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On Linear Hodge Newton Decomposition for Reductive Monoids

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Part of the Fields Institute Communications book series (FIC,volume 71)

Abstract

Let F be the field of fractions of a complete discrete valuation ring \(\mathfrak{o}\). Let \(\bar{\mathbf{G}}\) be an irreducible linear reductive monoid over F, such that its group G of units is split over \(\mathfrak{o}\). 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, Kottwitz and Viehmann had proved a relation between the Hodge point and the Newton point associated to an element \(\gamma \in \bar{\mathbf{G}}(F)\). Suppose F has characteristic zero. In [7], we had given a monoid theoretic generalization of this phenomenon. On the way, we had applied 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. In this article we give an exposition of these results.

Keywords

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

Subject Classifications:

  • Primary 11F85
  • Secondary 20Mxx
  • 20G25

To Professors Putcha and Renner, with admiration

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Correspondence to Sandeep Varma .

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Varma, S. (2014). On Linear Hodge Newton Decomposition for Reductive Monoids. In: Can, M., Li, Z., Steinberg, B., Wang, Q. (eds) Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics. Fields Institute Communications, vol 71. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0938-4_5

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