Complete Symbols of Equivariant Pseudodifferential Operators on Noncompact Symmetric Spaces

We study $G$-equivariant Hörmander pseudodifferential operators on a noncompact symmetric space $G/K$. We define a notion of a complete symbol function, called the Harish-Chandra symbol function, on the spherical tempered dual of $G$, for operators that satisfy a rapid off-diagonal decay condition on their Schwartz kernels, and we characterize the class of such operators in terms of their Harish-Chandra symbol function.