This version is called the " L 2 index theorem, " and was used by Atiyah and Schmid to give a geometric construction, using square integrable harmonic spinors, of Harish-Chandra's discrete series representations of semisimple Lie groups.
32.
Two such vectors " v " correspond to the same discrete series representation if and only if they are conjugate under the Weyl group " W " " K " of the maximal compact subgroup " K ".
33.
In terms of representation theory, modular forms correspond roughly to highest weight vectors of certain discrete series representations of SL 2 ( "'R "'), while almost holomorphic or quasimodular forms correspond roughly to other ( not necessarily highest weight ) vectors of these representations.
34.
If we fix a fundamental chamber for the Weyl group of " K ", then the discrete series representation are in 1 : 1 correspondence with the vectors of " L " + ? in this Weyl chamber that are not orthogonal to any root of " G ".
