On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras
Claudius Heyer1
1Einsteinstraße 62, 48149 Münster, Germany
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 941-997

We generalize a classical result of Andrianov on the decomposition of Hecke polynomials. If G is a connected reductive group defined over a non-archimedean local field 𝔉, we give a criterion for when a polynomial with coefficients in the spherical parahoric Hecke algebra of G(𝔉) decomposes over a parabolic Hecke algebra associated with a non-obtuse parabolic subgroup of 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 G contains factors of types E 6 or E 7 .

On généralise un résultat classique d’Andrianov sur la décomposition des polynômes de Hecke. Pour un groupe connexe réductif G défini sur un corps local non-archimédien 𝔉, on donne un critère pour déterminer sous quelles conditions un polynôme à coefficients dans une algèbre de Hecke sphérique parahorique de G(𝔉) se décompose sur une algèbre de Hecke parabolique associée à un groupe parabolique non obtus de G. On donne une classification des groupes paraboliques non obtus. Ceci montre alors que notre théorème de décomposition couvre tous les cas classiques dûs à Andrianov et Gritsenko. De plus, on obtient des cas nouveaux où le système de racines relatives de G contient des facteurs de types E 6 ou E 7 .

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DOI: 10.5802/jtnb.1235
Classification: 11C08, 20C08, 20G25
Keywords: Hecke Polynomials, Hecke algebras, reductive groups, Satake homomorphism, Iwasawa decomposition, Cartan decomposition

Claudius Heyer  1

1 Einsteinstraße 62, 48149 Münster, Germany
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Claudius Heyer. On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 941-997. doi: 10.5802/jtnb.1235
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