$\DeclareMathOperator\GL{GL}$Some context. In number theory, it is natural to study distribution questions for the family of elliptic curves over $\mathbb{Q}$ (or any fixed number field for that matter). Any elliptic curve $E_{/\mathbb{Q}}$ has a unique Weierstrass model $y^2=x^3+Ax+B$, where $(A,B)$ is a pair of integers such that for all primes $p$, either $p^4\nmid A$ and $p^6\nmid B$. The height of $E$ is then defined to be maximum of $|A|^3$ and $|B|^2$. The height function satisfies the Northcott property, i.e., the set of isomorphism classes of elliptic curves $E_{/\mathbb{Q}}$ with height $\leq X$ is finite. Moreover, as a function of $X$, it is asymptotic to $\frac{4}{\zeta(10)} X^{\frac{5}{6}}$. The rank distribution conjecture predicts that half of these elliptic curves have rank $0$ and the other half have rank $1$ (as $X\rightarrow \infty$). A density zero set is expected to have rank $\geq 2$. On the other hand, it is natural to also study density questions for elliptic curves with prescribed local conditions. For instance, the proportion of elliptic curves $E_{/\mathbb{Q}}$ that are semistable is somewhat close to $1/\zeta(2)$.
Automorphic forms. It is natural to study similar distribution questions for modular forms on a given group with prescribed weight. Here, the level of the modular form in question goes to infinity (as opposed to the "height"). These distribution results have been proven via dimension formulas. The question has also been generalized to the context of automorphic forms, cf. for instance, https://arxiv.org/abs/1610.07567
Local representations. This naturally motivates a similar question for irreducible local representations of a fixed group. For simplicity, consider the group $\GL_2(F)$, where $F$ is a fixed local field and consider the set $\text{Irr}(\GL_2(F))$ of all irreducible smooth admissible representations of $\GL_2(F)$ taking values in $\mathbb{C}$. These come in two families, the principal series and the supercuspidals. The principal series are easy to parametrize, the supercuspidals are parametrized by fundamental strata. Is there are good notion of a height function on $\text{Irr}(\GL_2(F))$ that satisfies the Northcott property? Given a representation $\pi$ of $\GL_2(F)$, since $\pi$ is smooth, the restriction of $\pi$ to $\GL_2(O_F)$ factors through a finite index subgroup $K$. Let $K$ be the largest such subgroup and set $\operatorname{ht}(\pi)$ to denote the index $[\GL_2(O_F):K]$. The problem with this definition is that the subset of $\text{Irr}(\GL_2(F))$ with height $<X$ is infinite. This is easy to see for principal series since the characters that are induced from the diagonal are free to take any values on the uniformizer of $O_F$. What would be an appropriate way to rectify this? Perhaps one can consider the subset of $\operatorname{Irr}(\GL_2(F))$ where the values of these uniformizers are prescribed. The same issue comes up when analyzing supercuspidals, but here it seems to be more subtle. Nonetheless, the parametrization is quite explicit.
Questions. Is there a good way to define a counting problem for local representations? What is the significance of "prescribing" the values of the uniformizers on the diagonal, or resort to any other modification of the problem? Maybe one has to further prescribe the $L$-factor?
Some musings. Fix a prime $p$. As we vary over all local representations in the family of modular forms of fixed weight (ordered by level), it is natural to study how the associated local representation varies in $\operatorname{Irr}(\GL_2(F))$. Perhaps there is a natural measure on $\operatorname{Irr}(\GL_2(F))$ which describes this distribution as the bound on the level approaches $\infty$. This is similar to the case of elliptic curves, where there is a natural measure on local models for Weierstrass equations.
