On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras

Heyer, Claudius

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 941-997.

On généralise un résultat classique d’Andrianov sur la décomposition des polynômes de Hecke. Pour un groupe connexe réductif $\mathbf{G}$ 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 $\mathbf{G}\left(𝔉\right)$ se décompose sur une algèbre de Hecke parabolique associée à un groupe parabolique non obtus de $\mathbf{G}$. 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 $\mathbf{G}$ contient des facteurs de types $E_6$ ou $E_7$.

We generalize a classical result of Andrianov on the decomposition of Hecke polynomials. If $\mathbf{G}$ 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 $\mathbf{G}\left(𝔉\right)$ 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 $E_7$.

Classification : 11C08, 20C08, 20G25
Mots-clés : Hecke Polynomials, Hecke algebras, reductive groups, Satake homomorphism, Iwasawa decomposition, Cartan decomposition

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