On a class of nearly holomorphic automorhic forms (Q1076061)

The ''nearly holomorphic'' automorphic forms discussed in this paper have, for their prototype, the ''non-analytic'' Eisenstein series \[ f(z,s)=\sum_{(0,0)\neq (c,d)\in {\mathbb{Z}}\times {\mathbb{Z}}}(cz+d)^{-k} | cz+d|^{-2s}\quad (with\quad z\in {\mathbb{C}},\quad Im z>0\quad and\quad k\in {\mathbb{N}}) \] encountered in Hecke's ''Grenzprozeß'', wherein, for \(k=2\), \(\lim_{s\to 0} f(z,s)\) involves a non-holomorphic function constant \(\times (Im z)^{-1}\). Continuing the deep investigations in his earlier papers on similar Eisenstein series corresponding to the symplectic group over a totally real algebraic number field or the unitary group over a CM-field and the ''inevitable non-holomorphy'' occurring in some cases, the author (develops a systematic calculus to deal with the ''non-holomorphy'' and) thoroughly analyses herein problems of the same kind for a more general family of series attached to orthogonal and unitary groups acting on bounded symmetric domains (besides providing a glimpse into arithmeticity and residues involved).