On the double-affine Bruhat order: the $\epsilon=1$ 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."


Introduction
1.1. The Bruhat order and previous work. Let G be a Kac-Moody group, let W be its Weyl group, and let T be the Tits cone of integral coweights. We can form the semi-direct product W T = T¸W , which will in general be a semi-group. In [BKP], Braverman, Kazhdan, and Patnaik consider the case when G is untwisted affine type, and they construct the Iwahori-Hecke algebra for the group G = G(F ) where F is a non-archimedian local field. A key property of their construction is that they need to restrict attention to functions supported on a subsemigroup (the Cartan semigroup) G + Ĺ G. Then they show that the Iwahori-double cosets on G + are exactly in bijection with the semi-group W T .
Additionally, they define a preorder on W T that we call the (double-affine) Bruhat order; they conjecture that this order is in fact a partial order. In [M], the first-named author constructs a function ℓ ε : W T → Z ⊕ Zε (1.1) that is strictly compatible with the preorder, where Z ⊕ Zε is ordered lexicographically. As a corollary, he proves that the double-affine Bruhat order is partial order. However, many questions remained open. In particular, because the intervals in Z ⊕ Zε are infinite in general, the results of [M] do not give strong finiteness results about the order W T .
1.2. Setting ε = 1. In [M, Question 5.10], the question is asked whether the composed function (the length function) obtained by setting ε = 1, is strictly compatible with the order on W T . Let us briefly describe the "single-affine" situation (see Section 1.5 below for the terminology), when G is finite-type and simply-connected. In this case, W T coincides with the usual notion of affine Weyl group, which by our simply-connected assumption is a Coxeter group. The order defined by Braverman, Kazhdan, and Patnaik exactly recovers the usual affine Bruhat order on this Coxeter group. Then, as explained in [M], the function ℓ : W T → Z exactly recovers the usual Coxeter length function; in particular, it is strictly compatible with the order.
In this paper, we give a positive answer to the above question.
Theorem 1.3 (Theorem 3.6 below). For any Kac-Moody group G, the length function ℓ is strictly compatible with the Bruhat order on W T .
Even more interesting than this positive answer is our method of proof and the role that inversion sets play. To explain this, we briefly recall some of the ingredients for the definition of the Bruhat order on W T . Let x ∈ W T . To every positive real root β for G and every n ∈ Z, there is an associated reflection element that we call s β [n] . In general, s β[n] / ∈ W T , but in the cases of interest, we will have xs β[n] ∈ W T . In this case, by definition, x and xs β [n] are comparable in the Bruhat order, and the order is generated by such relations.
First we construct a set Inv ++ x (s β [n] ), which is a certain subset of the inversion set of s β[n] (Inv ++ x (s β [n] ) is defined in Section 3.2) and we prove that xs β[n] > x if and only if Inv ++ x (s β[n] ) = ∅. Then, to prove our theorem we show that when xs β[n] > x: There is a similar statement when the inequality is reversed. To prove (1.4), we develop a generalized notion of inversion set and we relate ℓ(x) with the inversion set of x −1 (denoted Inv(x −1 )) and ℓ(xs β [n] ) with the inversion set of s β [n] x −1 (Inv(x −1 s β [n] ). However, this relationship is very subtle because these inversion sets are generally infinite and the function ℓ may take negative values. A particular manifestation of the subtlety of this relationship is that, unlike for Weyl groups, elements of different lengths may have identical inversion sets. Additionally, implicit in (1.4) is the finiteness of Inv ++ x (s β[n] ), which is not at all obvious. What we show is that there is a canonically defined injection from Inv(x −1 ) to Inv(x −1 s β[n] ), and we then construct a bijection between Inv ++ x (s β [n] ) and the complement of the image of this injection. We show that ℓ(x) can be computed by performing a weighted sum over certain finite subsets of Inv(x −1 ); similarly for ℓ(xs β [n] ). Finally, we need an analogue of the finite-type fact that 2ρ is the sum of positive roots. Putting these various ingredients together we get (1.4).
1.3. Classifying covers. As a consequence, given x, y ∈ W T with x < y, we know that a chain between x and y can have at most ℓ(y) − ℓ(x) elements. A natural question is whether this bound is always acheived. Equivalently, we can ask whether covers are classified by the function ℓ. Let us write x ⊳ y to denote that y covers x in the Bruhat order. Then we conjecture the following.
We note that in the finite and single-affine situations this conjecture is true because it is true for all Coxeter groups. In our double-affine situation those methods are not available. However, in untwisted affine type ADE, we have a positive result.
Theorem 1.6 (Theorem 6.12 below). Conjecture 1.5 is true for G of untwisted affine ADE type.
In untwisted affine ADE, using our explicit control of the pairings between roots and coroots, we reduce the problem to a calculation that is essentially the case of affine SL 2 . Then we verify the theorem in this situation by explicit computations.
1.4. Some remarks on further directions. Let G be an untwisted affine Kac-Moody group, and let F = k((π)) be the field of formal Laurent series over a field k. Let us take G = G(F ) , and let G + Ĺ G be the Cartan semigroup as in [BKP]. Let I ⊂ G be the Iwahori subgroup. Then we expect the quotient G + /I to be the k-points of the "double-affine flag variety". We expect the I-orbits on G + /I to be the "double-affine Schubert cells". By [BKP], we know that these I orbits are in bijection with W T . So we expect the closure order on "double-affine Schubert cells" to be precisely the double-affine Bruhat order. Unfortunately, in this double-affine situation we do not know how to properly work with G + /I as object of algebraic geometry. So the statement about the closure order is currently only a heuristic (or perhaps a definition). Of course, in the single-affine case when G is finite-type, all of the above has precise meaning and is well known.
In Section 8 we discuss some conjectures motivated by the above heuristics as well as directions for further work. In particular, we write down the sets we expect to be the transverse slices to one double-affine Schubert variety embedded in another. Motivated by these transverse slices, we give a purely group-theoretic but conjectural definition of the double-affine Bruhat order. We also discuss some further questions that are more combinatorial in nature.
1.5. A remark about terminology. We write k to refer to fields we consider without any valuation, and we write F to refer to a field with integer valuation and residue field k. We propose the following terminology.
We will use the adjective "finite" to refer to finite-type Kac-Moody groups over fields k and objects constructed from them. For example: finite roots, finite Weyl groups, finite flag varieties, finite Hecke algebras, finite Bruhat orders, etc.
We will use the the adjective "single-affine" to refer to affine-type Kac-Moody groups over fields k or finite-type Kac-Moody groups over valued fields F and the objects constructed from them. For example: single-affine roots, single-affine Weyl groups, single-affine flag varieties, single-affine Hecke algebras, single-affine Bruhat orders, etc.
Finally, we will use the adjective "double-affine" to refer to affine-type Kac-Moody groups over valued fields F and the objects constructed from them. For example: double-affine roots, doubleaffine Weyl groups, double-affine flag varieties, double-affine Hecke algebras, double-affine Bruhat orders, etc.
1.6. Acknowledgements. We would like to thank Manish Patnaik for many discussions and for carefully reading an early version of this manuscript. D.M. was supported by a PIMS postdoctoral fellowship. D.O. was supported by NSF grant DMS-1600653.

Notation
Let G be a Kac-Moody group, W its Weyl group, and ∆ the set of real roots of G. Define set of G-affine roots as (2.1) (When G is an affine Kac-Moody group, we will use the terminology "double-affine" synonymously with G-affine.) Let ∆ + ⊂ ∆ be the set of positive real roots of G. We call β + nπ ∈ r ∆ positive (and write β + nπ > 0) provided that β ∈ ∆ + and n ≥ 0 or β ∈ −∆ + and n > 0; otherwise we call β + nπ negative (and write β + nπ < 0). Let r ∆ + be the subset of positive elements of r ∆. For any β ∈ ∆ + and n ∈ Z, let us define where sgn : Z → {±1} is the signum function: Notice that β[n] always belongs to r ∆ + . Conversely, every element of r ∆ + is of this form for unique β ∈ ∆ + and n ∈ Z. We refer to β as the G-classical part of β[n]. (When G is an affine Kac-Moody group, we will use the terminology "single-affine" synonymously with G-classical.) Let P the coweight lattice of G and consider the semidirect product group W P = P¸W . We denote elements of W P by π µ w where µ ∈ P and w ∈ W . The group W P acts on the set r ∆ via the formula: π µ w(β + nπ) = w(β) + (n + µ, w(β) )π (2.4) where ·, · is the canonical pairing between the coweight lattice and the root lattice of G. We define the reflection s β[n] corresponding to β[n] ∈ r ∆ + as the following element of W P : where β ∨ is the real coroot of G associated with β. The action of s β[n] on affine roots is then given by the usual formula For any x ∈ W P , we define the inversion set Inv(x) to be the subset of elements of r ∆ + made negative under the action of x. If x ∈ W , then this definition coincides with the usual definition of inversion set (via the injection ∆ + ֒→ r ∆ + , β → β[0]). Let T ⊂ P be the Tits cone in the coweight lattice of G. Our main object of study is W T = T¸W , which is a subsemigroup of W P .
3. The length functions ℓ ε and ℓ 3.1. A height formula. Let ρ be the sum of fundamental weights of G (which we have chosen once and for all). For any β ∈ ∆ + , the height of the associated coroot β ∨ is ht(β ∨ ) = β ∨ , ρ , which is independent of the choice of ρ.
We apply the floor function for the psychological benefit of allowing the length function defined in the next section to take on only integral values. However, we will see that the invariant quantities we construct, i.e., independent of choice of ρ, will be the differences of lengths of elements that are comparable in the Bruhat order. These quantities will always be integral even if we allow the length to take on rational values. We note that in untwisted affine cases, ρ can be chosen so that µ, 2ρ ∈ Z for all µ ∈ P . For simplicity of notation, we shall assume below that µ, 2ρ ∈ Z and omit the ⌊·⌋, which can easily be added to the arguments below.
3.2. The length functions ℓ ε . Let us recall the definition of the length function ℓ ε : where µ + is the unique dominant translate of µ under the action W .
Let β[n] ∈ r ∆ + be a G-affine root such that π µ ws β[n] ∈ W T . If π µ w(β[n]) > 0, then we declare that π µ ws β[n] > π µ w in the Bruhat preorder. In [M], it is proved that π µ w(β[n]) > 0 is equivalent to ℓ ε (π µ ws β[n] ) > ℓ ε (π µ w), where Z ⊕ Zε is ordered lexicographically. So the map ℓ ε is strictly compatible with the order structure. In particular, this implies that the Bruhat preorder is in fact a partial order. However, because intervals in Z ⊕ Zε are not finite in general, one does not obtain any strong finiteness results.
3.3. Setting ε = 1. Let us define ℓ : W T → Z by composing ℓ ε with the map Z ⊕ Zε → Z given by sending ε to 1. We will prove the following, which was conjectured in [M]: Theorem 3.6. The map ℓ is strictly compatible with the Bruhat order on W T and the usual order on Z. That is, if x, y ∈ W T , x ≤ y and x = y, then ℓ(x) < ℓ(y).
Because Z does have finiteness of intervals, this implies that all chains between two fixed elements of W T must be finite and gives an explicit bound. In fact, we will prove the following stronger statement.
3.4. Some explicit formulas for length. Proposition 3.10. For any µ ∈ T , From the definition of ℓ and Proposition 3.10, we immediately obtain the following formula for the length of an arbitrary element π µ w ∈ W T : (3.14)

Maps between double-affine inversion sets
In order to prove Theorem 4.3, we need to introduce certain maps between inversion sets.
4.1. The analagous problem for Weyl groups. We believe that the discussion below is new even for Weyl groups, so let us first consider the analogous problem for the Weyl group W of G. Let w ∈ W and β ∈ ∆ + , and let us assume w(β) > 0, i.e., ws β > w in the Bruhat order. Then ℓ(ws β ) > ℓ(w). The problem that we wish to consider is: (1) Give an explicit injection Inv(w −1 ) ֒→ Inv(s β w −1 ).
4.1.1. The solution. The following gives the solution to the first part of the problem.
Proof. Let γ be as in the statement. By assumption, we have w(γ) > 0, and Proposition 4.3. The images of ϕ and ψ are disjoint.
Proof. Both sides have the same cardinality by the length formula mentioned above; this gives the proof immediately. However, we would like to give a proof that avoids counting because the relevant sets need not be finite in the double-affine case.

4.2.
The statements in the G-affine case. We can immediately generalize our solution to the G-affine case. Let π µ w ∈ W T and let β[n] ∈ r ∆ + be a positive G-affine root such that: The constructions and proofs of the previous subsection carry over without change once we substitute π µ w for w and β[n] for β. The explicit translations are as follows: This defines an injection ϕ : Inv(w −1 π −µ ) ֒→ Inv(s β[n] w −1 π −µ ).  Inv(s β[n] w −1 π −µ ) = image(ϕ) ⊔ image(ψ) (4.12) 5. Some results on inversion sets and proof of Theorem 3.7 5.1. Explicit computation of double-affine inversion sets. By direct computation, one finds that Let S ⊂ ∆ + be a finite subset of the positive single-affine real roots, and let us define Then Inv S (w −1 π −µ ) is finite and we have: ). Theorem 5.3. Let us assume that π µ w and β[n] are as in Section 4.2, i.e., π µ w(β[n]) > 0 and π µ ws β[n] ∈ W T . Then the set Inv ++ π µ w (s β[n] ) is finite. Proof. By Proposition 4.8, we can identify this set with its image under the map ψ. By the proof of Corollary 4.4, we can identify the image of ψ with the set of positive G-affine real roots θ[m] such that: ) is finite it suffices to show that only finitely many θ can occur. For the first condition, we compute: For the second condition: For the third condition: Because µ+nw(β ∨ ) ∈ T by assumption, µ+nw(β ∨ ), θ ≥ 0 for almost all θ. As we are interested in proving finiteness of the set of θ that occur, we can go ahead and assume µ + nw(β ∨ ), θ ≥ 0. Then the first condition necessitates that m ≥ 0. The second condition requires that µ, θ ≤ m.

Putting it all together.
Proof of Theorem 3.7. Let S ⊂ ∆ + be a finite subset of the positive single-affine roots such that: (1) S is invariant under |s w(β) | (2) S contains Inv(s w(β) ) and Inv(w −1 ) (3) S contains all η such that there exists m such that η[m] ∈ image(ψ).
The fact that such an S exists follows from the finiteness of Inv ++ π µ w (s β[n] ) and the assumption that µ, µ + nw(β ∨ ) ∈ T . Let us define S c = ∆ + \S. Then S c is also invariant under |s w(β) |.
6. Classification of covers 6.1. General conjecture. For x, y ∈ W T , let us write x ⊳ y to indicate a covering relation, i.e., x ⊳ y if and only x < y and {z | x < z < y} = ∅.
By the definition of the order, a necessary condition for x ⊳ y is that y = xs β[n] for some G-affine real root β[n]. Motivated by the well-known characterization of covering relations for the Bruhat order on a Coxeter group, we make the following: Conjecture 6.1. We have x ⊳ y if and only x < y and ℓ(y) = ℓ(x) + 1.
Below we will give a positive answer to this question when G is of untwisted affine ADE type. We will proceed by explicitly computing double-affine roots.
6.2. Explicit description of double-affine roots when G is untwisted affine. Let G 0 be a finite-type Kac-Moody group, and let G be the untwisted affinization of G 0 . Let us write δ for the minimal imaginary root for G. Now we refer to G-affine roots as double-affine roots.
Let us write (·, ·) for the Weyl-invariant Euclidean inner product on the root space of G 0 such that all roots have square length of 2. Then for any pair β and θ of positive roots for G 0 , we have: β ∨ , θ = (β, θ) (6.13) the right hand side of this equation. Below, we will abuse notation and simple write β, θ for the pairing (β, θ).
The following well-known fact is crucial for our argument.
Lemma 6.14. Let θ and β be distinct positive roots for G 0 which is finite-type ADE. Then: 1}. (6.15) In order to prove Theorem 6.12, we need to show that for x ∈ W T and β[r, n] a double-affine real root, x ⊳ xs β[r,n] implies #Inv ++ x (s β[r,n] ) = 1. Let us suppose that x → xs β [r,n] , and ℓ(xs β[r,n] ) − ℓ(x) > 1; we will show that xs β[r,n] is not a cover of x. This will be accomplished in Propositions 6. Therefore, we are now reduced to the "rank-one" case when x → xs β [r,n] , ℓ(xs β[r,n] ) − ℓ(x) > 1, and all elements of Inv ++ x (s β [r,n] ) are of the form β[s, m] for some (s, m) ∈ Z 2 . We will handle this in the next section.

The rank-one case
Let us consider x, β, and a pair (r, n) ∈ Z 2 as above, and let us suppose that x(β[r, n]) > 0 (7.1) ℓ(xs β[r,n] ) − ℓ(x) > 1 (7.2) and all double-affine roots in Inv ++ x (s β [r,n] ) are of the form β[s, m] for some (s, m) ∈ Z 2 . The second condition is equivalent by Theorem 3.7 to: To complete the proof of Theorem 6.12, we must prove: Proposition 7.4. The element xs β[r,n] is not a cover of x.
Let us first make the following simplifying assumptions: The other situations are handled by arguments similar to what we present below. We will divide the proof into the following cases.
Proof. For the first term of the chain, we have x(β[r − 1, n]) > 0 by assumption.
For the second term, we compute because of Lemma 7.7 and the fact that r ≥ 1.
Proof. Case: r − a = 0. This case does not happen because: #Inv ++ x (s β[r,n] ) = 1. (7.43) Case: r − a > 0. In this case, we must have: That is, the line through (r, n) and (a, b) has slope −ℓ. Using the involution on Inv ++ x (s β[r,n] ), we see that β[r r, r n] ∈ Inv ++ x (s β [r,n] ) only if (r r, r n) lies on this line passing through (r, n) and (a, b), which has slope −ℓ.

Further questions
Although we have developed the Bruhat order on W T and the length function in a combinatorial fashion, we expect both to have geometric and group-theoretic relevance. In this section we will describe some questions and conjectures about this perspective.
Let G be an untwisted affine Kac-Moody group with positive and negative Borel subgroups B and B − , and let k be a finite field. Likely we can relax these hypothesis to G being general Kac-Moody and k being an arbitrary field, but we retain these assumptions so that we can directly cite [BKP]. Let F = k((π)), the field of formal Laurent series over k, and let O = k [[π]] be the ring of integers in F . Let G = G(F ), let K = G(O), and let I = {g ∈ K | g ∈ B(k) mod π}. Let G + ⊂ G(F ) be the Cartan semi-group, i.e., the locus where the Cartan decomposition holds (see [BKP, M] for the details). Furthermore we have a set-theoretic (not homomorphic) embedding W T ⊂ G + that is uniquely specified up to right multiplication by I. Then we have the following decomposition of G + (see [BKP,Proposition 3.4

.2 and Lemma 3.4.3]).
Proposition 8.1. We have an equality of subsets: Rephrasing this, the I-orbits on G + /I are exactly indexed by W T . 8.1. Double-affine Schubert cells. If we momentarily consider the single-affine case of G being finite type, then G + /I is precisely the k-points of the (single) affine flag variety, and the I orbits on G + /I are precisely the (single) affine Schubert cells. And the (single) affine Bruhat order exactly describes the closure order on affine Schubert cells.
So following that, we will define G + /I to be the k-points of the double-affine flag variety and we will define the I orbits on G + /I to be the double-affine Schubert cells. Following the single-affine heuristic, let us make the following definition. We will call sets of the form Ix · I/I double-affine Schubert varieties.
Question 8.5. Can we define G + /I as an algebro-geometric object so that formula (8.4) coincides with the closure in the Zariski topology?
8.2. Transverse slices. One can easily see that Ix · I/I is an infinite set for most x ∈ W T , so there is no chance that Ix · I/I is equal to the k-points of a finite-type k-scheme. Moreover, because the Bruhat order on W T is unbounded below, it seems unlikely that Ix · I/I is an indscheme of ind-finite type. Unfortunately, it seems that giving G + /I a geometric structure will be comparably difficult to the problem of giving a geometric structure to the semi-infinite flag variety. So Question 8.5 may be too difficult.
However, what seems more reasonable is to work with transverse slices to a double-affine Schubert variety sitting inside one another. To define these objects (at the level of k-points), we need to introduce two other subgroups of G. Let K ∞ = G(k[π −1 ]) and let I ∞ = {g ∈ K ∞ | g ∈ B − (k) mod π −1 }.
Definition 8.6. Let x, y ∈ W T . Then let us define the transverse slice to Iy · I/I inside Ix · I/I as: Ix · I/I ∩ I ∞ y · I/I. We can also drop the closure, and make the following group theoretic conjecture Conjecture 8.10. Let x, y ∈ W T . Then Ix · I/I ∩ I ∞ y · I/I = ∅ (8.11) if and only if x ≤ y.
A positive answer to this conjecture would give a purely group-theoretic definition of the Bruhat order without having to discuss closures.
Question 8.12. Let x ≤ y. Give the transverse slice Ix · I/I ∩I ∞ y·I/I the structure of a finite-type affine scheme.
Following the situation in the single-affine case, we expect the transverse slice Ix · I/I ∩ I ∞ y · I/I to have dimension ℓ(y) − ℓ(x). Unfortunately, we do not how to currently make that precise. However, we can make the following precise conjecture.
Conjecture 8.13. Let x ≤ y. Then there exists a polynomial R x,y ∈ Z[v] of degree ℓ(y) − ℓ(x) independent of k such that #(Ix · I/I ∩ I ∞ y · I/I) = R x,y (q) (8.14) where q is the cardinality of k.
A positive answer to this would give a purely group-theoretic definition of the length function. 8.3. 2-dimensional phenomena. Because an untwisted affine Kac-Moody group is itself constructed as a central extension of a loop group of a finite-type group, the p-adic group G is a sort of double loop group. However, the two loops play very different roles in the discussion above. So a natural question is to understand G + /I from a purely 2-dimensional point of view where the two loops play symmetric roles.
In the single-affine case, the loop group perspective gives rise to a well-understood relationship between the affine flag variety and spaces of bundles on an algebraic curve. Therefore, in the double-affine case, we expect there should be a relationship with bundles on an algebraic surface.
Question 8.15. Describe G + /I and/or the transverse slices Ix · I/I ∩ I ∞ y · I/I in terms of bundles on an algebraic surface.
If one considers the double-affine Grassmannian G + /K instead of G + /I, a candidate definition for transverse slices to K-orbit closures is given in terms of bundles on an algebraic surface by Braverman and Finkelberg in [BF]. Even in that case, however, a precise bijection with the grouptheoretic slice is unknown.
On a combinatorial level, this double loop phenemonena manifests itself in the fact that W T contains two copies of the coroot lattice of finite-type group. The first copy lies in W , and the second copy arises because the Tits cone T roughly looks like the coroot lattice times the semigroup of natural numbers. Therefore 2-dimensional phenonemena from this point of view would be any non-trivial symmetry arising from interchanging these two lattices.
8.4.1. Deodhar's inequality. Recall that r ∆ + denotes the set of positive double-affine real roots.
Conjecture 8.16. Suppose x, y, z ∈ W T with x ≤ y ≤ z. Then we have the following inequality: (8.17) In finite and single-affine cases, the above inequality is a conjecture of Deodhar that has since been proved by many authors. Although the statement is purely combinatorial, many of the proofs are intimately related to singularities of Schubert varieties and transverse slices. In our doubleaffine situation, we hope that a proof of this conjecture will shed some light on the geometry of transverse slices. 8.4.2. Generalizing Coxeter group theory. The theory of Coxeter groups and Bruhat orders is very rich. Although we are slowly developing analogues of many results for W T and its Bruhat order, there are still many Coxeter-theoretic results that have not yet been generalized (see the book by Björner and Brenti [BB] for a nice exposition of many of these results). Below we list some problems that we think would be useful generalizations.
• Develop an analogue of reduced expressions and the subword criterion for the Bruhat order.
• Develop weak order.
• Develop a theory of Poincaré series.
• Develop a notion of parabolic sub-semigroups.