Universal Verma modules and the Misra-Miwa Fock space

0.47 MB·English

by  Arun Ram

UNIVERSAL VERMA MODULES AND THE MISRA-MIWA FOCK SPACE
ARUN RAM AND PETER TINGLEY
Abstract. The Misra-Miwa v-deformed Fock space is a representation of the quantized
1 affinealgebraUv(s(cid:98)l(cid:96)). Ithasastandardbasisindexedbypartitionsandthenon-zeromatrix
1 entries of the action of the Chevalley generators with respect to this basis are powers of v.
0 Partitions also index the polynomial Weyl modules for U (gl ) as N tends to infinity. We
2 q N
explainhowthepowersofvwhichappearintheMisra-MiwaFockspacealsoappearnaturally

n in the context of Weyl modules. The main tool we use is the Shapovalov determinant for a
a universal Verma module.
J

0
2

]
A 1. Introduction
Q
Fock space is an infinite dimensional vector space which is a representation of several im-
.
h portant algebras, as described in, for example, [14, Chapter 14]. Here we consider the charge
t
a zero part of Fock space, which we denote by F, and its v-deformation Fv. The space F has a
m standard Q-basis {|λ(cid:105) | λ is a partition} and Fv := F⊗QQ(v). Following Hayashi [11], Misra
[ and Miwa [23] define an action of the quantized universal enveloping algebra Uv(s(cid:98)l(cid:96)) on Fv.
2 The only non-zero matrix elements (cid:104)µ|F¯i|λ(cid:105) of the Chevalley generators F¯i in terms of the
v standard basis occur when µ is obtained by adding a single¯i-colored box to λ, and these are
8
powers of v.
5
We show that these powers of v also appear naturally in the following context: Partitions
5
0 with at most N parts index polynomial Weyl modules ∆(λ) for the integral quantum group
2. UqA(glN). Let V be the standard N dimensional representation of UqA(glN). If the matrix
0 element (cid:104)µ|F |λ(cid:105) is non-zero then, for sufficiently large N, (cid:0)∆A(λ)⊗ V(cid:1)⊗ Q(q) contains
¯i A A
0
a highest weight vector of weight µ. There is a unique such highest weight vector v which
1 µ
: satisfies a certain triangularity condition with respect to an integral basis of ∆A(λ)⊗AV. We
v
i show that the matrix element (cid:104)µ|F¯i|λ(cid:105) is equal to vvalφ2(cid:96)(vµ,vµ), where (·,·) is the Shapovalov
X
form and val is the valuation at the cyclotomic polynomial φ .
φ 2(cid:96)
r 2(cid:96)
Our proof is computational, making use of the Shapovalov determinant [26, 9, 20]. This is a
a
formulaforthedeterminantoftheShapovalovformonaweightspaceofaVermamodule. The
necessarycomputationismosteasilydoneintermsoftheuniversalVermamodulesintroduced
in the classical case by Kashiwara [17] and studied in the quantum case by Kamita [15]. The
statement for Weyl modules is then a straightforward consequence.
Before beginning, let us discuss some related work. In [19], Kleshchev carefully analyzed
the gl highest weight vectors in a Weyl module for gl , and used this information to give
N−1 N
modular branching rules for symmetric group representations. Brundan and Kleshchev [6]
have explained that highest weight vectors in the restriction of a Weyl module to gl give
N−1
information about highest weight vectors in a tensor product ∆(λ)⊗V of a Weyl module with
Date: Feb 2, 2010.
AMS Subject Classifications: Primary 17B37; Secondary 20G42.
1
2 ARUNRAMANDPETERTINGLEY
the standard N-dimensional representation of gl . Our computations put a new twist on the
N
analysis of the highest weight vectors in ∆(λ) ⊗ V, as we study them in their “universal”
versions and by the use of the Shapovalov determinant. Our techniques can be viewed as an
application of the theory of Jantzen [12] as extended to the quantum case by Wiesner [28].
Brundan [5] generalized Kleshchev’s [19] techniques and used this information to give mod-
ular branching rules for Hecke algebras. As discussed in [2, 21], these branching rules are
reflected in the fundamental representation of s(cid:98)lp and its crystal graph, recovering much of
the structure of the Misra-Miwa Fock space. Using Hecke algebras at a root of unity, Ryom-
Hansen[25]recoveredthefullUv(s(cid:98)l(cid:96))actiononFockspace. Tocompletethepictureoneshould
constructagradedcategory, wheremultiplicationbyv inthes(cid:98)l(cid:96) representationcorrespondsto
agradingshift. RecentworkofBrundan-Kleshchev[7]andAriki[1]explainsthatonesolution
to this problem is through the representation theory of Khovanov-Lauda-Rouquier algebras
[18, 24]. It would be interesting to explicitly describe the relationship between their category
and the present work. Another related construction due to Brundan-Stroppel considers the
case when the Fock space is replaced by ∧mV ⊗∧nV, where V is the natural gl module and
∞
m,n are fixed natural numbers.
We would also like to mention very recent work of Peng Shan [27] which independently
develops a similar story to the one presented here, but using representations of a quantum
SchuralgebrawhereweuserepresentationsofU (gl ). Theapproachtakenthereissomewhat
ε N
different, and in particular relies on localization techniques of Beilinson and Bernstein [4].
This paper is arranged as follows. Sections 2 and 3 are background on the quantum group
U (gl ) and the Fock space F . Sections 4 and 5 explain universal Verma modules and
q N v
the Shapovalov determinant. Section 6 contains the statement and proof of our main result
relating Fock space and Weyl modules.
1.1. Acknowledgments. We thank M. Kashiwara, A. Kleshchev, T. Tanisaki, R. Virk and
B.Websterforhelpfuldiscussions. ThefirstauthorwaspartlysupportedbyNSFGrantDMS-
0353038 and Australian Research Council Grants DP0986774 and DP0879951. The second
author was partly supported by the Australia Research Council grant DP0879951 and NSF
grant DMS-0902649.
2. The quantum group U (gl ) and its integral form UA(gl )
q N q N
This is a very brief review, intended mainly to fix notation. With slight modifications the
construction in this section works in the generality of symmetrizable Kac-Moody algebras.
See [8, Chapters 6 and 9] for details.
2.1. The rational quantum group. U (gl ) is the associative algebra over the field of
q N
rational functions Q(q) generated by
(2.1) X ,...,X , Y ,...,Y , and L±1,...,L±1,
1 N−1 1 N−1 1 N
with relations
L L−1 −L L−1
L L = L L , L L−1 = L−1L = 1, X Y −Y X = δ i i+1 i+1 i ,
i j j i i i i i i j j i i,j q−q−1
 
qX , if i = j, q−1Y , if i = j,
 j  j
 
(2.2) L X L−1 = q−1X , if i = j +1, L Y L−1 = qY , if i = j +1,
i j i j i j i j
 
X otherwise; Y , otherwise;
j j
UNIVERSAL VERMA MODULES AND THE MISRA-MIWA FOCK SPACE 3
X X = X X and Y Y = Y Y , if |i−j| ≥ 2,
i j j i i j j i
X2X −(q+q−1)X X X +X X2 = Y2Y −(q+q−1)Y Y Y +Y Y2 = 0, if |i−j| = 1.
i j i j i j i i j i j i j i
The algebra U (gl ) is a Hopf algebra with coproduct and antipode given by
q N
∆(L ) = L ⊗L , S(L ) = L−1,
i i i i i
(2.3) ∆(X ) = X ⊗L L−1 +1⊗X , and S(X ) = −X L−1L ,
i i i i+1 i i i i i+1
∆(Y ) = Y ⊗1+L−1L ⊗Y , S(Y ) = −L L−1 Y ,
i i i i+1 i i i i+1 i
respectively (see [8, Section 9.1]).
As a Q(q)-vector space, U (gl ) has a triangular decomposition
q N
(2.4) U (gl ) ∼= U (gl )<0⊗U (gl )0⊗U (gl )>0,
q N q N q N q N
where the inverse isomorphism is given by multiplication (see [8, Proposition 9.1.3]). Here
U (gl )<0 is the subalgebra generated by the Y for i = 1,...,N −1, U (gl )>0 is the subal-
q N i q N
gebra generated by the X for i = 1,...,N −1, and U (gl )0 is the subalgebra generated by
i q N
the L±1 for i = 1,...,N.
i
2.2. The integral quantum group. Let A = Z[q,q−1]. For n,k ∈ Z and c ∈ Z, let
>0
qn−q−n xk (cid:20) x;c (cid:21) (cid:89)k xqc+1−s−x−1qs−1−c
(2.5) [n] := , x(k) := , and := ,
q−q−1 [k][k−1]···[2][1] k qs−q−s
s=1
in Q(q,x). The restricted integral form UA(gl ) is the A-subalgebra of U (gl ) generated by
q N q N
(cid:20) (cid:21)
L ;c
X(k),Y(k), L±1 and i for 1 ≤ i ≤ N,c ∈ Z,k ∈ Z . As discussed in [22, Section 6],
i i i k >0
this is an integral form in the sense that
(2.6) UA(gl )⊗ Q(q) = U (gl ).
q N A q N
As with U (gl ), the algebra UA(gl ) has a triangular decomposition
q N q N
(2.7) UA(gl ) ∼= UA(gl )<0⊗UA(gl )0⊗UA(gl )>0,
q N q N q N q N
where the isomorphism is given by multiplication (see [8, Proposition 9.3.3]). In this case,
UA(gl )<0 is the subalgebra generated by the Y(k), UA(gl )>0 is the subalgebra generated
q N i q N
(cid:20) (cid:21)
L ;c
by the X(k), and UA(gl )0 is generated by L±1 and i for 1 ≤ i ≤ N, c ∈ Z, and
i q N i k
k ∈ Z .
>0
2.3. Rational representations. The Lie algebra gl = M (C) of N × N matrices has
N N
standard basis {E | 1 ≤ i,j ≤ N}, where E is the matrix with 1 in position (i,j) and 0
ij ij
everywhere else. Let h = span{E ,E ,...,E }. Let ε ∈ h∗ be the weight of gl given by
11 22 NN i N
ε (E ) = δ . Define
i jj i,j
h∗ := {λ = λ ε +λ ε +···+λ ε ∈ h∗ | λ ,...,λ ∈ Z},
Z 1 1 2 2 N N 1 N
(h∗)+ := {λ = λ ε +λ ε +···+λ ε ∈ h∗ | λ ≥ λ ≥ ··· ≥ λ },
Z 1 1 2 2 N N Z 1 2 N
(2.8) P+ := {λ = λ1ε1+λ2ε2+···+λNεN ∈ (h∗Z)+ | λN ≥ 0},
R+ := {ε −ε | 1 ≤ i < j ≤ N},
i j
Q := span (R+), Q+ := span (R+), and Q− := span (R+).
Z Z Z
≥0 ≤0
4 ARUNRAMANDPETERTINGLEY
to be the set of integral weights, the set of dominant integral weights, the set of dominant
polynomial weights, the set of positive roots, the root lattice, the positive part of the root
lattice, and the the negative part of the root lattice, respectively.
For an integral weight λ = λ ε + ··· + λ ε , the Verma module M(λ) for U (gl ) of
1 1 N N q N
highest weight λ is
(2.9) M(λ) := U (gl )⊗ Q(q) ,
q N Uq(glN)≥0 λ
where Q(q) = span {v } is the one dimensional vector space over Q(q) with U (gl )≥0
λ Q(q) λ q N
action given by
(2.10) X ·v = 0 and L ·v = qλjv , for 1 ≤ i ≤ N −1, 1 ≤ j ≤ N.
i λ j λ λ
Theorem 2.1. (see[8, Chapter10.1]) If λ ∈ (h∗)+ then M(λ) has a unique finite dimensional
Z
quotient ∆(λ) and the map λ (cid:55)→ ∆(λ) is a bijection between (h∗)+ and the set of isomorphism
Z
classes of irreducible finite dimensional U (gl )-modules.
q N
A singular vector in a representation of U (gl ) is a vector v such that X ·v = 0 for all i.
q N i
2.4. Integralrepresentations. TheintegralVermamodule MA(λ)istheUA(gl )-submodule
q N
ofM(λ)generatedbyv . Theintegral Weyl module ∆A(λ)istheUA(gl )-submoduleof∆(λ)
λ q N
generated by v . Using (2.6) and (2.4),
λ
(2.11) MA(λ)⊗ Q(q) = M(λ), and ∆A(λ)⊗ Q(q) = ∆(λ).
A A
In general, ∆A(λ) is not irreducible as a UA(gl ) module.
q N
3. Partitions and Fock space
We now describe the v-deformed Fock space representation of Uv(s(cid:98)l(cid:96)) constructed by Misra
and Miwa [23] following work of Hayashi [11]. Our presentation largely follows [3, Chapter
10].
3.1. Partitions. A partition λ is a finite length non-increasing sequence of positive integers.
Associated to a partition is its Ferrers diagram. We draw these diagrams as in Figure 1 so
that, if λ = (λ ,...,λ ), then λ is the number of boxes in row i (rows run southeast to
1 N i
northwest (cid:45) ). Say that λ is contained in µ if the diagram for λ fits inside the diagram for µ
and let µ/λ be the collection of boxes of µ that are not in λ. For each box b ∈ λ, the content
c(b) is the horizontal position of b and the color c(b) is the residue of c(b) modulo (cid:96). In Figure
1, the numbers c(b) are listed below the diagram. The size |λ| of a partition λ is the total
number of boxes in its Ferrers diagram.
The set P+ of dominant polynomial weights from Section 2.3 is naturally identified with
partitions with at most N parts. If λ ∈ P+ then
∼ (cid:77)
(3.1) ∆(λ)⊗∆(ε ) = ∆(λ+ε )
1 k
1≤k≤N
λ+εk∈P+
as U (gl )-modules. The diagram of λ+ε is obtained from the diagram of λ by adding a
q N k
box on row k, and ∆(λ+ε ) appears in the sum on the right side of (3.1) if and only if λ+ε
k k
is a partition. See, for example, [10, Section 6.1, Formula 6.8] for the classical statement, and
[8, Proposition 10.1.16] for the quantum case.
UNIVERSAL VERMA MODULES AND THE MISRA-MIWA FOCK SPACE 5
(cid:64) (cid:0)(cid:64) (cid:0)(cid:64) (cid:0)
(cid:64)(cid:64)(cid:64)(cid:64)(cid:0)(cid:0)¯0(cid:64)(cid:64)(cid:0)¯2(cid:64)(cid:0)¯1(cid:64)(cid:0)(cid:0)(cid:64)(cid:0)¯0¯0(cid:64)(cid:0)(cid:64)(cid:64)(cid:0)¯2(cid:64)(cid:0)(cid:0)(cid:64)(cid:0)¯1¯1(cid:64)(cid:64)(cid:0)(cid:64)(cid:0)¯0¯0(cid:0)(cid:64)(cid:0)(cid:64)¯2¯2(cid:64)(cid:0)(cid:64)(cid:0)¯1(cid:0)(cid:64)(cid:0)¯0¯0(cid:64)(cid:64)(cid:0)(cid:64)(cid:0)¯2¯2(cid:0)(cid:64)(cid:0)(cid:64)¯1¯1(cid:64)(cid:0)(cid:64)(cid:0)¯0(cid:0)(cid:64)(cid:0)(cid:0)¯2(cid:64)(cid:0)(cid:64)(cid:0)(cid:0)
(cid:64)(cid:0)¯1 (cid:64)(cid:0)¯2 (cid:64)(cid:0)¯0(cid:64)(cid:0)¯1(cid:64)(cid:0) ¯2(cid:64)(cid:0)
(cid:64)(cid:0)¯0 (cid:64)(cid:0)¯1(cid:64)(cid:0)¯2(cid:64)(cid:0)¯0(cid:64)(cid:0)
(cid:64)(cid:0)¯2(cid:64)(cid:0)¯0(cid:64)(cid:0)¯1(cid:64)(cid:0)
(cid:64)(cid:0)¯1(cid:64)(cid:0)¯2(cid:64)(cid:0)
(cid:64)(cid:0)¯0(cid:64)(cid:0)
(cid:64)(cid:0)
9 8 7 6 5 4 3 2 1 0−1−2−3−4−5−6−7−8−9
Figure 1. The partition (7,6,6,5,5,3,3,1) with each box containing its color
for (cid:96) = 3. The content c(b) of a box b is the horizontal position of b reading
right to left. The contents of boxes are listed beneath the diagram so that c(b)
is aligned with all boxes b of that content.
3.2. The quantum affine algebra. Let Uv(cid:48)(s(cid:98)l(cid:96)) be the quantized universal enveloping alge-
bra corresponding to the (cid:96)-node Dynkin diagram
(cid:117)
(cid:8)(cid:72)
(cid:8) (cid:72)
(cid:8) (cid:72)
(cid:117)(cid:8)(cid:8)(cid:117) (cid:117) ... (cid:117)(cid:72)(cid:117)(cid:72)(cid:117)
More precisely, Uv(cid:48)(s(cid:98)l(cid:96)) is the algebra generated by E¯i,F¯i,K¯i±1, for¯i ∈ Z/(cid:96)Z, with relations
K −K−1
K K = K K , K K−1 = K−1K = 1, E F −F E = δ ¯i ¯i ,
¯i ¯j ¯j ¯i ¯i ¯i ¯i ¯i ¯i ¯j ¯j ¯i ¯i,¯j v−v−1
 
v2E , if¯i =¯j, v−2F , if¯i =¯j,
 ¯j  ¯j
(3.2) K¯iE¯jK¯i−1 = v−1E¯j, if¯i =¯j ±1, K¯iF¯jK¯i−1 = vF¯j, if¯i =¯j ±1,
 
E otherwise; F , otherwise;
¯j ¯j
E E = E E and F F = F F , if |¯i−¯j| ≥ 2,
¯i ¯j ¯j ¯i ¯i ¯j ¯j ¯i
E2E −(v+v−1)E E E +E E2 = F2F −(v+v−1)F F F +F F2 = 0, if |¯i−¯j| = 1.
¯i ¯j ¯i ¯j ¯i ¯j ¯i ¯i ¯j ¯i ¯j ¯i ¯j ¯i
See [8, Definition Proposition 9.1.1]. The algebra Uv(cid:48)(s(cid:98)l(cid:96)) is the quantum group corresponding
(cid:48)
to the non-trivial central extension s(cid:98)l(cid:96) = sl(cid:96)[t,t−1]⊕Cc of the algebra of polynomial loops in
sl .
(cid:96)
3.3. Fock space. Define v-deformed Fock space to be the Q(v) vector space F with basis
v
{|λ(cid:105) | λ is a partition}. Our F is only the charge 0 part of Fock space described in [16]. Fix
v
¯i ∈ Z/(cid:96)Z and partitions λ ⊆ µ such that µ/λ is a single box. Define
A (λ):={boxes b : b ∈/ λ,b has color¯i and λ∪b is a partition},
¯i
R (λ):={boxes b : b ∈ λ,b has color¯i and λ\b is a partition},
(3.3) ¯i
Nl(µ/λ):=|{b ∈ R (λ) :b to the left of µ/λ}|−|{b ∈ A (λ) : b to the left of µ/λ}|,
¯i ¯i ¯i
Nr(µ/λ):=|{b ∈ R (λ) : b to the right of µ/λ}|−|{b ∈ A (λ) : b to the right of µ/λ}|
¯i ¯i ¯i
tobethesetofaddable boxes of color¯i,thesetofremovable boxes of color¯i,theleft removable-
addable difference, and the right removable-addable difference, respectively.
6 ARUNRAMANDPETERTINGLEY
Theorem 3.1. (see [3, Theorem 10.6]) There is an action of Uv(cid:48)(s(cid:98)l(cid:96)) on Fv determined by
(3.4) E¯i|λ(cid:105) := (cid:88) v−N¯ir(λ/µ)|µ(cid:105) and F¯i|λ(cid:105) := (cid:88) vN¯il(µ/λ)|µ(cid:105),
c(λ/µ)=¯i c(µ/λ)=¯i
where c(λ/µ) denotes the color of λ/µ and the sum is over partitions µ which differ from λ
by removing (respectively adding) a single ¯i-colored box.
As a Uv(cid:48)(s(cid:98)l(cid:96))-module, Fv is isomorphic to an infinite direct sum of copies of the basic
representation V(Λ ). Using the grading of F where |λ(cid:105) has degree |λ|, the highest weight
0 v
vectors in F occur in degrees divisible by (cid:96), and the number of highest weight vectors in
v
degree (cid:96)k is the number of partitions of k. Then F ∼= V(Λ )⊗C[x ,x ,...], where x has
v 0 1 2 k
degree (cid:96)k, and Uv(cid:48)(s(cid:98)l(cid:96)) acts trivially on the second factor (see [16, Prop. 2.3]). Note that we
are working with the ‘derived’ quantum group Uv(cid:48)(s(cid:98)l(cid:96)), not the ‘full’ quantum group Uv(s(cid:98)l(cid:96)),
which is why there are no δ-shifts in the summands of F .
v
Comment 1. Comparingwith[3,Chapter10],ourNl(µ/λ)isequaltoAriki’s−Na(µ/λ)and
¯i ¯i
our Nr(µ/λ) is equal to Ariki’s −Nb(µ/λ). However, these numbers play a slightly different
¯i ¯i
role in Ariki’s work, which is explained by a different choice of conventions.
4. Universal Verma modules
The purpose of this section is to construct a family of representations which are universal
Verma modules in the sense that each can be “evaluated” to obtain any given Verma module.
ThisnotionwasdefinedbyKashiwara[17]intheclassicalcase,andwasstudiedinthequantum
case by Kamita [15].
4.1. Rational universal Verma modules. Let K := Q(q,z ,z ,...,z ). This field is iso-
1 2 N
morphic to the field of fractions of U (gl )0 via the map
q N
(4.1) ψ : U (gl )0 → K defined by ψ(L±1) = z±1.
q N i i
For each µ ∈ h∗, define a Q(q)-linear automorphism σ : K → K by
Z µ
(4.2) σ (z ) := q(µ,εi)z , for 1 ≤ i ≤ N,
µ i i
where (·,·) is the inner product on h∗ defined by (ε ,ε ) = δ . Let K = span {v } be the
Z i j i,j µ K µ+
one dimensional vector space over K with basis vector v+ and U (gl )≥0 action given by
µ q N
(4.3) X ·v = 0, for 1 ≤ i ≤ N −1, and a·v = σ (ψ(a))v , for a ∈ U (gl )0.
i µ+ µ+ µ µ+ q N
The µ-shifted rational universal Verma module µM(cid:102) is the Uq(glN)-module
(4.4) µM(cid:102):= Uq(glN)⊗Uq(glN)≥0 Kµ.
The universal Verma module µM(cid:102)is actually a module over Uq(glN)⊗Uq(glN)0U(cid:101)q(glN)0, where
U(cid:101)q(glN)0 is the field of fractions of Uq(glN)0. However, if we identify U(cid:101)q(glN)0 with K using
the map ψ, the action of U(cid:101)q(glN)0 on µM(cid:102) is not by multiplication, but rather is twisted by
the automorphism σ . It is to keep track of the difference between the action of U (gl )0 and
µ q N
multiplication that we use different notation for the generators of K and U (gl )0 (that is, z
q N i
versus L ).
i
UNIVERSAL VERMA MODULES AND THE MISRA-MIWA FOCK SPACE 7
4.2. Integral universal Verma modules. The field K contains an A-subalgebra
(cid:20) (cid:21)
z ;c
(4.5) R generated by z±1 and i (1 ≤ i ≤ N,c ∈ Z,k ∈ Z ),
i k >0
which is isomorphic to UA(gl )0 via the restriction of the map ψ in (4.1). The integral uni-
q N
versal Verma module µM(cid:102)R is the UqA(glN)-submodule of µM(cid:102)generated by vµ+. By restricting
(4.4),
(4.6) µM(cid:102)R = UqA(glN)⊗UA(gl )≥0 Rµ,
q N
where Rµ is the R-submodule of Kµ spanned by vµ+. In particular, µM(cid:102)R is a free R-module.
4.3. Evaluation. Let evR : R → A be the map defined by
λ
(cid:20) (cid:21) (cid:20) (cid:21)
(4.7) evR(z ) = q(λ,εi) and evR zi;c = q(λ,εi);c ,
λ i λ n n
where (·,·) is the inner product on h∗ defined by (ε ,ε ) = δ .
i j i,j
There is a surjective UA(gl )-module homomorphism “evaluation at λ”
q N
(4.8) evλ : µM(cid:102)R → MA(µ+λ) defined by evλ(a·vµ+) := a·vµ+λ, for all a ∈ UqA(glN).
For fixed λ, the maps evRλ and evλ extend to a map from the subspace of K and µM(cid:102) =
µM(cid:102)R ⊗R K respectively where no denominators evaluate to 0. Where it is clear we denote
both these extended maps by ev .
λ
Example 4.1. Computing the action of L on v and v ,
i µ+ µ+λ
(4.9) L ·v = q(µ,εi)z v , and Li·vµ+λ = evλ(q(µ,εi)zi)vµ+λ
i µ+ i µ+ = q(µ,εi)q(λ,εi)vµ+λ = q(µ+λ,εi)vµ+λ.
4.4. Weight decompositions. LetV(cid:101) beaUq(glN)⊗AR-module. Foreachν ∈ h∗Z, wedefine
the ν-weight space of V(cid:101) to be
(4.10) V(cid:101)ν := {v ∈ V(cid:101) : Li·v = q(ν,εi)ziv}.
The universal Verma module µM(cid:102)R is a Uq(glN)⊗A R-module, where the second factor acts
as multiplication. The weight space µM(cid:102)η (cid:54)= 0 if and only if η = µ−ν with ν in the positive
part Q+ of the root lattice. These non-zero weight spaces and the weight decomposition of
µM(cid:102) can be described explicitly by
(cid:77)
(4.11) µM(cid:102)µR−ν = UqA(glN)<−0ν ·Rµ and µM(cid:102)R = µM(cid:102)µR−ν.
ν∈Q+
Here UA(gl )<0 is defined using the grading of U (gl )<0 with F ∈ U (gl )<0 .
q N −ν q N i q N −(εi−εi+1)
4.5. Tensor products. Let V(cid:101) be a UqA(glN)⊗A R-module and W a UqA(glN)-module. The
tensor product V(cid:101) ⊗AW is a UqA(glN)⊗AR-module, where the first factor acts via the usual
coproduct and the second factor acts by multiplication on V(cid:101). In the case when V(cid:101) and W both
have weight space decompositions, the weight spaces of V(cid:101) ⊗AW are
(cid:77)
(4.12) (V(cid:101) ⊗AW)ν = V(cid:101)γ ⊗AWη.
γ+η=ν
8 ARUNRAMANDPETERTINGLEY
We also need the following:
Proposition 4.2. The tensor product of a universal Verma module with a Weyl module
satisfies
(cid:32) (cid:33)
(4.13) (cid:16)µM(cid:102)R⊗A∆A(ν)(cid:17)⊗RK ∼= (cid:77)(µ+γM(cid:102)R)⊕dim∆A(ν)γ ⊗RK.
γ
Proof. Fix ν ∈ P+. In general, M(λ+µ)⊗∆(ν) has a Verma filtration (see, for example, [13,
Theorem 2.2]) and if λ+µ+γ is dominant for all γ such that ∆(ν) (cid:54)= 0 then
γ
(4.14) M(λ+µ)⊗∆(ν) ∼= (cid:77)M(λ+µ+γ)⊕dim∆(ν)γ,
γ
which can be seen by, for instance, taking central characters. The proposition follows since
this is true for a Zariski dense set of weights λ. (cid:3)
5. The Shapovalov form and the Shapovalov determinant
5.1. The Shapovalov form. The Cartan involution ω : U (gl ) → U (gl ) is the Q(q)-
q N q N
algebra anti-involution of U (gl ) defined by
q N
(5.1) ω(L±1) = L±1, ω(X ) = Y L L−1 , ω(Y ) = L−1L X .
i i i i i i+1 i i i+1 i
The map ω is also a co-algebra involution. An ω-contravariant form on a U (gl )-module V
q N
is a symmetric bilinear form (·,·) such that
(5.2) (u,a·v) = (ω(a)·u,v), for u,v ∈ V and a ∈ U (gl ).
q N
Itfollowsbythesameargumentusedintheclassicalcase[26]thatthereisanω-contravariant
form (the Shapovalov form) on each Verma module M(λ) and this is unique up to rescaling.
The radical of (·,·) is the maximal proper submodule of M(λ), so ∆(λ) = M(λ)/Rad(·,·) for
all λ ∈ P+. In particular, (·,·) descends to an ω-contravariant form on ∆(λ).
Since ω fixes UA(gl ) ⊆ U (gl ), there is a well defined notion of an ω-contravariant form
q N q N
on a UA(gl ) module. In particular, the restriction of the Shapovalov form on ∆(λ) to ∆A(λ)
q N
is ω-contravariant.
5.2. UniversalShapovalovforms. TherearesurjectivemapsofA-algebrasp : UA(gl )<0 →
− q N
Q(q) and p : UA(gl )>0 → Q(q) defined by p (F ) = 0 and p (E ) = 0, for 1 ≤ i ≤ N.
+ q N − i + i
Using the triangular decomposition (2.7), there is an A-linear surjection
(5.3) π := p ⊗Id⊗p : UA(gl ) ∼= UA(gl )<0⊗ UA(gl )0⊗ UA(gl )>0 → UA(gl )0.
0 − + q N q N A q N A q N q N
The standard universal Shapovalov form is the R-bilinear form (·,·) : µM(cid:102)R⊗µM(cid:102)R → R
µM(cid:102)R
defined by
(cid:0) (cid:1)
(5.4) (a ·v ,a ·v ) = σ ◦ψ◦π (ω(a )a )
1 µ+ 2 µ+ µM(cid:102)R µ 0 2 1
for all a ,a ∈ UR(gl )<0. Here ψ and σ are as in (4.1) and (4.2). Since
1 2 q N µ
(cid:0) (cid:1)
(5.5) (a a ·v ,a ·v ) = σ ◦ψ◦π (ω(a )ω(a )a ) = (a ·v ,ω(a )a ·v )
1 2 µ+ 3 µ+ µM(cid:102)R µ 0 2 1 3 2 µ+ 1 3 µ+ µM(cid:102)R
for a ,a ,a ∈ U (gl ), the form (·,·) is ω-contravariant. As with the usual Shapovalov
1 2 3 q N µM(cid:102)R
form, distinct weight spaces are orthogonal, where weight spaces are defined as in Section 4.4.
UNIVERSAL VERMA MODULES AND THE MISRA-MIWA FOCK SPACE 9
Evaluation at λ gives an A-valued ω-contravariant form (·,·) on MA(µ+λ) by
MA(µ+λ)
(5.6) (evλ(u1),evλ(u2))MA(µ+λ) = evλ(cid:0)(u1,u2)µM(cid:102)R(cid:1), for u1,u2 ∈ µM(cid:102)R.
The form (·,·) can be extended by linearity to an ω-contravariant form (·,·) on µM(cid:102).
µM(cid:102)R µM(cid:102)
5.3. The Shapovalov determinant. Let V(cid:101) be a (UqA(glN) ⊗A R)-module with a chosen
ω-contravariant form. Let Bη be an R basis for the η-weight space V(cid:101)η of V(cid:101). Let detV(cid:101)Bη be
the determinant of the form evaluated on the basis B . Changing the basis B changes the
η η
determinantbyaunitinRandwesometimeswritedetV(cid:101)η tomeanthedeterminantcalculated
on an unspecified basis (detV(cid:101)η which is only defined up to multiplication by unit in R). The
Shapovalov determinant is
(5.7) detM(cid:102)ηR := det((bi,bj)M(cid:102)R)bi,bj∈Bη.
Define the partition function p: h∗ → Z by
≥0
(5.8) p(γ) := dimM(0) .
γ
Then p(γ) = dimM(λ)γ+λ for any λ, and η (cid:54)∈ Q− implies that p(η) = 0 and detM(cid:102)ηR = 1.
Theorem 5.1. (see [9, Proposition 1.9A], [20, Theorem 3.4], [26]) For any weight η,
(5.9) detM(cid:102)ηR = cη (cid:89) (cid:16)zizj−1−q2m+2i−2jzi−1zj(cid:17)p(η+mεi−mεj),
1≤i<j≤N
m>0
where c is a unit in R⊗ Q(q) = Q(q)[z±1,...,z±1].
η A 1 N
Proposition 5.2. Fix µ,η ∈ h∗ with η−µ ∈ Q−. Choose an A-basis B for UA(gl ) .
Z η−µ q N η−µ
Consider the R-bases B(cid:101)η−µ := {b·v+ | b ∈ Bη−µ} for M(cid:102)ηR−µ and µB(cid:101)η := {b·vµ+ | b ∈ Bη−µ}
for µM(cid:102)ηR. Then detµM(cid:102)(RµB(cid:101)η) = σµ(cid:0)detM(cid:102)BR(cid:101)η−µ(cid:1).
Proof. For b,b(cid:48) ∈ B ,
η−µ
(5.10) (b·v ,b(cid:48)·v ) = σ ◦ψ◦π (ω(b(cid:48))b) = σ (cid:0)(b·v ,b(cid:48)·v ) (cid:1).
µ+ µ+ µM(cid:102)R µ 0 µ 0+ 0+ M(cid:102)R
The result follows by taking determinants. (cid:3)
5.4. Contravariant forms on tensor products. If V and W are UA(gl )-modules with
q N
ω-contravariant forms (·,·) and (·,·) , define an A-bilinear form (·,·) by (w ⊗v ,w ⊗
V W W⊗V 1 1 2
v2)W⊗V = (w1,w2)W(v1,v2)V. Similarly, for a UqA(glN) ⊗A R module W(cid:102) with R-bilinear
ω-contravariant form (·,·)W(cid:102), define a R-bilinear form (·,·)W(cid:102)⊗Q(q)V on W(cid:102)⊗Q(q)V by
(5.11) (u ⊗v ,u ⊗v ) = (u ,u ) (v ,v ) .
1 1 2 2 W(cid:102)⊗Q(q)V 1 2 W(cid:102) 1 2 V
Since ω is a coalgebra involution (i.e., ∆(ω(a)) = (ω ⊗ω)∆(a), for a ∈ U (gl )), the forms
q N
(·,·) and (·,·) are ω-contravariant.
V⊗W µM(cid:102)⊗Q(q)V
In the case when W(cid:102) = µM(cid:102)R, evaluation of the ω-contravariant form (·,·) at λ gives
µM(cid:102)R⊗AV
an ω-contravariant form (·,·) :
MA(µ+λ)⊗AV
(cid:16) (cid:17)
(u ⊗v ,u ⊗v ) = ev (u ⊗v ,u ⊗v )
(5.12) 1 1 2 2 MA(µ+λ)⊗AV λ 1 1 2 2 µM(cid:102)R⊗AV
= (ev (u )⊗v ,ev (u )⊗v ) ,
λ 1 1 λ 2 2 M(µ+λ)⊗AV
10 ARUNRAMANDPETERTINGLEY
for u1,u2 ∈ µM(cid:102) and v1,v2 ∈ V. As in Section 4.3, this evaluation can be extended to the
A-submodule of the rational module where no denominators evaluate to zero.
6. The Misra-Miwa formula for F from UA(gl ) representation theory
¯i q N
Let us prepare the setting for our main result (Theorem 6.1). Fix (cid:96) ≥ 2 and a partition λ.
Let N a positive integer greater than the number of parts of λ. All calculations below are in
terms of representations of UA(gl ).
q N
• Let V = ∆A(ε ) be the standard N-dimensional module. Since ∆A(λ)⊗ Q(q) = ∆(λ),
1 A
Equation (3.1) implies
(6.1) (cid:0)∆A(λ)⊗ V(cid:1)⊗ Q(q) (cid:39) (cid:77)∆A(λ+ε )⊗ Q(q),
A A kj A
where the sum is over those indices 1 = k < k < ··· < k ≤ N for which λ + ε is a
1 2 mλ kj
partition. For ease of notation let µ(j) = λ+ε .
kj
•FixanA-basis{v ,...,v }ofV wherev hasweightε andY (v ) = δ v . Recursively
1 N k k i k i,k k+1
define singular weight vectors v in (cid:0)∆A(λ)⊗V(cid:1)⊗ Q(q) by:
µ(j) A
(i) v = v ⊗v .
µ(1) λ 1
(ii) For each k, the submodule W of (∆(λ)⊗ V)⊗ Q(q) generated by {v ⊗v | 1 ≤
k A A λ i
i ≤ k} contains all weight vectors of (∆(λ)⊗ V)⊗ Q(q) of weight greater than or
A A
equal to λ+ε . Thus, using (6.1), for each 1 ≤ j ≤ m there is a one-dimensional
k λ
space of singular vectors of weight µ(j) in W , and this is not contained in W
kj kj−1
(since k > k ). This implies that there unique singular vector v of weight µ(j)
j j−1 µ(j)
in
(6.2) v ⊗v + (cid:77) U (gl )v ⊆ (cid:0)∆A(λ)⊗ V(cid:1)⊗ Q(q),
λ kj q N µ(i) A A
1≤i<j
where we recall that U (gl ) = UA(gl )⊗ Q(q).
q N q N A
• There is a unique ω-contravariant form on ∆A(λ) normalized so that (v ,v ) = 1 and a
λ λ
unique ω-contravariant form on V normalized so that (v ,v ) = 1. As in section 5.4, define a
1 1
ω-contravariant form on (cid:0)∆A(λ)⊗ V(cid:1)⊗ Q(q) by (u ⊗w ,u ⊗w ) = (u ,u )(w ,w ). For
A A 1 1 2 2 1 2 1 2
each 1 ≤ j ≤ m , define an element r (λ) ∈ Q(q) by
λ j
(6.3) r (λ) := (v ,v ).
j µ(j) µ(j)
Theorem 6.1. The Misra-Miwa operators F from Section 3.3 satisfy
¯i
(cid:88)
(6.4) F¯i|λ(cid:105) = vvalφ2(cid:96)rj(λ)|µ(j)(cid:105),
c¯(b(j))=¯i
where b(j) is the box µ(j)/λ, c¯(b(j)) is the color of box b(j) as in Figure 1, φ is the 2(cid:96)th
2(cid:96)
cyclotomic polynomial in q and val r is the number of factors of φ in the numerator of r
φ 2(cid:96)
2(cid:96)
minus the number of factors of φ in the denominator of r.
2(cid:96)
The proof of Theorem 6.1 will occupy the rest of this section. We will first prove a similar
statement, Proposition 6.6, where the role of the Weyl modules is played by the universal
Verma modules from Section 4. For ease of notation, let M(cid:102)R denote the module 0M(cid:102)R from
section 4.2.
(cid:16) (cid:17)
Definition 6.2. Recursively define singular weight vectors vεk+ ∈ M(cid:102)R⊗AV ⊗R K and
elements s ∈ K for 1 ≤ k ≤ N by
k