Let $F$ be a p-adic field and $GL_n(F)$ the general linear group over $F$. The irreducible complex finite length smooth representations are parametrized by multi-segements in the paper. A multi-segment is of the form $([a_1, b_1], \ldots, [a_m, b_m])$, $a_i \le b_i \in \mathbb{Z}$. My question is: what is the range of $m$ for $GL_n(F)$? Thank you very much.
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asked Feb 21, 2019 at 9:30
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I'm not sure I fully understand your notations but the numerology is as such.
If $\pi$ is an irreducible representation of $\operatorname{GL}_{m}(F)$ and if $\Delta(\pi)$ is a segment $[\pi,\cdots,\pi']$ of length $d$ (notations as in the paper of Zelevinsky) then the representation attached to $\Delta$ is a generalized Steinberg representation of $\operatorname{GL}_{md}$. The representation attached to a suitable multi-segment $\Delta_1,\cdots,\Delta_r$ of length $r$ of generalized Steinberg representations of $\operatorname{GL}_{n_i}$ is then constructed through parabolic induction from the parabolic subgroup with Levi component $\operatorname{GL}_{n_1}\times\cdots\times\operatorname{GL}_{n_r}$ so is a representation of $\operatorname{GL}_{n}$ with $n=n_1+\cdots+n_r$. The possible range of variation of the length of each individual segment depends on the $m_i$ such that the representation $\pi_i$ appearing in the definition of $\Delta_i$ is a representation of $\operatorname{GL}_{m_i}$ and on the number $r$ of segments in the multi-segment, with the maximum possible length of a segment attained for $r=1$ and $m=1$.
