Classical and adelic Eisenstein series

Abstract

We carry out "Hecke summation" for the classical Eisenstein series E k in an adelic setting. The connection between classical and adelic functions is made by explicit calculations of local and global intertwining operators and Whittaker functions. In the process we determine the automorphic representations generated by the E k , in particular for k = 2, where the representation is neither a pure tensor nor has finite length. We also consider Eisenstein series of weight 2 with level, and Eisenstein series with character.

The Bernstein-Gelfand Tensor Product Functor and the Weight-2 Eisenstein Series

“…Note that the discrete series D(1) in this sequence is the one for SL 2 (R) and thus different from the one in [11] for GL 2 (R).…”

“…In this section, we revisit some of theory of automorphic forms, principal series, and automorphic Eisenstein series. We base this section on recent work by Roy-Schmidt-Yi [11], who examined the automorphic representation associated with E 2 . The content of this section is mostly preparatory for Section 4.…”

“…The Eisenstein series of weight 2 Roy-Schmidt-Yi [11] determined the exact structure of (E 2 ) in the context of automorphic forms for GL 2 in their Theorem 5.11.…”

“…When giving up the assumption that f is cuspidal, this no longer holds in general. For example, Roy-Schmidt-Yi [11] determined the structure of (E 2 ) for the modular, nonholomorphic Eisenstein series E 2 of weight 2 and level 1. They found that it fits into an exact sequence…”