On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras

Heyer, Claudius

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 $\mathbf{G}$ 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 $E_7$.

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 $E_7$.

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

[1] Andrianov, Anatoli N. On factorization of Hecke polynomials for the symplectic groups of genus $n$, Math. USSR, Sb., Volume 33 (1977) no. 3, pp. 343-373 | DOI

[2] Andrianov, Anatoli N. The multiplicative arithmetic or Siegel modular forms, Russ. Math. Surv., Volume 34 (1979) no. 1, pp. 75-148 | DOI | Zbl

[3] Andrianov, Anatoli N.; Zhuravlev, Vladimir G. Modular Forms and Hecke Operators, Translations of Mathematical Monographs, 145, American Mathematical Society, 1995 | Zbl

[4] Bourbaki, Nicolas Groupes et algèbres de lie: Chapitres 4, 5 et 6, Éléments de Mathématiques, Masson, 1981

[5] Bruhat, François; Tits, Jacques Groupes réductifs sur un corps local. I: Données radicielles valuées, Publ. Math., Inst. Hautes Étud. Sci., Volume 41 (1972), pp. 5-251 | DOI | Numdam | Zbl

[6] Bruhat, François; Tits, Jacques Groupes réductifs sur un corps local. II: Schémas en groupes. Existence d’une donnée radicielle valuée, Publ. Math., Inst. Hautes Étud. Sci., Volume 60 (1984), pp. 5-184 | Zbl

[7] Bushnell, Colin J.; Kutzko, Philip C. Smooth representations of reductive $p$-adic groups: structure theory via types, Proc. Lond. Math. Soc., Volume 77 (1998) no. 3, pp. 582-634 | DOI | Zbl

[8] Dąbrowski, Romuald Comparison of the Bruhat and the Iwahori decompositions of a $p$-Adic Chevalley group, J. Algebra, Volume 167 (1994) no. 3, pp. 704-723 | DOI | Zbl

[9] Gritsenko, Valeri A. The action of modular operators on the Fourier–Jacobi coefficients of modular forms, Math. USSR, Sb., Volume 47 (1984) no. 1, pp. 237-268 | DOI | Zbl

[10] Gritsenko, Valeri A. Parabolic extensions of a Hecke ring of the general linear group, J. Sov. Math., Volume 43 (1988) no. 4, pp. 2533-2540 | DOI | Zbl

[11] Gritsenko, Valeri A. Expansion of Hecke polynomials of classical groups, Math. USSR, Sb., Volume 65 (1990) no. 2, pp. 333-356 | DOI | Zbl

[12] Gritsenko, Valeri A. Parabolic extensions of the Hecke ring of the general linear group. II, J. Sov. Math., Volume 62 (1992) no. 4, pp. 2869-2882 | DOI

[13] Haines, Thomas J.; Rostami, Sean The Satake isomorphism for special maximal parahoric Hecke algebras, Represent. Theory, Volume 14 (2010), pp. 264-284 | DOI | Zbl

[14] Henniart, Guy; Vignéras, Marie-France A Satake isomorphism for representations modulo $p$ of reductive groups over local fields, J. Reine Angew. Math., Volume 701 (2015), pp. 33-75 | Zbl

[15] Herzig, Florian A Satake isomorphism in characteristic $p$, Compos. Math., Volume 147 (2011) no. 1, pp. 263-283 | DOI | Zbl

[16] Heyer, Claudius Applications of parabolic Hecke algebras: parabolic induction and Hecke polynomials, Ph. D. Thesis, Humboldt-Universität zu Berlin (2019)

[17] Heyer, Claudius Localization of the Parabolic Hecke Algebra at a Strictly Positive Element (2021) (https://arxiv.org/abs/2103.16949)

[18] Heyer, Claudius Parabolic induction via the parabolic pro-$p$ Iwahori–Hecke algebra, Represent. Theory, Volume 25 (2021), pp. 807-843 | DOI | Zbl

[19] Humphreys, James E. Reflection groups and coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, 1990 | DOI

[20] Januszewski, Fabian On $p$-adic $L$-functions for $\mathrm{GL}\left(n\right)\times \mathrm{GL}(n-1)$ over totally real fields, Int. Math. Res. Not., Volume 2015 (2015) no. 17, pp. 7884-7949 | DOI | Zbl

[21] Lansky, Joshua M. Decomposition of double cosets in $p$-adic groups, Pac. J. Math., Volume 197 (2001) no. 1, pp. 97-117 | DOI | Zbl

[22] Macdonald, Ian G. Spherical functions on a group of $p$-adic type, Publ. Ramanujan Inst., 2, 1971

[23] Macdonald, Ian G. Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157, Cambridge University Press, 2003 | DOI

[24] Rapoport, Michael A positivity property of the Satake isomorphism, Manuscr. Math., Volume 101 (2000) no. 2, pp. 153-166 | DOI | Zbl

[25] Schneider, Peter; Stuhler, Ulrich Representation theory and sheaves on the Bruhat-Tits building, Publ. Math., Inst. Hautes Étud. Sci., Volume 85 (1997) no. 1, pp. 97-191 | DOI | Numdam | Zbl

[26] Vienney, Mathieu Construction de $(\phi ,\gamma )$-modules en caractéristique p, Ph. D. Thesis, École normale supérieure de Lyon – ENS LYON (2012)

[27] Vignéras, Marie-France Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$, Progress in Mathematics, 137, Birkhäuser, 1996 | Numdam

[28] Vigneras, Marie-France The pro-$p$ Iwahori Hecke algebra of a reductive $p$-adic group I, Compos. Math., Volume 152 (2016) no. 4, pp. 693-753 | DOI | Zbl