cdot \xi$ in the exponential term has anything to do with the pairing of elements from $T^*M$ with elements of the tangent bundle $TM$? some sources for example write the dot product in terms of a braket, $(x - y) \cdot \xi = \langle (x - y), \xi \rangle$ which makes this even more suggestive. but I struggle to make sense of this, I don't really know how to relate $(x - y)$ to the tangent bundle $TM$, making it very likely that this picture is wrong anyways.
Many thanks for your comments!