# Growth of dimension of fixed spaces in $GL_n(\mathbb{Q}_p)$-representations

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?

• I have never been an expert in the many and varied filtrations used to achieve unique (up to scalar) results of the type that you seek—it looks to me like you are using an affine version of the mirabolic, and so may want to seek out the analogues of mirabolics in finite reductive groups—but my impression is that, for general reductive groups, one is most interested in the Moy–Prasad depth, as in Theorem 5.2 of ams.org/mathscinet-getitem?mr=1253198 . Unfortunately, uniqueness need not hold in this setting. – LSpice Oct 13 '15 at 22:02
• What you write is not quite true: for example if $\pi$ is a 1-dimensional representation of $GL_2(L)$ then the growth is clearly not what you say it is. You need to assume $\pi$ is "generic". – eric Oct 13 '15 at 22:39
• Yes, that's correct. Thanks for the correction. – John Binder Oct 13 '15 at 22:42
• – Peter Humphries Oct 13 '15 at 23:49

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 $\mathrm{GL}_n$" 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 $\mathrm{GSp}_4$ ". 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".