# Bernstein-Zelevinsky classification: viewing a representation as a subrepresentation or a quotient

$\DeclareMathOperator{\GL}{GL}$ $\DeclareMathOperator{\Ind}{Ind}$I have a question on the details of the Bernstein-Zelevinsky classification. This classification allows us to obtain irreducible representations of $\operatorname{GL}_n(F)$ over a $p$-adic field $F$ as unique quotients of certain induced representations coming from supercuspidals, in an essentially unique way.

It also allows us to obtain these representations as unique subrepresentations in an essentially unique way.

Suppose $\pi$ is a smooth irreducible representation of $\GL_n(F)$, and we know how to obtain $\pi$ as a quotient $Q(\Delta_1, ... , \Delta_r)$ in the B.Z. classification. Then do we also know how to obtain $\pi$ as a subrepresentation $Z(\Delta_1', ... , \Delta_{r'}')$ in B.Z. classification?

Example: suppose $G = \operatorname{GL}_2(F)$, and $\chi = \chi_1$ is a character of $F^{\ast}$, $\chi_2(x) = \chi_1(x)|x|_F$. Then $\Ind_{TU}^G \chi_1 \otimes \chi_2$ has a unique irreducible quotient $\pi$. If we want to get $\pi$ as subrepresentation instead of a quotient, then we swap $\chi_1$ and $\chi_2$ and use $\Ind_{TU}^G \chi_2 \otimes \chi_1$.

• Have you tried reading Zelevinsky's paper? I haven't looked at it too closely, but I think he discusses the relation between the quotients and subrepresentations. – Kimball Jul 9 '18 at 5:12
• At least in the $r=1$ case, it follows from 9.15 in Zelevinsky's paper that, if we write $\Delta_1 = \{\pi,\dotsc,\pi[m-1]\}$, then we have $Q(\Delta_1) = Z(\{\pi[m-1]\},\dotsc,\{\pi\})$. – Charles Denis Jul 9 '18 at 13:57