Bianchi modular form (reviewed)

Let $K$ be an imaginary quadratic number field of class number $1$, and $\mathcal{O}_K$ be its ring of integers. Let $\Gamma$ be a congruence subgroup of the Bianchi group $\textrm{PSL}_2(\mathcal{O}_K)$. Let $\mathcal{H}_3$ be hyperbolic 3-space. A Bianchi modular form for $\Gamma$ with weight $k\ge0$ is a real analytic vector-valued function $F: \mathcal{H}_3 \rightarrow \mathbb{C}^{k+1}$ with the properties

1. $F|_k\gamma = F$ for every $\gamma \in \Gamma$;
2. $F$ is harmonic;
3. $F$ has at worst polynomial growth at each cusp of $\Gamma$.

The set $M(\Gamma,k)$ of Bianchi modular forms for $\Gamma$ with weight $k$ is a finite dimensional complex vector space. Bianchi modular forms have a Fourier-Bessel expansion., with coefficients indexed by elements of $\mathcal{O}_K$. Bianchi modular forms which vanish at all the cusps are called Bianchi cusp forms; their $0$th coefficient is $0$.

In condition 1, the weight $k$ "slash operator" is defined by $$ (F |_k\gamma)(z):=\mathrm{Sym}^k(J(\gamma, z)^{-1}) \ F(\gamma z), $$ where $\mathrm{Sym}^k$ is the symmetric $k^{th}$ power of the standard representation of $\textrm{PSL}_2(\mathbb{C})$ on $\mathbb{C}^2$, and for $\gamma=\begin{pmatrix} a & b \\ c&d \end{pmatrix} \in \textrm{PSL}_2(\mathbb{C})$ we define $$ J(\gamma, z):= \begin{pmatrix} cx+d & -cy \\ \bar{c}y & \overline{cx+d}\end{pmatrix}. $$

Condition 2, harmonicity, is the analogue of the usual condition of being holomorphic for classical and Hilbert modular forms. It means that $\Psi F=0$ and $\Psi' F=0$, where the differential operators $\Psi$, $\Psi'$ are the Casismir operators which generate the center of the universal enveloping algebra of the Lie algebra associated to the real Lie group $\textrm{PSL}_2(\mathbb{C})$.