Algorithms for parabolic inductions and Jacquet modules in $\mathrm{GL}_n$
Kei Yuen Chan, Basudev Pattanayak
TL;DR
The work delivers comprehensive, algorithmic tools for parabolic induction and Jacquet-modules analysis in GL_n(F) by intertwining Zelevinsky and Langlands frameworks. Central to the approach are derivative and integral procedures (ρ-derivatives, St-derivatives, and their left/right variants) and the MW algorithm, all controlled by the notions of minimally linked segments and removal/extension processes. A key innovation is the exotic duality connecting right integrals and left derivatives, enabling cross-transfer of results and reductions to more tractable ρ-integral cases. The paper culminates with practical algorithms for the highest Bernstein–Zelevinsky derivatives and Langlands parameters, providing concrete computational handles for branching laws and derivative–integral structures with broad implications for representation theory of GL_n.
Abstract
In this article, we present algorithms for computing parabolic inductions and Jacquet modules for the general linear group $G$ over a non-Archimedean local field. Given the Zelevinsky data or Langlands data of an irreducible smooth representation $π$ of $G$ and an essentially square-integrable representation $σ$, we explicitly determine the Jacquet module of $π$ with respect to $σ$ and the socle of the normalized parabolic induction $π\times σ$. Our result builds on and extends some previous work of Mœglin-Waldspurger, Jantzen, Mínguez, and Lapid-Mínguez, and also uses other methods such as sequences of derivatives and an exotic duality. As an application, we give a simple algorithm for computing the highest derivative multisegment and an algorithm for computing the Langlands parameter of the highest Bernstein-Zelevinsky derivatives.