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On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras

Claudius Heyer

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 $E_7$.

On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras

Abstract

We generalize a classical result of Andrianov on the decomposition of Hecke polynomials. Let be a non-archimedean local fied. For every connected reductive group , we give a criterion for when a polynomial with coefficients in the spherical parahoric Hecke algebra of decomposes over a parabolic Hecke algebra associated with a non-obtuse parabolic subgroup of . 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 contains factors of types or .

Paper Structure

This paper contains 23 sections, 29 theorems, 155 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Structure of the paper
  5. Acknowledgments
  6. Preliminaries
  7. Notations
  8. The standard apartment
  9. The associated reduced root system
  10. The Iwahori--Weyl group
  11. The positive monoid
  12. Double coset decompositions
  13. Abstract Hecke rings
  14. Non-obtuse parabolics
  15. The algorithm
  16. ...and 8 more sections

Key Result

Theorem 1

Assume that Hypothesis hyp-intro is satisfied. Let $d(t) \in \mathcal{H}_R(K,G)[t]$ be a polynomial such that $d^{\Theta^B_Z}(t)$, the polynomial obtained by applying $\Theta^B_Z$ to the coefficients of $d(t)$, decomposes in $R[Z/K_Z][t]$ as such that $\widetilde{g}(t)$ has coefficients in $\Theta^B_Z(C^-_P)$ with constant term $1$. Then there exist polynomials $f(t), g(t)\in \mathcal{H}_R(K_P,P)

Figures (5)

  • Figure 1: In both examples we choose $\Sigma_M = \{\pm\alpha\}$, and $\nu(\mu)$ can lie anywhere in the dotted region.
  • Figure 2: Type $A_2$: The translate of the dotted region by $\nu(\lambda)$ fits into the shaded area. Lemma \ref{['lem:no-cone']} holds in this case.
  • Figure 3: The black vertices are precisely the non-obtuse simple roots. Note that in types $E_8$, $F_4$, and $G_2$ there are no non-obtuse parabolics, while in type $A_n$ all maximal parabolics are non-obtuse.
  • Figure 4: The circular ordering in type $G_2$
  • Figure 5: A visual aid for the sets $\Sigma^{(k)}$ relative to the reduced decomposition $w_0 = s_1s_2s_1$, where $\alpha_1 = \beta_1$ and $\alpha_2 = \beta_3$.

Theorems & Definitions (70)

  • Theorem : Theorem \ref{['thm:decomp']}
  • Definition : See §\ref{['sec:non-obtuse']}
  • Theorem 1.2: Theorem \ref{['thm:hyp']}
  • Theorem 1.3: Theorem \ref{['thm:main']}
  • Lemma 2.1: Bruhat-Tits.1972
  • Definition : Bruhat-Tits.1972
  • Remark
  • Remark 2.4
  • Proposition 2.5: Andrianov.1995
  • Proposition 2.6: Andrianov.1995
  • ...and 60 more