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

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."

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