# Symmetrization of representations of $GL_N$

## TL;DR

This paper introduces a symmetrization process for irreducible admissible representations of $GL_N(Q_p)$, providing geometric insights into decomposition coefficients and proving Zelevinsky's conjecture.

## Contribution

It develops a novel symmetrization method for representations of $GL_N$ over $p$-adic fields and confirms a longstanding conjecture by Zelevinsky.

## Key findings

- A new symmetrization process for $GL_N$ representations.
- Geometric interpretation of decomposition coefficients.
- Proof of Zelevinsky's conjecture.

## Abstract

In this article, we develop a process to symmetrize the irreducible admissible representation of $GL_N(\mathbb{Q}_p)$, as a consequence we obtain a more geometric understanding of the coefficient $m(\mathbf{b}, \mathbf{a})$ appearing in the decomposition of parabolic inductions, which allows us to prove a conjecture posed by Zelevinsky.

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Source: https://tomesphere.com/paper/1905.05609