# Conjectures about certain parabolic Kazhdan--Lusztig polynomials

## TL;DR

This paper formulates conjectures about parabolic Kazhdan--Lusztig polynomials, inspired by irreducibility results in representation theory and computational evidence, suggesting they share properties with classical Kazhdan--Lusztig polynomials.

## Contribution

It introduces new conjectures on the properties of parabolic Kazhdan--Lusztig polynomials based on theoretical insights and computational experiments.

## Key findings

- Conjectures on the properties of parabolic Kazhdan--Lusztig polynomials.
- Evidence from computer calculations supporting the conjectures.
- Potential parallels between parabolic and ordinary Kazhdan--Lusztig polynomials.

## Abstract

Irreducibility results for parabolic induction of representations of the general linear group over a local non-archimedean field can be formulated in terms of Kazhdan--Lusztig polynomials of type $A$. Spurred by these results and some computer calculations, we conjecture that certain alternating sums of Kazhdan--Lusztig polynomials known as parabolic Kazhdan--Lusztig polynomials satisfy properties analogous to those of the ordinary ones.