Ergodic Theory with a view towards Number Theory, Chapter 9, Page 277, 278.
According to the attached image (below) it seems that $v$ and $w$ should be of the form $v=(z,\alpha)$ and $w=(z,\beta)$, where $\alpha$ and $\beta$ are two complex number and $z\in \mathbb{H}$ ($z$ is also a complex number).
The questions are:
How is the inner product $\langle \cdot,\cdot \rangle_z$ defined?
Is not it supposed to be a real inner product? (...dentos the usual inner product in $\mathbb{C}$ under the identification of ....).
A convenient model for the hyperbolic plane is the upper half-plane $$\mathbb{H}=\{x+\mathrm{i}y\in\mathbb{C}\mid y>0\}$$ with the hyperbolic metric. To define this metric, we need to introduce the tangent bundle* $\mathrm{T}\mathbb{H}=\mathbb{H}\times\mathbb{C}$ comprising the disjoint union of the tangent planes $\mathrm{T}_z\mathbb{H}=\{z\}\times\mathbb{C}$ for all $z\in\mathbb{H}$. One should think of $\mathrm{T}_z\mathbb{H}$ as a plane touching $\mathbb{H}$ tangentially at $z$ and having no other intersection with $\mathbb{H}$. This suggests that $\mathrm{T}_z\mathbb{H}$ is the natural space for derivatives in the following sense. If $\phi:[0,1]\longrightarrow\mathbb{H}$ is differentiable at $t\in[0,1]$ with $\phi(t)=z$, then we define the derivative of $\phi$ at $t$ by $$\mathrm{D}\phi(t)=(\phi(t),\phi'(t))\in\mathrm{T}_z\mathbb{H}.$$
Here $\phi'(t)$ is the derivative of $\phi$ as a map into $\mathbb{C}$. We give $\mathrm{T}_z\mathbb{H}$ the structure of a vector space inherited from the second component in $\mathrm{T}_z\mathbb{H}=\{z\}\times\mathbb{C}$.
The hyperbolic Riemannian metric is defined as the collection of inner products* $$\langle v,w\rangle_z=\frac{1}{y^2}(v\cdot w)$$ for $z=x+\mathrm{i}y\in\mathbb{H}$ and $v,w\in\mathrm{T}_z\mathbb{H}$. Here $(v\cdot w)$ denotes the usual inner product in $\mathbb{C}$ under the identification of $\mathbb{C}$ with $\mathbb{R}^2$ as real vector spaces.
The hyperbolic Riemannian metric induces the hyperbolic metric $\mathsf{d}(\cdot,\cdot)$ mentioned above as follows. If $\phi:[0,1]\longrightarrow\mathbb{H}$ is a continuous piecewise differentiable curve (we will refer to these as paths), then its length is defined by $$\mathrm{L}(\phi)=\int_0^1\|\mathrm{D}\phi(t)\|_{\phi(t)}\mathrm{d}t.$$ where $\|\mathrm{D}\phi(t)\|_{\phi(t)}$ denotes the length of the tangent vector $$\mathrm{D}(\phi(t),\phi'(t))\in\mathrm{T}_{\phi(t)}\mathbb{H}$$ with respect to the norm derived from $\langle\cdot,\cdot\rangle_{\phi(t)}$. We will refer to as the speed of the path $\phi$ at time $t$. The hyperbolic distance is now defined as $$\mathsf{d}(z_0,z_1)=\inf_\phi\mathrm{L}(\phi)$$ where the infimum is taken over all continuous piecewise differentiable curves $\phi$ with $\phi(0)=z_0$ and $\phi(1)=z_1$. It may be checked that this does indeed define a metric on $\mathbb{H}$.
(*) The tangent bundle can be defined abstractly on any manifold, but for our purposes we may think of it as the space in which derivatives live and use an ad hoc definition.
I do appreciate any help could be provided.
The inner product $\langle v,w\rangle_z$ for $v=(z,\alpha)$ and $w=(z,\beta)$ two tangent vectors in $T_z\mathbb{H}$ is defined as $$ \langle v,w\rangle_z=\frac{\operatorname{Re}(\alpha\bar\beta)}{y^2} $$ However, since we typically identify tangent spaces of a linear space $V$ with itself, we drop the $z$ and identify $v\in T_z\mathbb{H}$ with the complex number also called $v$ and similarly $w$ to get $v\cdot w=\operatorname{Re}(v\bar{w})$ instead of $\operatorname{Re}(\alpha\bar\beta)$.