A. Causal Symmetric Spaces
The original motivation for the study of spherical functions on ordered symmetric spaces came from a diagonalization procedure for the Bethe-Salpeter equation from scattering theory. The spherical functions give characters for algebras of invariant integral kernels satisfying causality conditions. Meanwhile there is a rich theory which parallels Harish-Chandra's theory for Riemannian symmetric spaces. In particular, for many examples one has integral representations and asymptotic estimates for spherical functions. There are interesting related questions about Laplace transforms, positive definiteness and connections with unitary representations.
[A17, A23, A34, A37, A38, A40, A41, A42, A43, A44, B04, C04]
B. Symmetric Super spaces
Some parts of Harish-Chandra's harmonic analysis for Riemannian symmetric spaces has been extended to symmetric super spaces in the sense of Zirnbauer. In particular one has results on the relevant invariant theory and Berezin integration. Moreover, necessary preparatory work on singular superspaces has been done.
[A58, A59, A63, A65, A67, A68]
C. Radon transforms on homogenen spaces
These are spezial integral operators which can be described via geometric substructures of given homogeneous spaces. A typical goal in this context is to establish inversion formulas.
[A47, A49, B06, B18]
D. Wiener-Hopf Operators
The starting point for this topic was the study of integral and differential equations satisfying causality and symmetry conditions (e.g. Lorentz invariance). One use the interplay of order and symmetry to derive properties of solutions. One of the problems is to build an index theory for Wiener-Hopf operators (certain integral operators taking the causal structure of the underlying manifold into account) on ordered symmetiric spaces. This requires a structure theory for the C*-algebra of all Wiener-Hopf operators. This algebra is a quotient of a well-understood groupoid C*-algebra. Trying to determine the ideal one has to factor out naturally leads Radon transforms on homogeneous spaces.
[A27, A29, B08, C04]