Steinberg’s cross-section of Newton strata

Abstract

In this note, we introduce a natural analogue of Steinberg’s cross-section in the loop group of a reductive group \(\textbf{G}\). We show this loop Steinberg’s cross-section provides a simple geometric model for the poset \(B(\textbf{G})\) of Frobenius-twisted conjugacy classes (referred to as Newton strata) of the loop group. As an application, we confirms a conjecture by Ivanov on decomposing 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 \(B(\textbf{G})\), and a key combinatorial identity in the study affine Deligne–Lusztig varieties with finite Coxeter parts.