The
philosophy of cusp forms was a slogan of
Harish-Chandra, expressing his idea of a kind of reverse engineering of
automorphic form theory, from the point of view of
representation theory. The
discrete group Γ fundamental to the classical theory disappears, superficially. What remains is the basic idea that representations in general are to be constructed by parabolic induction of
cuspidal representations. A similar philosophy was enunciated by
Israel Gelfand, and the philosophy is a precursor of the
Langlands program. A consequence for thinking about representation theory is that
cuspidal representations are the fundamental class of objects, from which other representations may be constructed by procedures of induction. According to
Nolan Wallach Put in the simplest terms the "philosophy of cusp forms" says that for each Γ-conjugacy classes of Q-rational parabolic subgroups one should construct automorphic functions (from objects from spaces of lower dimensions) whose constant terms are zero for other conjugacy classes and the constant terms for [an] element of the given class give all constant terms for this parabolic subgroup. This is almost possible and leads to a description of all automorphic forms in terms of these constructs and cusp forms. The construction that does this is the
Eisenstein series. ==Notes==