2023/07/24 by Sian Nie, Nie, Sian · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities
paper · pdf · doi:10.48550/arxiv.2307.12521
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.