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?



Growth of dimension of fixed spaces in $GL_n(\mathbb{Q}_p)$-representations
For question 1, you are asking about the theory of oldforms (or oldvectors). I think the canonical reference for $\mathrm{GL}\_n$ is "[Oldforms on $\mathrm{GL}\_n$](https://www.jstor.org/stable/2374790)" by Mark Reeder. In particular, he discusses how to find a basis of $\pi^{K(r')}$ if one already has a basis of $\pi^{K(r)}$ for all $r < r'$.


For question 2, I don't believe there is a particularly satisfactory answer, though this is an active area of research. For some groups of small rank this has been studied; for example, on $\mathrm{GSp}\_4$ (or rather $\mathrm{PGSp}\_4$), Brooks Roberts and Ralf Schmidt have studied this question extensively, with most of the results appearing in their book "[Local Newforms for $\mathrm{GSp}\_4$](http://link.springer.com/book/10.1007%2F978-3-540-73324-9) ". For split special orthogonal groups, the theory has been developed in Pei-Yu Tsai's recent thesis, "[On Newforms for Split Special Odd Orthogonal Groups](http://math.harvard.edu/~pytsai/FixedVectors.pdf)".


