Bianchi modular forms (reviewed)

Given $\gamma=\begin{pmatrix} a & b \\ c&d \end{pmatrix} \in \textrm{PSL}_2(\mathbb{C})$ and $z=(x,y) \in \mathcal{H}_3$, hyperbolic 3-space, let us introduce the multiplier system $$J(\gamma, z):= \begin{pmatrix} cx+d & -cy \\ \bar{c}y & \overline{cx+d}\end{pmatrix}.$$

Given a function $F: \mathcal{H}_3 \rightarrow \C^{k+1}$ and $\gamma \in \textrm{PSL}_2(\mathbb{C})$, we define the "slash operator" $$(F |_k\gamma)(z):=\mathrm{Sym}^k(J(\gamma, z)^{-1}) \ F(\gamma z),$$ where $\mathrm{Sym}^k$ is the symmetric $k^{th}$ power of the standard representation of $\textrm{PSL}_2(\mathbb{C})$ on $\mathbb{C}^2$.

The center of the universal enveloping algebra of the Lie algebra associated to the real Lie group $\textrm{PSL}_2(\mathbb{C})$ is generated by two elements (Casimir operators) $\Psi, \Psi'$. These act on real analytic functions $F: \mathcal{H}_3 \rightarrow \mathbb{C}^{k+1}$ as differential operators.

Let $K$ an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Let $\Gamma$ be a congruence subgroup of a Bianchi group $\textrm{PSL}_2(\mathcal{O}_K)$. A Bianchi modular form for $\Gamma$ with weight $k$ is a real analytic function $F: \mathcal{H}_3 \rightarrow \mathbb{C}^{k+1}$ with the properties

1. $F|_k\gamma = F$ for every $\gamma \in \Gamma$,

2. $\Psi F = 0$ and $\Psi' F = 0$,

3. $F$ has at worst polynomial growth at each cusp of $\Gamma$.

A Bianchi modular form $F$ has a Fourier-Bessel expansion.

The set $M_k(\Gamma)$ of Bianchi modular forms for $\Gamma$ with weight $k$ is a finite dimensional complex vector space.