Given a principal congruence subgroup $\Gamma(N)$ of $SL(2,\mathbb{R})$, since $\Gamma(N)$ is free, consider a probability distribution $\mu_1$ of a simple random walk on $\Gamma(N)$ and consider its associated Poisson boundary $(\Omega,\nu)$. Define representation of $\Gamma(N)$ acting on $L^2(\Omega,\nu)$ by $(\pi_z(x)\eta)(\omega) := P^z(x,\omega)\eta(x^{-1}\omega)$, where $\omega\in\Omega,x\in\Gamma(N),\eta\in L^2(\Omega,\nu),P^z(x,\omega) = (\frac{d\nu_x}{d\nu})^{z}$ is the Poisson kernel to the power z. For $Re(z) = \frac{1}{2}$, $\pi_z$ is a unitary representation (acting on a Hilbert space $\mathcal{H}$, which is completion of characteristic function on $\Omega$ with a norm different from $L^2$ norm). It is known that such representations are irreducible.
Now we may also consider principal series representation of $SL(2,\mathbb{R})$ and consider its restriction to $\Gamma(N)$. The question is: what is the relation between representation $\pi_z$ and the restriction of principal series representation.
Any information / reference / theorems are very welcome. I thank you in advance. I'm a mathematical physicists working mostly in operator algebra, and I have very limited knowledge on modular forms /automorphic forms etc. So any help is greatly appreciated. Thanks.
