Let $o$ be the ring of integers in a local field $F$ with prime-ideal $p$. Let $K$ be either $GL_2(o)$ or the normalizer of the Iwahori subgroup. Let $\sigma$ be a representation of $K$ times the center $Z$ of $GL_2(F)$ such that the induced representation is supercuspidal and irreducible. Fix $k\geq 0$. $$N(p^k) := \{ \left( \begin{smallmatrix}1 & x \\ 0 & 1 \end{smallmatrix} \right) : x \in p^k\}$$

Q1: Can we compute the dimension $d_k$ of $N(p^k)$-invariant vectors in $\rho$?

Is the result either $d_k = dim(\rho)$ or zero depending whether $k \geq \ell(\pi)$ or $k < \ell(\pi)$? Here, $$ \ell(\pi) = \inf \{ k : \pi\; admits\; N(p^k)-invariant\;vector\}.$$

Q2: What is known for induced characters $\mu: \left(\begin{smallmatrix}a & * \\ * & * \end{smallmatrix} \right)\mapsto \mu(a)$ from $\Gamma_0(p^N)$ to $GL_2(o)$ with conductor $p^N$?

At least for the second question, I have some computations: let $\rho$ be that representation from Q2.

If $k\geq N$, then $d_k = dim(\rho)$, because $\rho$ has conductor $\Gamma(p^N)$.

For $k=0$, then $d_0=1$ for $\mu=1$ and $d_0=2$ else by Frobenius reciprocity and some other arguments.

For the other cases, note that the Mackey induction restriction formula yields $$ Res_{N(p^N)} Ind_{\Gamma_0(p^N)} \mu = \bigoplus_{g \in N(p^N) \backslash GL_2(o) / \Gamma_0(p^N)} Ind_{g^{-1}\Gamma_0(p^N)g \cap N(p^N)}^{N(p^N)} \mu( g \; \ast\; g^{-1}). $$ The coset decomposition follows easily by using Bruhat over the residue field and Iwahori decomposition $$ N(p^N) \backslash GL_2(o) / \Gamma_0(p^N) = \coprod_{\delta \in p \bmod p^N} \left(\begin{smallmatrix}1 & 0 \\ \delta & 1 \end{smallmatrix} \right) \amalg \coprod_{\tau\in o \bmod p^N} \left(\begin{smallmatrix}\tau & 1 \\ 1 & 0 \end{smallmatrix} \right).$$ So lets look at the contribution of $g =\left(\begin{smallmatrix}1 & 0 \\ \delta & 1 \end{smallmatrix} \right)$: $$ Ind_{g^{-1}\Gamma_0(p^N)g \cap N(p^N)}^{N(p^N)} \mu( g \; \ast\; g^{-1}) = 1.$$ What about the contribution of $g=\left(\begin{smallmatrix}\tau & 1 \\ 1 & 0 \end{smallmatrix} \right)$?