Modules with Demazure Flags and Character Formulae

In this paper we study a family of finite-dimensional graded representations of the current algebra of $\mathfrak{sl}_2$ which are indexed by partitions. We show that these representations admit a flag where the successive quotients are Demazure modules which occur in a level $\ell$-integrable module for $A_1^1$ as long as $\ell$ is large. We associate to each partition and to each $\ell$ an edge-labeled directed graph which allows us to describe in a combinatorial way the graded multiplicity of a given level $\ell$-Demazure module in the filtration. In the special case of the partition $1^s$ and $\ell=2$, we give a closed formula for the graded multiplicity of level two Demazure modules in a level one Demazure module. As an application, we use our result along with the results of Naoi and Lenart et al., to give the character of a $\mathfrak{g}$-stable level one Demazure module associated to $B_n^1$ as an explicit combination of suitably specialized Macdonald polynomials. In the case of $\mathfrak{sl}_2$, we also study the filtration of the level two Demazure module by level three Demazure modules and compute the numerical filtration multiplicities and show that the graded multiplicites are related to (variants of) partial theta series.


Introduction
The current algebra associated to a simple Lie algebra g is just the Lie algebra of polynomial maps C → g or equivalently, the special maximal parabolic subalgebra of the untwisted affine Lie algebraĝ associated to g. Given any integrable highest weight representation of the affine Lie algebra, one can define a certain family of modules for the appropriate Borel subalgebra inĝ. If the center of the affine algebra acts by the positive integer , these modules are referred to as the Demazure module of level . In certain special cases, the Demazure modules are actually modules for the parabolic subalgebra, i.e., for the current algebra and from now on, we shall only be concerned with such Demazure modules.
The notion of a local Weyl module for the current algebra was introduced and first studied in [6] and the definition was motivated by the representation theory of quantum affine algebras. It was proved later (see [5,7,10]) that in the simply-laced case a local Weyl module is isomorphic to a Demazure module. In the non-simply laced case, it was proved in [15] that a local Weyl module has a filtration by Demazure modules. One of the important ingredients in [15] is the following: a level -Demazure module for A 1 n has a filtration by level m Demazure modules for all m ≥ . The proof of this statement requires taking the q = 1 limit of a result due to A. Joseph [12] in the quantum case using the theory of canonical bases. However, Joseph's result was only proved for simply-laced Kac-Moody Lie algebras and hence did not include A 1 1 . It is remarked in [15,Remark 4.15] that an inspection of Joseph's proof along with a positivity This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html result proved in [14] showed that Joseph's proof worked worked for A 1 1 as well. Our paper was motivated by an attempt to see if it was possible to give a direct proof of this result for A 1 1 . This is achieved as a particular case of Theorem 3.3. Our methods also allow us to determine the multiplicity of the level two Demazure module in the local Weyl module. Using [15,Sections 4.19 and 9] this means that we give the filtration multiplicities (see Proposition 3.5) for a Demazure module occurring in the filtration of a local Weyl module for B n . In Subsection 3.10, we study the case of the Demazure module of level two and show that the multiplicities of the level three Demazure flag are related to partial theta series.
The paper is organized as follows. In Section 2, we recall the definitions and collect the necessary results from [5,10,15] on the relationship between the g-stable level one Demazure modules and local Weyl modules. In the next section we restrict our attention to sl 2 and recall the definition from [7] of the modules V (ξ). We prove that these modules admit a Demazure flag of level for all sufficiently large. In the case when = 2 we give a closed formula for the multiplicities. We then use results in [15] to give a character formula for the local Weyl module in terms of the Demazure module when g is of type B n . Together with the results of [13] we can give the character of the Demazure module for B 1 n as a combination of Macdonald polynomials. In the last section, we define a directed, edge labeled, graph H (ξ) and show that the multiplicity of a level Demazure module in V (ξ) is given by natural polynomials associated to specific paths of this graph.

Local Weyl modules and Demazure modules of level
We summarize the main definitions and results needed for our paper. We refer the reader to [3] for further information on the local Weyl modules, to [5,10,15] for the definition of Demazure modules and their connection with local Weyl modules.
2.1. Throughout this paper C denotes the field of complex numbers and Z, Z + , N denote the integers, non-negative integers and positive integers respectively. Given a complex Lie algebra a denote by U(a) the corresponding universal enveloping algebra. The associated current Lie algebra is denoted a[t]: as a vector space it is just a ⊗ C[t] and the Lie bracket is given in the natural way (i.e., [a ⊗ f, b ⊗ g] = [a, b] ⊗ f g, for all a, b ∈ a and f, g ∈ C[t]). (Here C[t] is the polynomial algebra in an indeterminate t). The degree grading on C[t] defines a natural Z + -grading on a[t] and hence also on U(a[t]): an element of the form (a 1 ⊗ t r 1 ) · · · (a s ⊗ t rs ) has grade r 1 + · · · + r s .
A graded representation of a[t] is a Z-graded vector space which admits a compatible Lie algebra action of a[t], i.e., If V and V are graded a[t]-modules, we say that π : V → V is a morphism of graded a[t]modules if π is a degree zero morphism of a[t]-modules. In particular, this means that V [r] is a a-submodule of V . For r ∈ Z, let τ r be the grading shift operator: if V is a graded a[t]module, then τ r V is the graded a[t]-module with the graded pieces shifted uniformly by r and the action of a[t] unchanged. If M is an a-module define a a[t]-module structure on M by: (a ⊗ t r )m = δ r,0 am where a ∈ a and r ∈ Z + and δ r,0 is the Kronecker delta symbol. We denote this module as ev 0 M and observe that ev 0 M is a graded a[t]-module with (ev 0 M ) [0] = M . 2.2. From now on, g is an arbitrary simple finite-dimensional complex Lie algebra of rank n, h is a fixed Cartan subalgebra of g and R is the set of roots of g with respect to h. The restriction of the Killing form of g to h induces an isomorphism between h and h * and hence also a symmetric non-degenerate form ( , ) on h * . We shall assume that this form on h * is normalized so that the square length of a long root is two and for α ∈ R set d α = 2 (α,α) .
Let I = {1, . . . , n} and fix a set {α i : i ∈ I} of simple roots for R and let {ω i : i ∈ I} ⊂ h * be the set of fundamental weights. Let Q (resp. Q + ) be the Z span (resp. the Z + span) of {α i : i ∈ I} and similarly define P (resp. P + ) to be the Z (resp. Z + ) span of {ω i : i ∈ I} and set R + = R ∩ Q + . Define a partial order on P + by λ ≥ µ if and only if λ − µ ∈ Q + .
Finally, let {x ± α , h i : α ∈ R + , i ∈ I} be a Chevalley basis for g and set x ± i = x ± α i for i ∈ I. 2.3. For µ ∈ P + , let V (µ) be the irreducible finite-dimensional g-module generated by an element v µ with defining relations It is well-known that any finite-dimensional g-module V is isomorphic to a direct sum of irreducible modules V (µ), µ ∈ P + . Further, we may write and we set wt V = {ν ∈ P : V ν = 0}. Let Z[P ] denote the group ring of P with integer coefficients and basis e(µ), µ ∈ P . The character of a finite-dimensional g-module V is the element of Z[P ] defined by Since wt V (µ) ⊂ µ − Q + , it follows that the characters ch g V (µ), µ ∈ P + form a linearly independent subset of Z[P ]. Given a graded g[t]-module and an indeterminate q, we let

2.4.
For λ ∈ P + , the local Weyl module W loc (λ) is the g[t]-module generated by an element w λ with defining relations, for i ∈ I and s ∈ Z + , It is trivial to see that wt W loc (λ) ⊂ λ − Q + . The local Weyl module is clearly Z + -graded once we declare the grade of w λ to be zero, and Finally, ev 0 V (λ) is the unique graded irreducible quotient of W loc (λ) and any irreducible graded finite-dimensional module is isomorphic to τ s ev 0 V (λ) for some (λ, s) ∈ P + × Z + . 2.5. The Demazure module of level and weight λ is denoted by D( , λ) and for our purposes the definition is as follows: it is the graded quotient of W loc (λ) by the submodule generated by the elements (2.1) By abuse of notation, we continue to denote the image of w λ in D( , λ) by w λ . Notice that for ch gr D( , λ) = ch g V (λ) + We refer the reader to [10] and [15] for a more traditional definition of a Demazure module: namely as a module (for a Borel subalgebra) which is generated by an extremal vector in a level -highest weight module for an affine Kac-Moody algebra. We remark also for later use, that this approach allows one to prove that the bounds given in (2.1) are tight, i.e., 2.6. The following is now clear. In particular the lemma shows that the character of a local Weyl module is a Z[q] linear combination of Demazure characters. We shall see in what follows that this is more than a formal linear combination.
2.7. The next result gives the relationship between W loc (λ) and the Demazure modules D(1, λ). It was established in [6] for sl 2 . This was then used in [5] and [10] to prove the result for sl r+1 and the general simply-laced case respectively. In the non-simply-laced case the result was proved in [15]. A different approach to part (i) of this theorem along with a generalization to the higher level Demazure modules is developed in [7, Section 3].
We have an isomorphism of g[t]-modules, (ii) Let λ(h i ) = 0 for some i ∈ I with α i a short root. Then we have a decreasing sequence of graded g[t]-submodules Remark 2.3. We make some comments on the proof given in [15] of part (ii) of the preceding theorem. An important ingredient of the proof is the following statement. Assume that g is of type sl n+1 . For ≥ 1, the module W loc (λ) (or equivalently, by part (i) of the theorem D(1, λ)), has a filtration in which the successive quotients are isomorphic to D( , µ) for some µ ∈ P + . The result is only needed when = 2 if n > 2 and for = 2, 3 if n = 1. However, Naoi establishes this result for an arbitrary simply-laced Lie algebra and arbitrary . As we remarked in the introduction the proof of this result is indirect, using a result of A. Joseph [12] for Demazure modules in the quantum case. His result uses, in a serious way, the theory of canonical bases and is proved only for simply-laced affine Lie algebras, i.e., is not proved for A 1 1 . Our paper gives a direct proof in the case of Demazure modules of affine sl 2 . This is essential in Naoi's work and is also needed in the recent preprint [2].
2.8. Let V be a graded g[t]-module. Motivated by the previous discussion, we call a decreasing sequence Let [F(V ) : τ s D( , µ)] be the number of times τ s D( , µ) occurs in this flag. The graded multiplicity of D( , µ) in this flag is defined to be Using Lemma 2.1, one can show easily that [F(V ) : D( , µ)] q is independent of the choice of the flag and hence from now on we will denote this polynomial by [V : D( , µ)] q . The following is trivial.
-submodule of V such that U and V /U both admit a Demazure flag of level . Then V also admits a Demazure flag of level and (ii) If V admits a Demazure flag of level , we have an equality of graded characters 2.9. The next result is a reformulation of the result proved in [15,Sections 4 and 9] of which the special cases will be of interest to us in this paper.
Proposition 2.5. Suppose that g is of type B n and that = 2 or that g is of type G 2 and = 3. Let λ ∈ P + be such that λ(h n ) = r ∈ Z + where α n is the unique short root. Then, W loc (λ) has a Demazure flag of level one and for µ ∈ P + , otherwise.
Here W sl 2 loc (λ(h n )) is the local Weyl module for sl 2 of weight λ(h n ) and D sl 2 ( , µ(h n )) is defined similarly.
We shall revisit this result later in the paper and give a closed formula for the graded multiplicities when g is of type B n and generating series for the numerical multiplicities (q = 1) when g is of type G 2 . We remark that the analogous result when g is of type F 4 or C n is also proved in [15]: here one takes = 2 and replaced sl 2 by sl 3 and sl n−1 respectively.

The modules V (ξ) and Demazure f lags
In this section we shall, for the most part, be interested only in the case when g is sl 2 . We shall denote the unique simple root by α and the corresponding fundamental weight by ω. We also set x + α = x, x − α = y and h = [x, y] so that {x, y, h} is the standard basis of sl 2 . 3.1. Recall that a partition with s parts is a finite decreasing sequence of positive integers ξ = (ξ 1 ≥ ξ 2 ≥ · · · ≥ ξ s > 0) and that |ξ| = r≥1 ξ r . We shall sometimes also write ξ as the Following [7], given a partition ξ we define a sl 2 [t]-module V (ξ) as follows. It is the sl 2 [t]module generated by an element v ξ with defining relations: where s ∈ Z + in (3.1) and (3.3) holds for all s, r ∈ N for which there exists k ∈ N with of v ξ to be zero, and that we have a surjective map of Observe also that if ξ is the empty partition then V (ξ) is the trivial sl 2 [t]-module.
The following was proved in [7, Proposition 2.7] and will be needed later.
The structure of the modules V (ξ) were studied in great detail in [7] for simple Lie algebras. In the case of sl 2 , these modules are known to be the fusion product of simple irreducible finitedimensional representations with dimensions ξ i + 1. These were first studied in [9] and [8] where an alternate presentation can be found.
3.2. In certain special cases the defining relations of V (ξ) can be greatly simplified. Thus let ξ = 1 and, suppose that s, r, k ∈ N are such that s + r ≥ 1 + rk + j≥k+1 ξ j . It is clear that s + r ≥ |ξ| + 1 and hence (3.3) is a consequence of (3.2), i.e., If ξ = ( ) and if we take s = r = 1, we find that (3.3) holds for k = 1, and so, Using the commutation relations in U(sl 2 [t]) and the defining relations of V (ξ) it is simple to prove that (y ⊗ t)v ξ = 0. In turn, this implies that Moreover we have an isomorphism of sl 2 [t]-modules, 3.3. Given m, ∈ N with m ≥ , set The main result of this section is the following.   In the case when ξ = 1 m the Jordan-Holder series is known (see [5]) and that In the next section, we associate to each partition ξ and an integer m ≥ ξ 1 , a directed edge labeled graph H m (ξ). The graded multiplicities [V (ξ) : D(m, sω)] q can then be described by natural polynomials associated to this graph. The graph H m (1 ) allows one to do computations with Sage to generate information on the graded multiplicities. But, even in the case when q = 1, where one can use the OEIS database it seems difficult to guess the appropriate generating series for the multiplicities in general. See however Subsection 3.10, where we discuss the level three Demazure flag of the local Weyl module.

3.4.
The following is now immediate from Proposition 2.5 and Theorem 3.3.
Proposition 3.5. Suppose that g is of type B n and that λ ∈ P + . Then we have an equality of characters ch gr W loc (λ) = r≥0 q r λ(hn)/2 λ(h n )/2 r q ch gr D(1, λ − rα n ), 3.5. The proof of Theorem 3.3 relies on a further result proved in [7] which we now recall. Given a partition ξ = (ξ 1 ≥ · · · ≥ ξ s > 0) with s parts we define an associated pair of partitions ξ ± as follows. If s = 1, then ξ + = ξ and ξ − is the empty partition. If s > 1, then we take and ξ + to be the unique partition associated to the s-tuple (ξ 1 , . . . , ξ s−2 , ξ s−1 + 1, ξ s − 1). Theorem 3.6. Let ξ = (ξ 1 ≥ · · · ≥ ξ s > 0) be a partition with s parts. For s > 1, there exists a short exact sequence of sl 2 [t]-modules 3.6. Define a partial order on the set of all partitions as follows: given partitions ξ and ξ with s and s parts respectively, we say that ξ > ξ if and only if s ≥ s and if s = s , then ξ s > ξ s (where ξ s , ξ s is the smallest part of ξ and ξ respectively). Note that ξ > ξ + > ξ − .
We first prove by an induction (given by the partial order) that if m ≥ ξ 1 , then V (ξ) has a Demazure flag of level m. If ξ = ( ), then Theorem 3.3(i) is trivially true by (3.4). Assume now that we have proved the result for all partitions with at most (s − 1) parts and that ξ is a partition with s parts. The inductive hypothesis applies to ξ ± provided that m ≥ ξ + 1 and hence τ s−1 V (ξ − ) and V (ξ + ) both have a Demazure flag of level m. Lemma 2.4 now shows that V (ξ) also admits a Demazure flag of level m.

3.7.
To complete the proof of part (i), we must show that if V (ξ) has a Demazure flag of level m, then m ≥ ξ 1 . This clearly follows from: Lemma 3.7. Suppose that r, m ∈ Z + and ξ is a partition. If D(m, rω) is a quotient of V (ξ), then r = |ξ| and m ≥ ξ 1 .

3.9.
We now prove part (ii) of Theorem 3.3. Since W loc (sω) ∼ = V (1 s ), we have and hence we get where the second equality follows from Lemma 3.8. Setting we get the following recurrence relation, p s,r (q) = q (s−r)/2 p s−2,r−2 (q) + q s−1 p s−2,r (q), s ≥ r ≥ 2, with the initial conditions p 0,0 (q) = p 1,1 (q) = 1 and p s,r (q) = 0 if r > s or (s − r) / ∈ 2Z + . This recurrence has a unique solution and a simple calculation shows that taking p s,s−2 = q s/2 s/2 q , satisfies the recurrence and the initial conditions and part (ii) is proved. We may regard part (ii) as giving the change of basis matrix (see Lemma 2.1) from the basis of local Weyl modules to the basis of level two Demazure modules. Part (iii) is then just given by the inverse of this matrix and is a straightforward calculation.
In the case when q = 1, the recurrences can be solved and we find or equivalently with initial conditions , Summarizing, we have Proposition 3.9. The module D(2, sω) has a flag by level three Demazure modules and the numerical multiplicity is given by Together with part (ii) of Theorem 3.3 this allows us to calculate the numerical multiplicities in the filtration of the local Weyl module by the level three Demazure modules and hence in turn the character of the local Weyl module when g is of type G 2 in terms of Demazure modules.
3.11. For arbitrary m ∈ Z + , one can set p s,r (q) = [D(m − 1, sω) : D(m, rω)] q and write down recurrence relations with initial conditions which are similar to those in (3.9). To define generating functions it is not enough to just index by the weight sω of the level m-Demazure module, we have to also keep track of the smallest part of the partition, i.e., the value of s 0 where s = (m − 1)s 1 + s 0 , 0 ≤ s 0 ≤ m − 2. This leads to a large number of dependent recurrence relations and makes solving even for the numerical multiplicities very difficult.
3.12. We conclude this section by explaining the connection of our work with Macdonald polynomials. We avoid introducing all the notation related to Macdonald polynomials, and refer the interested reader to [4, Section 3] for a quick exposition which is adequate for our purposes. It was shown in [16] for sl r and in [11] for an arbitrary simply-laced Lie algebra that the character of the Demazure module D(1, λ), λ ∈ P + in the level one representation of the associated untwisted affine Lie algebra, is given by the specialized Macdonald polynomial P w 0 (λ) , where w 0 is the longest element of the Weyl group of g. In the non-simply laced case however this is not necessarily true. In fact, by combining the recent work of [13] and of [15] we find that the character of W loc (λ) is given by the specialized Macdonald polynomial P w 0 (λ) (q). In other words, the character of W loc (λ) is always given by P w 0 λ (q). Proposition 3.5 can therefore be regarded as giving the character of the Demazure module in a level one representation of the affine algebra B

The graph H (ξ)
The goal of this section is to give a combinatorial way to calculate the multiplicities of a specific Demazure module occurring in a Demazure flag of V (ξ) of level . To do this we define a graph H (ξ). The vertices of the graph are a finite subset of Z 3 + and are in bijective correspondence with the set of partitions obtained by an iteration of the operations ξ ± (see Theorem 3.6) with the restriction that the maximal part of the partition never exceeds . The edges of the graph are defined to match up with the short exact sequence in Theorem 3.6 and have suitable labels. The sinks of the graph correspond to the Demazure modules of level which could occur with non-zero multiplicity in the flag. The graded multiplicity is then obtained by summing over all "weighted paths" from the vertex in H (ξ) corresponding to ξ and the vertex corresponding to the desired Demazure module.
4.1. Given integers s, r ∈ N let res r s be the remainder if s is not divisible by r and r otherwise. Writing s = (res r s) + rp for some p ≥ 0, we see that s ≥ r + 1 =⇒ p ≥ 1 =⇒ s ≥ 2 res r s. (4.1) For ∈ N and a partition ξ = ( It is convenient to introduce the following notation: for i = (i, j, k) ∈ V (ξ), let r(i), s(i) ∈ Z + be defined by Lemma 4.1. Let ξ = 1 b 1 2 b 2 · · · b and i = (i, j, k) ∈ V (ξ) be such that either k < or k = and i − j ≥ .
Proof . The proofs of part (i) and (ii) are identical. Notice that if Otherwise, the choice of (i, j, k) forces k < and so k ≤ i − jk < 2k. Then resp. by using equation (4.1), i.e.

Set
where m = m(i) ± is as in Lemma 4.1. The graph H (ξ) is defined to be the directed graph with vertices V (ξ) and directed edges are i → i ± if i = (i, j, k) is such that either k < or k = and i − j ≥ . In particular, the elements (i, i/ , ) are precisely the sinks of the graph. The label of i → i + is zero while the label of i → i − is

4.3.
Let q be an indeterminate. Given vertices i, j ∈ V (ξ) define a polynomial p i,j (q) as follows. Set p i,i = 1. If i is not a sink, set p i,i + = 1, p i,i − = q (i) . Given a directed path p from (i = i 0 → i 1 → · · · → i s = j) with labels a 1 , . . . , a s set r(p) = s m=1 a m and define where the sum is over all directed paths from i to j. In all other cases we take p i,j (q) = 0. The following is immediate:

4.4.
We give an example. Suppose that = 3 and ξ = 1 8 . Then H 3 (1 8 ) is the following edge labeled directed graph. For convenience we have color coded the edges to reflect the type of edge. Thus the green (resp. blue) arrow is an edge corresponding to i → i + coming from the first (resp. second) possibility in Lemma 4.1(i) while the black and red arrows correspond to the two possibilities for i → i − coming from Lemma 4.1(ii).

4.6.
To prove the rest of the theorem, we relate the vertices of the graph H (ξ) to partitions and the edges to the short exact sequence in Theorem 3.6. Given i = (i, j, k) ∈ V (ξ), Since i − jk ≥ k(1 − δ k, ) we have by equation (4.1) that s(i) ≥ (1 − δ k, ) and we assume this from now on. Let ξ(i) be the partition It is trivially checked that the assignment i → ξ(i) is one-one. Note also that ξ(i ξ ) = ξ, ξ(j r ) = r(i) r , | r, r , otherwise, where i ξ and j r are as defined in Theorem 4.3. In particular, V (ξ(j r )) ∼ = D( , rω).
If r(i) > 1, then it follows immediately that i − (j + 1)k is not divisible by (k − 1)and hence we get while if r(i) = 1, then we get that r(i + ) = k − 1, s(i + ) = s(i) − 2.
Since i − jk = res k−1 (i − jk) + k − 1 =⇒ i + b k k = (res k−1 (i − jk) − 1) + k(j + b k + 1), we see that i + b k k is not divisible by k and so 0 < res k (i + b k k) = res k−1 (i − jk) − 1 as needed. The proof that ξ(i) − = ξ(i − ) is a similar computation and we omit the details. Proof . We proceed by induction on the length of the longest path connecting i and j r . If i = j m for some m ≥ b , then the proposition is true V (ξ(j r )) ∼ = D( , rω), V (ξ(j m )) ∼ = D( , mω).
In other words induction begins when the length is zero. The inductive step follows by using using Theorem 3.