There are two notions of $G$-equivariant pseudodifferential operators on a Riemannian symmetric space of non-compact type $G/K$. One is the traditional Hörmander class of pseudodifferential operators defined using the Fourier inversion formula on $\mathbb{R}^n$ that are $G$-equivariant. These operators do not have a well-defined global symbol function. The other is the class of $G$-equivariant multiplier operators defined using the Harish-Chandra Fourier transform on $G/K$ satisfying the appropriate symbol-type estimates. I will call them Harish-Chandra pseudodifferential operators. We will see that these two notions coincide identically for a complex semisimple Lie group $G$.

If time permits, we will see that the Hörmander principal symbol of a $G$-equivariant pseudodifferential operator can be interpreted as an appropriate limit of its Harish-Chandra symbol, under a deformation of $G$ to its Cartan motion group. This is a joint work with Nigel Higson.