This chapter goes to the transcendental level, i.e., take an embedding ι : E → ℂ, and extend the ground field to ℂ. The entirety of this chapter works over ℂ and therefore suppresses the subscript ℂ. It begins with the cuspidal parameters and the representation 𝔻λ at infinity. Next, the chapter defines the square-integrable cohomology as well as the de Rham complex. Finally, cuspidal cohomology is addressed. Here, the chapter looks at the cohomological cuspidal spectrum and the consequence of multiplicity one and strong multiplicity one. It also shows the character of the component group I, before dropping the assumption that we are working over ℂ and go back to our coefficient system 𝓜̃λ,E defined over E.
Computations of cuspidal cohomology of congruence subgroups of SL(3, Z)