Double affine galleries

Abstract

We develop a theory of galleries for double affine hyperplane arrangements. A gallery is an infinite sequence of chambers, indexed by an ordered set, which is maximal with respect to a finiteness condition on the multiset of wall-crossings.

We study the possible order types of galleries. We also use galleries to define a double affine Bruhat order which generalizes the one introduced by Braverman, Kazhdan, and Patnaik, and studied by Muthiah and Orr. We prove an analogue of the classical characterization of the Bruhat order in terms of subexpressions of reduced expressions, and we define an analogue of the Demazure product.

We also study tours, which are certain finite sequences of chambers. Using the previous results, we show that tours form a category which behaves similarly to the category of generalized galleries defined in the classical setting. We construct a functor from tours to schemes, whose image consists of double affine analogues of Demazure varieties. We show that the colimit of this functor recovers the double affine flag variety at the level of sets, but we do not think that the colimit of schemes is well-behaved. Instead, we describe a different way of equipping the colimit set with a ringed space structure, and we conjecture that this ringed space is a scheme.

Our main result is that the category of tours (with fixed start and end chambers, and subject to certain constraints) is contractible. We call this result 'homotopical deletion' because it generalizes the Coxeter deletion lemma.