About the convolution of distributions on groupoids

We review the properties of transversality of distributions with respect to submersions. This allows us to construct a convolution product for a large class of distributions on Lie groupoids. We get a unital involutive algebra $\mathcal E_{r,s}'(G,\Omega^{1/2})$ enlarging the convolution algebra $C^\infty_c(G,\Omega^{1/2})$ associated with any Lie groupoid $G$. We prove that $G$-operators are convolution operators by transversal distributions. We also investigate the microlocal aspects of the convolution product. We give sufficient conditions on wave front sets to compute the convolution product and we show that the wave front set of the convolution product of two distributions is essentially the product of their wave front sets in the symplectic groupoid $T^*G$ of Coste–Dazord–Weinstein. This also leads to a subalgebra $\mathcal E_{a}'(G,\Omega^{1/2})$ of $\mathcal E_{r,s}'(G,\Omega^{1/2})$ which contains for instance the algebra of pseudodifferential $G$-operators and a class of Fourier integral $G$-operators which will be the central theme of a forthcoming paper.