On the double-affine Bruhat order: the $\epsilon=1$ conjecture and classification of covers in ADE type

TL;DR

This paper investigates the structure of the double-affine Bruhat order in Kac-Moody groups, proving compatibility with a length function and establishing grading in affine ADE cases, with conjectures on geometric intersections.

Contribution

It proves the compatibility of the Bruhat order with a length function for Kac-Moody groups and establishes grading in affine ADE types, advancing understanding of double-affine structures.

Findings

Bruhat order is compatible with a length function in Kac-Moody groups.

Bruhat order is graded by this length function in affine ADE types.

Formulation of conjectures relating length functions to geometric intersections.

Abstract

For any Kac-Moody group $\mathbf{G}$, we prove that the Bruhat order on the semidirect product of the Weyl group and the Tits cone for $\mathbf{G}$ is strictly compatible with a $\mathbb{Z}$-valued length function. We conjecture in general and prove for $\mathbf{G}$ of affine ADE type that the Bruhat order is graded by this length function. We also formulate and discuss conjectures relating the length function to intersections of "double-affine Schubert varieties."