I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for fixed eigenvalue should be at most one. 


Since Maass cusp forms always are defined for a Fuchsian lattice, I wonder 
1) for which lattices this conjecture had been conjectured?
2) what is the motivation for this conjecture?
3) to whom this conjecture is due?
4) is it published somewhere?
5) is it proven in some cases?



Multiplicity one conjecture
I rethought your question and have discovered a partial answer for 1) and 2). I add this as a disjoint answer, since my other answer adresses a totally different (negative) issue.


In Sarnak's article <http://web.math.princeton.edu/sarnak/baltimore.pdf>, he recalls one famous conjecture (Conjecture I, due to himself) that $\Gamma \backslash \mathbb{H}$ should have very few Maass forms for "most" Fuchsian lattices $\Gamma$.


Btw, he attributes the simplicity conjecture (Conjecture 3) for $SL\_2(\mathbb{Z})$ to Cartier.


Wolpert (Theorem I) has shown that Sarnak's conjecture would follow if the simplicity conjecture holds for the congruence subgroup $\Gamma(2)$.


Also GH last conjecture that the multiplicity is uniformly bounded in $N$ would suffice for Sarnak's conjecture, but current knowledge is that the multiplicity of an eigenvalue of magnitude $T$ is at most $\ll\_N \sqrt{T}/ \log T$ and not $\ll\_N 1$. 


In fact, Sarnak conjectures that there exist $\Gamma$ with only finitely many Maass wave forms, but this does not follow from the simplicity conjecture for $\Gamma(2)$.


