# Haar measure on $PGL(2,\mathbb{Q}_p)$, the local Jacquet-Langlands correspondence, and Ihara's theorem

Goal. I would like to calculate the product of the formal dimension of a discrete series representation of $GL(2,\mathbb{Q}_p)$ with trivial central character (so, an irreducible unitary representation of $PGL(2,\mathbb{Q}_p)$ that is a subrepresentation of $L^2(PGL(2,\mathbb{Q}_p))$) and the covolume of a torsion-free (cocompact) lattice in $PGL(2,\mathbb{Q}_p)$. First, the formal dimension and the covolume must be calculated with respect to the same Haar measure.

Background and attempt. The local Jacquet-Langlands correspondence (Section 15 of , or Section 4 of ) gives a bijection between irreducible representations of the unit group of the quaternion algebra over a local field $F$ and the discrete series representations of $GL(2,F)$, in which the central characters on both groups are the same. Let $Z$ denote the center of $GL(2,F)$, which we may identify with $F^\times$. As explained in , if the Haar measure on $GL(2,F)/Z$ is normalized so that the formal dimension of the Steinberg representation of $GL(2,F)$ is $1$, then the formal dimension of the discrete series representation of $GL(2,F)$ is equal to the actual dimension of the corresponding representation of the unit group of the quaternion algebra over $F$. For the rest of this post, specialize to the case $F=\mathbb{Q}_p$, where $p \neq 2$. If I understand Section 2.2 of  correctly, normalizing the Haar measure on $GL(2,\mathbb{Q}_p)/Z$ so that the measure of $Z . GL(2,\mathbb{Z}_p) / Z=\frac{1}{2}(p-1)$ will force the formal dimension of the Steinberg representation to be $1$.

Suppose we have an irreducible representation of the unit group of the quaternion algebra over $\mathbb{Q}_p$ of dimension $m$ with trivial central character. (I haven't looked closely at the dimensions calculated in  yet, but these would supply the main examples of formal dimensions for my project.) If the Haar measure on $GL(2,\mathbb{Q}_p)/Z$ is normalized so that the measure of $Z . GL(2,\mathbb{Z}_p) / Z=\frac{1}{2}(p-1)$, as in the previous paragraph, then the formal dimension of the corresponding representation of $GL(2,\mathbb{Q}_p)$ will be $m$. Since the central character of the representation of the unit group of the quaternions is trivial, the central character of the corresponding representation of $GL(2,\mathbb{Q}_p)$ is trivial, and we have an irreducible unitary representation of $PGL(2,\mathbb{Q}_p)$ that is a subrepresentation of $L^2(PGL(2,\mathbb{Q}_p))$, with formal dimension $m$.

Ihara showed (Corollary to Theorem 1 in ) that any torsion-free discrete subgroup $\Gamma$ of $PGL(2,\mathbb{Q}_p)$ (or any other locally compact field under a discrete valuation) is isomorphic to a free group, over a set of at most countable generators; and if $\Gamma \backslash PGL(2,\mathbb{Q}_p)$ is compact, then the number of free generators $n$ of $\Gamma$ is equal to $\frac{1}{2} (p-1) h + 1$, where $h$ is the cardinality of $\Gamma \backslash PGL(2,\mathbb{Q}_p) / PGL(2,\mathbb{Z}_p)$. (Observations: According to Ihara's results, the free group on $n$ generators can be found in $PGL(2,\mathbb{Q}_p)$ only if $h=2(n-1)/(p-1)$ is an integer, since $h$ is the cardinality of a (finite) set of double coset representatives. As for existence, he gives a construction in Section 4.)

Let $\Gamma$ be a free group situated as a lattice in $PGL(2,\mathbb{Q}_p)$, and let $n$ be the number of generators of $\Gamma$. Normalize the Haar measure on $GL(2,\mathbb{Q}_p)/Z$ so that the measure of $Z . GL(2,\mathbb{Z}_p) / Z \cong GL(2,\mathbb{Z}_p) / \mathbb{Z}_p^\times = PGL(2,\mathbb{Z}_p)$ is $1$. Then the measure of $\Gamma \backslash PGL(2,\mathbb{Q}_p)$ is just the cardinality of $\Gamma \backslash PGL(2,\mathbb{Q}_p) / PGL(2,\mathbb{Z}_p)$, which is $h=2(n-1)/(p-1)$. Next, multiply this measure by $\frac{1}{2}(p-1)$ to get the measure on $GL(2,\mathbb{Q}_p)/Z$ in Section 2.2 of . Now the measure of $\Gamma \backslash PGL(2,\mathbb{Q}_p)$ is just $n-1$.

So, if my understanding is correct, the product of the formal dimension of the discrete series representation of $PGL(2,\mathbb{Q}_p)$ in the second paragraph of this "Background and attempt" section --- an integer $m$, in the normalization of Section 2.2 of  --- and the measure of $\Gamma \backslash PGL(2,\mathbb{Q}_p)$ --- $n-1$, in the same normalization --- is equal to $m(n-1)$.

Question. Have I correctly dealt with the Haar measure involved in calculating the formal dimension and the covolume? According to the remarks in Section 4 of , if the representation is supercuspidal, the product should indeed be an integer.

(Apologies for this "Can someone check my work?"-type question! I thought others might benefit from seeing the product calculation worked out.)

References.

 Jacquet and Langlands, Automorphic forms on $GL(2)$, 1970

 Gérardin and Li, Fourier transforms of representations of quaternions, 1985

 Corwin, Moy, and Sally, Degrees and formal degrees for division algebras and $GL_n$ over a $p$-adic field, 1990

 The Jacquet-Langlands correspondence for $GL(2)$ math.stanford.edu/~conrad/JLseminar/Notes/L20.pdf

 Ihara, On discrete subgroups of the two by two projective linear group over $\mathfrak{p}$-adic fields, 1966

 Borel and Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, 1978

EDIT: I don't know if the central character of the irreducible representation of the unit quaternions given in Section 1.4.2 of the notes  can be trivial, so for now, I cannot use $2$ (the dimension there) as the dimension in my example. I have replaced "$2$" in the first version of this post with some unknown dimension "$m$". Maybe I will ask about the possible dimensions of irreducible representations of unit quaternions with trivial central character in a separate question later.