# Principal series representation of $SL(2,\mathbb{R})$ restricted to principal congruence subgroup

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.