Question

I have one issue with the Jacquet Langlands correspondance. 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 correspondance explicitely 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 an 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, i.e.

Risager, Morten S. Asymptotic densities of Maass newforms. J. Number Theory 109 (2004), no. 1, 96–119.

Answer 1

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.