ABOUT THE CONVOLUTION OF DISTRIBUTIONS ON GROUPOIDS
Résumé
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 algebra $\cE_{r,s}'(G,\Omega^{1/2})$ 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 conditions on wave front sets sufficient 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 cotangent groupoid $T^*G$ of Costes-Dazord-Weinstein. This also leads to a subalgebra $\cE_{a}'(G,\Omega^{1/2})$ of $\cE_{r,s}'(G,\Omega^{1/2})$ which contains for instance the algebra of pseudodifferential operators on $G$.
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