Looking into the Wigner-Weyl transformation mapping Hilbert space operators to functions on phase-space, I've run up against the need for a symplectic convolution

$$[F\star G](x,p) = \int \!dy\,dk\, F(y,k)G(x-y,p-k)e^{i (xk-yp)}$$

or more compactly

$$[F\star G](\alpha) = \int \!d\beta\, F(\beta)G(\alpha-\beta)e^{i \alpha \wedge \beta}$$

where $\alpha = (x,p)$ and $\wedge$ is the symplectic form. Unlike the symplectic Fourier transform,

$$\hat{F}(\xi) = \int \!d\alpha\, F(\alpha)e^{i\alpha\wedge\xi}$$

which can be completely understood as normal Fourier transform followed by a change of variables $(\xi_x,\xi_p) \to (-\xi_p,\xi_x)$, the symplectic convolution appears to have nontrivial properties. However, I have not been able to find almost any references that discuss it. The most clear reference to it I could find is in "Toeplitz and Hankel operators and Dixmier traces on the unit ball of $\mathbb{C}^n$" by Englis et al., but they seem to use it without much discussion.

Does the symplectic convolution go by another name? What introductory references discuss its basic properties?