What is the relationship between (g,K)-module and Maass forms for GL(2)?


(g,K)-modules are defined in chapter 2 of Bump, Automorphic forms and representations.


There is a classification of (g,K)-modules.


What is the relationship between (g,K)-module and Maass forms for GL(2)? and what does the classification of (g,K)-module imply for GL(2) Maass forms?



What is the relationship between (g,K)-module and Maass forms?
So you've seen that there are essentially three types of (g, K)-modules: finite-dimensional ones; principal series; and discrete series. The finite-dimensional ones don't interest us, since they are never unitary. So that leaves two cases. And these correspond precisely to the two kinds of cuspidal automorphic representations for $GL(2, \mathbf{A}\_\mathbf{Q})$: those whose factor at $\infty$ is discrete series (or a limit of discrete series) correspond to classical modular eigenforms; and those whose factor at $\infty$ is principal series correspond to Maass forms.


You could say that the classification doesn't directly tell you anything new about Maass forms themselves, but rather it tells you about how Maass forms fit into the general picture of automorphic forms.


