Generic Newton points and the Newton poset in Iwahori double cosets

TL;DR

This paper investigates the structure of Newton stratification on Iwahori double cosets in loop groups, introducing a group-theoretic condition called cordiality that ensures the Newton poset is saturated and supports Grothendieck's conjecture.

Contribution

It introduces the concept of cordiality, a condition that guarantees the saturation of the Newton poset and the validity of Grothendieck's conjecture in this context.

Findings

Cordiality condition ensures Newton poset saturation

Grothendieck's conjecture holds under the cordiality condition

Identifies classes of cosets satisfying the condition via quantum Bruhat graph paths

Abstract

We consider the Newton stratification on Iwahori double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (i.e. the index set for non-empty Newton strata) is saturated and Grothendieck's conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.