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

For any Kac–Moody group G, we prove that the Bruhat order on the semidirect product of the Weyl group and the Tits cone for G is strictly compatible with a Z-valued length function. We conjecture in general and prove for 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”