TITLE: Jacquet Langlands correspondence
QUESTION [6 upvotes]: I have one issue with the Jacquet Langlands correspondence. The Weyl law for $H$ modulo a congruence subgroup and the Weyl law for cocompact groups are different. So why does this not contradict this functoriality? What am I missing?
I have not yet studied the Jacquet Langlands correspondence explicitly yet. How explicit are the lifts, about the level etc.? I know that there is not an expansion formula for cocompact groups available as we have it for groups with a parabolic element.
Update: After a reading a little bit, I found a paper which focuses exactly on the first part of the question and also gives references for the second part of the question:
Risager, Morten S. Asymptotic densities of Maass newforms. J. Number Theory 109 (2004), no. 1, 96–119.

REPLY [4 votes]: In what sense is the Weyl law different for congruence subgroups and cocompact groups? 
At any rate, the Jacquet-Langlands correspondence is not a bijection between the two cuspidal spectra. More precisely, let $D$ be a quaternion algebra over a number field $F$, and consider the groups $G=PD^\times$ and $G'=PGL_2$. Then the Jacquet-Langlands correspondence injects the automorphic representations of $G(\mathbb{A}_F)$ into those of $G'(\mathbb{A}_F)$. A cuspidal representation $\pi$ of $G'(\mathbb{A}_F)$ lies in the image of this map if and only if $\pi_v$ is a discrete series representation of $G'(F_v)$ at all places $v$ where $D$ ramifies. So unless $G'=G$, the image will miss several cuspidal representations of $G'(\mathbb{A}_F)$.
I think the lifts are not explicit in the sense that they are not given by an explicit construction.