Functions associated with group representations

Given a linear representation of a group, the usual approach is to consider numerical functions on the group that are associated with the representation. These may be generalized functions, that is, distributions. The simplest functions are matrix elements. They are given by a pair of vectors whose choice is arbitrary and random. However, we are interested in functions which are free of this arbitrariness (and thus are naturally associated with the representation); we refer to them as the modified traces of the representation. The usual trace of the representation (if it exists, possibly as a distribution) and spherical functions generated by a fixed vector of some subgroup give examples of these functions. However, it is possible that neither the trace nor spherical functions can be defined for a given representation. It is still desirable to introduce functions on the group that are naturally associated with the representation. We solve this problem for diffeomorphism groups and some representations of these groups.
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