Steinberg's cross-section of Newton strata

TL;DR

This paper introduces a geometric model called loop Steinberg's cross-section for Newton strata in the loop group of an unramified reductive group, confirming a conjecture and providing new proofs of classical results.

Contribution

It develops a novel geometric model for Newton strata in loop groups, enabling proofs of conjectures and classical theorems in the theory of affine Deligne-Lusztig varieties.

Findings

Confirmed Ivanov's conjecture on loop Deligne-Lusztig varieties

Provided new proofs of Mazur's inequality converse and Chai's length formula

Established a geometric framework for Newton strata using loop Steinberg's cross-section

Abstract

In this note, we introduce a natural analogue of Steinberg's cross-section in the loop group of an unramified reductive group $\mathbf G$. We show this loop Steinberg's cross-section provides a simple geometric model for the poset $B(\mathbf G)$ of Frobenius-twisted conjugacy classes (referred to as Newton strata) of the loop group. As an application, we confirm a conjecture by Ivanov on loop Delgine-Lusztig varieties of Coxeter type. This geometric model also leads to new and direct proofs of several classical results, including the converse to Mazur's inequality, Chai's length formula on $B(\mathbf G)$, and a key combinatorial identity in the study affine Deligne-Lusztig varieties with finite Coxeter parts.