CARTAN DETERMINANTS AND SHAPOVALOV FORMS

JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Abstract. We compute the determinant of the Gram matrix of the Shapovalov form on weight spaces of the basic representation of an affine Kac-Moody algebra of ADE type (possibly twisted). As a consequence, we obtain explicit formulae for the determinants of the Cartan matrices of p-blocks of the symmetric group and its double cover, and of the associated Hecke algebras at roots of unity.

1. Introduction

Let 𝔤 be an affine Kac-Moody algebra of type X_{N}^{(r)} as in the table:

\begin{matrix}X_{N}^{(r)} & A_{ℓ}^{(1)} & D_{ℓ}^{(1)} & E_{ℓ}^{(1)} & A_{2ℓ−1}^{(2)} & A_{2ℓ}^{(2)} & D_{ℓ+1}^{(2)} & E_{6}^{(2)} & D_{4}^{(3)} \\ ℓ & ≥ 1 & ≥ 4 & 6, 7 or 8 & ≥ 3 & ≥ 1 & ≥ 2 & 4 & 2 \\ k & 0 & 0 & 0 & ℓ − 1 & ℓ & 1 & 2 & 1 \\ α & ℓ + 1 & 4 & 9 − ℓ & 2 & 1 & 2 & 1 & 1 \\ β & 1 & 1 & 1 & ℓ & 2ℓ + 1 & 2 & 3 & 2 \end{matrix}

We are interested here in the basic representation V = V (Λ_{0}) of 𝔤, see [11]. Let |0⟩ be a vacuum vector and define the lattice V_{ℤ} := U_{ℤ}|0⟩ in V, where U_{ℤ} is the ℤ-subalgebra of the universal enveloping algebra of 𝔤 generated by the divided powers

e_{i}^{n}/n!, f_{i}^{n}/n! (i = 0, 1, …, ℓ, n ≥ 1)

in the Chevalley generators. Let (. , .)_{S} denote the Shapovalov form, the unique Hermitian form on V satisfying (|0⟩, |0⟩)_{S} = 1 and (e_{i}v, v′)_{S} = (v, f_{i}v′)_{S} for i = 0, …, ℓ and all v, v′ ∈ V. Its restriction to V_{ℤ} gives a symmetric bilinear form

(. , .)_{S} : V_{ℤ} × V_{ℤ} → ℤ.

Our Main Theorem gives an explicit formula for the determinant of the Gram matrix of this form on each weight space of V_{ℤ}.

To state the result precisely, recall the description of the weights of V [11, §12.6]: every weight is of the form wΛ_{0} − dδ for some w in the Weyl group W associated to 𝔤 and some integer d ≥ 0. Also let 𝒫(d) denote the set of all partitions λ = (λ_{1} ≥ λ_{2} ≥ …) of d. Given λ ∈ 𝒫(d), we can gather together its equal parts to represent it as λ = (1^{r_{1}}2^{r_{2}} …). Also recall the number r ∈ {1, 2, 3} which comes from the type X_{N}^{(r)}. Then:

Main Theorem. The determinant of the Gram matrix of the Shapovalov form on the (wΛ_{0} − dδ)-weight space of V_{ℤ} is α^{a(d)}β^{b(d)} where a(d) = ∑_{λ∈𝒫(d)} a_{λ}, b(d) = ∑_{λ∈𝒫(d)} b_{λ} and for λ = (1^{r_{1}}2^{r_{2}} …),

Second author partially supported by the NSF (grant no. DMS-9900134).

a_{λ} = ∏_{i with r|i} \binom{ℓ + r_{i} − 1}{r_{i}} · ∏_{i with r∤i} \binom{k + r_{i} − 1}{r_{i}} · ∑_{i with r|i} \frac{r_{i}}{ℓ} ,

b_{λ} = ∏_{i with r|i} \binom{ℓ + r_{i} − 1}{r_{i}} · ∏_{i with r∤i} \binom{k + r_{i} − 1}{r_{i}} · ∑_{i with r∤i} \frac{r_{i}}{k} ,

ℓ, k, α, β being as in the above table. The generating functions a(q) = ∑_{d≥0} a(d)q^{d} and b(q) = ∑_{d≥0} b(d)q^{d} are given by the formulae

a(q) = T (q^{r})P (q)^{k}P (q^{r})^{ℓ−k},
b(q) = (T (q) − T (q^{r}))P (q)^{k}P (q^{r})^{ℓ−k}

where P (q) = ∏_{i≥1} \frac{1}{1−q^{i}} is the generating function for the number of partitions of d and

T (q) = ∑_{i≥1} \frac{q^{i}}{1−q^{i}} is the generating function for the number of divisors of d.

In [5], De Concini, Kac and Kazhdan constructed the basic representation over ℤ (at least in the untwisted cases) using an integral version of the vertex operator construction of [8]. They showed in particular that the basic representation remains irreducible on reduction modulo p if and only if p ∤ det X_{N}, where X_{N} is the Cartan matrix of the underlying finite root system; this also follows immediately from our Main Theorem on noting that det X_{N} = αβ^{r−1}.

Our interest in the theorem comes instead from modular representation theory. Suppose now that 𝔤 is of type A_{ℓ}^{(1)} and set p = (ℓ + 1). Let F S_{n} denote the group algebra of the symmetric group over a field F of characteristic p (assuming in this case that p is prime), and let H_{n} denote the Iwahori-Hecke algebra associated to S_{n} over an arbitrary field but at a primitive pth root of 1 (this case making sense for arbitrary p ≥ 2). By [1, 9], there is an isomorphism between the basic representation V_{ℤ} of 𝔤 and the direct sum K = ⨁_{n≥0} K_{n} of the Grothendieck groups K_{n} of finitely generated projective F S_{n}- (resp. H_{n}-) modules for all n. Under the isomorphism, the weight spaces of V_{ℤ} are in 1–1 correspondence with the block components of K, a weight space of the form wΛ − dδ corresponding to a block of p-weight d (see e.g. [13, §5.3] for the definition of the p-weight of a block). Moreover, according to [9, Theorem 14.2], the Shapovalov form corresponds to the usual Cartan pairing ([P], [Q]) = dim Hom(P, Q) between projective modules P, Q. Thus the theorem has the following immediate corollary:

Corollary 1. Let B be a block of p-weight d of either the group algebra F S_{n} of the symmetric group over a field of prime characteristic p, or the Hecke algebra H_{n} over an arbitrary field but at a primitive pth root of unity, in which case p ≥ 2 is an arbitrary integer. Then the determinant of the Cartan matrix of B is p^{N(d)} where

N(d) = ∑_{λ=(1^{r_{1}}2^{r_{2}}...)∈𝒫(d)} \frac{r_{1} + r_{2} + …}{p − 1}\binom{p − 2 + r_{1}}{r_{1}}\binom{p − 2 + r_{2}}{r_{2}} ….

The generating function N(q) = ∑_{d≥0} N(d)q^{d} equals T (q)P (q)^{p−1}.

It is a classical result of Brauer that the determinant of the Cartan matrix of a block of F S_{n} is a power of p (see [6, 84.17]). Donkin [7] has proved similarly that the determinant of the Cartan matrix of a block of H_{n} divides a power of p. The corollary shows in particular that the determinant is exactly a power of p, even in those cases where p is not prime, as had been conjectured by Mathas. We remark that in the case of blocks of F S_{n}, but not of H_{n}, the explicit generating function given in the corollary has also recently been obtained by Bessenrodt and Olsson [2] using methods from block theory.

Finally suppose that 𝔤 is of type A_{2ℓ}^{(2)} and set p = (2ℓ + 1). In this case, the Main Theorem can be reinterpreted as a computation of Cartan determinants of the p-blocks of the double covers Ŝ_{n} of the symmetric group. Following [3, §9-c] for notation, let S(n) be the twisted group algebra of S_{n} over an algebraically closed field F of characteristic p (assuming p is an odd prime in this case), and let W (n) be the Hecke-Clifford superalgebra over an algebraically closed field of characteristic different from 2 at a primitive pth root of unity (for arbitrary odd p ≥ 3). By [3, 7.16, 8.13, 9.9], there is an isomorphism between the basic representation V_{ℤ} and the direct sum K = ⨁_{n≥0} K_{n} of the Grothendieck groups of finitely generated projective S(n)- (resp. W (n)-) supermodules, under which a weight space of the form wΛ − dδ maps to a superblock of p-bar weight d (see [3, §9-a] for the definition of p-bar weight of a superblock), and the Shapovalov form corresponds to the Cartan pairing on projective supermodules (see [3, §7-c]). So:

Corollary 2. Let B be a superblock of p-bar weight d of either S(n) in odd characteristic p, or W (n) at a primitive pth root of unity, in which case p ≥ 3 is an arbitrary odd integer. Then the determinant of the Cartan matrix of B is p^{N(d)} where

N(d) = ∑_{λ=(1^{r_{1}}2^{r_{2}}...)∈𝒫(d)} \frac{2r_{1} + 2r_{3} + 2r_{5} + …}{p − 1}\binom{\frac{p−3}{2} + r_{1}}{r_{1}}\binom{\frac{p−3}{2} + r_{2}}{r_{2}}\binom{\frac{p−3}{2} + r_{3}}{r_{3}} ….

The generating function N(q) = ∑_{d≥0} N(d)q^{d} equals (T (q) − T (q^{2}))P (q)^{(p−1)/2}.

It is more natural from the point of view of finite group theory to ask for the Cartan determinant of a block B of the twisted group algebra S(n) in the usual ungraded sense. According to Humphreys’ classification [10], see also [3, 9.16], we can associated to B its p-bar weight d and a type ε ∈ {M, Q}. In case ε = M, B coincides with a superblock of p-bar weight d and it is immediate that its Cartan determinant is as in Corollary 2. But in the cases when ε = Q and d > 0, the Cartan matrix of B has twice as many rows and columns as the Cartan matrix of the corresponding superblock. Nevertheless, we believe the Cartan determinant is the same, based on explicit computations for small d. In other words, we conjecture that Cartan determinants of p-blocks of S(n) depend only on the p-bar weight d, not on the type ε, of the block.

2. The affine algebras

We begin by recalling the construction of the affine Lie algebras from [11, Chapter 8]. Let X_{N}^{(r)} be an affine Dynkin diagram of ADE type as in the introduction, and let X_{N} be the underlying finite Dynkin diagram. We use the same numbering of Dynkin diagrams as [11, §4.8] with two exceptions: in the case X_{N}^{(r)} = E_{6}^{(2)} we will number the vertices of the finite Dynkin diagram X_{N} = E_{6} by

[figure: 1; 2; 3; 5; 6; 4]

and in the case X_{N}^{(r)} = A_{2ℓ}^{(2)} we will number the vertices of the finite Dynkin diagram X_{N} = A_{2ℓ} by

[figure: ℓ−1; ℓ−2; 0; ℓ+1; 2ℓ−1; 2ℓ]

Let Q′ denote the root lattice of type X_{N}, with simple roots α′_{i} and invariant bilinear form (.|.)′ normalized so that each (α′_{i}|α′_{i})′ = 2. Let μ : Q′ → Q′ be a graph automorphism of order r, as in e.g. [11, §7.9]. Let

ε : Q′ × Q′ → {±1}

be an asymmetry function as in [11, §7.8] chosen so that ε(μ(α′), μ(β′)) = ε(α′, β′). In case X_{N}^{(r)} = A_{2ℓ}^{(2)} this is not possible so we instead require here that ε(μ(α′), μ(β′)) = ε(β′, α′). Let 𝔥′ = ℂ ⊗_{ℤ} Q′ viewed as an abelian Lie algebra, and extend μ and (.|.)′ linearly to 𝔥′. Then we can construct the finite dimensional simple Lie algebra 𝔤′ of type X_{N} as the vector space

𝔤′ = 𝔥′ ⊕ ⨁_{α′ a root} ℂE_{α′}

viewed as a Lie algebra so that 𝔥′ is abelian and

[α′, E_{β′}] = (α′|β′)′E_{β′},  [E_{α′}, E_{−α′}] = −α′,
[E_{α′}, E_{β′}] = \{ \begin{matrix}ε(α′, β′)E_{α′+β′} & if α′ + β′ is a root, \\ 0 & otherwise. \end{matrix}

The invariant form on 𝔥′ extends to 𝔤′ by (𝔥′|E_{α′})′ = 0 and (E_{α′}|E_{β′})′ = −δ_{α′,−β′} for all roots α′, β′.

Let a_{i}, a_{i}^{∨} (i = 0, …, ℓ) be the numerical labels on the Dynkin diagram X_{N}^{(r)} and its dual as in [11, §4.8]. We note especially that a_{0} = 1 if X_{N}^{(r)} ≠ A_{2ℓ}^{(2)} and a_{0} = 2 if X_{N}^{(r)} = A_{2ℓ}^{(2)}. It will also be convenient to define

c_{i} = \{ \begin{matrix}2 & if X_{N}^{(r)} = A_{2ℓ}^{(2)} and i = 0, \\ 1 & otherwise; \end{matrix}
d_{i} = c_{i}a_{i}^{∨}a_{i}^{−1} ∈ {1, r}

for i = 0, 1, …, ℓ. Let m = a_{0}r and fix a primitive mth root of unity ω ∈ ℂ. In all types other than A_{2ℓ}^{(2)}, let η : Q′ → ℂ^{×} denote the constant function with η(α′) = 1 for all α′ ∈ Q′; in type A_{2ℓ}^{(2)}, define η instead by the rules

η(0) = 1, η(α′ + β′) = η(α′)η(β′)(−1)^{(α′|β′)′}, η(α′_{j}) = \{ \begin{matrix}1 & j ≠ 0, ℓ + 1, \\ ω & j = 0, ℓ + 1. \end{matrix}

Now extend μ from 𝔥′ to 𝔤′ by declaring that μ(E_{α′}) = η(α′)E_{μ(α′)} for all roots α′ ∈ Q′. The order of the resulting automorphism μ of 𝔤′ is equal to m in all cases.

Decompose

𝔤′ = ⨁_{n∈ℤ/m}𝔤′_{n}  where  𝔤′_{n} = {X ∈ 𝔤′ | μ(X) = ω^{n}X}.

Also write 𝔥′_{n} = 𝔥′ ∩ 𝔤′_{n}. Introduce the infinite dimensional Lie algebras

𝔤 = ⨁_{n∈ℤ}𝔤′_{n} ⊗ t^{n} ⊕ ℂc ⊕ ℂd ⊆ 𝔤′ ⊗ ℂ[t, t^{−1}] ⊕ ℂc ⊕ ℂd,
𝔥 = 𝔥′_{0} ⊗ 1 ⊕ ℂc ⊕ ℂd ⊂ 𝔤,
𝔱 = 𝔱^{+} ⊕ ℂc ⊕ 𝔱^{−} ⊂ 𝔤 where 𝔱^{±} = ⨁_{±n>0} 𝔥′_{n} ⊗ t^{n}.

Multiplication is defined by the rules

[d, X ⊗ t^{n}] = nX ⊗ t^{n}, [c, 𝔤] = 0,

[X ⊗ t^{n}, Y ⊗ t^{k}] = [X, Y] ⊗ t^{n+k} + δ_{n,−k}n\frac{(X|Y)′}{m} c.

Then 𝔤 is the affine Lie algebra of type X_{N}^{(r)} with canonical central element c and scaling element d, and 𝔥 is a Cartan subalgebra. As a matter of notation, we will write

X(n) := ∑_{j=0}^{m−1} ω^{−nj}μ^{j}(X) ⊗ t^{n} ∈ 𝔤′_{n} ⊗ t^{n}

for X ∈ 𝔤′ and n ∈ ℤ. The normalized invariant form on 𝔤 will be denoted (.|.), and is defined by

(X ⊗ t^{n}|Y ⊗ t^{k}) = δ_{n,−k}(X|Y)′/r

for all X ∈ 𝔤_{n}, Y ∈ 𝔤_{k}.

In order to write down a choice of Chevalley generators for 𝔤, let ℓ denote the number of μ-orbits on the simple roots in Q′. Let

ε = \{ \begin{matrix}0 & if X_{N}^{(r)} ≠ A_{2ℓ}^{(2)}, \\ ℓ & if X_{N}^{(r)} = A_{2ℓ}^{(2)}, \end{matrix}

and set

I = {0, 1, …, ℓ} − {ε}.

Then, the α′_{i} for i ∈ I give a set of representatives for the μ-orbits on the simple roots. Define

−α′_{ε} = \{ \begin{matrix}the longest root in Q′ & if r = 1 or X_{N}^{(r)} = A_{2ℓ}^{(2)}, \\ α′_{1} + ⋯ + α′_{2ℓ−2} & if X_{N}^{(r)} = A_{2ℓ−1}^{(2)}, \\ α′_{1} + ⋯ + α′_{ℓ} & if X_{N}^{(r)} = D_{ℓ+1}^{(2)}, \\ α′_{2} + α′_{3} + α′_{4} & if X_{N}^{(r)} = D_{4}^{(3)}, \\ α′_{1} + 2α′_{2} + 2α′_{3} + α′_{4} + α′_{5} + α′_{6} & if X_{N}^{(r)} = E_{6}^{(2)}. \end{matrix}

For i = 0, 1, …, ℓ, write

e_{i}(n) = \frac{\sqrt{c_{i}}}{a_{0}d_{i}} E_{α′_{i}}(n) and f_{i}(n) = −\frac{\sqrt{c_{i}}}{a_{0}d_{i}} E_{−α′_{i}}(n).

The Chevalley generators of 𝔤 are e_{0} = e_{0}(1), e_{i} = e_{i}(0) and f_{0} = f_{0}(−1), f_{i} = f_{i}(0) for i = 1, …, ℓ, as is proved in [11, §8.7] (taking s_{0} = 1, s_{1} = ⋯ = s_{ℓ} = 0). We also define

h_{i} = [e_{i}, f_{i}] = δ_{i,0}c + \frac{c_{i}}{a_{0}d_{i}} α′_{i}(0).

Next let Q ⊂ 𝔥^{∗} denote the root lattice associated to 𝔤. So following [11, §6.2],

Q = ⨁_{i=0}^{ℓ} ℤα_{i} ⊕ ℤΛ_{0}

where α_{0}, …, α_{ℓ} are the simple roots corresponding to h_{0}, …, h_{ℓ} and Λ_{0} is the zeroth fundamental dominant weight, i.e.

⟨h_{i}, α_{j}⟩ = the ij-entry of the Cartan matrix of type X_{N}^{(r)},
⟨h_{i}, Λ_{0}⟩ = ⟨d, α_{i}⟩ = δ_{i,0},
⟨d, Λ_{0}⟩ = 0,

for i, j = 0, …, ℓ. Also as in [11, §6.2], we have the normalized invariant form (.|.) on 𝔥^{∗} and the element δ = ∑_{i=0}^{ℓ} a_{i}α_{i} ∈ Q.

To conclude, we explain the relationship between the form (.|.)′ on Q′ and the form (.|.) on Q. Introduce the new symmetric bilinear form (.|.)_{μ} on Q′ defined by

(α′|β′)_{μ} = (α′| ∑_{j=0}^{r−1} μ^{j}(β′))′

for all α′, β′ ∈ Q′. There is an orthogonal decomposition

𝔥^{∗} =𝔥^{∘}^{∗} ⊕ (ℂδ + ℂΛ_{0})

where 𝔥^{∘} ^{∗} = ⨁_{i=1}^{ℓ} ℂα_{i}, see [11, §6.2]. As in loc. cit. we write − : 𝔥^{∗} →𝔥^{∘} ^{∗} for the orthogonal projection, in particular \overline{Q} denotes the orthogonal projection of Q onto 𝔥^{∘}^{∗}. Define a ℤ-linear map

ι : Q′ → \overline{Q}  (2.1)

by ι(μ^{j}(α′_{i})) = \overline{α}_{i} for each i ∈ I and j ≥ 0. The kernel of ι is the space

M′ = {α′ − μ(α′) | α′ ∈ Q′}  (2.2)

which is precisely the radical of the bilinear form (.|.)_{μ}. Moreover, ι induces an isometry between Q′/M′ and \overline{Q} with respect to the forms induced by (.|.)_{μ} and (.|.) respectively.

3. The basic representation

Next we recall the construction of the basic representation V = V (Λ_{0}) of 𝔤, following Lepowsky [12]. Let Z = ⟨−1, ω⟩ ⊂ ℂ^{×} be the multiplicative group generated by −1 and ω. Form the central extension

1 ⟶ Z ⟶ Q̂ ⟶^{π} Q′ ⟶ 1,

namely, Q̂ = {e_{x}^{α′} | α′ ∈ Q′, x ∈ Z} with multiplication

e_{x}^{α′}e_{y}^{β′} = \{ \begin{matrix}e_{xyε(α′,β′)}^{α′+β′} & if X_{N}^{(r)} ≠ A_{2ℓ}^{(2)}, D_{4}^{(3)}, \\ e_{xyε(α′,β′)(−ω)^{−(α′|μ(β′))′}}^{α′+β′} & if X_{N}^{(r)} = A_{2ℓ}^{(2)} or D_{4}^{(3)}, \end{matrix}

for α′, β′ ∈ Q′, x, y ∈ Z. The map π : Q̂ → Q′ here is defined by π(e_{x}^{α′}) = α′. Let M̂ = π^{−1}(M′), where M′ is as in (2.2). There is a well-defined multiplicative character τ : M̂ → ℂ^{×} defined in [12, Proposition 6.1] by

τ(e_{x}^{α′−μ(α′)}) = (−1)^{(α′|α′)′/2}xη(α′)ε(α′, μ(α′))ω^{−a_{0}^{2}(α′|α′)_{μ}/2}.

So we can form the induced Q̂-module ℂ[Q̂] ⊗_{ℂ[M̂]} τ. We note the useful formula

e_{1}^{α′} ⊗ τ = η(α′)ω^{a_{0}(α′|α′)_{μ}/2}e_{1}^{μ(α′)} ⊗ τ (α′ ∈ Q′).

View the symmetric algebra S(𝔱^{−}) as a 𝔱-module in the unique way so that c acts as 1, elements of 𝔱^{−} act by multiplication, and elements of 𝔱^{+} annihilate 1. It is ℤ-graded by declaring that

deg(h ⊗ t^{−n}) = \frac{n}{a_{0}}

for each h ∈ 𝔥′_{−n}, n ≥ 1. Let

V = S(𝔱^{−}) ⊗ ℂ[Q̂] ⊗_{ℂ[M̂]} τ.

Let 𝔱 act on S(𝔱^{−}) as given and trivially on ℂ[Q̂] ⊗_{ℂ[M̂]} τ, let h ⊗ t^{0} for h ∈ 𝔥′_{0} act by

(h ⊗ t^{0})(f ⊗ e_{x}^{α′} ⊗ τ) = (h|α′)′f ⊗ e_{x}^{α′} ⊗ τ,

and let d act by

d(f ⊗ e_{x}^{α′} ⊗ τ) = −a_{0} (deg(f) + (α′|α′)_{μ}/2) f ⊗ e_{x}^{α′} ⊗ τ.

We have now defined the action of 𝔥 + 𝔱 on V. To extend the action to all of 𝔤, let α′ ∈ Q′ be a root. As in [12, (4.8)], let

σ(α′) = \{ \begin{matrix}1 & r = 1, \\ \sqrt{2}^{(α′|μ(α′))′} & if X_{N}^{(r)} = A_{2ℓ−1}^{(2)}, D_{ℓ+1}^{(2)} or E_{6}^{(2)}, \\ (1 − ω^{−1})^{(α′|μ(α′))′} & if X_{N}^{(r)} = D_{4}^{(3)}, \\ 2(1 + ω)^{(α′|μ(α′))′} & if X_{N}^{(r)} = A_{2ℓ}^{(2)}. \end{matrix}

Also define

P_{α′}(z) = exp (∑_{n≥1} \frac{α′(−n)z^{n}}{n} ) , Q_{α′}(z) = exp (− ∑_{n≥1} \frac{α′(n)z^{n}}{n} ) ,

viewed as elements of End(V)[[z^{±1}]]. Let

E_{α′}(z) = σ(α′)P_{α′}(z)Q_{α′}(z^{−1})e_{1}^{α′}z^{a_{0}α′}z^{a_{0}(α′|α′)_{μ}/2−1}.

Here, z^{a_{0}α′} denotes the operator with

z^{a_{0}α′}(f ⊗ e_{x}^{β′} ⊗ τ) = z^{(a_{0}α′|β′)_{μ}}f ⊗ e_{x}^{β′} ⊗ τ

for each f ∈ S(𝔱^{−}) and β ∈ Q′, and

e_{1}^{α′}(f ⊗ e_{x}^{β′} ⊗ τ) = f ⊗ (e_{1}^{α′}e_{x}^{β′}) ⊗ τ.

Expanding E_{α′}(z) in powers of z we get the required action of E_{α′}(n) ∈ 𝔤 on V for each root α′ ∈ Q′ and each n ∈ ℤ:

E_{α′}(z) = ∑_{n∈ℤ}E_{α′}(n)z^{−n−1}.

For a proof that this is a well-defined irreducible representation of 𝔤 in case r = 1 see [11, §14.8]; the general case is due to Lepowsky [12].

Let ℂ[\overline{Q}] denote the group algebra of \overline{Q}, with natural basis e^{α} for α ∈ \overline{Q} and multiplication e^{α}e^{β} = e^{α+β}. Note ℂ[Q̂] ⊗_{ℂ[M̂]} τ has a basis given by the elements e_{1}^{α′} ⊗ τ for all α′ ∈ ∑_{i∈I} ℤα′_{i}. For such an α′, let

ι(e_{1}^{α′} ⊗ τ) = \{ \begin{matrix}e^{ι(α′)} & if X_{N}^{(r)} ≠ A_{2ℓ}^{(2)}, D_{4}^{(3)}, \\ (−ω)^{(α′|μ(α′))′/2}e^{ι(α′)} & if X_{N}^{(r)} = D_{4}^{(3)}, \\ (\frac{1−ω}{\sqrt{2}} )^{(α′|μ(α′))′} e^{ι(α′)} & if X_{N}^{(r)} = A_{2ℓ}^{(2)}, \end{matrix}

recalling the map ι : Q′ → \overline{Q} defined in (2.1). Extending linearly, we obtain a vector space isomorphism ι : ℂ[Q̂]⊗_{ℂ[M̂]}τ → ℂ[\overline{Q}]. For i = 0, 1, …, ℓ, we define functions σ_{i}^{±} : \overline{Q} → ℂ^{×} by the equation

σ_{i}^{±}(α)e^{α±\overline{α}_{i}} = ±\frac{\sqrt{c_{i}}}{a_{0}d_{i}} σ(α′_{i})ι(e_{1}^{±α′_{i}}ι^{−1}(e^{α}))

for all α ∈ \overline{Q}. The choice of the renormalization map ι above ensures:

Lemma 3.1. For all i = 0, 1, …, ℓ and α ∈ \overline{Q}, σ_{i}^{±}(α) ∈ \{±1\}. Moreover, for i ∈ I, we have that σ_{i}^{−} = −σ_{i}^{+}, and σ_{i}^{+} : \overline{Q} → \{±1\} is a group homomorphism such that σ_{i}^{+}(\overline{α}_{j}) = ε(α′_{i}, α′_{j}) for each j ∈ I.

Now we can rewrite the construction of the basic representation V in terms of the Chevalley generators. We will identify

V = S(𝔱^{−}) ⊗ ℂ[Q̂] ⊗_{ℂ[M̂]} τ = S(𝔱^{−}) ⊗ ℂ[\overline{Q}]

via the map id ⊗ι. Then, the actions of h_{i} for i = 0, …, ℓ and of d are as

h_{i}(f ⊗ e^{α}) = (δ_{i,0} + ⟨h_{i}, α⟩)f ⊗ e^{α},
d(f ⊗ e^{α}) = −a_{0} (deg(f) + (α|α)/2) f ⊗ e^{α}

for all α ∈ \overline{Q}. In particular, we note from this that

wt(f ⊗ e^{α}) = Λ_{0} + α − (deg(f) + (α|α)/2) δ  (3.2)

for each homogeneous f ∈ S(𝔱^{−}) and α ∈ \overline{Q}. This shows that 1 ⊗ e^{0} is a highest weight vector in V of highest weight Λ_{0} (cf. [11, Lemma 12.6]), identifying V with the irreducible highest weight module V (Λ_{0}). Finally, for i = 0, …, ℓ,

e_{i}(z) = ∑_{n∈ℤ}e_{i}(n) ⊗ z^{−n−1} = P_{α′_{i}}(z)Q_{α′_{i}}(z^{−1})e^{\overline{α}^{i}}z^{a_{0}α_{i}}z^{a_{0}(α_{i}|α_{i})/2−1}s_{i}^{+},  (3.3)
f_{i}(z) = ∑_{n∈ℤ}f_{i}(n) ⊗ z^{−n−1} = P_{−α′_{i}}(z)Q_{−α′_{i}}(z^{−1})e^{−\overline{α}_{i}}z^{−a_{0}α_{i}}z^{a_{0}(α_{i}|α_{i})/2−1}s_{i}^{−},  (3.4)

where

z^{±a_{0}α_{i}}(f ⊗ e^{β}) = z^{(±a_{0}α_{i}|β)}f ⊗ e^{β},
s_{i}^{±}(f ⊗ e^{β}) = σ_{i}^{±}(β)f ⊗ e^{β},
e^{±\overline{α}_{i}}(f ⊗ e^{β}) = f ⊗ e^{β±\overline{α}_{i}}.

The following lemma will be needed later on:

Lemma 3.5. For i_{1}, …, i_{s} ∈ \{0, …, ℓ\}, roots β′_{1}, …, β′_{t} ∈ Q′ and γ ∈ \overline{Q}, we have that

e_{i_{1}}(z_{1})e_{i_{2}}(z_{2}) … e_{i_{s}}(z_{s})P_{β′_{1}}(w_{1}) … P_{β′_{t}}(w_{t}) ⊗ e^{γ} =

± ∏_{1≤u≤s} z_{u}^{\frac{a_{0}}{2} (α_{iu}|α_{iu})−1+a_{0}(α_{iu}|γ)}

× ∏_{1≤u<v≤s} z_{u}^{a_{0}(α_{iu}|α_{iv})} ∏_{k∈ℤ/m}(1 − ω^{−k} \frac{z_{v}}{z_{u}} )^{(μ^{k}(α′_{iu})|α′_{iv})′}

× ∏_{1≤u≤s} ∏_{1≤v≤t} ∏_{k∈ℤ/m}(1 − ω^{−k} \frac{w_{v}}{z_{u}} )^{(μ^{k}(α′_{iu})|β′_{v})′}

×P_{α′_{i_{1}}} (z_{1}) … P_{α′_{is}} (z_{s})P_{β′_{1}}(w_{1}) … P_{β′_{t}}(w_{t}) ⊗ e^{γ+\overline{α}_{i_{1}}+···+\overline{α}_{is}} .

A similar formula holds for f_{i_{1}}(z_{1}) … f_{i_{s}}(z_{s})P_{β′_{1}}(w_{1}) … P_{β′_{s}}(w_{s}) ⊗ e^{γ}, replacing α_{i_{u}} by −α_{i_{u}}, α′_{i_{u}} by −α′_{i_{u}}, and ᾱ_{i_{u}} by −ᾱ_{i_{u}} everywhere.

Proof. This follows from the following commutation relation obtained in [12, 3.4]: for α′, β′ ∈ Q′,

Q_{α′}(z^{−1})P_{β′}(w) = P_{β′}(w)Q_{α′}(z^{−1}) ∏_{k∈ℤ/m}(1 − ω^{−k} \frac{w}{z})^{(μ^{k}(α′)|β′)′},

which is a consequence of the Campbell-Hausdorff formula, cf. [11, (14.8.12)]. □

4. The Integral form

As in the introduction, let U_{ℤ} denote the ℤ-subalgebra of the universal enveloping algebra of 𝔤 generated by the elements e_{i}^{r}/r!, f_{i}^{r}/r! for i = 0, …, ℓ and r ≥ 0, and let

V_{ℤ} := U_{ℤ}(1 ⊗ e^{0}) ⊂ V.

In this section, we will give an explicit description of V_{ℤ}.

To start with, let τ : 𝔤 → 𝔤 be the antilinear Chevalley antiautomorphism, so

τ(d) = d, τ(e_{i}(n)) = f_{i}(−n), τ(f_{i}(n)) = e_{i}(−n)

for each i = 0, …, ℓ and n ∈ ℤ, cf. [11, §§7.6, 8.3]. The Shapovalov form (.|.)_{S} on V is the unique Hermitian form such that

(1 ⊗ e^{0}, 1 ⊗ e^{0})_{S} = 1 and (xv, w)_{S} = (v, τ(x)w)_{S}

for all v, w ∈ V, x ∈ 𝔤. The restriction of τ to 𝔱 gives the antilinear Chevalley antiautomorphism of 𝔱, and we can also consider the Shapovalov form on S(𝔱^{−}), satisfying (1, 1)_{S} = 1 and (xf, g)_{S} = (f, τ(x)g)_{S} for all f, g ∈ S(𝔱^{−}), x ∈ 𝔱.

Lemma 4.1. For all f, g ∈ S(𝔱^{−}) and α, β ∈ \overline{Q}, (f ⊗ e^{α}, g ⊗ e^{β})_{S} = (f, g)_{S}.

Proof. Since different weight spaces are orthogonal and in view of (3.2), this reduces to checking that (1⊗e^{α}, 1⊗e^{α})_{S} = 1 for all α ∈ \overline{Q}. Proceeding by induction, we may assume that there is some β ∈ \overline{Q} and i ∈ I such that (1 ⊗ e^{β}, 1 ⊗ e^{β})_{S} = 1 and either α = β + \overline{α}_{i} or α = β − \overline{α}_{i}.

Suppose that α = β + \overline{α}_{i}. Letting n = −a_{0}(α_{i}|β) − a_{0}(α_{i}|α_{i})/2, one checks easily using (3.3), (3.4) that

e_{i}(n)(1 ⊗ e^{β}) = σ_{i}^{+}(α)(1 ⊗ e^{α}),  f_{i}(−n)(1 ⊗ e^{α}) = σ_{i}^{−}(β)(1 ⊗ e^{β}).  (4.2)

Hence,

(1 ⊗ e^{α}, 1 ⊗ e^{α})_{S} = σ_{i}^{+}(β)(e_{i}(n)(1 ⊗ e^{β}), 1 ⊗ e^{α})_{S}
= σ_{i}^{+}(β)(1 ⊗ e^{β}, f_{i}(−n)(1 ⊗ e^{α}))_{S}
= σ_{i}^{−}(α)σ_{i}^{+}(β)(1 ⊗ e^{β}, 1 ⊗ e^{β})_{S} = 1,

since σ_{i}^{−}(α) = σ_{i}^{+}(β) by Lemma 3.1.

A similar argument in the case that α = β − \overline{α}_{i} completes the proof. □

Lemma 4.3. For all i = 0, 1, …, ℓ and n ∈ ℤ, the elements e_{i}(n) and f_{i}(n) belong to U_{ℤ}.

Proof. Suppose that e_{i}(n) ≠ 0. Then, wt(e_{i}(n)) = \overline{α}_{i}+ \frac{n}{a_{0}} δ is a real root, hence is conjugate under the Weyl group W associated to 𝔤 to some simple root α_{j}. So we can find simple reflections s_{i_{1}}, …, s_{i_{t}} ∈ W such that \overline{α}_{i} + \frac{n}{a_{0}} δ = s_{i_{1}} … s_{i_{t}}α_{j}. Let r_{i}^{ad} be the automorphism of 𝔤 defined by r_{i}^{ad} = exp(adf_{i}) exp(−ade_{i}) exp(adf_{i}), for i = 0, 1, …, ℓ. Since real root spaces of 𝔤 are one dimensional,

r_{i_{1}}^{ad} … r_{i_{t}}^{ad}e_{j} = ce_{i}(n)

for some non-zero scalar c. Now, τ(exp(ady)(x)) = exp(−adτ(y))(τ(x)), whence by an SL_{2}-calculation we have r_{i}^{ad}(x) = τ(r_{i}^{ad}(τ(x))) for all x ∈ 𝔤, we also get that

r_{i_{1}}^{ad} … r_{i_{t}}^{ad}f_{j} = cf_{i}(−n).

But the r_{i}^{ad} preserve the normalized invariant form on 𝔤, so

a_{j}(a_{j}^{∨})^{−1} = (e_{j}|f_{j}) = (ce_{i}(n)|cf_{i}(−n)) = c^{2}a_{i}(a_{i}^{∨})^{−1}.

Clearly, α_{i} and α_{j} are roots of the same length, i.e. a_{j}(a_{j}^{∨})^{−1} = a_{i}(a_{i}^{∨})^{−1}, so this gives that c = ±1. Finally, the action of r_{i}^{ad} on 𝔤 leaves U_{ℤ} ∩ 𝔤 invariant, so

e_{i}(n) = ±r_{i_{1}}^{ad} … r_{i_{s}}^{ad}e_{j} ∈ U_{ℤ},

and similarly f_{i}(n) ∈ U_{ℤ} too. □

For n ≥ 1 and i = 0, 1, …, ℓ, define

y_{nd_{i}}^{(i)} = \frac{α′_{i}(−a_{0}nd_{i})}{a_{0}nd_{i}}, x_{nd_{i}}^{(i)} = ∑_{k_{1}+2k_{2}+···=n} \frac{y_{d_{i}}^{(i)}^{k_{1}}}{k_{1}!} \frac{y_{2d_{i}}^{(i)}^{k_{2}}}{k_{2}!} \frac{y_{3d_{i}}^{(i)}^{k_{3}}}{k_{3}!} · · · .

Observe that

P_{α′_{i}}(z) = exp (∑_{n≥1} y_{nd_{i}}^{(i)}z^{a_{0}nd_{i}} ) = 1 + ∑_{n≥1} x_{nd_{i}}^{(i)}z^{a_{0}nd_{i}}.  (4.4)

The y_{nd_{i}}^{(i)} for n ≥ 1 and i ∈ I give a basis for 𝔱^{−}. So S(𝔱^{−}) is equal to the free polynomial algebra

B := ℂ[y_{nd_{i}}^{(i)} | n ≥ 1, i ∈ I].

Since the x_{nd_{i}}^{(i)} are related to the y_{nd_{i}}^{(i)} in a unitriangular way, we obtain a ℤ-form

B_{ℤ} := ℤ[x_{nd_{i}}^{(i)} | n ≥ 1, i ∈ I] ⊂ B

for B. As α′_{ε} is an integral linear combination of the α′_{i} with i ∈ I, it follows from (4.4) that the elements x_{nd_{ε}}^{(ε)} also belong to the lattice B_{ℤ}. The ℤ-grading on B_{ℤ} induced by the

grading on S(𝔱^{−}) is determined by deg(y_{n}^{(i)}) = deg(x_{n}^{(i)}) = n.

The following theorem (or rather its q-analogue) for the non-twisted case has been proved in [4]. Our argument for the general case is similar.

Theorem 4.5. V_{ℤ} = B_{ℤ} ⊗_{ℤ} ℤ[\overline{Q}].

Proof. Let us first show that B_{ℤ} ⊗_{ℤ} ℤ[\overline{Q}] ⊆ V_{ℤ}. Fix i_{1}, …, i_{s} ∈ I, and let

M(i_{1}, …, i_{s}) = {(n_{1}, …, n_{s}) | n_{1} ≥ ⋯ ≥ n_{s} ≥ 0 and d_{i_{j}}|n_{j} for all j = 1, …, s}.

Denote by > the dominance ordering on partitions belonging to M(i_{1}, …, i_{s}). We will show that x_{n_{1}}^{(i_{1})} … x_{n_{s}}^{(i_{s})} ⊗ e^{β} ∈ V_{ℤ} for all (n_{1}, …, n_{s}) ∈ M(i_{1}, …, i_{s}) and each β ∈ \overline{Q}. Clearly every monomial in B_{ℤ} is of the form x_{n_{1}}^{(i_{1})} … x_{n_{s}}^{(i_{s})} for some choice of i_{1}, …, i_{s} and (n_{1}, …, n_{s}) ∈ M(i_{1}, …, i_{s}), so this is good enough.

To start with, each e_{i}(n), f_{i}(−n) ∈ U_{ℤ} by Lemma 4.3. So an obvious inductive argument using (4.2) gives that 1 ⊗ e^{γ} ∈ V_{ℤ} for each γ ∈ \overline{Q}. Hence, letting γ = β − ᾱ_{i_{1}} − ⋯ − ᾱ_{i_{s}}, Lemma 4.3 implies that all coefficients of e_{i_{1}}(z_{1}) … e_{i_{s}}(z_{s}) ⊗ e^{γ} belong to V_{ℤ}. Applying Lemma 3.5, we deduce that all the coefficients of

X := ( ∏_{1≤u<v≤s} ∏_{k∈ℤ/m}(1 − ω^{−k} \frac{z_{v}}{z_{u}} )^{(μ^{k}(α′_{iu})|α′_{iv})′} ) P_{α′_{i_{1}}} (z_{1}) … P_{α′_{is}} (z_{s}) ⊗ e^{β}

belong to V_{ℤ}. One checks that in all cases,

∏_{k∈ℤ/m}(1 − ω^{−k} \frac{z_{v}}{z_{u}} )^{(μ^{k}(α′_{iu})|α′_{iv})′} = 1 + (∗),

where (∗) is a ℤ-linear combination of (\frac{z_{v}}{z_{u}} )^{p} for p ≥ 1. It follows that the z_{1}^{a_{0}n_{1}} … z_{s}^{a_{0}n_{s}}- coefficient of X equals

x_{n_{1}}^{(i_{1})} … x_{n_{s}}^{(i_{s})} ⊗ e^{β} + (∗∗),

where (∗∗) is a ℤ-linear combination of x_{n′_{1}}^{(i_{1})} … x_{n′_{s}}^{(i_{s})} ⊗ e^{β} for (n′_{1}, …, n′_{s}) > (n_{1}, …, n_{s}).

Using downward induction on this ordering, we deduce that x_{n_{1}}^{(i_{1})} … x_{n_{s}}^{(i_{s})} ∈ V_{ℤ}.

Finally, we prove that B_{ℤ} ⊗_{ℤ} ℤ[\overline{Q}] ⊇ V_{ℤ}. As the high weight vector 1 ⊗ e^{0} belongs to B_{ℤ} ⊗_{ℤ} ℤ[\overline{Q}] ⊇ V_{ℤ}, it suffices to show that B_{ℤ} ⊗_{ℤ} ℤ[\overline{Q}] is invariant under each of the operators f_{i}(n)^{s}/s! for n ∈ ℤ, s ≥ 1 and i = 0, 1, …, ℓ. Fix i ∈ {0, 1, …, ℓ} and consider

Y := f_{i}(z_{1}) … f_{i}(z_{s})P_{α′_{i_{1}}} (w_{1}) … P_{α′_{i_{t}}} (w_{t}) ⊗ e^{γ+s\overline{α}_{i}}

for i_{1}, …, i_{t} ∈ I and γ ∈ \overline{Q}. Applying Lemma 3.5 and simplifying,

Y = ±(z_{1} … z_{s})^{\frac{a_{0}}{2} (α_{i}|α_{i})−1−a_{0}(α_{i}|γ+s\overline{α}_{i})}

× ∏_{1≤u<v≤s} ∏_{k∈ℤ/m}(z_{u} − ω^{−k}z_{v})^{(μ^{k}(α′_{i})|α′_{i})′}

× ∏_{{1≤u≤s \atop 1≤v≤t}} ∏_{k∈ℤ/m}(1 − ω^{−k} \frac{w_{v}}{z_{u}} )^{−(μ^{k}(α′_{i})|α′_{iv})′}

×P_{−α′_{i}}(z_{1}) … P_{−α′_{i}}(z_{s})P_{α′_{i_{1}}} (w_{1}) … P_{α′_{i_{t}}} (w_{t}) ⊗ e^{γ}.

Certainly, each coefficient of P_{−α′_{i}}(z_{1}) … P_{−α′_{i}}(z_{s})P_{α′_{i_{1}}} (w_{1}) … P_{α′_{i_{t}}} (w_{t}) ⊗ e^{γ} belongs to V_{ℤ}. One checks that

∏_{k∈ℤ/m}(z_{u} − ω^{−k}z_{v})^{(μ^{k}(α′_{i})|α′_{i})′} = \{ \begin{matrix}(z_{u} − z_{v})^{2} \frac{(z_{u}+z_{v})^{2}}{z_{u}^{2}+z_{v}^{2}} & if X_{N}^{(r)} = A_{2ℓ}^{(2)} and i = 0, \\ (z_{u}^{a_{0}d_{i}} − z_{v}^{a_{0}d_{i}})^{2} & otherwise. \end{matrix}

Hence in all cases, Y looks like ∏_{1≤u<v≤s}(z_{u}−z_{v})^{2} times an expression that is symmetric in z_{1}, …, z_{s}. Hence, by [4, Lemma 2.5(ii)], the coefficient of (z_{1} … z_{s})^{−n−1} in Y is divisible by s!. Hence, all coefficients of (f_{i}(n)^{s}/s!)P_{α′_{i_{1}}} (w_{1}) … P_{α′_{is}} (w_{s}) ⊗ e^{γ−s\overline{α}_{i}} belong to V_{ℤ}, which completes the proof. □

5. The determinant

Fix now some d ≥ 0. Lemma 4.1 and Theorem 4.5 reduce the problem of computing the determinant of the Shapovalov form on the (wΛ_{0} − dδ) weight space of V_{ℤ} for any w ∈ W to the problem of computing the determinant of the Shapovalov form on the degree d component of B_{ℤ}. To tackle the latter question, observe

B = ⨂_{i∈I} ℂ[y_{d_{i}}^{(i)}, y_{2d_{i}}^{(i)}, …], B_{ℤ} = ⨂_{i∈I} ℤ[x_{d_{i}}^{(i)}, x_{2d_{i}}^{(i)}, …].

So according to [5, Corollary 2.1], there is a well-defined Hermitian form on B determined by the rules

(1, 1)_{K} = 1, (ny_{nd_{i}}^{(i)}f, g)_{K} = (f, \frac{∂}{∂y_{nd_{i}}^{(i)}} g)_{K}

for all i ∈ I, n ≥ 1 and f, g ∈ B. Moreover, there is a homogeneous basis for the lattice B_{ℤ} (given by Schur polynomials) that is orthonormal with respect to the form (. , .)_{K}. In particular, the determinant of the form (. , .)_{K} on the degree d component of B_{ℤ} is equal to 1. Our strategy will therefore be to relate the Shapovalov form (. , .)_{S} to the form (. , .)_{K}.

Define I(n) = {i ∈ I | d_{i}|n}. Introduce the matrices A^{(n)} = (a_{i,j}^{(n)})_{i,j∈I(n)} with

a_{i,j}^{(n)} = \frac{1}{d_{i}} (α′_{i}| ∑_{k=0}^{r−1} ω^{a_{0}nk}μ^{k}(α′_{j}))′.

Recall α, β from the introduction. One verifies:

Lemma 5.1. For any n ≥ 0, we have det A^{(n)} = \{ \begin{matrix}α & if r | n, \\ β & if r ∤ n, \end{matrix}

The significance of the matrices A^{(n)} is that for i ∈ I(n), the element τ(ny_{n}^{(i)}/d_{i}) = α′_{i}(a_{0}n)/a_{0}d_{i} acts on B as the operator ∑_{j∈I(n)} a_{i,j}^{(n)}_{\frac{∂}{∂y_{n}^{(j)}}} . We set

z_{n}^{(i)} = ∑_{j∈I(n)} a_{i,j}^{(n)}y_{n}^{(j)}.

For a partition λ = (λ_{1} ≥ ⋯ ≥ λ_{h} > 0) we let I(λ) = I(λ_{1}) × ⋯ × I(λ_{h}). Given i = (i_{1}, …, i_{h}) ∈ I(λ), let

x_{λ}^{(i)} = x_{λ_{1}}^{(i_{1})} … x_{λ_{h}}^{(i_{h})},  y_{λ}^{(i)} = y_{λ_{1}}^{(i_{1})} … x_{λ_{h}}^{(i_{h})},  z_{λ}^{(i)} = z_{λ_{1}}^{(i_{1})} … x_{λ_{h}}^{(i_{h})},

all elements of B of degree |λ|.

Lemma 5.2. For i ∈ I(λ) and any f ∈ B, we have (y_{λ}^{(i)}, f)_{S} = (z_{λ}^{(i)}, f)_{K}.

Proof. Proceed by induction on the number of non-zero parts of λ, starting induction from the obvious fact that (1, f)_{S} = (1, f)_{K}. For the induction step, note that for i ∈ I(n),

(y_{n}^{(i)}y_{λ}^{(i)}, f)_{S} = (\frac{d_{i}}{n} y_{λ}^{(i)}, ∑_{j∈I(n)} a_{i,j}^{(n)} \frac{∂}{∂y_{n}^{(j)}} f)_{S}

= (\frac{d_{i}}{n} z_{λ}^{(i)}, ∑_{j∈I(n)} a_{i,j}^{(n)} \frac{∂}{∂y_{n}^{(j)}} f)_{K} = (z_{n}^{(i)}z_{λ}^{(i)}, f)_{K}.

□

Let

Ω(λ) = {i ∈ I(λ) | i_{j} ≤ i_{j+1} whenever λ_{j} = λ_{j+1}}.

Then {x_{λ}^{(i)} |λ ∈ 𝒫(d), i ∈ Ω(λ)}, {y_{λ}^{(i)} |λ ∈ 𝒫(d), i ∈ Ω(λ)} and {z_{λ}^{(i)} |λ ∈ 𝒫(d), i ∈ Ω(λ)} give three different bases for the degree d component of B. Consider the transition matrices P = (p_{λ,μ}^{i,j}) and Q = (q_{λ,μ}^{i,j}) where λ, μ ∈ 𝒫(d), i ∈ Ω(λ), j ∈ Ω(μ) defined from

x_{λ}^{(i)} = ∑_{μ∈𝒫(d), j∈Ω(μ)} p_{λ,μ}^{i,j}y_{μ}^{(j)},  z_{λ}^{(i)} = ∑_{μ∈𝒫(d), j∈Ω(μ)} q_{λ,μ}^{i,j}y_{μ}^{(j)}.

Lemma 5.3. The matrix Q is block diagonal, i.e. q_{λ,μ}^{i,j} = 0 for λ ≠ μ. Moreover, the determinant of the λ-block Q_{λ} = (q_{λ,λ}^{i,j})_{i,j∈Ω(λ)} of Q is α^{a_{λ}}β^{b_{λ}}, notation as in the introduction.

Proof. It is immediate from the definition that q_{λ,μ}^{i,j} = 0 for λ ≠ μ. So consider the λ-block Q_{λ} of Q. Represent λ = (λ_{1} ≥ ⋯ ≥ λ_{h} > 0) instead as (1^{r_{1}}2^{r_{2}} … s^{r_{s}}). By definition,

z_{λ}^{(i)} = ∑_{j∈I(λ)} a_{i_{1},j_{1}}^{(λ_{1})} … a_{i_{h},j_{h}}^{(λ_{h})}y_{λ}^{(j)}.

Thus Q_{λ} is the matrix S^{r_{1}}(A^{(1)}) ⊗ ⋯ ⊗ S^{r_{s}}(A^{(s)}), a tensor product of symmetric powers of the matrices A^{(n)}. Now, note that for an n × n matrix A,

det S^{m}(A) = (det A)^{\binom{n+m−1}{n}},

while for an n × n matrix B and an m × m matrix C,

det(B ⊗ C) = (det B)^{m}(det C)^{n}.

These are both proved by reducing to the case that the matrices are diagonal. Combining the formulae with Lemma 5.1, one computes det Q_{λ} = α^{a_{λ}}β^{b_{λ}}. □

Now we can prove the main theorem:

Theorem 5.4. The determinant of the restriction of the Shapovalov form to the degree d part of B_{ℤ} is ∏_{λ∈𝒫(d)} α^{a_{λ}}β^{b_{λ}}.

Proof. Consider the matrices M = (m_{λ,μ}^{i,j}) and N = (n_{λ,μ}^{i,j}) for λ, μ ∈ 𝒫(d), i ∈ Ω(λ), j ∈ Ω(μ) defined from

m_{λ,μ}^{i,j} = (x_{λ}^{(i)}, x_{μ}^{(j)})_{S}, n_{λ,μ}^{i,j} = (x_{λ}^{(i)}, x_{μ}^{(j)})_{K}.

Recalling the transition matrices P and Q introduced above, Lemma 5.2 gives at once that M = P QP^{−1}N. On the other hand, N has determinant 1, since as we observed above the degree d component of B_{ℤ} admits an orthonormal basis with respect to the contravariant form (. , .)_{K} (see [5, Corollary 2.1]). So we can compute det M at once using Lemma 5.3. □

We finally indicate how to deduce the generating functions a(q) and b(q) stated in the introduction. By definition, a_{λ} is

\frac{h}{ℓ} × ( \begin{matrix}the number of ways of coloring \\ the parts λ_{i} ≡ 0 (r) with ℓ colors \end{matrix} ) × ( \begin{matrix}the number of ways of coloring \\ the parts λ_{i} ≢ 0 (r) with k colors \end{matrix} ) where h is the number of λ_{i} ≡ 0 (r). Consider

G(q, t, u) = (∏_{n≥1} \frac{1}{1 − q^{nr}t})^{ℓ} (∏_{n≥1} \frac{1 − q^{nr}u}{1 − q^{n}u})^{k} .

The coefficient of q^{d}t^{h}u^{i} is equal to the number of partitions of d with h parts divisible by r colored by ℓ different colors and with i parts not divisible by r colored by k different colors. Hence the generating function a(q) for a(d) = ∑_{λ∈𝒫(d)} a_{λ} is equal to \frac{1}{ℓ} \frac{d}{dt}G(q, t, u)|_{t=u=1}. Similarly, b(q) = \frac{1}{k} \frac{d}{du}G(q, t, u)|_{t=u=1}.

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Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

E-mail address: brundan@darkwing.uoregon.edu, klesh@math.uoregon.edu