On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras

TL;DR

This paper extends classical results on Hecke polynomial decomposition to a broad class of reductive groups over non-archimedean fields, providing a criterion for decomposition over parabolic Hecke algebras and classifying relevant parabolics.

Contribution

It generalizes the decomposition theorem for Hecke polynomials to all classical cases and new cases involving groups with $E_6$ or $E_7$ root systems.

Findings

Provides a criterion for polynomial decomposition over parabolic Hecke algebras.

Classifies non-obtuse parabolic subgroups relevant to the decomposition.

Extends known results to new cases with exceptional root systems.

Abstract

We generalize a classical result of Andrianov on the decomposition of Hecke polynomials. Let $\mathfrak{F}$ be a non-archimedean local fied. For every connected reductive group $\mathbf{G}$, we give a criterion for when a polynomial with coefficients in the spherical parahoric Hecke algebra of $\mathbf{G}(\mathfrak{F})$ decomposes over a parabolic Hecke algebra associated with a non-obtuse parabolic subgroup of $\mathbf{G}$. We classify the non-obtuse parabolics. This then shows that our decomposition theorem covers all the classical cases due to Andrianov and Gritsenko. We also obtain new cases when the relative root system of $\mathbf{G}$ contains factors of types $E_6$ or $E_7$.