Let $\pi$ be a generic irreducible admissible representation of $GL_n(L)$, where $L$ is a $p$-adic field, $R$ is its ring of integers, and $\mathfrak{p}$ is its prime ideal. The conductor of $\pi$ has the following definition (follwoing Jacquet, Pietetski-Shapiro, and Shalika): fix a non-negative integer $r$, and let $K(r)$ be the subgroup of matrices:
\begin{pmatrix} x & y \\ u & v \end{pmatrix}
where $x\in GL_{n-1}(R)$, $y\in R^{n-1}$, $u$ is a vector of elements of $\mathfrak{p}^r$ and $v\in 1 + \mathfrak{p}^r$. The conductor of $\pi$ is the minimal $r$ such that $\pi$ has a $K(r)$-fixed vector. In this case, $\pi$ has a unique $K(r)$-fixed vector (up to scalar multiplication).
For $n = 1$ or $2$, the growth of the $K(r)$-fixed vectors is known. Clearly, if $\pi$ is an irreducible $GL_1(L)$-representation of conductor $r$, then $\dim(\pi^{K(r')}) = 1$ if $r' \geq r$ and $0$ otherwise. If $\pi$ is an irreducible $GL_2(K)$-representation and $r' \geq r$ then $\dim(\pi^{K(r')}) = r' - r + 1$.
I have two questions, the first of which I assume is known:
1). Let $n \geq 3$ and let $\pi$ be a $GL_n(L)$-representation of conductor $r$. Is there a formula for computing $\dim(\pi^{K_(r')})$, when $r' \geq r$?
2). If $G$ is a reductive group defined over $L$, is there a sequence of open-compact subgroups $K'(r)$ that mimic the behavior of the $K(r)$ in $GL_n$, in the sense that for every irreducible admissible representation $\pi$ of $G(L)$, there is an $r$ such that $\pi$ has a unique $K'(r)$-fixed vector?
