Geometric Schur Duality of Classical Type, II

We establish algebraically and geometrically a duality between the Iwahori-Hecke algebra of type D and two new quantum algebras arising from the geometry of N-step isotropic flag varieties of type D. This duality is a type D counterpart of the Schur-Jimbo duality of type A and the Schur-like duality of type B/C discovered by Bao-Wang. The new algebras play a role in the type D duality similar to the modified quantum gl(N) in type A, and the modified coideal subalgebras of quantum gl(N) in type B/C. We construct canonical bases for these two algebras.


Introduction
Let G be a classical linear algebraic group over an algebraically closed field. One of the milestones in geometric representation theory is the geometric realization of the associated Iwahori-Hecke algebra of G, by using the bounded derived category of G-equivariant constructible sheaves on the product variety of two copies of the complete flag variety of G. Via this realization, many problems related to the Iwahori-Hecke algebra of G are solved. For example, the positivity conjecture for the structure constants of the Kazhdan-Lusztig bases ( [KL79]) are settled by interpreting the basis elements as the (shifted) intersection cohomology complexes attached to G-orbit closures in the product variety.
One may wonder if the geometric approach can be adapted to study other objects in representation theory, besides Iwahori-Hecke algebras. Indeed, a modification by replacing the adjective 'complete' in the construction by 'partial' already yields highly nontrivial results, as we explain in the following.
If G is of type A, i.e., G = GL(d), and the complete flag variety is replaced by the N-step partial flag variety of GL(d) with N bearing not relation to d, then an analogous construction provides a geometric realization of the v-Schur quotient of the quantum gl(N) in the classic work [BLM90]. Moreover, the quantum gl(N) can then be realized in the projective limit of the v-Schur quotients (as d goes to infinity). Remarkably, an idempotented version of quantum gl(N) is discovered inside the projective limit as well admitting a canonical basis. The role of the canonical basis for the modified quantum gl(N) is similar to that of Kazhdan-Lusztig bases for Iwahori-Hecke algebras. Subsequently, the Schur-Jimbo duality, as a bridge connecting the Iwahori-Hecke algebra of GL(d) and (modified) quantum gl(N), is realized geometrically by considering the product variety of the complete flag variety and the N-step partial flag variety of GL(d) in [GL92]. The modified quantum sl(N) (a variant of quantum gl(N)) and its canonical basis are further categorified in the works [La10] and [KhLa10], which play a fundamental role in higher representation theory and the categorification of knot invariants.
If G is of type B/C, i.e., G = SO(2d + 1)/SP(2d), and the variety involved is replaced by the N-step isotropic flag variety of SO(2d + 1)/SP(2d), then one gets a geometric realization of the modified forms of two coideal subalgebras U ı and U  of quantum gl(N) in [BKLW14] by mimicking the approach in [BLM90]. Moreover, the canonical bases of these modified coideal subalgebras are constructed and studied for the first time. Along the way, a duality of Bao-Wang in [BW13] relating the (modified) coideal subalgebras and the Iwahori-Hecke algebra of type B/C associated to SO(2d + 1)/SP(2d) is also geometrically realized in a similar manner as the type-A case. (See also [G97] for a duality closely related to the duality of Bao-Wang.) The canonical basis theory for these coideal subalgebras is initiated in the seminal work [BW13], and is used substantially to give simultaneously a new formulation of the Kazhdan-Lusztig conjecture of type B/C on the irreducible character problem and the resolution of the analogous problem for the ortho-symplectic Lie superalgebras.
To this end, it is compelling to ask what happens to the remaining classical case: G = SO(2d) of type D. The purpose of this paper is to provide an answer to this question, as a sequel to [BKLW14]. More precisely, we obtain two quantum algebras K and K m via the geometry of the N-step isotropic flag variety of type D and a stabilization process following [BLM90] and [BKLW14]. We show that both algebras possess three distinguished bases, i.e., the standard, monomial and canonical bases, similar to the results in type ABC. We further establish new dualities between these two algebras and the Iwahori-Hecke algebra of type D attached to SO(2d) algebraically and geometrically.
Unlike type ABC, the algebras K and K m are not modified forms of some known quantum algebras in literature, even though they resemble the modified formsU ı ,U  of coideal subalgebras of quantum gl(N). It is natural to ask for a presentation of the two algebras by generators and relations. We have a complete answer for the algebra K m , and partial results for K. We show that the algebra K m admits defining relations similar to those ofU ı , but with the size of the set of idempotent generators doubled, after extending the underlying ring to the field of rational functions. Despite all the similarities, we caution the reader thatU ı is not a subalgebra of K m . The presentation for K m is obtained by showing that (the complexification of) K m is isomorphic to the modified form of a new unital associative algebra U m containing the coideal subalgebra U ı and two additional idempotents. The appearance of the new idempotents reflects the geometric fact that there are two connected components for maximal isotropic Grassmannians in the type D geometry. As for the bigger algebra K, we formulate another new unital associative algebra U containing the coideal subalgebra U  and three extra idempotents, and expect its modified form to be isomorphic to K after a suitable field extension. As an evidence in support of this expectation, we show that U and the Iwahori-Hecke algebra of type D satisfy a double centralizer property. Notice that the commuting actions between U ı , U  and the Iwahori-Hecke algebra of type D are first observed in [ES13b,7.8] (see also [ES13a]), so this result can be thought of as an enhancement of those in loc. cit.
As an application, we expect that the type-D duality and the canonical basis theory for the new algebras K and K m developed in this paper will shed light on the type-D problems similar to those addressed in [BW13], currently under investigation by H. Bao. Since our results are governed in principle by the (parabolic) Kazhdan-Lusztig polynomials of type D, they are obviously different from those in type ABC in [BLM90] and [BKLW14]. Furthermore, the geometry of type D is more challenging to handle. In particular, there are mainly three new technical barriers in our type D setting that we overcomed. The first one is that there are two connected components for the maximal isotropic Grassmannian associated to SO(2d). This forces us to parameterize the SO(2d)-orbits by using signed matrices instead of matrices in type ABC. The second one is that the number of isotropic lines in a given quadratic space of even dimension over a finite field depends on its isometric class, we choose to work in the case when the group is split, which has a remarkable hereditary property (see Lemma 3.1.1). The last one is that during the stabilization process, one can not subtract/add by an (even) multiple of the identity matrix as in loc. cit., because the signs of the matrices may change. To circumvent this difficulty, we subtract/add an even multiple of a matrix obtained by changing the middle entry to be zero in the diagonal of the identity matrix. All these factors make the computations and arguments more involved than those in previous cases.
As this paper provides a complete picture for the cases of the classical groups, the problem of whether a similar picture exists for exceptional groups is still wide open. Meanwhile, for G replaced by a loop group of type A, there exists a similar geometric theory involving affine Iwahori-Hecke algebras of type A and affine quantum gl(N) in [Lu99], [GV93], [SV00] and [M10]. The investigation for G being a loop group of type BCD will be presented in a separate article.
Acknowledgement. Y. Li thanks Huanchen Bao, Jonathan Kujawa and Weiqiang Wang for fruitful collaborations, which pave the way for the current project. We thank Weiqiang Wang for comments on an earlier version of this article. Y. Li is partially supported by the NSF grant DMS 1160351.

Schur dualities of type D d
In this section, we shall introduce the algebras U and U m , and formulate algebraically the dualities between these two algebras and the Iwahori-Hecke algebras of type D d .
Let V be a vector space over Q(v) of dimension 2n + 1. We fix a basis (v i ) 1≤i≤2n+1 for V. Let V ⊗d be the d-th tensor space of V. Thus we have a basis (v r 1 ⊗ · · · ⊗ v r d ), where r 1 , · · · , r d ∈ [1, 2n + 1], for the tensor space V ⊗d .
For a sequence r and a fixed integer p ∈ [1, 2d], we define the sequence r ′ p and r ′′ p by Lemma 2.1.1. We have a left U-action on V ⊗d defined by, for any i ∈ [1, n], a ∈ [1, n + 1], The lemma follows from (13), Proposition 4.1.1, and Corollary 4.4.1.
Recall that the Iwahori-Hecke algebra H d of type D d is a unital associative algebra over Q(v) generated by τ i for i ∈ [1, d] and subject to the following relations.
Lemma 2.1.2. We have a right H d -action on V ⊗d given by, for (2) Here we identify the sequence r with the associated sequencer.
This lemma follows from (13) and Lemmas 3.4.1 and 3.4.2. We now can state the first duality.
The proposition follows from the previous two lemmas, Lemma 3.2.1, Proposition 4.1.1 and Corollary 4.6.6.
2.2. The algebra U m and the second duality. Let U m be an associative Q(v)-algebra with unit generated by the symbols a , T, J α , ∀i ∈ [1, n − 1], a ∈ [1, n], α ∈ {+, −}, and subject to the following defining relations.
Note that the subalgebra U ı generated by E i , F i , H ±1 a and T is the algebra in the same notation in [BKLW14,5.3]. See also [Le02] and [ES13b].
Let W be the subspace of V spanned by the basis elements v i for i = n+1. Its d-th tensor space W ⊗d is naturally a subspace of V ⊗d spanned by the vectors v r such that r i = n + 1 for any i. Then we see that W ⊗d is a vector space of dimension (2n) d . By

A geometric setting
We now turn to the geometric setting in order to prove the above results among others.
3.1. Preliminary. We start by recalling some results on counting isotropic subspaces in an even dimensional quadratic space over a finite field. We refer to [W93] and the references therein for more details.
Let F q be a finite field of q elements and of odd characteristic. Recall that d is a fixed positive integer, and we set D = 2d. On the D-dimensional vector space F D q , we fix a non-degenerate symmetric bilinear form Q whose associated matrix is under the standard basis of F D q . By convention, W ⊥ stands for the orthogonal complement of a vector subspace W in F D q . Moreover, we call W isotropic if W ⊆ W ⊥ . We write |W | for the dimension of W .
For any isotropic subspace W , the bilinear form Q induces a non-degenerate symmetric bilinear form Q| W ⊥ /W on W ⊥ /W . One of the reasons that we fix Q of the form (3) is its hereditary property in the following lemma, which can be proved inductively.
We remark that the order of the set S d with respect to a symmetric bilinear form on F D q not isometric to Q may not be the same as the number in the above lemma. We will need the following lemma later. We write W a ⊂ V if W ⊆ V and |V /W | = a.
We have Proof. To prove (i), we consider the set Z ′ 3 be the map defined by U → V 2 + U. Clearly, the map φ is surjective. Observe that the order of each fiber is q |V 2 | = q a 1 +a 2 and, moreover, Z ′ 3 gets identified with the set of all isotropic lines in V 3 /V 2 . By Lemma 3.1.2, we have #Z ′ 3 = q a 3 −1 −1 2 −1 . This proves (i).
We now prove (ii). Consider the set This is a well-defined and surjective map and the cardinality of each fiber is q |V 1 | = q a 1 . To calculate the cardinality of Z ′ 4 , we set By a similar calculation in (i), we have This proves (ii).
3.2. The first double centralizer. We fix another positive integer n and let N = 2n + 1. We fix a maximal isotropic vector subspace M d in F D q (of dimension d). Consider the following sets.
• The set X of N- be the special orthogonal group attached to Q. The sets X and Y admit naturally G-action from the left. Moreoever, G acts transitively on Y thanks to the condition |F d ∩ M d | ≡ d mod 2. Let G act diagonally on the product X × X (resp. X × Y and Y × Y ). Set be the set of all A-valued G-invariant functions on X × X . Clearly, the set S X is a free A-module. Moreover, S X admits an associative A-algebra structure ' * ' under a standard convolution product as discussed in [BKLW14,2.3]. In particular, when v is specialized to √ q, we have Similarly, we define the free A-modules A similar convolution product gives an associative algebra structure on H Y and a left S Xaction and a right H Y -action on V. Moreover, these two actions commute and hence we have the following A-algebra homomorphisms.
By [P09, Theorem 2.1], we have the following double centralizer property.
We note that the result in [P09, Theorem 2.1] is obtained over the field C of complex numbers, but the proof can be adapted to our setting over the ring A.
3.3. G-orbits on X × Y and Y × Y . We shall give a description of the G-orbits on X × Y and Y × Y . The description of the G-orbits on X × X is more complicated, and postponed until Section 4.2.
We start by introducing the following notations associated to a matrix M = (m ij ) 1≤i,j≤c .
A similar assignment yields a bijection where the set Π consists of all matrices Moreover, we have is the characteristic function of the G-orbit corresponding to the permutation matrix (d − 1, d + 1)(d, d + 2). Then we have the following well-known result. Given B = (b ij ) ∈ Π, let r c be the unique number in [1, N] such that b rc,c = 1 for each c ∈ [1, D]. The correspondence B →r = (r 1 , · · · , r D ) defines a bijection between Π and the set of all sequences (r 1 , · · · , r d ) such that r i + r D+1−i = N + 1 for any i ∈ [1, D]. Denote by e r 1 ...r D the characteristic function of the G-orbit corresponding to the matrix B in V. It is clear that the collection of these characteristic functions provides a basis for V.
Recall from Section 2.1 that we have the space V ⊗d spanned by vectors v r and to each sequence r a sequencer is uniquely defined. Thus we have an isomorphism of vector spaces over Q(v): Moreover, we have Proof. Formula (14) agrees with the one in [GL92, 1.12], whose proof is also the same as the one for type-A case. We shall prove (15). It suffices to show the result by specializing v to √ q. By the definition of convolution product, we have So the calculation is reduced to the case when D = 4. Note also that it is enough to calculate the case when n = 2, which we will assume.
If two of r 1 , r 2 , r 3 , r 4 are equal, then the calculation can be reduced further to the case when n = 1. In this case, we have For the case when r 1 , r 2 , r 3 , r 4 are all distinct, we have Formula (15) follows from the above computations.

Calculus of the algebra S
Recall from the previous section that S X is the convolution algebra on X × X defined in (5). For simplicity, we shall denote S instead of S X . In this section, we determine the generators for S and the associated multiplication formula. Furthermore, we provide with a (conjectural) algebraic presentation of S and deduce various bases. 4.1. Defining relations of S. For any i ∈ [1, n], a ∈ [1, n + 1], we set It is clear that these functions are elements in S.
a and J α in S, for any i ∈ [1, n], a ∈ [1, n + 1] and α ∈ {±, 0}, satisfy the defining relations of the algebra U in Section 2.1, together with the following ones.
Proof. The proofs of the identities in the first four rows of the defining relations of U are straightforward. We show the identity in the fifth row. Let It is easy to check that the right hand side is equal to . We now show the penultimate identity. By a direct calculation, we have otherwise. The penultimate identity follows.
To prove the last identity, we define a map ρ : . It is clear that ρ is an anti-automorphism. Moreover, we have Applying ρ to both sides of the penultimate identity, we get the last identity. The rest relations are reduced to type-A case, and will not be reproduced here.

4.2.
Parametrization of G-orbits on X × X . In order to describe the structure of the algebra S, we need to parametrize the G-orbits in X × X . Recall from Section 3.2 that X is the set of N-step flags in For any pair (V, V ′ ) of flags in X , we can assign an N by N matrix as (9) whose (i, j)-entry equal to dim where Ξ is the set of all N × N matrices A with entries in N subject to This map is surjective, but not injective. We need to refine it.
Recall that M d is a fixed maximal isotropic subspace in F D q and We have a partition of X : Let O(D) be the orthogonal group associated to Q. For any g ∈ O(D) \ G, the map ψ g : X 2 → X 3 , defined by V → g · V , is a bijection, which yields the following bijections.
Moreover, corresponding pairs on both sides under the bijections in (20) get sent to the same matrix byΦ. In corresponding to (20), we define a sign function Recall the notation ro(A) and co(A) from (8), we set For convenience, we sometimes write A ± for (A, ±) ∈ Ξ ± and A 0 for (A, 0) in Ξ 0 . We further set Elements in Ξ D will be called signed matrices. We have Proof. The first part follows from (20) and the definition of Ξ D . We now calculate #Ξ D . (26) Lemma follows from (26) and (27).

Multiplication formulas in S.
For each signed matrix a ∈ Ξ D , we denote by O a the associated G-orbit. We introduce the following notations.
We note that a + B is a matrix instead of a signed matrix. For a signed matrix a ∈ Ξ D , we define   For any n ∈ Z, k ∈ N, we set where E ij is the N × N matrix whose (i, j)-entry is 1 and all other entries are 0. Let e a be the characteristic function of the G-orbit corresponding to a ∈ Ξ D . It is clear that the set {e a |a ∈ Ξ D } forms a basis of S. For convenience, we set Recall the notations, such as a + B, from (28). We have Remark 4.3.3. Although the formulas (30), (31) and (32) look similar to formulas (3.9), (3.10) and (3.11) in [BKLW14], they are different in many ways. For example, if we take h = n and a in (30) to be such that A − E θ n+1,n is diagonal, then we have otherwise.
While in [BKLW14], the product e B * e A is a sum of the terms in the right-hand side of the above identity in a similar situation.
Proof. The proof of (a) and (b) is the same as the one for Lemma 3.2 in [BLM90]. We must show (c), which can be reduced to analogous results at the specialization of v to √ q.
We first deal with the case when ro(a) n+1 > 0.
then a n,p = a ′ n,p + 1 and a n+1,p = a ′ n+1,p − 1. Hence a ′ = a(n, p). In particular, we have e c * e a = p #Z p e a(n,p) .
Observe that This matches with the coefficient of the first term on the right-hand side of (32) for p ≤ n.
We now compute the number #Z p for p ≥ n + 1. We set W i = Vn+V ⊥ n ∩V ′ i Vn and consider the following flags From this observation and applying Lemma 3.1.3, we have that #Z p matches with the coefficients of the terms in (32). Therefore, we have (c) when ro(a) n+1 > 0. Finally, we assume that ro(a) n+1 = 0. In this case, sgn(a) = + or −, and hence e c e a (V, V ′ ) = V U e c (V, V U )e a (V U , V ′ ) and V U runs as follows.
where M d is the fixed maximum isotropic subspace in Section 3.2. In this case, the coefficient of e a(n,n+1) is equal to 1 in both cases. Therefore, we have (c) for the case ro(a) n+1 = 0.
Recall the notations from (28). By Proposition 4.3.2 and an induction process, we have the following corollary.
Note that L i ∈ A since a n+1,n+1 is even.
Proof. The proof of (a) and (b) is the same as the one for Lemma 3.4 in [BLM90]. We now show (c) by induction on r. We rewrite c as c r to emphasize the dependence on r. Let d t ′ (a) be the coefficient of a(n, t ′ ) in the product e cr e a . Let p ∈ N n be the vector whose p-th entry is 1 and 0 elsewhere. The statement (c) is reduced to show that for any t = (t 1 , t 2 , · · · , t N ) ∈ N n such that u t u = r + 1, we have where the sum runs over pairs (t ′ , p) such that t ′ u = r and t ′ + p = t. We shall prove (36) by induction. When r = 0, the statement (36) holds automatically. We first deal with the case when ro(a) n+1 > 0. By the induction assumption, we have Since t ′ + p = t, we have t i = t ′ i + δ ip . We can compute the quotient t ′ ,p d t ′ (a)d p (a(n, t ′ ))/d t (a). We first calculate the power of v 2 for each p in this quotient, which is We then calculate the coefficients containing v-numbers for each p in the above quotient, which can be broken into the following three cases. If p < n+1, then the coefficient involving v-numbers is If p = n + 1, then the term is If p > n + 1, then the term is Summing up, we have This proves (36) under the assumption that ro(a) n+1 > 0. The proof of (36) for the case of ro(a) n+1 = 0 is similar and skipped.
4.4. S-action on V. A degenerate version of Proposition 4.3.2 gives us an explicit description of the S-action on V = A G (X ×Y ) as follows. For any r j ∈ [1, N], we denoteř j = r j +1 andr j = r j − 1.
Proof. Since the number of columns of the matrix associated to e r 1 ···r D is D = 2d, the second term in (32) disappears when we calculate the E n action on e r 1 ···r D . The first two identities follow directly from Proposition 4.3.2. The last four identities are straightforward.
4.5. Standard basis of S. In this subsection, we assume that the ground field is an algebraic closure F q of F q when we talk about the dimension of a G-orbit or its stabilizer. We set where b = (b ij ) ǫ is the signed diagonal matrix such that b ii = k a ik and ǫ = sgn(s l (a), s l (a)). Denote by C G (V, V ′ ) the stabilizer of (V, V ′ ) in G.
Lemma 4.5.1. We have Notice that the above dimensions are independent of the sign of a. By using dim G = 1 2 D(D − 1) = 1 2 ( i,j,k,l a ij a kl − i,j a ij ), we have the second equality. The third equality follows from the previous two equalities.
For any a ∈ Ξ D , let We define a bar involution '−' on A byv = v −1 . By Lemma 4.5.1, Corollary 4.3.4 can be rewritten in the following form.

(c) If the condition h = n in (b) is replaced by h = n, then we have
The proof involves lengthy mechanical computations and is hence skipped.

Generators of S. Define a partial order "
For any a = (a ij ) α and b = (b ij ) ǫ in Ξ D , we say that a b if and only if α = ǫ and the following two conditions hold. r≤i,s≥j The relation " " defines a second partial order on Ξ D . We say that a ≺ b if a b and at least one of the inequalities in (40)    (a) Assume that a = (a ij ) α ∈ Ξ D satisfies one of the following two conditions: (1) a hj = 0, ∀j ≥ k, a h+1,k = r, a h+1,j = 0, ∀j > k, if h < n; (2) a nj = 0, ∀j ≥ k, a n+1,k = r + (r + c)δ n+1,k , a n+1,j = 0, ∀j > k, if h = n, k ≥ n + 1. (b) Assume that a = (a ij ) α ∈ Ξ D satisfies one of the following conditions: (1) a hj = 0, ∀j < k, a hk = r, a h+1,j = 0, ∀j ≤ k, if h < n, or (2) a nj = 0, ∀j < k, a nk = r, a n+1,j = 0, ∀j ≤ k, if h = n, k ≤ n. We define an order on N × N by By using Corollary 4.6.2, we are able to prove the following theorem. Proof. We show the theorem for n = 2. Let B 10 be a diagonal matrix with diagonal entries being (co(a) 1 , · · · , co(a) 5 ). We set b 10 = (B 10 , sgn(s r (a), s r (a))).
For i = 1, · · · , 4, let B 1i be the matrix such that B 1i − R i,i+1 E θ i,i+1 is a diagonal matrix and co(B 1i ) = ro(B 1,i−1 ). We set b 1i = (B 1,i , sgn(s l (b 1,i ), s r (b 1,i ))), where s l (b 1,i ) and s r (b 1,i ) are defined inductively by We note that s l (b 1,i ) has multiple choices in some cases. When this case happens, we always set s l (b 1,i ) = 2. By Corollary 4.6.2, we have and the * s in the diagonal are some nonnegative integers uniquely determined by co(A 1 ) = co(B 10 ). Now let B ji be the matrices such that B ji − R i,i+j E θ i,i+1 is a diagonal matrix and co(B j,i ) = ro(B j,i−1 ) for all i ∈ [1, 5 − j], j ∈ [2, 4]. Here we assume that B j0 = B j−1,6−j . We set b ji = (B ji , sgn(s l (b ji ), s r (b ji ))), where s l (b ji ) and s r (b ji ) are defined in a similar way as (43) and s l (b ji ) = 2 if it has multiple choices. By repeating the above process, we have Theorem follows for n = 2. The general case can be shown similarly.
We have immediately    , and d a is the (a, a)-entry of the matrix in d. We have the following corollary by Corollary 4.6.5.
Remark 4.6.7. The order (42) in Theorem 4.6.3 is different from the ones in [BKLW14, Theorem 3.6.1] and [BLM90,3.9]. It can be shown that using the latter orders, one can construct a different monomial basis for the algebra S.

4.7.
Canonical basis of S. In this subsection, we assume that the ground field is an algebraic closure F q of the finite field F q . Let IC a be the intersection cohomology complex of O a , normalized so that the restriction of IC a to O a is the constant sheaf on O a . Since IC a is a G-equivariant complex and the stabilizers of the points in O a are connected, the restriction of the i-th cohomology sheaf is a trivial local system. We denote n b,a,i the rank of this local system. We set The polynomials P b,a satisfy (45) P a,a = 1 and P b, (44) and (45), we have For any a ∈ Ξ D , we set In particular, where V is any element in X such that |V i /V i−1 | = co(a) i . By the definition of d a and Lemma 4.5.1, we have This implies that if ro(a) = ro(c), co(a) = ro(b) and co(b) = co(c). By using (47) and the same argument as the one proving Proposition 3.2 in [M10], we have the following proposition.
Proposition 4.8.1. For any a, b, c ∈ Ξ D , we have Moreover, the following proposition holds from Proposition 4.8.1.
Proposition 4.8.2. For any b, c ∈ Ξ D , we have where ρ is defined in (18).
We define a bar involution¯: S → S bȳ

The limit algebra K and its canonical basis
We shall apply the stabilization process to the algebras S in (5) as D goes to ∞, following [BLM90]. We write S D to emphasize the dependence on D, and Ξ D (D) for the set Ξ D in (23) for the same reason. 5.1. Stabilization. Let I ′ = I − E n+1,n+1 , where I is the identity matrix. We set For any matrix a ∈ Ξ D , the notations introduced in (28) and (29) are still well-defined and will be used freely in the following. Moreover, we observe that sgn(a) = sgn( p a). Let where the notation [a] is a formal symbol bearing no geometric meaning. Let v ′ be a second indeterminate, and We have Proposition 5. 1.1. Suppose that a 1 , a 2 , · · · , a r (r ≥ 2) are signed matrices in Ξ D such that co(a i ) = ro(a i+1 ) and s r (a i ) = s l (a i+1 ) for 1 ≤ i ≤ r − 1. There exist z 1 , · · · , z m ∈ Ξ D , G j (v, v ′ ) ∈ R and p 0 ∈ N such that in S D for some D, we have Proof. The proof is essentially the same as the one for Proposition 4.2 in [BLM90] by using Corollary 4.5.2 and Theorem 4.6.3. The main difference is that when h = n, the twists β(t) and β ′′ (t) in (37) and (39), respectively, change when a is replaced by p a. To remedy this difference, we adjust these two twists as follows.
Then the new twists γ(t) and γ ′′ (t) remain the same when a is replaced by p a. For example, when r = 2 and a 1 is chosen such that a 1 − RE θ n,n+1 is a diagonal with R ∈ N, the structure Similaryly, if r = 2 and a 1 is chosen such that a 1 − RE θ n+1,n is diagonal with R ∈ N, the structure constant G t (v, v ′ ) is defined by For the case when a 1 is chosen such that a 1 − RE θ h,h+1 or a 1 − RE θ h+1,h is diagonal for some h < n, then the structural constant G t (v, v ′ ) is defined similarly as that in the proof of Proposition 4.2 in [BLM90], i.e., for a 1 such that a 1 − RE θ h,h+1 is diagonal for some h < n, and for a 1 such that a 1 − RE θ h+1,h is diagonal for some h < n. Bearing in mind the above modifications, the rest of the proof for Proposition 4.2 in [BLM90] can be repeated here.
By specialization v ′ at v ′ = 1, we have Corollary 5.1.2. There is a unique associative A-algebra structure on K, without unit, where the product is given by if a 1 , · · · , a r are as in Proposition 5.1.1.
By corollary 5.1.2 and comparing the G t (v, 1)'s with (37), (38) and (39), the structure of K can be determined by the following multiplication formulas. Recall the notations from (28).
Let a and b ∈ Ξ D be chosen such that b − rE θ h,h+1 is diagonal for some 1 ≤ h ≤ n, r ∈ N satisfying co(b) = ro(a) and s r (b) = s l (a). Then we have where the sum is taken over all t = (t u ) ∈ N N such that N u=1 t u = r, β(t) is defined in (37), and a t ∈ Ξ D is in (33).
Similarly, if a, c ∈ Ξ D are chosen such that c−rE θ h+1,h is diagonal for some 1 ≤ h < n, r ∈ N satisfying co(c) = ro(a) and s r (c) = s l (a), then we have where the sum is taken over all t = (t u ) ∈ N N such that N u=1 t u = r, β ′ (t) is defined in (38), and a(h, t) ∈ Ξ D is in (34).
If a, c ∈ Ξ D are chosen such that c − rE θ n+1,n is diagonal for some r ∈ N satisfying co(c) = ro(a) and s r (c) = s l (a), then we have where the sum is taken over all t = (t u ) ∈ N N such that N u=1 t u = r, G and β ′′ (t) are in (39) and a(n, t) ∈ Ξ D .
Given a, a ′ ∈ Ξ D , we shall denote a ′ ⊑ a if a ′ a, co(a ′ ) = co(a), ro(a ′ ) = ro(a), s l (a ′ ) = s l (a) and s r (a ′ ) = s r (a).
By using (50), (51) and (52) and arguing in a similar way as the proof of Theorem 4.6.3, we have where γ a ′ ,a ∈ A and the product is taken in the order (42).
As a consequence of the above proposition, we have Proposition 5.1.4. The algebra K is generated by the elements [e] such that e − rE θ i,i+1 is diagonal for some r ∈ N and i such that 1 ≤ i < N.
Corollary 5.1.5. The algebra Q(v) ⊗ A K is generated by the elements [e] such that either e or e − E θ i,i+1 is diagonal for some i such that 1 ≤ i < N.

Bases of K.
We define a bar involution − : for any e such that e − RE θ i,i+1 is diagonal for some R ∈ N and i ∈ [1, N − 1]. By using (53), we have By a standard argument similar to the proof of Proposition 4.7 in [BLM90], we have the following proposition.
Now the algebra K acts on the A-module V in (7) via Ψ and the S-action. By Lemma 3.2.1, we have Proposition 5.3.2. The algebra K and H Y form a double centralizer, i.e.,

Towards a presentation of Q(v)⊗ A K.
We make an observation of the signed diagonal matrices in Ξ D in (48). We denote by D λ the diagonal matrix whose i-th diagonal entry is λ i , for any λ = (λ i ) ∈ Z N . We have For any signed diagonal matrix d, we set . For a signed diagonal matrix d = (D λ , 0) of sign 0, we set For any element y ∈ U in Section 2.1 and singed diagonal matrix d, we shall define the notation yd. We may assume that y is homogeneous. We assume that xd is defined for all homogenous x ∈ U of degree strictly less than y , then we define where the sum runs over all signed matrices e j in Ξ D such that e j − E θ j+1,j is diagonal. Although an infinite sum, there is only finitely many nonzero terms, hence well-defined. Similarly, we can define F j xd for any j ∈ [1, n]. Therefore, the notation yd for y ∈ U is well-defined.
For the remaining relations, they can be proved by the following principle. Suppose that xd = C xd,a a with C xd,a ∈ A. We can pick a large enough p such that p d and p a all have non-negative entries. For an appropriate D ′ , we have an element in S D ′ of the form x p d defined in a similar way as that in K. We can write (49). If x is of the form in the remaining relations, we have This follows from the comparison of (50), (51) and (52) in K with (37), (38) and (39) in S D ′ , respectively. Now the remaining relations all hold in S D ′ for all D ′ large enough by Proposition 4.1.1, so are those relations without specializing v ′ . Now relations in K are obtained by specializing v ′ = 1.
5.5. The algebra U. In this section, we shall define a new algebra U in the completion of K similar to [BLM90, Section 5]. We show that U is a quotient of the algebra U defined in Section 2.1.
LetK be the Q(v)-vector space of all formal sum a∈Ξ D ξ a [a] with ξ a ∈ Q(v) and a locally finite property, i.e., for any t ∈ Z N , the sets {a ∈Ξ D |ro(a) = t, ξ a = 0} and {a ∈ Ξ D |co(a) = t, ξ a = 0} are finite. The spaceK becomes an associative algebra over Q(v) when equipped with the following multiplication: where the product [a] · [b] is taken in K. This is shown in exactly the same as [BLM90,Section 5].
Observe that the algebraK has a unit element d, the summation of all diagonal signed matrices.
Proposition 5.5.1. The following relations hold in U.
Proof. We show (67). By checking the values of functions s l and s r defined in (29) at 0(j) and F n , we have , where the sums run through in an obvious range by the definition in (65).
. So we have the first identity in (67) for the case of h = n. Other cases for the first identity and all other identities in (66) and (67) can be shown similarly.
We show (68). By the definition of 0 + (0) and F h for h < n, we have . The other identities in (68) can be shown similarly.
We show (69). By Proposition 5.4.2 (68), we have . We now show (72). By definition, we have v0(n + 1 − n)F n = F n 0(n + 1 − n) = λ v λ n+1 −λn−2 F n D 0 λ . Similarly, v −1 0(n − n + 1)F n = λ v λn−λ n+1 +2 F n D 0 λ . Moreover, Proof. Under the map Υ, all defining relations of U map to the corresponding relations in U given in Proposition 5.5.1 except the commutator relation between J ± and F n E n . Since This shows that Υ is an algebra homomorphism. The surjectivity is clear.

Case II
In this section, we turn to the case when all flags at the n-th step are assumed to be maximal isotropic.
6.1. The second double centralizer. We define X m to be the subset of X in Section 3.2 subject to the condition that the n-th step of the flags is maximal isotropic. In particular, we have V n = V n+1 for any V ∈ X m , and thus Similar to the definition of the algebra S in Section 4.2, we consider the convolution algebra on X m × X m and the free A-module Under the convolution product, W has a S m -H Y -bimodule structure. By [P09], we have Let Π m = {B ∈ Π|b n+1,j = 0, ∀j}, where Π is defined in Section 3.3. A restriction of the bijection (11) in Section 3.3 yields a bijection Moreover, the isomorphism (13) restricts to an isomorphism where W ⊗d is defined in Section 2.2. which implies again by (79) that Now apply the map ρ in (18) to (80) and (81), we get The other defining equations of U m are straightforward to check and skipped. This finishes the proof of Proposition 6.2.1.
6.3. Generators and bases for S m . We consider the following subset of Ξ D .
Recall from Theorem 4.6.3 that we set R ij = i k=1 a kj for a signed matrix a = (A, ǫ). Let e i,t denote a signed matrix such that e i,t − R i,i+t E θ i,i+1 is diagonal. For a sequence a s , a s+1 , · · · , a r with s ≤ r, we set s ⊓ i=r a i = a r a r−1 · · · a s . Theorem 6.3.1. For any a = A ǫ ∈ Ξ ′ D , there exists a product of signed matrices e i,t where the matrices e i,t are completely determined by the conditions ro(e 1,N −1 ) = ro(a) and co(e 1,1 ) = co(a) and the signs of e i,t are inductively determined by the conditions that s r (e 1,1 ) = s r (a) and s l (e i,t ) = s r (e i,t ) + (−1) sr(e i,t ) p(e i,t ).
Proof. The proof is a modification of the one of Theorem 4.6.3. We show it for n = 2. We consider a signed matrix a = (A, +1) in Ξ ′ D . Without lost of generality, we assume that ur(a) is even, i.e. a 14 + a 24 + a 15 + a 25 is even. Let B 10 be a diagonal matrix with diagonal entries being the entries of co(a). Let B 11 be the matrix such that B 11 − R 12 E θ 12 is a diagonal matrix and co(B 11 ) = ro(B 10 ).
and the * 's are some positive numbers unique determined by co(A 1 ) = co (B 10   6.4. The algebra K m . Recall that Ξ ′ D = {a ∈ Ξ D |ro(a) n+1 = co(a) n+1 = 0}. Let Ξ ′ D = {a ∈ Ξ D |a n+1,j = a j,n+1 = 0, ∀j}. Let K m be the subalgebra of K spanned by the elements [a] for any a ∈ Ξ ′ D . Notice that K m can be obtained via a stabilization similar to Section 5.1 by using the algebras S m . Similar to Theorem 6.3.1, we have n a = a + lower terms, ∀a ∈ Ξ ′ D , where n a is defined in (83). Moreover, by (50) T where F n E n d is defined in (57) and lies in K m . Note that λ n+1 = 0 in this case.
Proof. Proposition can be shown by using (85) and Proposition 5.4.2. One could prove them directly by using the same argument as we make for Proposition 5.4.2. More precisely, all identities can be reduced into S m by replacing [a] by [ p a]. Proposition then follows from Proposition 6.2.1.
6.6. The identification K m =U m . Recall the algebra U m from Section 2.2. Following [Lu93, Section 23], we shall define the modified formU m of U m . We set For any λ, λ ′ ∈ Λ m , we define Let π λ,λ ′ : U m → λ U m λ ′ be the canonical projection. We set sgn(π λ,λ (J + )) = + and sgn(π λ,λ (J − )) = −. SetU m = ⊕ λ,λ ′ ∈Λ m λ U m λ ′ . Similarly, we can defineU ı by replacing U m by its subalgebra as vector spaces, where the sum runs over all elements d of the form π λ,λ (J + ) or π λ,λ (J − ) for λ ∈ Λ m . Let A D be the associative Q(v)-algebra without unit generated by E i d, F i d, T d and d for all i ∈ [1, n − 1] and d runs over all diagonal signed matrices in Ξ ′ D , subjects to the relations (i)-(viii) in Proposition 6.5.1. We have Proposition 6.6.1. The map φ : A D →U m sending generators in A D to the respective elements inU m is an algebra isomorphism.
Proof. Observe that all relations in U m can be transformed into corresponding relations iṅ U m by adjoining diagonal signed matrixes. By comparing the defining relations of U m and those in Proposition 6.5.1, we have thatU m is an associative Q(v)-algebra generated by E i d, F i d, T d and d for all i ∈ [1, n − 1] and d diagonal signed matrices in Ξ ′ D and subject to the defining relations of A D . So we see that the map φ is a surjective algebra homomorphism.
It is left to show φ is injective. By using the same argument of (86), we have A D ≃U ı ⊕U ı , as vector spaces. So the map φ is injective. We are done.
Theorem 6.6.2. The assignment of sending generators inU m to the respective generators in K m defines an algebra isomorphism Υ ′ :U m → Q(v) ⊗ A K m .
Proof. By Propositions 6.5.1 and 6.6.1, we see that Υ ′ is a surjective algebra homomorphism. We observe that Q(v) ⊗ A K m is a direct sum of two copies ofU ı as Q(v) vector spaces. So we have the injectivity. 6.7. The algebra U m . Recall the algebraK and the notations 0 ± from Section 5.5 and the notationâ(j) from (65). We consider the following elements inK. O(j) = 0 + (j) + 0 − (j), ∀j ∈ Z N , λ + E θ n,n+2 ] + [D − λ + E θ n,n+2 ]). Let U m be the subalgebra ofK generated by E i , F i , T, O(j), 0 + (0) and 0 − (0) for all i ∈ [1, n−1] and j ∈ Z N . By a similar argument as Proposition 6.7.1, we have the following proposition.