Modules with Demazure Flags and Character Formulae
In this paper we study a family of finite-dimensional graded representations of the current algebra of sl₂ 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 ℓ-integrable module for A₁¹ as long a...
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| Цитувати: | Modules with Demazure Flags and Character Formulae / V. Chari, L. Schneider, P. Shereen, J. Wand // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 16 назв. — англ. |
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irk-123456789-1468202019-02-12T01:23:04Z Modules with Demazure Flags and Character Formulae Chari, V. Schneider, L. Shereen, P. Wand, J. In this paper we study a family of finite-dimensional graded representations of the current algebra of sl₂ 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 ℓ-integrable module for A₁¹ as long as ℓ is large. We associate to each partition and to each ℓ an edge-labeled directed graph which allows us to describe in a combinatorial way the graded multiplicity of a given level ℓ-Demazure module in the filtration. In the special case of the partition 1s and ℓ=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 g-stable level one Demazure module associated to B¹n as an explicit combination of suitably specialized Macdonald polynomials. In the case of sl₂, 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. 2014 Article Modules with Demazure Flags and Character Formulae / V. Chari, L. Schneider, P. Shereen, J. Wand // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 16 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 06B15 ; 05E10; 14H42 DOI:10.3842/SIGMA.2014.032 http://dspace.nbuv.gov.ua/handle/123456789/146820 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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In this paper we study a family of finite-dimensional graded representations of the current algebra of sl₂ 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 ℓ-integrable module for A₁¹ as long as ℓ is large. We associate to each partition and to each ℓ an edge-labeled directed graph which allows us to describe in a combinatorial way the graded multiplicity of a given level ℓ-Demazure module in the filtration. In the special case of the partition 1s and ℓ=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 g-stable level one Demazure module associated to B¹n as an explicit combination of suitably specialized Macdonald polynomials. In the case of sl₂, 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. |
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Chari, V. Schneider, L. Shereen, P. Wand, J. |
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Chari, V. Schneider, L. Shereen, P. Wand, J. Modules with Demazure Flags and Character Formulae Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Chari, V. Schneider, L. Shereen, P. Wand, J. |
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Chari, V. |
| title |
Modules with Demazure Flags and Character Formulae |
| title_short |
Modules with Demazure Flags and Character Formulae |
| title_full |
Modules with Demazure Flags and Character Formulae |
| title_fullStr |
Modules with Demazure Flags and Character Formulae |
| title_full_unstemmed |
Modules with Demazure Flags and Character Formulae |
| title_sort |
modules with demazure flags and character formulae |
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Інститут математики НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/146820 |
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Modules with Demazure Flags and Character Formulae / V. Chari, L. Schneider, P. Shereen, J. Wand // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 16 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT chariv moduleswithdemazureflagsandcharacterformulae AT schneiderl moduleswithdemazureflagsandcharacterformulae AT shereenp moduleswithdemazureflagsandcharacterformulae AT wandj moduleswithdemazureflagsandcharacterformulae |
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2025-07-11T00:41:15Z |
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2025-07-11T00:41:15Z |
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1837309094567870464 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 032, 16 pages
Modules with Demazure Flags
and Character Formulae?
Vyjayanthi CHARI, Lisa SCHNEIDER, Peri SHEREEN and Jeffrey WAND
Department of Mathematics, University of California, Riverside, CA 92521, USA
E-mail: chari@math.ucr.edu, lschn005@ucr.edu, psher001@ucr.edu, wand@math.ucr.edu
Received October 22, 2013, in final form March 17, 2014; Published online March 27, 2014
http://dx.doi.org/10.3842/SIGMA.2014.032
Abstract. In this paper we study a family of finite-dimensional graded representations of
the current algebra of sl2 which are indexed by partitions. We show that these represen-
tations admit a flag where the successive quotients are Demazure modules which occur in
a level `-integrable module for A1
1 as long as ` is large. We associate to each partition and
to each ` an edge-labeled directed graph which allows us to describe in a combinatorial way
the graded multiplicity of a given level `-Demazure module in the filtration. In the special
case of the partition 1s and ` = 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 g-
stable level one Demazure module associated to B1
n as an explicit combination of suitably
specialized Macdonald polynomials. In the case of sl2, 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.
Key words: Demazure flags; Demazure modules; theta series
2010 Mathematics Subject Classification: 06B15 ; 05E10; 14H42
1 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 algebra ĝ 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 in ĝ.
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 A1
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 A1
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
mailto:chari@math.ucr.edu
mailto:lschn005@ucr.edu
mailto:psher001@ucr.edu
mailto:wand@math.ucr.edu
http://dx.doi.org/10.3842/SIGMA.2014.032
http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html
2 V. Chari, L. Schneider, P. Shereen and J. Wand
result proved in [14] showed that Joseph’s proof worked worked for A1
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 A1
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 Bn. 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 sl2 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 Bn. Together with the results
of [13] we can give the character of the Demazure module for B1
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.
2 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] ⊗ fg, 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 (a1⊗ tr1) · · · (as⊗ trs) has
grade r1 + · · ·+ rs.
A graded representation of a[t] is a Z-graded vector space which admits a compatible Lie
algebra action of a[t], i.e.,
V =
⊕
r∈Z
V [r], (a⊗ ts)V [r] ⊂ V [r + s], r ∈ Z, s ∈ Z+.
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 τr be the grading shift operator: if V is a graded a[t]-
module, then τrV 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⊗ tr)m = δr,0am where a ∈ a and r ∈ Z+ and δr,0 is the Kronecker delta symbol. We denote
this module as ev0M and observe that ev0M is a graded a[t]-module with (ev0M) [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 dα = 2
(α,α) .
Modules with Demazure Flags and Character Formulae 3
Let I = {1, . . . , n} and fix a set {αi : i ∈ I} of simple roots for R and let {ωi : i ∈ I} ⊂ h∗
be the set of fundamental weights. Let Q (resp. Q+) be the Z span (resp. the Z+ span) of
{αi : i ∈ I} and similarly define P (resp. P+) to be the Z (resp. Z+) span of {ωi : 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 , hi : α ∈ R+, i ∈ I} be a Chevalley basis for g and set x±i = x±αi for i ∈ I.
2.3. For µ ∈ P+, let V (µ) be the irreducible finite-dimensional g-module generated by an
element vµ with defining relations
x+i vµ = 0, hivµ = µ(hi)vµ, (x−i )µ(hi)+1vµ = 0, i ∈ I.
It is well-known that any finite-dimensional g-module V is isomorphic to a direct sum of irre-
ducible modules V (µ), µ ∈ P+. Further, we may write
V =
⊕
ν∈P
Vν , Vν = {v ∈ V : hv = ν(h)v, h ∈ h},
and we set wtV = {ν ∈ P : Vν 6= 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
chg(V ) =
∑
µ∈P
dimVµe(µ).
Since wtV (µ) ⊂ µ − Q+, it follows that the characters chg V (µ), µ ∈ P+ form a linearly
independent subset of Z[P ]. Given a graded g[t]-module and an indeterminate q, we let
chgr(V ) =
∑
r≥0
chg V [r]qr ∈ Z[P ][q].
2.4. For λ ∈ P+, the local Weyl moduleWloc(λ) is the g[t]-module generated by an element wλ
with defining relations, for i ∈ I and s ∈ Z+,
(x+i ⊗C[t])wλ = 0, (hi ⊗ ts)wλ = λ(hi)δs,0wλ, (x−i ⊗ 1)λ(hi)+1wλ = 0.
It is trivial to see that wtWloc(λ) ⊂ λ−Q+. The local Weyl module is clearly Z+-graded once
we declare the grade of wλ to be zero, and
Wloc(λ)[0] ∼=g V (λ), chgrWloc(λ) = chg V (λ) +
∑
(µ,r)∈P+×N
µ<λ
dim Homg(V (µ),Wloc(λ)[r])qr.
Finally, ev0 V (λ) is the unique graded irreducible quotient of Wloc(λ) and any irreducible graded
finite-dimensional module is isomorphic to τs ev0 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 Wloc(λ) by the submodule generated by
the elements
{(x−α ⊗ tp)r+1wλ : p ∈ Z+, r ≥ max{0, λ(hα)− dα`p}, for α ∈ R+}. (2.1)
By abuse of notation, we continue to denote the image of wλ in D(`, λ) by wλ. Notice that for
all α ∈ R+, p ≥ 1, (x−α ⊗ tp)wλ = 0 if ` is such that ` ≥ λ(hα).
Again, we have
D(`, λ)[0] ∼=g V (λ), (2.2)
4 V. Chari, L. Schneider, P. Shereen and J. Wand
chgrD(`, λ) = chg V (λ) +
∑
(µ,r)∈P+×N
µ<λ
dim Homg(V (µ), D(`, λ)[r])qr.
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.,
r < max{0, λ(hα)− dα`p} =⇒ 0 6= (x−α ⊗ tp)r+1wλ ∈ D(`, λ). (2.3)
2.6. The following is now clear.
Lemma 2.1. The elements of the set {chgrWloc(λ) : λ ∈ P+} (resp. {chgrD(`, λ) : λ ∈ P+}, for
a fixed ` ∈ N) are a Z[q]-basis for the subspace of Z[P ][q] spanned by {chgV (λ) : λ ∈ P+}.
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 Wloc(λ) and the Demazure modules
D(1, λ). It was established in [6] for sl2. This was then used in [5] and [10] to prove the result
for slr+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].
Theorem 2.2. Let λ ∈ P+.
(i) Assume that either g is simply-laced or that λ(hi) = 0 for all i ∈ I such that αi is short.
We have an isomorphism of g[t]-modules,
Wloc(λ) ∼= D(1, λ).
(ii) Let λ(hi) 6= 0 for some i ∈ I with αi a short root. Then we have a decreasing sequence of
graded g[t]-submodules
Wloc(λ) = W0 ⊃W1 ⊃ · · · ⊃Wk ⊃Wk+1 = 0,
such that Wj/Wj+1
∼= τsjD(1, µj) for some (µj , sj) ∈ P+ × Z+, 0 ≤ j ≤ k.
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 sln+1. For ` ≥ 1, the module Wloc(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 A1
1. Our paper
gives a direct proof in the case of Demazure modules of affine sl2. 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
F(V ) = (V = V0 ⊇ V1 ⊇ · · · ⊇ Vk ⊇ Vk+1 = 0),
Modules with Demazure Flags and Character Formulae 5
of graded g[t]-submodules of V , a Demazure flag F(V ) of level `, if
Vj/Vj+1
∼= τsjD(`, µj), (µj , sj) ∈ P+ × Z+, 0 ≤ j ≤ k.
Let [F(V ) : τsD(`, µ)] be the number of times τsD(`, µ) occurs in this flag. The graded multi-
plicity of D(`, µ) in this flag is defined to be
[F(V ) : D(`, µ)]q =
∑
s≥0
[F(V ) : τsD(`, µ)]qs.
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.
Lemma 2.4. Let V be a graded g[t]-module.
(i) Suppose that U is a graded g[t]-submodule of V such that U and V/U both admit a De-
mazure flag of level `. Then V also admits a Demazure flag of level ` and
[V : D(`, µ)]q = [U : D(`, µ)]q + [V/U : D(`, µ)]q.
(ii) If V admits a Demazure flag of level `, we have an equality of graded characters
chgrV =
∑
µ∈P+
[V : D(`, µ)]qchgrD(`, µ).
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 Bn and that ` = 2 or that g is of type G2 and ` = 3.
Let λ ∈ P+ be such that λ(hn) = r ∈ Z+ where αn is the unique short root. Then, Wloc(λ) has
a Demazure flag of level one and for µ ∈ P+,
[Wloc(λ)) : D(1, µ)]q =
{
[W sl2
loc (λ(hn)) : Dsl2(`, µ(hn))]q, (λ− µ) ∈ Z+αn,
0, otherwise.
Here W sl2
loc (λ(hn)) is the local Weyl module for sl2 of weight λ(hn) and Dsl2(`, µ(hn)) is defined
similarly.
We shall revisit this result later in the paper and give a closed formula for the graded mul-
tiplicities when g is of type Bn and generating series for the numerical multiplicities (q = 1)
when g is of type G2. We remark that the analogous result when g is of type F4 or Cn is also
proved in [15]: here one takes ` = 2 and replaced sl2 by sl3 and sln−1 respectively.
3 The modules V (ξ) and Demazure flags
In this section we shall, for the most part, be interested only in the case when g is sl2. 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 sl2.
3.1. Recall that a partition with s parts is a finite decreasing sequence of positive integers
ξ = (ξ1 ≥ ξ2 ≥ · · · ≥ ξs > 0) and that |ξ| =
∑
r≥1
ξr. We shall sometimes also write ξ as the
sequence 1b12b2 · · · sbs , where bj is the number of times the integer j occurs in ξ.
6 V. Chari, L. Schneider, P. Shereen and J. Wand
Following [7], given a partition ξ we define a sl2[t]-module V (ξ) as follows. It is the sl2[t]-
module generated by an element vξ with defining relations:
(x⊗C[t])vξ = 0, (h⊗ ts)vξ = |ξ|δs,0vξ, (3.1)
(y ⊗ 1)|ξ|+1vξ = 0, (3.2)
(x⊗ t)s(y ⊗ 1)s+rvξ = 0, (3.3)
where s ∈ Z+ in (3.1) and (3.3) holds for all s, r ∈ N for which there exists k ∈ N with
s + r ≥ 1 + rk +
∑
j≥k+1
ξj . It is clear that V (ξ) is a graded sl2[t]-module once we set the grade
of vξ to be zero, and that we have a surjective map of Wloc(|ξ|ω)→ V (ξ) of graded sl2[t]-modules.
Observe also that if ξ is the empty partition then V (ξ) is the trivial sl2[t]-module.
The following was proved in [7, Proposition 2.7] and will be needed later.
Lemma 3.1. For all k ∈ Z+, we have(
y ⊗ tk
)r+1
vξ = 0, r ≥
∑
j≥k+1
ξj .
The structure of the modules V (ξ) were studied in great detail in [7] for simple Lie algebras.
In the case of sl2, these modules are known to be the fusion product of simple irreducible finite-
dimensional representations with dimensions ξi+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
ξj . It is clear that
s+ r ≥ |ξ|+ 1 and hence (3.3) is a consequence of (3.2), i.e.,
V (1`) ∼= Wloc(`ω).
If ξ = (`) and if we take s = r = 1, we find that (3.3) holds for k = 1, and so,
(x⊗ t)(y ⊗ 1)2vξ = 0.
Using the commutation relations in U(sl2[t]) and the defining relations of V (ξ) it is simple to
prove that (y ⊗ t)vξ = 0. In turn, this implies that
(sl2 ⊗ tC[t])vξ = 0, i.e. V (ξ) ∼= ev0 V (`ω), (3.4)
as sl2[t]-modules. More generally, the following was proved in [7, Theorem 2].
Theorem 3.2. Suppose that ξ = sm` for some m, s ∈ N with 0 < s ≤ m and ` ∈ Z+.
Then V (ξ) is isomorphic to the quotient of Wloc(λ) (where λ = (m`+ s)ω) obtained by imposing
the additional relations(
y ⊗ t`+1
)
wλ = 0, (y ⊗ t`)s+1wλ = 0, s < m.
Moreover we have an isomorphism of sl2[t]-modules,
V (ξ) ∼= D(m,λ).
3.3. Given m, ` ∈ N with m ≥ `, set[
m
`
]
q
=
(1− qm)(1− qm−1) · · · (1− qm−`+1)
(1− q`)(1− q`−1) · · · (1− q)
.
The main result of this section is the following.
Modules with Demazure Flags and Character Formulae 7
Theorem 3.3.
(i) For all ξ = (ξ1 ≥ ξ2 ≥ · · · ≥ ξs > 0) and m ∈ N the module V (ξ) has a Demazure flag of
level m if and only if m ≥ ξ1. In particular, a level k-Demazure module has a Demazure
flag of level m if and only if m ≥ k.
(ii) The local Weyl module Wloc(sω) has a level two Demazure flag and
[Wloc(sω) : D(2, rω)]q =
q
pds/2e
[
bs/2c
p
]
q
, r = s− 2p ≥ 0,
0, otherwise.
(iii) We have an equality of characters
chgrD(2, sω) =
∑
p≥0
(−1)pqp(ds/2e−(p−1)/2)
[
bs/2c
p
]
q
chgrWloc((s− 2p)ω).
Remark 3.4. Recall from equation (2.2) thatD(m, sω) ∼= ev0 V (sω) for allm ≥ s. In particular,
if we take m = |ξ|, a level m-Demazure flag is just a Jordan–Holder series of V (ξ). In the case
when ξ = 1m the Jordan–Holder series is known (see [5]) and that
[Wloc(mω) : ev0 V ((m− 2`)ω)]q =
[
m
`
]
q
−
[
m
`− 1
]
q
.
In the next section, we associate to each partition ξ and an integer m ≥ ξ1, a directed edge
labeled graph Hm(ξ). The graded multiplicities [V (ξ) : D(m, sω)]q can then be described by
natural polynomials associated to this graph. The graph Hm(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 Bn and that λ ∈ P+. Then we have an equality of
characters
chgrWloc(λ) =
∑
r≥0
qrdλ(hn)/2e
[
bλ(hn)/2c
r
]
q
chgrD(1, λ− rαn),
chgrD(1, λ) =
∑
s≥0
(−1)sqs(dλ(hn)/2e−(s−1)/2)
[
bλ(hn)/2c
s
]
q
chgrWloc(λ− sαn).
3.5. The proof of Theorem 3.3 relies on a further result proved in [7] which we now recall.
Given a partition ξ = (ξ1 ≥ · · · ≥ ξs > 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
ξ− = (ξ1 ≥ ξ2 ≥ · · · ≥ ξs−2 ≥ ξs−1 − ξs ≥ 0),
and ξ+ to be the unique partition associated to the s-tuple (ξ1, . . . , ξs−2, ξs−1 + 1, ξs − 1).
Theorem 3.6. Let ξ = (ξ1 ≥ · · · ≥ ξs > 0) be a partition with s parts. For s > 1, there exists
a short exact sequence of sl2[t]-modules
0→ τ(s−1)ξsV (ξ−)
ϕ−−→ V (ξ)
ϕ+
−→V (ξ+)→ 0,
and
ϕ+(vξ) = vξ+ , ϕ−(vξ−) =
(
y ⊗ ts−1
)ξsvξ.
8 V. Chari, L. Schneider, P. Shereen and J. Wand
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
ξs > ξ′s (where ξs, 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 ≥ ξ1, 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 τs−1V (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.
It remains to consider the case when m < ξ+1 which can only happen when m = ξ1 and
ξ1 = ξs−1, i.e., ξ = (ξs−11 , ξs). But now, Theorem 3.2 implies that D(m, |ξ|ω) ∼= V (ξ) and there
is nothing to prove. We note the following consequence of our discussion:
[V (ξ) : D(m, rω)]q = [V (ξ+) : D(m, rω)]q + q(s−1)ξs [V (ξ−) : D(m, rω)]q. (3.5)
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 ≥ ξ1. 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 ≥ ξ1.
Proof. Since hvξ = |ξ|vξ, and D(m, rω) is a quotient of V (ξ) we must have r = |ξ| and the
element vξ maps to the the generator wrω of D(m, rω). To prove that m ≥ ξ1 we see from
Lemma 3.1 that
(y ⊗ t)
1+
∑
j≥2
ξj
vξ = 0,
and hence we must have
(y ⊗ t)
1+
∑
j≥2
ξj
wrω = 0.
Using (2.3), we see that this means that∑
j≥2
ξj ≥ |ξ| −m,
which forces m ≥ ξ1 as required. �
3.8. To prove part (ii) and for later use we establish the following result. Note that
[V (ξ) : D(m, rω)]q 6= 0 =⇒ (|ξ| − r) ∈ 2Z+.
Lemma 3.8. Given a partition ξ = (ξ1 ≥ ξ2 ≥ · · · ≥ ξs > 0) with ξ1 = m and s ≥ 2, we have
[V (ξ) : D(m, rω)]q = q(|ξ|−r)/2[V (ξ(m)) : D(m, (r −m)ω)]q,
where ξ(m) = (ξ2 ≥ · · · ≥ ξs > 0).
Proof. The proof is by an induction very similar to the one used in Subsection 3.6. If ξs−1 = m,
then by Theorem 3.2 we have
V (ξ) ∼= D(m, |ξ|ω), V (ξ(m)) ∼= D(m, (|ξ| −m)ω),
Modules with Demazure Flags and Character Formulae 9
and there is nothing to prove. In particular induction begins at s = 2 and we may assume also
that ξs−1 < m. Since ξ− = (m ≥ ξ2 ≥ · · · ≥ ξs−1 − ξs) and ξ+ is the partition associated to
(m, ξ2, . . . , ξs−1 + 1, ξs − 1) we get
ξ±(m) = ξ(m)±.
Equation (3.5) gives
[V (ξ(m)) : D(m, rω)]q = [V (ξ+(m)) : D(m, rω)]q + q(s−2)ξs [V (ξ−(m)) : D(m, rω)]q. (3.6)
Applying the induction hypothesis to ξ±, and noting that |ξ| = |ξ+|, we get
[V (ξ−) : D(m, rω)]q = q(|ξ
−|−r)/2[V (ξ−(m)) : D(m, (r −m)ω)]q, (3.7)
[V (ξ+) : D(m, rω)]q = q(|ξ|−r)/2[V (ξ+(m) : D(m, (r −m)ω)]q. (3.8)
Substituting (3.7) and (3.8) in (3.5), recall that |ξ−| = |ξ| − 2ξs and then use (3.6) to complete
induction. �
3.9. We now prove part (ii) of Theorem 3.3. Since Wloc(sω) ∼= V (1s), we have
ξ = 1s, ξ+ = 21s−2, ξ− = 1s−2,
and hence we get
[Wloc(sω) : D(2, rω)]q = [V (ξ+) : D(2, rω)]q + qs−1[Wloc((s− 2)ω) : D(2, rω)]q
= q(s−r)/2[Wloc((s− 2)ω) : D(2, (r − 2)ω)]q + qs−1[Wloc((s− 2)ω) : D(2, rω)]q,
where the second equality follows from Lemma 3.8. Setting
ps,r(q) = [Wloc(sω) : D(2, rω)]q,
we get the following recurrence relation,
ps,r(q) = q(s−r)/2ps−2,r−2(q) + qs−1ps−2,r(q), s ≥ r ≥ 2,
with the initial conditions p0,0(q) = p1,1(q) = 1 and ps,r(q) = 0 if r > s or (s− r) /∈ 2Z+. This
recurrence has a unique solution and a simple calculation shows that taking
ps,s−2` = q`ds/2e
[
bs/2c
`
]
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.
3.10. We make some comments about the general case. Consider an arbitrary partition
ξ = (ξ1 ≥ · · · ≥ ξs ≥ 0) and suppose that we are interested in computing the graded multiplicity
of D(m, rω) for some m ≥ ξ1 and r ∈ Z+. One way to calculate this is to use Theorem 3.6
repeatedly. In the first iteration, one uses it to calculate the multiplicities of the Demazure
flag of level ξ1, then one computes the multiplicity of D(ξ1 + 1, rω) in a Demazure flag of level
(ξ1 + 1) in D(ξ1, sω) and so on. In other words, we have, that [V (ξ) : D(m, rω)]q is equal to∑
p∈Zm−ξ1+
[V (ξ) : D(ξ1, p1ω)]q[D(ξ1, p1ω) : D(ξ1 +1, p2ω)]q · · · [D(m−1 : pm−ξ1ω) : D(m, rω)]q.
10 V. Chari, L. Schneider, P. Shereen and J. Wand
This brings us to the natural question of calculating the multiplicity occurring in a Demazure
flag of D(m − 1, sω) of level m. In principle this can be done by using Lemma 3.8 to set up
(m− 1) recursive formulae. We illustrate this in the case of m = 3 which is of interest to us and
already shows the complexity of the recursions. Set
ps,r(q) = [D(2, sω) : D(3, rω)]q,
and note that ps,r = 0 if s < r and also if s and r do not have the same parity. We shall use
this freely. Writing s = 2s1 + s0, 0 ≤ s0 ≤ 1 and using Lemma 3.8 and Theorem 3.6, we get
ps,r(q) =
q(s−r)/2ps−3,r−3(q) + q(s1+s0−1)(2−s0)ps−4+2s0,r(q), s ≥ r ≥ 3,
q(s1+s0−1)(2−s0)ps−4+2s0,r(q), s ≥ 3, r = 0, 1, 2,
1, (r, s) ∈ {(0, 0), (1, 1), (2, 2)},
0, otherwise.
(3.9)
Setting
P2r(q, u) =
∑
s≥0
p2s,2r(q)u
s, P2r+1(q, u) =
∑
s≥0
p2s+1,2r+1(q)u
s+1, r ≥ 0,
we get the following equality of generating functions,
qrP2r(q, u) = uqP2r−3(q, qu) + u2qr+2P2r
(
q, uq2
)
, r ≥ 2,
qrP2r+1(q, u) = u2qP2r−2(q, uq) + uqrP2r+1(q, uq), r ≥ 1,
and
P0(q, u) =
∑
s≥0
q2s
2
u2s, P1(q, u) =
∑
s≥0
q
s(s+1)
2 us+1, P2(q, u) =
∑
s≥0
q2s(s+1)u2s+1.
If we define the partial theta series (see [1] for instance) as
Θ(q, u) =
∑
s≥0
qs
2
us,
then note that
P0(q, u) = Θ
(
q2, u2
)
, P2(q, u) = uΘ
(
q2, q2u2
)
,
P1(q, u) = uΘ
(
q2, qu2
)
+ qu2Θ
(
q2, q3u2
)
.
In the case when q = 1, the recurrences can be solved and we find(
1− u2
)
P2r = uP2r−3, (1− u)P2r+1 = u2P2r−2,
or equivalently
Pr(u) =
u3
(1− u)(1− u2)
Pr−6, r ≥ 6,
with initial conditions
P0 =
1
1− u2
, P1 =
u
1− u
, P2 =
u
1− u2
,
P3 =
u2
(1− u)(1− u2)
, P4 =
u2
(1− u)(1− u2)
, P5 =
u3
(1− u)(1− u2)
.
Summarizing, we have
Modules with Demazure Flags and Character Formulae 11
Proposition 3.9. The module D(2, sω) has a flag by level three Demazure modules and the
numerical multiplicity is given by
[D(2, sω) : D(3, rω)] = coefficient of ur in
(
u3r0Pr1
(1− u)r0(1− u2)r0
)
,
r = 6r0 + r1, 0 ≤ r1 < 6.
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 G2 in terms of Demazure modules.
3.11. For arbitrary m ∈ Z+, one can set ps,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 s0
where s = (m−1)s1 +s0, 0 ≤ s0 ≤ 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 slr 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 Pw0(λ),
where w0 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 Wloc(λ) is given by the specialized Macdonald polynomial Pw0(λ)(q). In
other words, the character of Wloc(λ) is always given by Pw0λ(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
(1)
n explicitly as a combination of Macdonald polynomials.
4 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 Z3
+ and are in bijective correspon-
dence 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 resr s be the remainder if s is not divisible by r and r
otherwise. Writing s = (resr s) + rp for some p ≥ 0, we see that
s ≥ r + 1 =⇒ p ≥ 1 =⇒ s ≥ 2 resr s. (4.1)
For ` ∈ N and a partition ξ = (1b12b2 · · · ) where bs = 0 if s > ` set
V`(ξ) =
{
(i, j, k) ∈ Z3
+ : i ≤
k−1∑
m=1
mbm, i− jk ≥ k(1− δk,`), 2 ≤ k ≤ `
}
.
12 V. Chari, L. Schneider, P. Shereen and J. Wand
It is convenient to introduce the following notation: for i = (i, j, k) ∈ V`(ξ), let r(i), s(i) ∈ Z+
be defined by
r(i) = resk−1(i− jk), i− jk = r(i) + s(i)(k − 1).
Lemma 4.1. Let ξ = 1b12b2 · · · `b` and i = (i, j, k) ∈ V`(ξ) be such that either k < ` or k = `
and i− j` ≥ `.
(i) Either (i, j + 1, k) ∈ V`(ξ) or, there exists m(i)+ ∈ Z+ with k ≤ m(i)+ < ` minimal such
that (i+ bmm, 0,m+ 1) ∈ V`(ξ) where m = m(i)+.
(ii) Either (i − 2r(i), j, k) ∈ V`(ξ) or there exists m(i)− ∈ Z+ with k ≤ m(i)− < ` minimal
such that (i− 2r(i) + bmm, 0,m+ 1) ∈ V`(ξ) where m = m(i)−.
Proof. The proofs of part (i) and (ii) are identical. Notice that if
i− jk ≥ k(2− δk,`) =⇒ i− (j + 1)k ≥ k(1− δk,`) =⇒ (i, j + 1, k) ∈ V`(ξ),
resp.
i− 2r(i)− jk ≥ k(1− δk,`) =⇒ (i− 2r(i), j, k) ∈ V`(ξ).
Otherwise, the choice of (i, j, k) forces k < ` and so k ≤ i− jk < 2k. Then
i+ b`−1(`− 1) ≥ 0 and i+ b`−1(`− 1) ≤
`−1∑
m=1
mbm,
resp. by using equation (4.1),
i− 2r(i) + b`−1(`− 1) ≥ jk ≥ 0, and i− 2r(i) + b`−1(`− 1) ≤
`−1∑
m=1
mbm),
i.e.
(i+ (`− 1)b`−1, 0, `) ∈ V`(ξ), resp. (i− 2r(i) + b`−1(`− 1), 0, `) ∈ V`(ξ).
Hence there exists a minimal k ≤ m = m(i)± < ` such that (i + bmm, 0,m + 1) ∈ V`(ξ) (resp.
(i− 2r(i) + bmm, 0,m+ 1) ∈ V`(ξ)) and the lemma is established. �
4.2. Set
i+ =
{
(i, j + 1, k), i− jk ≥ k(2− δk,`),
(i+mbm, 0,m+ 1), otherwise, m = m(i)+
and
i− =
{
(i− 2r(i), j, k), i− 2r(i)− jk ≥ k(1− δk,`),
(i− 2r(i) +mbm, 0,m+ 1), otherwise, m = m(i)−,
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, bi/`c, `) are precisely the sinks of the graph. The label
of i→ i+ is zero while the label of i→ i− is
`(i) = r(i) (j + bk + · · ·+ b` + s(i)) .
Modules with Demazure Flags and Character Formulae 13
(2, 0, 2)
(8, 0, 2) (8, 2, 2) (8, 3, 2)
(0,0,3)
(2, 0, 3)
(8, 0, 3) (8, 2, 3)
7
5
3
1
6
4
2
5
3
4
6
4
2
5
3
4
6
3
4
2
2
1
4
4
2
Figure 1. H3(18).
4.3. Let q be an indeterminate. Given vertices i, j ∈ V`(ξ) define a polynomial pi,j(q) as
follows. Set pi,i = 1. If i is not a sink, set pi,i+ = 1, pi,i− = q`(i). Given a directed path p from
(i = i0 → i1 → · · · → is = j) with labels a1, . . . , as set r(p) =
s∑
m=1
am and define
pi,j(q) =
∑
p
qr(p),
where the sum is over all directed paths from i to j. In all other cases we take pi,j(q) = 0. The
following is immediate:
Lemma 4.2. Given i, j ∈ V`(ξ), we have
pi,j = pi+,j + q`(i)pi−,j.
4.4. We give an example. Suppose that ` = 3 and ξ = 18. Then H3(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).
We give a few examples of pi,j(q) for i, j ∈ V3(1
8) which can be read off easily from the graph.
pi,j(q) =
1, i = (8, 0, 2), j = (8, 0, 3),
q15, i = (8, 0, 2), j = (2, 0, 2),
q10 + q8, i = (8, 2, 2), j = (0, 0, 3).
4.5. The main result of this section is the following.
Theorem 4.3. Given a partition ξ = 1b1 · · · sbs with bs = 0 if s > `, we have
[V (ξ) : D(`, rω)]q =
{
0, r < b``,
piξ,jr(q), iξ = (b1, 0, 2), jr = (r − b``,
⌊
r
`
⌋
− b`, `).
14 V. Chari, L. Schneider, P. Shereen and J. Wand
To illustrate Theorem 4.3 we give an example for [V (18) : D(3, 2ω)]q. In this case, iξ = (8, 0, 2)
and jr = (2, 0, 3). Thus,
[V (18) : D(3, 2ω)]q = p(8,0,2),(2,0,3) = q15 + q14 + q13 + q12 + q11 + q10 + q9 + q8 = q8[8]q.
The theorem is proved in the rest of the section. If ξ = m`b` for some 0 ≤ m < `, then
V (ξ) ∼= D(`,m + b``) and hence [V (ξ) : D(`, r)]q = 0 if r < b``. Otherwise, writing ξ = (ξ1 ≥
· · · ≥ ξs > 0) we see by using the definitions of ξ± that
b` = #{j : ξj = `} = #{j : ξ−j = `} ≤ #{j : ξ+j = `}.
Recall that
[V (ξ) : D(`, rω)]q = [V (ξ+) : D(`, rω)]q + q(s−1)ξs [V (ξ−) : D(`, rω)]q.
A similar induction used in Section 2 proves the assertion that
[V (ξ) : D(`, rω)]q = 0, r < b``.
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 − δk,`) we have by equation (4.1) that s(i) ≥ (1 − δk,`) and we assume this from
now on. Let ξ(i) be the partition
ξ(i) =
{
r(i)(k − 1)s(i)kj+bk(k + 1)bk+1 · · · `b` , i 6=
(
i, i` , `
)
,
`b`+
i
` , i =
(
i, i` , `
)
.
It is trivially checked that the assignment i→ ξ(i) is one-one. Note also that
ξ(iξ) = ξ, ξ(jr) =
{
r(i)`b
r
` c, ` 6 | r,
`
r
` , otherwise,
where iξ and jr are as defined in Theorem 4.3. In particular,
V (ξ(jr)) ∼= D(`, rω).
Lemma 4.4. Let i ∈ V`(ξ) and assume that either k < ` or k = ` and i− j` ≥ `. Then,
ξ(i)± = ξ(i±).
Proof. By definition, we have
ξ(i)+ = (r(i)− 1)(k − 1)s(i)−1kj+bk+1(k + 1)bk+1 · · · `b` .
Suppose first that i− jk ≥ k(2− δk,`) in which case s(i) ≥ 2− δk,` and i+ = (i, j + 1, k) and we
have
i− (j + 1)k = (r(i)− 1) + (s(i)− 1)(k − 1).
If r(i) > 1, then it follows immediately that i− (j + 1)k is not divisible by (k− 1)and hence we
get
r(i+) = r(i)− 1, s(i+) = s(i)− 1,
Modules with Demazure Flags and Character Formulae 15
while if r(i) = 1, then we get that
r(i+) = k − 1, s(i+) = s(i)− 2.
In either case, we have now proved the assertion of the lemma.
It remains to consider the case when k ≤ i − jk < 2k. We have that either s(i) = 2 and
r(i) = 1, i.e., i − jk = 2k − 1 or s(i) = 1. If s(i) = 2, then i ≥ 2k − 1 ≥ k + 1 and hence
i+ = (i+ bkk, 0, k + 1). Moreover,
i+ bkk = 1 + jk + 2(k − 1) + bkk, i.e., i+ bkk = k − 1 + k(j + bk + 1),
i.e. s(i+) = j + bk + 1, r(i+) = k − 1, and we have
ξ(i)+ = (k − 1)kj+bk+1(k + 1)bk+1 · · · `b` = ξ(i+).
Assume now that s(i) = 1. If resk−1(i− jk) = 1, then we get
ξ(i)+ = (resk k)kj+bk(k + 1)bk+1 · · · `b` .
By Lemma 4.1 there exists k ≤ m < ` minimal such that i + bmm ≥ (m + 1)(1 − δm+1,`). If
j > 0 then m = k, i+ = (i + bkk, 0, k + 1) and we see that ξ(i)+ = ξ(i+) while if j = 0 then
either k < m < ` minimal with bm > 0 or m = `. Noting that resk k = resm k, it follows again
that ξ(i+) = ξ(i+).
Finally, we must consider the case when s(i) = 1 and r(i) > 1. Here we have i− jk ≥ k + 1
and hence i+ = (i+ bkk, 0, k + 1). This time we just need to check that
r(i)− 1 = r(i+), s(i+) = bk + j + 1.
Since
i− jk = resk−1(i− jk) + k − 1 =⇒ i+ bkk = (resk−1(i− jk)− 1) + k(j + bk + 1),
we see that i+ bkk is not divisible by k and so 0 < resk(i+ bkk) = resk−1(i− jk)− 1 as needed.
The proof that ξ(i)− = ξ(i−) is a similar computation and we omit the details. �
4.7. Theorem 4.3 is clearly established if we prove the following
Proposition 4.5. For all i ∈ V`(ξ) and jr = (r− b``, b r` c− b`, `) ∈ V`(ξ), with r ≥ b``, we have
[V (ξ(i)) : V (ξ(jr))]q = pi,jr(q).
Proof. We proceed by induction on the length of the longest path connecting i and jr. If i = jm
for some m ≥ b``, then the proposition is true
V (ξ(jr)) ∼= D(`, rω), V (ξ(jm)) ∼= D(`,mω).
In other words induction begins when the length is zero. The inductive step follows by using
using Theorem 3.6, Lemma 4.4 and Lemma 4.2 which together give
[V (ξ(i)) : V (ξ(jr)]q = [V (ξ(i)+) : V (ξ(jr)]q + q`(i)[V (ξ(i)−) : V (ξ(jr)]q
= [V (ξ(i+)) : V (ξ(jr)]q + q`(i)[V (ξ(i−)) : V (ξ(jr)]q = pi+,jr + q`(i)pi−,jr = pi,jr . �
16 V. Chari, L. Schneider, P. Shereen and J. Wand
Acknowledgements
The authors thank S. Viswanath for discussions regarding the graph H`(ξ) and for drawing their
attention to the connection of the results of Subsection 3.10 to partial theta series. The first and
third authors acknowledge the hospitality and excellent working conditions at the Institute of
Mathematical Sciences, Chennai, India where part of this work was done. They also thank the
referees of the paper for their careful reading of the paper and for their many valuable comments.
The first author was partially supported by DMS-0901253 and DMS-1303052.
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1 Introduction
2 Local Weyl modules and Demazure modules of level
3 The modules V() and Demazure flags
4 The graph H()
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