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On linear Hodge-Newton decomposition for reductive monoids

Abstract

Let \(\bar{\mathbf{G}}\) be an irreducible linear reductive monoid over a characteristic zero field F of fractions of a complete discrete valuation ring \(\mathfrak{o}\), such that its group G of units is split over \(\mathfrak{o}\). This paper concerns a relation between the Hodge point and the Newton point associated to an element \(\gamma\in\bar{\mathbf{G}}(F)\), proved by Kottwitz and Viehmann when \(\bar{\mathbf{G}}\) is either a connected reductive \(\mathfrak{o}\)-split linear algebraic group over F or the monoid of n×n matrices over F. On the way to proving this relation, we apply the Putcha-Renner theory of linear algebraic monoids over algebraically closed fields to study \(\bar{\mathbf{G}}(F)\) by generalizing various results for linear algebraic groups over F such as the Iwasawa, Cartan and affine Bruhat decompositions.

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Correspondence to Sandeep Varma.

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Communicated by Mohan S. Putcha.

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Varma, S. On linear Hodge-Newton decomposition for reductive monoids. Semigroup Forum 85, 381–416 (2012). https://doi.org/10.1007/s00233-011-9366-y

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Keywords

  • Hodge-Newton decomposition
  • Linear algebraic monoids
  • Mazur’s inequality