REPRESENTATIONS OF QUANTUM GROUPS

DEFINED OVER COMMUTATIVE RINGS

Ben Cox*and Thomas J. Enright**

Department of Mathematical Sciences, University of Montana, Missoula MT. 59812-1032 ; Department of Mathematics, University of California at San Diego, La Jolla, CA. 92093

Abstract. In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules. The notion of an induced invariant form is introduced and a setting in described where all invariant forms are induced.

§1. Introduction and Summary of Results.

(1.1) Highest weight modules play a central role in the representation theory of semisimple Lie groups and algebras. Their role is quite apparent in the study of the character theory and cohomology of the finite dimensional representations. For the construction of the discrete series group representations and the study of the interplay between the principal series representations and the Jacquet modules, their role is somewhat more indirect. Still in these cases their structure theory governs the behavior of the general theory in many ways. At a somewhat more refined level we can analyze highest weight modules for Lie algebras defined over a commutative ring. This was first done by Shapovalov in his article [S] where the determinant formula for the natural invariant form on a Verma module was computed. An especially important application of the role to be played by this fine structure theory is given by Jantzen in the proof of his Character Sum Formula [J]. Subsequent work using highest weight modules and their Shapovalov forms in this setting is found in

Typeset by AMS-TEX

*This author has been supported in part by MONTS Grant 291533

**This author has been supported in part by NSF Grant DMS92-06941

1991 Mathematics Subject Classification. 17 B67.

the work of Gabber and Joseph [GJ] which concerns the relationships between two Verma modules related by a Weyl group reflection. The recent work by Andersen, Jantzen and Soergel [AJS] on the relationship between reductive algebraic groups in finite characteristic and quantum groups at roots of unity includes among many other points a development of the theory of modules for quantum enveloping algebras defined over commutative algebras.

In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules. Sections two and three set our notation for quantum groups U defined over commutative rings and Verma modules in this setting and present the basics of localization with regard to some distinguished elements of U. Section four contains the first main result, a lifting theorem for invariant bilinear forms from a module to its localization. The next two sections study the functorial properties of this lifting. Results here describe the behavior of tensor products and invariance with regard to an ambient quantum group. These parts may be regarded as a extension of similar results found in [E2] from the Kac-Moody Lie algebra setting to the domain of quantum groups. In section seven we introduce the notion of an induced invariant bilinear form on the tensor product A ⊗ F for an integrable module F and a Verma module A. The main result here asserts that all invariant forms are in fact induced. This gives a natural set of parameters for the space of invariant forms. These results, which are stated in the context of quantum groups of finite type, are new even in the Lie algebra setting.

The role of deformations has for some years now been a useful technique in representation theory. We hope that its role in the representation theory of quantum groups will also make an important contribution.

(1.2) In the remainder of this section we will introduce some basic notation and give a more detailed summary of the results of the article. Readers familiar with the standard notation of Lusztig [L] for quantum groups should go directly to (1.4) and refer back to this section when needed.

A pair (I, ·) where I is a finite set and · denotes a symmetric bilinear form on the free abelian group ℤ[I] with values in ℤ is called a Cartan datum if

i·i ∈ \{2, 4, 6, … \} for any i ∈ I; 2\frac{i · j}{i · i} ∈ \{0, −1, −2, … \} for any i ≠ j ∈ I.

(1.2.1)

We let v denote an indeterminate and if P ∈ ℚ(v) is a rational function then P_{i} denotes P (v_{i}) where v_{i} = v^{i·i/2}. ′f will denote the free associative ℚ(v)- algebra with 1 and generators θ_{i} (i ∈ I). For any ν ∈ ℕ[I] we let ′f_{ν} denote the subspace of ′f spanned by the monomials θ_{i_{1}} ⋯ θ_{i_{r}} where ∑_{j=1}^{r} i_{j} = ν. We say that x ∈′ f is homogeneous if x ∈′ f_{ν} for some ν. In this case we set |x| = ν. Let r :′ f →′ f ⊗′ f denote the unique algebra homomorphism such that r(θ_{i}) = θ_{i} ⊗ 1 + 1 ⊗ θ_{i} for all i. Then there exists a unique ℚ(v)-values bilinear inner product on ′f such that

(θ_{i}, θ_{i}) = δ_{i,j}(1 − v_{i}^{−2})^{−1} for all i, j ∈ I; (x, y′y′′) = (r(x), y′ ⊗ y′′) for all x,

y′, y′′ ∈′ f; and (xx′, y) = (x ⊗ x′, r(y)) for all x, x′, y ∈′ f.  (1.2.2)

The product bilinear form on ′f ⊗′ f is again denoted by ( , ) (cf. [L, 1.2.3]). Let I denote the radical of the form ( , ) on ′f. f is defined to be the quotient algebra ′f/I where the notions of homogeneity and |x| still make sense.

Suppose that (I, ·) is a Cartan datum. If i ≠ j in I are such that (i · i)(j · j) − (i · j)^{2} > 0, define h(i, j) ∈ \{2, 3, 4, … \} by cos^{2} \frac{π}{h(i,j)} = \frac{i·j}{i·i} \frac{j·i}{j·j}. If i ≠ j in I is such that (i · i)(j · j) − (i · j)^{2} ≤ 0 then set h(i, j) = ∞. The braid group defined by (I, ·) is the group defined by generators s_{i} and relations s_{i}s_{j}s_{i} ⋯ = s_{j}s_{i}s_{j} ⋯ with h(i, j) factors for h(i, j) < ∞. The Weyl group W is the quotient of the braid group by the normal subgroup generated by s_{i}^{2} for all i.

Define a root datum of type (I, ·) following Lusztig to be: finitely generated free abelian groups X, Y, and a perfect bilinear pairing ⟨, ⟩ : Y × X → ℤ; an injection I ⊂ X(i ↦ i′) and an injection I ⊂ Y (i ↦ i) such that ⟨i, j′⟩ = 2\frac{i·j}{i·i} for all i, j ∈ I. The root datum is Y -regular if the image of I under the injection I ⊂ Y is linearly independent in Y. We will assume throughout the article that the root datum is Y -regular. Set X^{+} = {λ ∈ X| ⟨i, λ⟩ ∈ ℕ} where ℕ denotes the set of nonnegative integers. If (Y, X, …) is a root datum, then the Weyl group W acts on Y by s_{i}(μ) = μ − ⟨μ, i′⟩i and on X by s_{i}(λ) = λ − ⟨i, λ⟩i′. A Cartan datum (I, ·) is of finite type if the symmetric matrix (i · j) is positive definite.

(1.3) Suppose that the root datum (Y, X, ⟨, ⟩, …) is of type (I, ·). ′U is defined to be the associative ℚ(v)-algebra with 1 defined by generators E_{i}, F_{i}, K_{μ}, where i ∈ I and μ ∈ Y, subject to the relations

K_{0} = 1, K_{μ}K_{ν} = K_{μ+ν} for all μ, ν ∈ Y,  (1.3.1)
K_{μ}E_{i} = v^{⟨μ,i′⟩}E_{i}K_{μ} , K_{μ}F_{i} = v^{−⟨μ,i′⟩}F_{i}K_{μ} for all μ ∈ Y, i ∈ I,

(1.3.2)

E_{i}F_{j} − F_{j}E_{i} = δ_{i,j} \frac{K̃_{i} − K̃_{−i}}{v_{i} − v_{i}^{−1}} for all i, j ∈ I.  (1.3.3)

For μ = ∑_{i} μ_{i}i ∈ ℤ[I], define K̃_{μ} = ∏_{i} K_{(i·i/2)μ_{i}i}, K̃_{±i} = K_{±(i·i/2)i}.

Now define the quantum group U to be the associative ℚ(v)-algebra with 1 defined by the same set of generators and relations as ′U with the additional relation ; for any f(θ_{i}) ∈ I , f(E_{i}) = 0 = f(F_{i}) in U. Here we could have used the equivalent definition in which passing from ′U to U is given by the analogues of the Serre elements (cf. [L,(33.1.3)]).

Define the subalgebra U^{0} of U to be the associative ℚ(v)-subalgebra with 1 generated by K_{μ}, μ ∈ Y. Extend the action of W on Y to all of U^{0} by wK_{μ} = K_{wμ}.

Recall the standard notation for Gaussian binomial coefficients. For n ∈ ℤ, we set

[n] = \left[{n \atop 1}\right] = \frac{v^{n} − v^{−n}}{v − v^{−1}} , [n]^{!} = ∏_{s=1}^{n}[s] and \left[{a \atop t}\right] = \frac{[a]^{!}}{[t]^{!}[a − t]^{!}} . (1.3.4)

For p ∈ ℤ, p ≥ 0 , set θ_{i}^{(p)} = θ_{i}^{p}/[p]_{i}^{!} , otherwise set θ_{i}^{(p)} = 0.

Let ρ_{1} : U → U be the algebra isomorphism determined by the assignment

ρ_{1}(E_{i}) = −v_{i}F_{i}, ρ_{1}(F_{i}) = −v_{i}^{−1}E_{i}, ρ_{1}(K_{μ}) = K_{−μ}  (1.3.5)

for all i ∈ I and μ ∈ Y. Define also an algebra anti-automorphism ϱ : U → U by

ϱ(E_{i}) = v_{i}K̃_{i}F_{i}, ϱ(F_{i}) = v_{i}K̃_{−i}E_{i}, ϱ(K_{μ}) = K_{μ}.  (1.3.6)

These maps are related through the antipode S of U by ϱ = ρ_{1}S . This corrects a misprint in [L,(19.1.1)] where the order is transposed.

For each i ∈ I and e = ±1, Lusztig defines algebra automorphisms of U, T′′_{i,−e} and T′_{i,e} by the formulas: (cf. [L, 37.1.3])

T′_{i,e}(E_{i}) = −K̃_{ei}F_{i}, T′_{i,e}(F_{i}) = −E_{i}K̃_{−ei};
T′_{i,e}(E_{j}) = ∑_{r+s=−⟨i,j′⟩}(−1)^{r}v_{i}^{er}E_{i}^{(r)}E_{j}E_{i}^{(s)} for i ≠ j;
T′_{i,e}(F_{j}) = ∑_{r+s=−⟨i,j′⟩}(−1)^{r}v_{i}^{−er}F_{i}^{(s)}F_{j}F_{i}^{(r)} for i ≠ j;
T′_{i,e}(K_{μ}) = K_{μ−⟨μ,i′⟩i};  (1.3.7)

T′′_{i,−e}(E_{i}) = −F_{i}K̃_{−ei}, T′′_{i,−e}(F_{i}) = −K̃_{ei}E_{i};
T′′_{i,−e}(E_{j}) = ∑_{r+s=−⟨i,j′⟩}(−1)^{r}v_{i}^{er}E_{i}^{(s)}E_{j}E_{i}^{(r)} for i ≠ j;
T′′_{i,−e}(F_{j}) = ∑_{r+s=−⟨i,j′⟩}(−1)^{r}v_{i}^{−er}F_{i}^{(r)}F_{j}F_{i}^{(s)} for i ≠ j;
T′′_{i,−e}(K_{μ}) = K_{μ−⟨μ,i′⟩i}.  (1.3.8)

For any reduced expression w = s_{i_{1}} ⋯ s_{i_{N}} ∈ W, define isomorphisms T′′_{w,e} : U → U and T′_{w,e} : U → U by (cf. [L, 39.4.4]) ; T′_{w,e} = T′_{i_{1},e}T′_{i_{2},e} ⋯ T′_{i_{N} ,e}, and T′′_{w,e} = T′′_{i_{1},e}T′′_{i_{2},e} ⋯ T′′_{i_{N} ,e}.

(1.4) Let R be a commutative algebra over ℚ(v) with no zero divisors, containing U^{0}. Suppose J is a maximal ideal in R for which J ∩ U^{0} is a maximal ideal of U^{0}. Set K equal to the field of fractions of R , k equal to the residue field R/J and let π_{k} denote the projection of R onto k. Then k is an extension field of ℚ(v) and of course frequently they will coincide. Let the subscript R denote the extension of scalars from ℚ(v) to R , i.e. RU = R ⊗_{ℚ(v)} U. For fixed i ∈ I set s = s_{i} and let T = T′_{i,e} or T = T′′_{i,e}. We assume that the involution s of U^{0} extends to an involution of R which we also denote by s. For any representation (π, A) of U we can twist the representation in two ways by composing with automorphisms of _{R}U. The first is π∘(s⊗1) while the second is π∘(1⊗T). We designate the corresponding RU-modules by A^{s} and A^{T}. Twisting the action by both s and T we obtain the composite (A^{s})^{T} = (A^{T})^{s} which we denote by A^{sT}.

(1.5) The principal theme in this article is the extension of the idea of a cycle and the determination of its properties from the setting of semisimple Lie algebras [E2] to that of quantum groups. In both settings we find that these ideas are connected to questions about crossing Weyl chamber walls. If J ⊂ I, let U_{J} denote the Hopf subalgebra of U generated over ℚ(v) by U^{0} and the E_{i} and F_{i} for all i ∈ J. Suppose now that J = {i}. The multiplicative set generated by F := F_{i}, S = {F^{n}| n ∈ ℕ}, is an Ore subset of U so we can localize U with respect to S. Denote these localizations of modules by a subscript F. For any U-module A (resp. U_{i}-module B) we set

A_{F} = U_{F} ⊗_{U} A , B_{F} = U_{i}_{F} ⊗_{U_{i}} B .  (1.5.1)

Then setting A_{π} = A_{F} /ιA we have an exact sequence

A→^{ι} A_{F} →^{π} A_{π} → 0 ,  (1.5.2)

where ι and π are the natural maps.

(1.6) We define a cycle ( for A ) to be a pair (A, Ψ) where A is a U (or RU) module and Ψ is a module homomorphism

Ψ : A_{π}^{sT} → A.  (1.6.1)

We say that the cycle is nondegenerate whenever Ψ is an isomorphism and that A admits a nondegenerate cycle when such a pair exists with Ψ an isomorphism. Remarks. (i) The modules A_{π} and A_{π}^{sT} are equal as sets but in general very different as _{R}U-modules. So with π as in (1.5.2) the map Ψ ∘ π is not a homomorphism of _{R}U-modules in general. (ii) Let Z be the center of U. We say a module A has infinitesimal character χ : Z → R, where χ is an algebra homomorphism, if for all a ∈ A and z ∈ Z one has z.a = χ(z)a. If A is a _{R}U-module with infinitesimal character, then A_{F} has the same infinitesimal character as does A_{π}. But then in general A_{π}^{sT} will have a different infinitesimal character and so (A, Ψ) with Ψ = 0 will be the only cycle for A. In section three we prove there are an ample supply of nondegenerate cycles.

These properties of cycles will be used to introduce an action of the braid group on certain spaces of invariant bilinear forms.

(1.7) The triangular decomposition of the quantum group leads as in the enveloping algebra case to the basic induced modules, Verma modules. We let ρ ∈ X be an element that satisfies ⟨i, ρ⟩ = 1 for all i ∈ I. In the case of positive definite Cartan matrix for (I, ·), 2ρ is the sum of positive roots, cf. [L, 2.3.1]. We assume that there is such an element ρ in X. Let ι : U^{0} → R denote the canonical inclusion and let m denote the algebra homomorphism of _{R}U^{0} onto R given by multiplication m(r ⊗ K_{μ}) = ι(K_{μ})r. For any λ ∈ X there is a unique _{R}U-module structure on the R-module _{R}f such that for any homogeneous y ∈ _{R}f, μ ∈ Y and any i ∈ I, we have K_{μ}(y) = ι(K_{μ})v^{⟨μ,λ−ρ−|y|⟩}y, F_{i}(y) = θ_{i}y, and E_{i}(1) = 0 (cf. [L, 3.4.5]). Thus _{R}U^{0} acts through the algebra homomorphism m + λ − ρ. We denote this module by _{R}M_{m+λ} and call it the Verma module of highest weight m + λ − ρ. This notation is a slight departure from Lusztig’s as ours involves a shift by −ρ whereas his does not.

Let U^{+} (resp. U^{−}) denote the subalgebra generated by E_{i} (resp. F_{i}) i ∈ I and let B be the subalgebra generated by U^{+} and U^{0}. For any λ ∈ Y, let R_{λ−ρ} denote the free rank one R-module which we make into a _{R}U^{0}-module by letting _{R}U^{0} act through the homomorphism m + λ − ρ. We let R_{m+λ−ρ} denote this new _{R}U^{0}-module. In particular r ⊗ K̃_{i} acts by left multiplication by ι(K̃_{i})v_{i}^{⟨i,λ−ρ⟩}r. Now for each w ∈ W and T = T′_{w,e} or T = T′′_{w,e} define Verma modules alternatively by the formula

_{R}^{T}M_{m+λ} = _{R}U ⊗_{T (_{R}B)} R_{m+λ−wρ}.  (1.7.1)

where T (E_{j})R_{m+λ−ρ} = 0 for all j ∈ I. When T equals the identity _{R}^{T}M_{m+λ} = RM_{m+λ} as defined in the previous paragraph.

(1.8) Let M be a _{R}U-module. We say that M is _{R}U^{0}-semisimple if M is the direct sum of R-modules M_{m+μ} where K̃_{i} acts by ι(K̃_{i}) v_{i}^{⟨i,μ⟩} ;i.e. by weight m + μ . Let C_{R} (resp. C_{K} ) denote the category of _{R}U (resp. _{K}U)) modules M for which (i) E_{i} acts locally nilpotently for all i and (ii) M is RU^{0}-semisimple with all nonzero weight spaces having weights m + μ with μ ∈ X and ⟨i, μ⟩ ∈ ℤ for all i ∈ I. Let C (resp. C_{k}) denote the corresponding category of U (resp. _{k}U)-modules where condition (i) is as above and, in condition (ii), K̃_{i} acts by v_{i}^{⟨i,μ⟩} (resp. π_{k} ∘ (ιK̃_{i}v_{i}^{⟨i,μ⟩})). For any category A whose objects are R-modules we let A_{f} designate the subcategory (necessarily not abelian) of A whose objects are the free R-modules in A. For any positive integer j and element r ∈ R we set

[r]_{(0)} = 1 , [r]_{(j)} = [r][r − 1]...[r − j + 1] , \left[{r \atop j}\right] = \frac{[r]_{(j)}}{[j]^{!}} .  (1.8.1)

For r ∈ ℤ and j ∈ ℕ define as elements of U^{0}

[K̃_{i}; r] := \frac{v_{i}^{r}K̃_{i} − v_{i}^{−r}K̃_{i}^{−1}}{v_{i} − v_{i}^{−1}} and [K̃_{i}; r]_{(j)} := [K̃_{i}; r][K̃_{i}; r−1] ⋯ [K̃_{i}; r−j+1].

(1.8.2)

(1.9) Fix i ∈ I. Let _{R}U_{i}^{+} (resp._{R}U_{i}^{−}) denote the subalgebra of _{R}U generated over R by E_{i} (resp. F_{i}) and let _{R}B_{i} be the subalgebra generated by RU_{i}^{+} and _{R}U^{0}. _{R}U_{i} is defined to be the subalgebra of _{R}U generated by _{R}U^{0}, E_{i} and F_{i}. We will assume ρ_{i} ∈ X satisfies ⟨j, ρ_{i}⟩ = δ_{i,j} for all j ∈ I, where δ_{i,j} is the Kronecker delta. By letting E_{i} act by zero we can make R_{m+λ−ρ_{i}} into a _{R}B_{i}-module. We define also the Verma module

M_{i,m+λ} =_{R} U_{i} ⊗_{_{R}B_{i}} R_{m+λ−ρ_{i}}.

Define categories C_{R}^{i} (resp. Ci_{K} , C^{i} , Ci_{k}) exactly as in (1.8) for the categories C_{R} (resp. C_{K} , C , C_{k}) with the subalgebra _{R}U_{i} replacing _{R}U in the definition.

For_{R}U-modules A and B let ℙ_{ϱ}(A, B) denote the set of all R-bilinear maps ϕ : A × B → R which have the invariance property; ϕ(ua, b) = ϕ(a, ϱu b) , for a ∈ A, b ∈ B and u ∈ _{R}U. Our first result is what we call the lifting theorem for invariant forms. Although it is a precise analogue of the corresponding theorem in the enveloping algebra setting (cf. [E2]) this cleaner formulation through identity (1.9.1) is new even in the Lie setting.

THEOREM (Lifting Theorem). Set F = F_{i} with i ∈ I fixed. Let A and B be modules in C_{R}^{i} and ϕ ∈ ℙ_{ϱ}(A, B). Then ϕ uniquely determines an invariant form ϕ_{F} ∈ ℙ_{ϱ}(A_{F} , B_{F}) which is determined by the following properties:

(i) ϕ_{F} vanishes on the subspaces ιA × B_{F} and A_{F} × ιB .

(ii) For each μ ∈ X with ⟨i, μ⟩ + 1 = r ∈ ℕ, and any vectors a ∈ A and b ∈ B both of weight m + ϵμ with ϵ ∈ {1, s_{i}} and E_{i} a = E_{i} b = 0 ,

ϕ_{F} (F^{−1}a, F^{−1}b) = v_{i}^{−r+1} \frac{ιϵ[K̃_{i}; 0]}{ιϵ[K̃_{i}; r − 1]} ϕ(a, b) .  (1.9.1)

COROLLARY. ϕ_{F} induces an _{R}U_{i}-invariant bilinear map on A_{π} × B_{π} which we denote by ϕ_{π}.

(1.10) The next result indicates that although liftings were defined by the RU_{i} identities in (1.9.1), liftings preserve U-invariance.

THEOREM (Invariance Theorem). Suppose A and B are objects in C_{R}. Then ϕ_{F} and ϕ_{π} are _{R}U-invariant; i.e. ϕ_{F} ∈ ℙ_{ϱ}(A_{F} , B_{F}) , ϕ_{π} ∈ ℙ_{ϱ}(A_{π}, B_{π}) .

(1.11) The two theorems above are the first two of three main results of this article. To indicate some of their significance as well as to link them with our third result we now briefly describe their connection with the notion of a cycle and preview some of the results of the article which follows this. Suppose ϕ is an invariant form on a U_{R}-module A as in Theorem 2 and A admits a nondegenerate cycle (A, Ψ). We obtain a second invariant form on A denoted ϕ_{Ψ} through the formula: for a, b ∈ A_{π} ,

ϕ_{Ψ}(Ψa, Ψb) = sϕ_{π}(a, b) .  (1.11.1)

Under suitable normalizations this action ϕ → ϕ_{Ψ} induces an action of the braid group on the space of invariant forms. We describe the symmetry properties satisfied by invariant forms which transform by a character under this action in a subsequent article. These symmetry properties will have something to say about crossing Weyl chamber walls.

(1.12) These symmetries are presented in the context of what we call induced forms. Let E and F be finite dimensional U-modules and τ a Umodule homomorphism into U; τ :_{R} E ⊗_{R} F^{ρ_{1}} → U. Suppose ϕ is a pairing of M and N. Define ψ_{τ,ϕ} to be the invariant pairing of M ⊗_{R} E and N ⊗_{R} F defined by the formula, for e ∈ E, f ∈ F, m ∈ M, and n ∈ N,

ψ_{τ,ϕ}(m ⊗ e, n ⊗ f) = ∑ ϕ(τ(e_{ij} ⊗ f_{ij})m_{ij}, n_{j}) ,  (1.12.1)

where we transpose the entries using the Drinfeld ”R-matrix” and notation as in [L,32.1.5] and section seven below; _{f}R(n ⊗ f) = ∑ f_{j} ⊗ n_{j} and _{f}R(m ⊗ e ⊗ f_{j}) = ∑ e_{ij} ⊗ f_{ij} ⊗ m_{ij}. Here F^{ρ_{1}} is a twist of the representation F by ρ_{1}. We call the pairing ψ_{τ,ϕ} the pairing induced by τ and ϕ. In the cases when M, N and ϕ are fixed we write ψ_{τ} in place of ψ_{τ,ϕ} and say this pairing is induced by τ. Roughly speaking the pairing induced by τ is the pullback by τ of the pairing of M and N by ϕ.

In the setting of Verma modules this notion of induced form gives the natural set of parameters for invariant forms.

PROPOSITION. Suppose U is of finite type and E and F are finite dimensional U-modules. Let M be an _{R}U-Verma module and ϕ the Shapovalov form on M. Suppose the Shapovalov form on M is nondegenerate. Then every invariant pairing of M ⊗_{R} E and M ⊗_{R} F is induced by ϕ.

The proof of this result is a consequence of the recent work by Joseph and Letzter (cf. [JL2,3]) on annihilators of Verma modules for quantum groups.

§2. Notation and Some Preliminary Lemmas.

(2.1) The comultiplication, counit and antipode of U are defined on generators by

Δ(E_{i}) = E_{i} ⊗ 1 + K̃_{i} ⊗ E_{i}, Δ(F_{i}) = F_{i} ⊗ K̃_{−i} + 1 ⊗ F_{i}, Δ(K_{μ}) = K_{μ} ⊗ K_{μ}

e(E_{i}) = e(F_{i}) = 0, e(K̃_{i}) = 1
S(E_{i}) = −K̃_{−i}E_{i}, S(F_{i}) = −F_{i}K̃_{i}, S(K_{μ}) = K_{μ}^{−1},

for all μ ∈ Y and i ∈ I.

(2.2) We keep in force the notation and conventions from section one. Fix i ∈ I and set K = K̃_{i} and s = s_{i}. Some of our calculations will require from time to time a choice of basis for M_{i,m+λ}. Set n = ⟨i, λ⟩ + 1 ∈ ℕ^{∗} and define indexing sets I_{λ} and I_{sλ} by I_{λ} = {n−2, n−4, ...} and I_{sλ} = {−n, −n−2, ...}.

LEMMA. Now for integers j ∈ I_{λ} (resp. I_{sλ}) set k_{j} = \frac{n−2−j}{2} and l_{j} = \frac{−n−j}{2} and define basis vectors for M_{i,m+λ} and M_{i,m+sλ} by w_{λ,j} = F_{i}^{k_{j}} ⊗ 1_{m+λ−ρ} and w_{sλ,j} = F_{i}^{l_{j}} ⊗ 1_{m+sλ−ρ}. The action of _{R}U_{i} is given by

K_{μ}w_{λ,j} = ιK_{μ}v^{⟨μ,λ−ρ−k_{j}i′⟩}w_{λ,j} , F_{i}w_{λ,j} = w_{λ,j−2} ,
K_{μ}w_{sλ,j} = ιK_{μ}v^{⟨μ,sλ−ρ−l_{j}i′⟩}w_{sλ,j} , F_{i}w_{sλ,j} = w_{sλ,j−2} ,  (2.2.1)
E_{i}w_{λ,j} = [k_{j}]_{i}[ιK; −l_{j}]w_{λ,j+2} , E_{i}w_{sλ,j} = [l_{j}]_{i}[ιK; −k_{j}]w_{sλ,j+2} .

Proof. The first four identities are clear while the last two follow from the basic relation

[E_{i}, F_{i}^{r}] = [r]_{i}F_{i}^{r−1}[K; −(r − 1)]  (2.2.2)

for all r ≥ 1 (see [L, 3.1.9]). □

(2.3) We define the Casimir element Ω_{0} of _{R}U_{i} by

Ω_{0} = F_{i}E_{i} + \frac{v_{i}K̃_{i} − 2 + v_{i}^{−1}K̃_{i}^{−1}}{(v_{i} − v_{i}^{−1})^{2}} .  (2.3.1)

Then Ω_{0} acts on the Verma modules M_{i,m+λ} and M_{i,m+sλ} respectively by the scalars c(λ) and c(sλ) defined by the expressions:

c(λ) = \frac{v_{i}^{n−1}ιK̃_{i} − 2 + v_{i}^{−n+1}ιK̃_{i}^{−1}}{(v_{i} − v_{i}^{−1})^{2}} and c(sλ) = \frac{v_{i}^{−n+1}ιK̃_{i} − 2 + v_{i}^{n−1}ιK̃_{i}^{−1}}{(v_{i} − v_{i}^{−1})^{2}} .

(2.4) Set I_{∘} equal to the ideal in R generated by [ιK; 0] and let R_{∘} = R/I_{∘}.

LEMMA. Assume that the elements [ιK̃_{i}; r], r ∈ ℤ^{∗}, are units in R. Suppose λ ∈ X^{+} with ⟨i, λ⟩ ∈ ℕ^{∗}. Set M =_{R} M_{m+λ} and M_{s} =_{R} M_{m+sλ}. Then there exists a nonsplit extension P =_{R} P_{m+λ} with short exact sequences;

0 → M → P → M_{s} → 0 ,
0 → M + M_{s} → P →_{R_{∘}} M_{m+sλ} → 0 .

Moreover P is a free R-module with basis {[ιK̃_{i}; 0]w_{λ,j} : j ∈ I_{λ}}∪{a_{j} : j ∈ I_{sλ}} . The action of _{R}U_{i} is given on the w_{λ,j} by the identities in (2.2) and the action on the a_{j} is given as follows:

K_{μ}a_{j} = ιK_{μ} v^{⟨μ,sλ−ρ−l_{j}i′⟩}a_{j} , F_{i}a_{j} = a_{j−2}
E_{i}a_{j} = [l_{j}][ιK̃_{i}; −k_{j}]a_{j+2} + [k_{j} − l_{j}][ιK̃_{i}; 0]w_{λ,j+2} .

Proof. Let U(i) denote the R-subalgebra of U generated by _{R}B and F_{i}. Since tensoring by U over U(i) is an exact functor and since this operation carries the _{R}U_{i}-module _{R}M_{i,m+λ} to the _{R}U-module _{R}M_{m+λ−ρ_{i}+ρ} and _{R}M_{i,m+sλ} to M_{m+sλ−ρ_{i}+ρ} it is sufficient to prove the lemma for _{R}U_{i}.

Set A = M ⊕ M_{s} and for j ∈ I_{sλ}, set a_{j} = (w_{λ,j}, w_{sλ,j}) and let P denote the _{R}U module generated by a_{−n}. Since the generator is a weight vector, P is spanned by the monomial powers of E_{i} and F_{i} acting on a_{−n} (see 3.4). From (2.2) we obtain the identities, for t ∈ I_{sλ} ,

E_{i}a_{t−2} = [l_{t−2}][ιK; −k_{t−2}]a_{t} + [k_{t−2} − l_{t−2}][ιK_{i}; 0]w_{λ,t} ,
E_{i}a_{−n} = [−n + 1][ιK; 0]w_{λ,−n+2} and F_{i}a_{t} = a_{t−2} .

From these, our hypothesis about units and (2.2) we conclude that P contains I_{∘}A and is spanned as an R-module by this subspace and the vectors a_{j} for j ∈ I_{sλ}. Moreover for each j ∈ I_{sλ} the (m + λ − ρ − k_{j}i′)-th weight subspace of P is a free R-module with basis {[ιK̃_{i}; 0]w_{λ,j} , a_{j}}. For j ∈ I_{λ} \ I_{sλ}, the (m + λ − ρ − k_{j}i′)-th weight subspace of P is a free R-module with basis {[ιK̃_{i}; 0]w_{λ,j}}.

Set P_{1} equal to the _{R}U module generated by the vector [ιK; 0]w_{λ,n−2}. Then P_{1} is isomorphic to M and the quotient P/P_{1} is generated by \overline{a_{−n}}, the image of a_{−n}, which is a highest weight vector of weight m + sλ − ρ. This vector is torsion free under the action of F_{i} and so P/P_{1} ≃ M_{s}. This gives the first short exact sequence.

With no zero divisors in R clearly I_{∘}A ≃ M ⊕ M_{s} and from our hypothesis regarding units this submodule is contained in P. For each t ∈ I_{sλ} the m + sλ − ρ − l_{t}i′ weight subspace of P is a rank two free R module with basis a_{t} and [ιK_{i}; 0]w_{sλ,t}. Since R has no zero divisors the quotient of this R module by the submodule with basis [ιK_{i}; 0]w_{λ,t} and [ιK_{i}; 0]w_{sλ,t} is isomorphic to R/I_{∘}. In turn the image of a_{−n} in P/I_{∘}A is a highest weight vector for the quantized enveloping algebra _{R_{∘}}U and we find that P/I_{∘}A ≃ _{R^{0}}M_{m+sλ−ρ}. This this gives the second exact sequence. The identities in (2.2) complete the proof. □

(2.5) Essentially the same proof as in (2.4) goes through to prove that T M = _{R}^{T}M_{m+λ} and^{T} M_{s} = _{R}^{T}M_{m+sλ} have a nonsplit extension^{T} P = _{R}^{T}P_{m+λ}.

(2.6) Let Z denote the center of the quantized enveloping algebra _{R}U. For each z ∈ Z and λ ∈ X, we have left multiplication by z acts by a scalar, χ_{λ}(z), on M_{m+λ}. This defines an algebra homomorphism χ_{λ} : Z → R. Set J_{χ_{λ}} = Ann_{Z}M_{m+λ}. For any U-module M and any λ ∈ X, set M_{χ_{λ}} = {m ∈ M| J_{χ_{λ}}m = 0}. On the other hand J_{χ_{λ}} has codimension one in Z so that J_{χ_{λ}} + J_{χ_{μ}} = ℤ for J_{χ_{λ}} ≠ J_{χ_{μ}}.

LEMMA [GJ, 1.4.2 and 1.4.3]. Let A be an object in C_{R}. Then A is the direct sum of submodules A = ∑_{χ_{λ}} A_{χ_{λ}}.

Remark: One can be more specific about the decomposition in the case that (I, ·) is of finite type (see [JL3, 4.6]).

(2.7) LEMMA. (i) Suppose A is a module in C_{K}^{i}. Then A is the direct sum of submodules A_{t} which are isomorphic to Verma modules of the form _{K}M_{m+μ} with ⟨i, μ⟩ ∈ Z.

Proof. This decomposition is merely the decomposition into generalized eigenspaces for_{K}Z since the_{K}M_{m+μ} are irreducible with distinct infinitesimal characters (cf. (2.3), [DK, 1.9.(a)]).

(2.8) If τ is an anti-automorphism of _{R}U and A is any _{R}U^{0}-semi-simple RU-module define the dual module (or τ-dual if more precision is required) A^{τ} to be the R-module

A^{τ} = ⨁_{μ∈_{R}U^{0∗}} Hom_{R}(A_{μ}, R)

with action given by x · f(a) = f(τ(x)a) for x ∈ _{R}U and a ∈ A.

On the other hand, if τ is an automorphism of_{R}U and A is any_{R}U-module, let A^{τ} be the same R-module as A but with _{R}U-action given by x·a = τ(x)a, for x ∈ _{R}U and a ∈ A.

(2.9) As a convenience we record here the standard identities for the action of finite dimensional U_{i}-modules. For n ∈ ℕ , set I_{n} = {j ∈ ℤ| − n ≤ j ≤ n , j ≡ n mod 2}.

LEMMA. Suppose μ ∈ X , n ∈ ℕ. Then there is an irreducible U_{i}–module F_{n} with highest weight μ and ⟨i, μ⟩ = n, having a basis {f_{n,j}}, j ∈ I_{n}, of F_{n} with the properties: for j ∈ I_{n} and t = \frac{n−j}{2} , F_{i}f_{n,j} = f_{n,j−2} , (j > −n), F_{i}f_{n,−n} = 0 , Kf_{n,j} = v_{i}^{j}f_{n,j}, and E_{i}f_{n,j} = [t]_{i}[n + 1 − t]_{i}f_{n,j+2}.

Remark: Suppose F is an irreducible finite dimensional U_{i}-module with a highest weight vector w. Applying the relation (2.2.2) to w we see that K acts through multiplication by ±v^{n} for some nonnegative integer n. Consequently there are exactly two non-isomorphic irreducible modules of dimension n + 1 for what Lusztig calls the simply connected quantum analogue of sl(2) . In particular tensoring F_{n} in the lemma above with one dimensional modules gives us the other irreducible U_{i}-modules of dimension n + 1.

§3. Localization

(3.1) Localizations and rings of fractions are standard constructions in both commutative and noncommutative ring theory. We now consider a role for localization in the theory of U_{i} modules and more generally modules for a quantized enveloping algebra.

(3.2) Let S be a multiplicative subset of _{R}U. We say that S is an Ore set if (i) for each x in _{R}U and s in S there is an s_{1} in S and y in _{R}U with s_{1}x = ys and (ii) if xs = 0 for some x in _{R}U and s in S then there exists s_{1} in S with s_{1}x = 0. A localization of _{R}U with respect to S is a ring Q and a ring homomorphism ν : _{R}U → Q such that (i) νs is a unit for each s in S and ker ν = {x ∈ _{R}U : sx = 0 for some s ∈ S} and (ii) every element in Q has the form (νs)^{−1}νx. When elements of S act without torsion as is almost always the case in this article ν is an injection and we shall frequently drop the symbol ν and view _{R}U as a subring of Q.

(3.3) Keeping with the notation from the previous sections i ∈ I. Let S denote the set of positive integral powers of F_{i}. Using (1.3), (2.2.2) and the quantum Serre relations for U, one can check that S is an Ore set. Let ν : _{R}U → _{R}U_{F} denote a localization of _{R}U with respect to S. By our convention we consider _{R}U as a subring of _{R}U_{F}. For any _{R}U module M define the localization M_{F} of M by M_{F} = _{R}U_{F} ⊗_{_{R}U} M.

LEMMA. _{R}U_{F} is a flat _{R}U-module and so the map M ↦ M_{F} is exact on the category of _{R}U-modules.

(3.4) The version of the Poincaré-Witt-Birkhoff Theorem in the setting of RU_{i} is:

LEMMA. Let F = F_{i}, E = E_{i} and K_{μ_{1}}, …, K_{μ_{l}} be a ℤ-basis of Y. The monomials

F^{m}K_{μ_{1}}^{n_{1}} ⋯ K_{μ_{l}}^{n_{l}}E^{p},

where m, n_{i} ∈ ℤ and p ∈ ℕ, forms a basis for _{R}U_{i}_{F} as a free R-module.

Proof. Clearly these monomials span the localization. Suppose there is an R-linear relation among them. Then multiplying on the left by a suitably high power of F will produce a relation within _{R}U_{i}. So the PWB Theorem of [JL1], [L], or [R] implies this relation is trivial and since F acts injectively on the localization we conclude that the original relation was trivial. □

(3.5) We now record the effects of this localization on Verma modules for _{R}U = _{R}U_{i}. Recalling the notation of (2.2) suppose I = {i} and set M = M_{i,m+λ} , M_{s} = M_{i,m+sλ} and n = ⟨i, λ⟩ + 1. Let ℤ_{λ} denote the even or odd integers depending as n is even or odd. Then for any j in ℤ_{λ} let k_{j} and l_{j} be given by the formulas in (2.2); i.e. k_{j} = \frac{n−2−j}{2} and l_{j} = \frac{−n−j}{2} . Then the formulas in (2.2) are meaningful for all j in ℤ_{λ} if interpreted as elements of M_{F} = _{R}U_{F} ⊗_{_{R}U} M and M_{sF} = _{R}U_{F} ⊗_{_{R}U} M_{s}.

LEMMA. Let E = E_{i}, K = K̃_{i}, F = F_{i}, M = M_{i,m+λ} , M_{s} = M_{i,m+sλ} and let other notation be as above. Then with w_{λ,j} = F^{k_{j}} ⊗ 1_{m+λ−ρ} and w_{sλ,j} = F^{l_{j}} ⊗ 1_{m+sλ−ρ} for all j in ℤ_{λ}, the sets {w_{λ,j} : j ∈ ℤ_{λ}} and {w_{sλ,j} : j ∈ ℤ_{λ}} are bases for the localizations M_{F} and M_{sF} respectively. Moreover the action of _{R}U_{i} is given by the formulas below, for all j ∈ ℤ_{λ} and all i ∈ I,

K_{μ}w_{λ,j} = ιK_{μ}v^{⟨μ,λ−ρ−k_{j}i′⟩}w_{λ,j} , F w_{λ,j} = w_{λ,j−2}
K_{μ}w_{sλ,j} = ιK_{μ}v^{⟨μ,sλ−ρ−l_{j}i′⟩}w_{sλ,j} , F w_{sλ,j} = w_{sλ,j−2}
Ew_{λ,j} = [k_{j}][ιK; −l_{j}]w_{λ,j+2} , Ew_{sλ,j} = [l_{j}][ιK; −k_{j}]w_{sλ,j+2}.

Proof. By the previous lemma the set of monomials F^{m}K_{μ_{1}}^{n_{1}} ⋯ K_{μ_{l}}^{n_{l}}E^{p} where m, n_{i} ∈ ℤ, and p ∈ ℕ, is an R basis for _{R}U_{F}. Then since both M and M_{s} are free modules for ℂ[F], the sets {F^{j}⊗1_{m+λ−ρ}|j ∈ ℤ} and {F^{j}⊗1_{m+sλ−ρ}|j ∈ ℤ} give R bases for M and M_{s}. This implies the assertion regarding bases. Since the commutator is a derivation we obtain : for all r ∈ ℤ, 2.2.2 implies

EF^{r} ≡ [r]F^{r−1}[K; −r + 1] mod U_{i}E,  (3.5.1)

from which the identities follow. □

(3.6) COROLLARY. Let P = P_{m+λ} and for j ∈ ℤ_{λ} , set a_{j} = w_{λ,j} +w_{sλ,j}. Then the set {[ιK]w_{λ,j} : j ∈ ℤ_{λ}} ∪ {a_{j} : j ∈ ℤ_{λ}} is an R basis for the localization P_{F}. Also the action of _{R}U_{i} is given by the formulas in (3.5) as well as the formulas : for all indices j ∈ ℤ_{λ} ,

Ka_{j} = ιKv^{⟨i,sλ−ρ−l_{j}i′⟩}a_{j} , F a_{j} = a_{j−2}
Ea_{j} = [l_{j}][ιK; −k_{j}]a_{j+2} + [k_{j} − l_{j}][ιK; 0]w_{λ,j+2} .

Proof. The formulas established in (3.5) imply that the action described for P in (2.4) extends to all indices j ∈ ℤ_{λ}. □

(3.7) For convenience in later sections we summarize some of the calculations which follow from the formulas above. For ϵ ∈ {1, s}, r ∈ ℤ_{λ} and j ∈ ℕ define coefficients c[ϵλ, r, j] by the formula;

E_{i}^{j}w_{ϵλ,r} = c[ϵλ, r, j]w_{ϵλ,r+2j}.  (3.7.1)

Then from the identities (1.10.1) and (3.5) we obtain:

LEMMA. Let s = r + 2j. Then c[λ, r, j] = [\frac{n−r}{2} − 1]_{(j)} [ιK; \frac{n+s}{2} − 1]_{(j)} and c[sλ, r, j] = [\frac{n+s}{2} − 1]_{(j)} [ιK^{−1}; \frac{n−r}{2} − 1]_{(j)} .

We extend the definition of the constants c[·, ·, ·] for some other indices as follows: for r ∈ ℤ_{λ} and j ∈ ℕ set s = r + 2j and define c[ϵλ, s, −j] = c[ϵλ, r, j]^{−1} whenever c[ϵλ, r, j] is nonzero.

(3.8) For any study of induced modules the universal properties of the enveloping algebra are an essential ingredient in many of the proofs. With this in mind we next discuss some of the universal properties of the localization RU.

An _{R}U-module W is _{R}U^{0}-admissible (resp. X-admissible) if it is a direct sum of finite rank free R-submodules on which each K_{j} acts by a scalar of the form ιK_{j}v^{⟨j,λ⟩} (resp. v^{⟨j,λ⟩}) for some λ ∈ X. We define an X-multiplicative subset of _{R}U^{±} to be a multiplicative subset S of _{R}U^{±} generated by homogeneous elements. Then for example F in Corollary 2.9 and any _{R}U-submodule of _{R}U under the adjoint action ad u(b) = ∑ u_{(1)}bS(u_{(2)}) is X-admissible while M_{m+λ} is _{R}U^{0}-admissible. All tensor products below will be over R unless otherwise stated.

LEMMA. Let S = {F_{i}^{n}| n ∈ ℕ} so that S is a X-multiplicative subset of RU^{−}. Let W be an _{R}U^{0}-admissible S-divisible (resp. S-torsion free) _{R}Umodule, and let E be an X-admissible _{R}U-module that is free of finite rank as an R-module. Then W ⊗ E and E ⊗ W are _{R}U^{0}-admissible S-divisible (resp. S-torsion free) _{R}U-modules.

Proof. The proof is almost identical to that of [C, 4.14 & 4.15] and is thus omitted. □

If H is a Hopf algebra as in section 2 and M and N are two H-modules, then there is a canonical H-module structure on Hom_{R}(M, N) given by

(aϕ)(m) = ∑ a_{(1)}ϕ(S(a_{(2)})m)

for all m ∈ M and a ∈ H where Δ(a) = ∑ a_{(1)} ⊗ a_{(2)}. We also make R into an H-module via the augmentation e; u.r = e(u)r.

3.9 LEMMA. (a) If E and W are _{R}U-modules with E a free R-module of finite rank, then Hom_{R}(E, W) ≅ W⊗E^{∗} . (b) If M, N and P are_{R}U-modules then Hom_{_{R}U}(M ⊗ N, P) ≅ Hom_{_{R}U}(M, Hom_{R}(N, P)) as R-modules.

THEOREM. (Mackey Isomorphism Theorem) Suppose that V and E are RU-modules with E an X-admissible free R-module of finite rank and V, _{R}U^{0}- admissible. Let S = {F_{i}^{n}| n ∈ ℕ}. Then we have a natural isomorphism S^{−1}(V ⊗ E) ≅ (S^{−1}V) ⊗ E of _{R}U-modules. In particular the isomorphism is induced by

F^{−r} ⊗ a ⊗ e ↦ ∑_{j∈ℕ}(−1)^{j}\left[{j + r − 1 \atop r − 1}\right]_{i}v_{i}^{−j(j+r)} F^{−j−r}a ⊗ F^{j}K^{j+r}e , (3.9.1)

and for r = 0, 1 ⊗ a ⊗ e ↦ a ⊗ e .

Proof. Let W be an _{R}U^{0}-admissible S^{−1}_{R}U-module. Now consider the following sequence of isomorphisms:

Hom_{S^{−1}_{R}U}(S^{−1}(V ⊗E), W) ≅ Hom_{_{R}U}(V ⊗E, W) ≅ Hom_{_{R}U}(V, Hom_{R}(E, W))

≅ Hom_{S^{−1}_{R}U}(S^{−1}V, Hom_{R}(E, W)) ≅ Hom_{S^{−1}_{R}U}(S^{−1}V ⊗ E, W) .

The first isomorphism is just the universal property of the module of quotients (see [GW]). The second and fourth isomorphisms are derived from the adjoint associativity property of Hom given in Lemma 3.9.(b). Now an _{R}U-module is an S^{−1}_{R}U-module if and only if it is S-divisible and Storsion free (see [GW, 9.12]). Thus Lemma 3.8 and Lemma 3.9.(a) imply that Hom_{R}(E, W) is a S^{−1}_{R}U-module so that the third map is an isomorphism. We now proceed as in [Kn, 5.14] and let W = S^{−1}(E ⊗ V). The identity in Hom_{S^{−1}_{R}U}(S^{−1}(E ⊗ V), S^{−1}(E ⊗ V)) induces a map in Hom_{S^{−1}_{R}U}(E ⊗ S^{−1}V, S^{−1}(E⊗V)) which we will denote by φ. Similarly the identity morphism in Hom_{S^{−1}_{R}U}(E ⊗ S^{−1}V, E ⊗ S^{−1}V) induces a map ψ ∈ Hom_{S^{−1}_{R}U}(S^{−1}(E ⊗ V), E ⊗ S^{−1}V). One can check as in [Kn, 5.14] that ψ ∘ φ = 1 and φ ∘ ψ = 1. Now [Kn, 5.14] tells us that the isomorphism is induced by 1 ⊗ a ⊗ e ↦ 1 ⊗ a ⊗ e. Then 3.9.1 is proven by induction on r and by using the formula: v^{j}\left[{m \atop j}\right]_{i} + v_{i}^{(j−1−m)}\left[{m \atop j−1}\right]_{i} = \left[{m+1 \atop j}\right]_{i}. □

(3.10) Localizing at the left denominator set {F_{i}^{n}| n ∈ ℕ} we get an (_{R}U_{i},_{R}U)-module isomorphism j : _{R}U_{i}_{F} ⊗_{_{R}U_{i}} _{R}U → _{R}U_{F} . which is induced from the map given by a ⊗ g ↦ ag for a ∈ _{R}U_{i} and g ∈ _{R}U_{i}.

LEMMA. Let f denote the forgetful functor from the category of_{R}U-modules to the category of _{R}U_{i}-modules.

(i) Suppose A is a _{R}U-module. Then the natural bimodule isomorphism

j induces an isomorphism of _{R}U_{i}-modules ,

j ⊗ 1 : _{R}U_{i}_{F} ⊗_{_{R}U_{i}} fA → f(_{R}U_{F} ⊗_{_{R}U} A) .

(ii) Suppose A and B are _{R}U-modules and γ : A → B is a _{R}U-module homomorphism. Then f(γ_{F}) = (fγ)_{F} .

We omit the proof which is similar to the proof of [E2,3.10].

(3.11) For J ⊂ I, let U(J) denote the R-subalgebra of _{R}U generated by RB and by the F_{i} with i ∈ J. When J = {i} we write U(i) in place of U({i}). Induction in stages can be expressed as follows in our setting.

LEMMA. Suppose J ⊂ I = {1, …, n}. Fix i ∈ J, set F = F_{i} and suppose A is a U(J)-module. Then the natural isomorphism of (_{R}U, U(J))-bimodules Ω : _{R}U_{F} ≅ _{R}U ⊗_{U(J)} U(J)_{F} induces _{R}U-module isomorphisms of induced modules :

(_{R}U ⊗_{U(J)} A)_{F} ≅ _{R}U ⊗_{U(J)} A_{F} .

Proof. Identify (_{R}U ⊗_{U(J)} A)_{F} ≅ _{R}U_{F} ⊗_{U(J)} A and combine this isomorphism with Ω. □

(3.12) Recall the definition of A_{π} and the exact sequence (1.5.2) as well as the contravariant dual module A^{ϱ} as in (2.8).

COROLLARY. Let i ∈ I, s = s_{i}, T = T′_{i,e} or T = T′′_{i,e}, e = ±1, K = K̃_{i} and F = F_{i}. Assume [ιK; r] for all r ∈ ℤ^{∗} is a unit in R. Then

M_{m+λ,π} ≅ ^{T} M_{m+λ} , M_{m+sλ,π} ≅ _{R}U⊗_{U(i)}^{T} M_{i,m+sλ}^{ϱ} , P_{m+λ,π} ≅ ^{T} P_{m+λ}.

The proof is straightforward using (3.11) and the exactness of induction.

§4. Lifting Bilinear Pairings

(4.1) We now turn to the main topic of this article, the correspondence between forms on a module A in C_{R} and those on the localization A_{F}. Let ϱ denote the involutive antiautomorphism of _{R}U_{i} defined by the identities:

ϱ|_{_{R}U}^{0} ≡ 1 , ϱ(E_{i}) = v_{i}K̃_{i}F_{i}, ϱ(F_{i}) = v_{i}K̃_{−i}E_{i}.  (4.1.1)

For _{R}U_{i}-modules A and B let ℙ_{ϱ}(A, B) denote the set of all R-bilinear maps ϕ : A × B → R which have the usual invariance property; i.e.

ϕ(ua, b) = ϕ(a, ϱub), for a ∈ A, b ∈ B and u ∈ _{R}U_{i}.  (4.1.2)

(4.2) It will frequently be useful to extend scalars in order to simplify the modules under consideration. With this in mind the following lemma will be a great technical convenience.

LEMMA [E2]. Suppose ϕ ∈ ℙ_{ϱ}(A, B). Then ϕ is the pullback of a unique K-bilinear pairing ϕ_{K} : _{K}A ×_{K}B → K. □

(4.3) If A is any U_{i} or _{R}U_{i}-module let ι denote the canonical map of A into A_{F}. For ϕ ∈ ℙ_{ϱ}(A, B) define a lifting of ϕ to be a bilinear pairing ϕ_{F} ∈ ℙ_{ϱ}(A_{F} , B_{F}) with the following two properties:

ϕ_{F} vanishes on A_{F} × ιB and ιA × B_{F}  (4.3.1)

For ϵ equal to either 1 or s in the Weyl group, any n ∈ ℕ^{∗} and λ ∈ X with ⟨i, λ⟩+1 = n and any_{R}U_{i}-module maps S : M_{m+ϵλ} → A and T : M_{m+ϵλ} → B we have the identity:

ιK^{−n+1}[n − 1]_{i}^{!}[ϵιK; n − 1]_{(n−1)} ϕ_{F} (S_{F} w_{ϵλ,n} , T_{F} w_{ϵλ,n})
= ϕ(Sw_{ϵλ,−n}, T w_{ϵλ,−n}).  (4.3.2)

Here of course the vectors w_{•,•} are the basis vectors given in (2.2) , S_{F} and T_{F} are the maps induced from S and T by localization and other notation is as in (1.3) and (1.8).

This definition appears to be different and more complicated than the definition in section one. However they are the same as a short calculation using (i) and (iii) of the following lemma will show. Here we have kept this earlier version of the definition so that the arguments in this section and the next conform to those in [E2].

LEMMA. Assume ϕ is an invariant form on A admitting a lifting ϕ_{F} of ϕ. Let the constants c[ϵλ, t, j] be determined as in (3.7) and set K = K̃_{i}.

(i) Suppose t ∈ I_{ϵλ} and j ∈ ℕ. Then

ϕ(Sw_{ϵλ,t}, T w_{ϵλ,t}) = v_{i}^{−j(j+t+2)}ιK^{−j}c[ϵλ, t, j] ϕ(Sw_{ϵλ,t+2j}, T w_{ϵλ,t+2j}).

(ii) Suppose s ∈ ℤ_{λ} and j ∈ ℕ. Then

ϕ_{F} (S_{F} w_{ϵλ,s}, T_{F} w_{ϵλ,s}) = v_{i}^{−j(j+s)}ιK^{−j}c[ϵλ, s, j] ϕ_{F} (S_{F} w_{ϵλ,s+2j}, T_{F} w_{ϵλ,s+2j}).

(iii) Suppose s ∈ ℤ_{λ} \ I_{ϵλ} and set h = \frac{n−s}{2}. Then

ϕ_{F} (S_{F} w_{ϵλ,s}, T_{F} w_{ϵλ,s}) = \frac{v_{i}^{−h(h+s)}ιK^{−h+n−1}c[ϵλ, s, h]}{[n − 1]_{i}^{!}[ιϵK; n − 1]_{(n−1)}}ϕ(Sw_{ϵλ,−n}, T w_{ϵλ,−n}).

(iv) For j ∈ ℕ set s = n + 2j. Then ϕ_{F} (S_{F} w_{ϵλ,s}, T_{F} w_{ϵλ,s}) =

\frac{ιK^{n−1+2j}}{[n + j − 1]_{i}^{!}[ιϵK; n + j − 1]_{(n+j−1)} [j]_{i}^{!}[ιϵK^{−1}; j]_{(j)}} ϕ(Sw_{ϵλ,−s}, T w_{ϵλ,−s}).

Proof. Invariance gives the identities: for t ∈ I_{ϵλ} and j ∈ ℕ ,

ϕ(w_{ϵλ,t}, w_{ϵλ,t}) = ϕ(w_{ϵλ,t}, F^{j}w_{ϵλ,t+2j}) = v_{i}^{−j}ϕ((K^{−1}E)^{j}w_{ϵλ,t}w_{ϵλ,t+2j})
= v_{i}^{−j(j+t)}ιK^{−j}c[ϵλ, t, j]ϕ(w_{ϵλ,t+2j}, w_{ϵλ,t+2j}).

This implies (i). The same calculation for ϕ_{F} gives (ii). Combining (ii) and (4.3.2) yields (iii). To prove (iv) we combine (i) and (iii) to show that the ratio of ϕ_{F} (S_{F} w_{ϵλ,s}, T_{F} w_{ϵλ,s}) by ϕ(Sw_{ϵλ,−s}, T w_{ϵλ,−s}) equals

\frac{v_{i}^{j(s−j)}ιK^{−j+n−1}c[ϵλ, n + 2j, −j] c[ϵλ, −n − 2j, j]^{−1}}{v_{i}^{−j(−s+j)}ιK^{−j}[n − 1]![ιϵK; n − 1]_{(n−1)}}

= \frac{ιK^{n+2j−1} c[ϵλ, n, j]^{−1} c[ϵλ, −n − 2j, j]^{−1}}{[n − 1]![ιϵK; n − 1]_{(n−1)}} .

A short calculation using (3.7) completes the proof. □

(4.4) We now establish the main existence theorem.

THEOREM. Suppose A and B are modules in C_{R}^{i}, the elements [ιK̃_{i}; n] are units in R, for all n ∈ ℤ^{∗}, and ϕ ∈ ℙ_{ϱ}(A, B). Then ϕ admits a unique lifting.

Proof. Let j denote the natural maps of A and B into _{K}A and _{K}B. By (4.2) the pairing ϕ is the pullback of this induced K-valued pairing on _{K}A × _{K}B restricted to jA × jB. Any object A in the category C_{K} is the direct sum of copies of the irreducible _{K}U-modules _{K}M_{i,m+μ} while A_{π} is the direct sum of copies of the irreducible _{K}U_{i}-modules _{K}^{T}M_{i,m+μ}. Therefore the properties (4.3.1) and (4.3.2) uniquely determine an invariant K-valued pairing _{K}ϕ_{F}. This implies uniqueness for ϕ_{F}.

From our hypothesis on units and (2.6) we may assume that _{K}A splits as the direct sum of copies of _{K}M_{i,,m+λ} and _{K}M_{i,,m+sλ}. Let j denote the map j : A → _{K}A. Localization commutes with extension of scalars giving the commutative diagrams:

\begin{matrix}A & ⟶^{j} & _{K}A \\ ^{ι}↓ & & ^{ι}↓ \\ A_{F} & ⟶^{j_{F}} & _{K}A_{F} \end{matrix}  \begin{matrix}B & ⟶^{j} & _{K}B \\ ^{ι}↓ & & ^{ι}↓ \\ B_{F} & ⟶^{j_{F}} & _{K}B_{F} \end{matrix}

Applying (4.2), ϕ_{F} is the pullback of _{K}ϕ_{F} and so it is sufficient to show that Kϕ_{F} is R-valued on jA_{F} × jB_{F}. Applying (3.5) we see that A_{π} is generated by its weight subspace A_{π,n}. Then by the invariance of _{K}ϕ_{F} we are reduced to the verification that _{K}ϕ_{F} is R-valued on jA_{F,n} × jB_{F,n}.

Choose a′ ∈ jA_{F,n} and b′ ∈ jB_{F,n} and set a = F^{n}a′ and b = F^{n}b′. Set P = P_{m+λ} and let τ denote the _{R}U_{i}-module map of P × P into jA_{F} × jB_{F} with τ(a_{−n}, 0) = a and τ(0, a_{−n}) = b. Let ψ denote the pullback of _{K}ϕ to P × P. Then

Kϕ_{F} (a′, b′) = ψ_{F} (a_{n}, a_{n}) = ψ_{F} (w_{λ,n}, w_{λ,n}) + ψ_{F} (w_{sλ,n}, w_{sλ,n})

= ∑_{ϵ∈\{1,s\}} \frac{ιK^{n−1}}{[n − 1]_{i}^{!}[ιϵK; n − 1]_{(n−1)}} ψ(w_{ϵλ,−n}, w_{ϵλ,−n})

≡ \frac{ιK^{n−1}}{[n − 1]_{i}^{!}[ιϵK; n − 1]_{(n−1)}} ψ(a_{−n}, a_{−n}) mod Rψ(a_{−n}, [ιK; 0]w_{λ,−n}) .

Since a_{−n} and [ιK; 0]w_{λ,−n} both lie in P we conclude_{K}ϕ_{F} (a′, b′) is an element of R. □

§5 Liftings and Tensor Products

(5.1) In this section we describe the behavior of liftings of pairings on certain tensor products. For any finite dimensional X-admissible U_{i}-module F, define _{R}F = R ⊗_{ℚ(v)} F. Then _{R}F is an _{R}U_{i}-module and we have natural isomorphisms: for A in C_{R}^{i} , (A ⊗_{R} F)_{F} ≃ A_{F} ⊗_{R} F , given by

F^{−1} ⊗ a ⊗ e ↦ ∑_{j∈ℕ}(−1)^{j}v_{i}^{−j(j+1)} F^{−j−1}a ⊗ F^{j}K^{j+1}e ,  (5.1.1)

which are compatible with tensoring. Note that the nilpotence of the action of F on F insures that this formula is well defined. Also note that since comultiplication is not symmetric the analogue of (5.1.1) is different if we transpose A and F.

Suppose τ : _{R}U → _{R}U is an anti-automorphism and M (resp. N) is an _{R}U-module with bilinear τ-invariant form (−, −)_{M} (resp. (−, −)_{N}) i.e. (x.m, n) = (m, τ(x).n) for all x ∈ _{R}U, m, n ∈ M (resp. N). As Kashiwara observed, if τ is also a coalgebra automorphism, then M ⊗N has a τ-invariant form defined by: for all m, m′ ∈ M and n, n′ ∈ N ,

(m ⊗ n, m′ ⊗ n′)_{M⊗N} = (m, m′)_{M}(n, n′)_{N} .  (5.1.2 )

One can check that ϱ is a coalgebra homomorphism and thus the tensor product of two ϱ-invariant forms is again ϱ-invariant.

THEOREM. Suppose F is a finite dimensional X-admissible U_{i}-module and A and B are objects in C_{R}^{i}. For ϕ ∈ ℙ_{ϱ|_{_{R}U_{i}}} (A, B) and ψ ∈ ℙ_{ϱ|_{_{R}U_{i}}} (F_{R}, F_{R}), ϕ ⊗ ψ ∈ ℙ_{ϱ|_{_{R}U_{i}}} (A ⊗ F_{R}, B ⊗ F_{R}) and the liftings (ψ ⊗ ϕ)_{F} and ψ ⊗ ϕ_{F} are equal.

The proof of this proceeds through a reduction to the case of two dimensional F and an explicit verification in that case. We now setup the detailed notation necessary to handle the two dimensional case.

(5.2) Let E denote the two dimensional U_{i}-module with highest weight v_{i}. Set E = E_{i}, F_{i} = F and K = K̃_{i} where α = α_{i}. Fix a basis e_{+}, e_{−} with properties: Ke_{±} = v^{±1}e_{±} , Ee_{+} = 0 , Ee_{−} = e_{+} , F e_{+} = e_{−} , F e_{−} = 0. Define an invariant R-valued symmetric bilinear form ψ_{E} on E_{R} by the identities: 1 = ψ_{E}(e_{+}, e_{+}) = ψ_{E}(e_{−}, e_{−}) , 0 = ψ_{E}(e_{−}, e_{+}). Continue with the notation ϵ ∈ {1, s_{i}}, λ ∈ X with ⟨i, λ⟩ ∈ ℕ and n = ⟨i, λ⟩ + 1. Set δ equal to the determinant of ϵ. We fix M_{ϵλ} = M_{i,m+ϵλ} and as usual let the subscript K denote the extension of scalars from R to K. Then we have isomorphisms

Φ : _{K}M_{ϵλ+(i′/2)} ⊕_{K}M_{ϵλ−(i′/2)} → _{K}M_{ϵλ,K} ⊗ _{K}E.  (5.2.1)

Here we replace U_{i} by its simply connected quantum group if necessary to insure i′/2 ∈ X. The formulas which define liftings indicate that we shall need formulas for certain distinguished vectors in this tensor product. Next we localize the modules and map in (5.2.1) to define vectors u_{ϵλ,±} and u_{ϵλ,±}^{♯}, for n ≥ 2, by : u_{ϵλ,±} = Φ(w_{ϵ(λ± \frac{i′}{2} ),−n∓δ}), u_{ϵλ,±}^{♯} = Φ_{F} (w_{ϵ(λ± \frac{i′}{2} ),n±δ}) and u_{λ,±}

and u_{λ,±}^{♯}, for n = 1 by : u_{λ,±} = Φ(w_{λ± \frac{i′}{2} ,−2}), u_{λ,±}^{♯} = Φ_{F} (w_{λ± \frac{i′}{2} ,2}).

LEMMA. The isomorphisms Φ can be chosen such that the following identities hold: for n ≥ 2

u_{λ,+} = v_{i}^{−n}w_{λ,−n−2} ⊗ e_{+} + [n]_{i} w_{λ,−n} ⊗ e_{−},
u_{λ,−} = v_{i}^{−n+2} (w_{λ,−n} ⊗ e_{+} − ιK^{−1}[ιK; 0]w_{λ,−n+2} ⊗ e_{−}) ,  (5.2.2)

u_{s_{i}λ,−} = w_{s_{i}λ,−n}⊗e_{+}, u_{s_{i}λ,+} = w_{s_{i}λ,−n−2}⊗e_{+}+ιK^{−1}v_{i}^{n}[ιK^{−1}; n]w_{s_{i}λ,−n}⊗e_{−}.

For the case n = 1, λ = s_{i}λ , u_{λ,+} = v_{i}^{−1}w_{λ,−3} ⊗ e_{+} + w_{λ,−1} ⊗ e_{−}

and u_{λ,−} = w_{λ,−3} ⊗ e_{+} + v_{i}ιK^{−1}[ιK^{−1}; 1] w_{λ,−1} ⊗ e_{−} .  (5.2.3)

Proof. First suppose n ≥ 2. The third and fourth vectors in (5.2.2) are themselves highest weight vectors as is easily verified using identities from (2.2).

Similarly the vectors w_{λ,n−2} ⊗ e_{+} and w_{λ,n−4} ⊗ e_{+} + ιK^{−1}v_{i}^{−n+2}[ιK^{−1}; 2 − n]w_{λ,n−2} ⊗ e_{−} are highest weight vectors and the first two vectors in (5.2.2) are obtained by applying F_{i}^{n} and F_{i}^{n−2} to these highest weight vectors respectively.

In the case n = 1 to obtain the expression in (5.2.3) for u_{λ,+} we apply F_{i} to the highest weight vector w_{λ,−1} ⊗e_{+}. Finally u_{λ,−} is itself a highest weight vector.

(5.3) The corresponding formulas for u_{ϵλ,±}^{♯} are given in the following lemma.

LEMMA. With Φ chosen as in (5.2) we have: for n ≥ 2

u_{λ,+}^{♯} = v_{i} w_{λ,n} ⊗ e_{+} − w_{λ,n+2} ⊗ e_{−} ,
u_{λ,−}^{♯} = v_{i} w_{λ,n−2} ⊗ e_{+} − ιK^{−1}v_{i}^{−n+2}[ιK; n − 1]w_{λ,n} ⊗ e_{−} ,  (5.3.1)
u_{s_{i}λ,−}^{♯} = v_{i}^{n−1}w_{s_{i}λ,n−2} ⊗ e_{+} − [n − 1] w_{s_{i}λ,n} ⊗ e_{−} ,
u_{s_{i}λ,+}^{♯} = v_{i}^{n} (v_{i}w_{s_{i}λ,n} ⊗ e_{+} − ιK^{−1} [ιK; 1]w_{s_{i}λ,n+2} ⊗ e_{−}) .

For n = 1, λ = s_{i}λ and

u_{λ,+}^{♯} = v_{i} w_{λ,1}⊗e_{+}− w_{λ,3}⊗e_{−}, u_{λ,−}^{♯} = v_{i}^{2}w_{λ,1}⊗e_{+}−ιK^{−1} v_{i} [ιK; 1] w_{λ,3}⊗e_{−} . (5.3.2)

Proof. From (5.1) we apply the action of F_{i}^{−n−1} and F_{i}^{−n+1} to the first and fourth (respectively second and third) elements in (5.2.2) to obtain formulas for u_{ϵλ,±}^{♯} for n ≥ 2. In the case n = 1, we apply F_{i}^{−2} to the expressions in (5.2.3).

(5.4) Let ϕ ∈ ℙ_{ϱ|_{𝔞_{R}}} (M_{ϵλ}, M_{ϵλ}) and set θ = ψ_{E} ⊗ ϕ and θ^{♯} = ψ_{E} ⊗ ϕ_{F}. Define constants a_{ϵλ,±} by the ratios: a_{ϵλ,±} θ^{♯}(u_{ϵλ,±}^{♯}, u_{ϵλ,±}^{♯}) = θ(u_{ϵλ,±}, u_{ϵλ,±}) . The defining relation for ϕ_{F} (4.3.2), gives rise to the following formulas.

LEMMA. For all n ≥ 2,

a_{ϵλ,+} = ιK^{−n}[n]_{i}^{!} [ιϵK; n]_{(n)} , a_{ϵλ,−} = ιK^{−(n−2)}[n − 2]_{i}^{!} [ιϵK; n − 2]_{(n−2)}

For n = 1 , a_{λ,+} = ιK^{−1}[ιK; 1] , a_{λ,−} = ιK^{−1}[ιs_{i}K; 1].

Proof. For convenience we set ϕ_{ϵλ,j} = ϕ(w_{ϵλ,j}, w_{ϵλ,j}) for j ∈ I_{ϵλ} and ϕ_{F,ϵλ,j} = ϕ_{F} (w_{ϵλ,j}, w_{ϵλ,j}) for j ∈ ℤ_{ϵλ}. From the basic relations (2.2) and the defining relations for ψ, we obtain:

θ(u_{λ,+}, u_{λ,+}) = v_{i}^{−2n} ϕ_{λ,−n−2} + [n]_{i}^{2}ϕ_{λ,−n} = v_{i}ιK^{−1}[n]_{i}[ιK; n − 1] ϕ_{λ,−n} . (5.4.1)

Similarly we have: θ^{♯}(u_{λ,+}^{♯}, u_{λ,+}^{♯}) = v_{i}^{2}ϕ_{F,λ,n} + ϕ_{F,λ,n+2}

= \frac{1}{[n − 1]_{i}^{!}[ιK; n − 1]_{(n−1)}} (v_{i}^{2}ιK^{n−1}c[λ, n, 0] + ιK^{n} v_{i}^{n+1}c[λ, n + 2, −1])ϕ_{λ,−n} .

Here c[λ, n, 0] = 1 and c[λ, n + 2, −1] = \frac{1}{c[λ,n,1]} = \frac{−1}{[ιK;n]} and so we obtain:

θ^{♯}(u_{λ,+}^{♯}, u_{λ,+}^{♯}) = \frac{v_{i}ιK^{n−1}}{[n − 1]_{i}^{!}[ιK; n − 1]_{(n−1)}} \frac{[ιK; n − 1]}{[ιK; n]}ϕ_{λ,−n} .  (5.4.2)

Taking the ratio of (5.4.1) and (5.4.2) gives the first identity in the lemma. The other three identities are obtained in a similar fashion. We list below the corresponding necessary identities:

θ(u_{λ,−}, u_{λ,−}) = v_{i}^{−2n+4}ϕ_{λ,−n} + ιK^{−2}v_{i}^{−2n+4}[ιK : 0]^{2}ϕ_{λ,−n+2}
= \frac{ιK^{−1}v_{i}^{−2n+4}[ιK; n − 1]}{[n − 1]}ϕ_{λ,−n}  (5.4.3)

θ^{♯}(u_{λ,−}^{♯}, u_{λ,−}^{♯}) = v_{i}^{2}ϕ_{F,λ,n−2} + ιK^{−2}v_{i}^{−2n+4}[ιK; n − 1]^{2}ϕ_{F,λ,n}
= \frac{ιK^{n−3}v_{i}^{−2n+4}[ιK; n − 1]}{[n − 1]_{i}^{!}[ιK; n − 2]_{(n−2)}} ϕ_{λ,−n}  (5.4.4)

θ(u_{s_{i}λ,−}, u_{s_{i}λ,−}) = ϕ_{s_{i}λ,−n}  (5.4.5)

θ^{♯}(u_{s_{i}λ,−}^{♯}, u_{s_{i}λ,−}^{♯}) = v_{i}^{2n−2}ϕ_{F,s_{i}λ,n−2} + [n − 1]^{2} ϕ_{F,s_{i}λ,n}
= \frac{ιK^{n−2}}{[n − 2]_{i}^{!}[ιs_{i}t; n − 2]_{(n−2)}} ϕ_{s_{i}λ,−n} .  (5.4.6)

θ(u_{s_{i}λ,+}, u_{s_{i}λ,+}) = ϕ_{s_{i}λ,−n−2} + ιK^{−2}v_{i}^{2n}[ιK^{−1}; n]^{2} ϕ_{s_{i}λ,−n}
= ιK^{−2}v_{i}^{2n+1}[ιs_{i}K; n][ιs_{i}K; n − 1] ϕ_{s_{i}λ,−n} (5.4.7)

θ^{♯}(u_{s_{i}λ,+}^{♯}, u_{s_{i}λ,+}^{♯}) = v_{i}^{2n+2}ϕ_{F,s_{i}λ,n} + ιK^{−2}v_{i}^{2n}[ιK; 1]^{2} ϕ_{F,s_{i}λ,n+2}
= \frac{ιK^{n−2}v_{i}^{2n+1}}{[n]_{i}^{!}[ιs_{i}K; n − 2]_{(n−2)}} ϕ_{s_{i}λ,−n} .  (5.4.8)

Finally in the case n = 1 we have : θ(u_{λ,−}, u_{λ,−})

= ϕ_{λ,−3} + v_{i}^{2}ιK^{−2}[ιK^{−1}; 1]^{2} ϕ_{λ,−1} = v_{i}^{3}ιK^{−2}[ιK^{−1}; 0][ιK^{−1}; −1] ϕ_{λ,−1} , (5.4.9)

θ^{♯}(u_{λ,−}^{♯}, u_{λ,−}^{♯}) = v_{i}^{4}ϕ_{F,λ,1} + ιK^{−2}v_{i}^{2}[ιK; 1]^{2} ϕ_{F,λ,3} = v_{i}^{3}ιK^{−1}[ιK^{−1}; 0] ϕ_{λ,−1} (5.4.10)

θ(u_{λ,+}, u_{λ,+}) = v_{i}^{−2}ϕ_{λ,−3} + ϕ_{λ,−1} = v_{i}ιK^{−1}[ιK; 0] ϕ_{λ,−1} ,  (5.4.11)

θ^{♯}(u_{λ,+}^{♯}, u_{λ,+}^{♯}) = v_{i}^{2}ϕ_{F,λ,1} + ϕ_{F,λ,3} = \frac{v_{i}[ιK; 0]}{[ιK; 1]} ϕ_{λ,−1}  (5.4.12)

These identities combine to give the ratios of the lemma. □

(5.5) We now consider the quantum symmetric algebra.

THEOREM [MS, 4.3]. Let A = ℚ(v)[x, y] where xy = v yx. Define an action on A by: E_{i}x = 0, E_{i}y = x, F_{i}y = 0, F_{i}x = y, K_{μ}x = v^{⟨μ,i′⟩}x, K_{μ}y = v^{−⟨μ,i′⟩}y, and set A_{n} = ⊕_{0≤i≤n}ℚ(v)x^{i}y^{n−i}. Then this induces a U_{i}-module action on A and A_{n} is (up to isomorphism) the unique X-admissible irreducible U_{i}-module of highest weight ni′ of dimension n + 1.

(5.6) Proof of (5.1) To prove that (ϕ ⊗ ψ)_{F} and ϕ_{F} ⊗ ψ are equal it is sufficient to prove the identity when scalars have been extended from R to K. Then since _{K}A and _{K}B are direct sums of the U_{i}-modules _{K}M_{i,m+ϵλ} and these modules all have distinct infinitesimal characters we may assume KA = _{K}B = _{K}M_{i,m+ϵλ}.

The map (5.1.1) is additive in F and so it is sufficient to prove the identity for X-admissible irreducible F. For two dimensional irreducible F by the remark in (2.9) there is a one dimensional module K_{ξ} with F = E ⊗ K_{ξ}. Tensoring all elements by K_{ξ} in (5.2) and (5.3) we find that the ratios given in (5.4) imply Theorem (5.1) for two dimensional F. For the general case we begin with the algebra A in (5.5). We identify A as a submodule of the full tensor algebra T (E) of E by actions of the symmetric groups. The validity of (5.1) for E implies the same for T (E), then A and finally the homogeneous summands of A. Since F ≅ A_{n}⊗K_{ξ} for some ξ, this proves (5.1) in general. □

§6 Invariance of Liftings

(6.1) Here we consider the ambient setting where_{R}U_{i} is a Hopf subalgebra of another quantum group _{R}U and the bilinear pairing ϕ has ϱ-invariance not merely ϱ|_{_{R}U_{i}}-invariance. Let ϱ be the involutive anti-automorphism of RU as defined in (1.3). Assume the restriction of ϱ to _{R}U_{i} equals the antiautomorphism given in (4.1). Also as introduced in (1.9), for _{R}U-modules A and B, let ℙ_{ϱ}(A, B) denote the space of R-bilinear maps ϕ : A×B → R which are ϱ-invariant.

THEOREM. Let A and B be objects in C_{R} and suppose ϕ ∈ ℙ_{ϱ}(A, B). Then the lifted pairing ϕ_{F_{i}} is _{R}U-invariant as well ; i.e. ϕ_{Y} ∈ ℙ_{ϱ}(A_{F_{i}}, B_{F_{i}}).

This proof is an adaptation of the proof of 𝔤-invariance in [E2] given in the Lie algebra setting. There are some delicate points in the extension to the quantum setting. The presence of the R-matrix, _{f}R is the main complication as we shall see. We begin with several results on vector valued pairings and the resulting two notions of liftings in this setting.

(6.2) A U-module M is said to be integrable if for any m ∈ M and all i ∈ I, there exists a positive integer N such that E_{i}^{(n)}m = 0 = F_{i}^{(n)}m for all n ≥ N , and M = ⊕_{λ∈X}M^{λ} where for any μ ∈ Y, λ ∈ X and m ∈ M^{λ} one has K_{μ}m = v^{⟨μ,λ⟩}m. Let f : X × X → ℚ be a function such that

f(ζ + ν, ζ′ + ν′) − f(ζ, ζ′) = − ∑ ν_{i}⟨i, ζ′⟩(i · i/2) − ∑ ν′_{i}⟨i, ζ⟩(i · i/2) − ν · ν′

(6.2.1)

for all ζ, ζ′ ∈ X and all ν, ν ∈ ℤ[I]. For any positive integer d, there is always such an f with im f ⊂ \frac{1}{d}ℤ, see [L, 32.1.3].

THEOREM [L, 32.1.5]. Fix an integer d ≥ 1. If M is an integrable _{R}U module and A ∈ C_{R}, then for each f satisfying (1) with im f ∈ \frac{1}{d}ℤ, there exists an isomorphism _{f}R : A ⊗ M → M ⊗ A.

Remark: This theorem is stated for M and A integrable. However the proof applies under the hypothesis above. Moreover from the explicit form of the map [L, 32.1] , (_{f}R)_{Y} =_{f} R.

Let F be a finite dimensional U-module and A and B objects in C_{R}. An R-bilinear map ϕ : A × B →_{R} F is called ϱ-invariant if for all x ∈ _{R}U, a ∈ A and b ∈ B,

∑ ϕ(x_{(1)} · a, ρ_{1}(x_{(2)})b) = x · ϕ(a, b)  (6.2.2)

where Δ(x) = ∑ x_{(1)} ⊗ x_{(2)}. Let ℙ_{ϱ}(A, B, F) denote the space of all such bilinear pairings. If F is the trivial module, then a calculation on generators shows ℙ_{ϱ}(A, B, F) = ℙ_{ϱ}(A, B). Let ℙ_{_{K}ϱ}(_{K}A,_{K} B,_{K} F) denote the corresponding space of K-bilinear _{K}F-valued _{K}U-invariant forms. For ϕ ∈ ℙ_{ϱ}(A, B, F), (4.2) implies that ϕ is the pullback of a pairing _{K}ϕ in ℙ_{_{K}ϱ}(_{K}A,_{K} B,_{K} F). The K vector spaces _{K}F and _{K}F^{ϱ} are naturally paired by evaluation and so we can define K-valued bilinear mappings ϕ^{r} : _{K}A × (_{K}B ⊗ _{K}F^{ϱ}) → K and ϕ^{l} : (_{K}A ⊗ (_{K}F^{ϱ})^{ρ_{1}}) × _{K}B → K by the formulas: for λ ∈_{K} F^{ϱ}, a ∈ A and b ∈ B

ϕ^{r}(a, b ⊗ λ) = λ(ϕ(a, b)), ϕ^{l}(a ⊗ λ, b) = ∑ λ_{i}(ϕ(a, b_{i}))  (6.2.3)

where ∑ b_{i} ⊗ λ_{i} = _{f}R^{−1}(λ ⊗ b). Clearly ϕ^{r} (resp. ϕ^{l}) is determined by ϕ and vice versa.

LEMMA. The maps θ_{r} : ℙ_{ϱ}(A, B, F) → ℙ_{ϱ}(A, B⊗F^{ϱ}) and θ_{l} : ℙ_{ϱ}(A, B, F) → ℙ_{ϱ}(A ⊗ (F^{ϱ})^{ρ_{1}}, B), given by θ_{r}ϕ = ϕ^{r} and θ_{l}ϕ = ϕ^{l}, are isomorphisms.

Proof. Let C be another U-module. First one shows that since m∘(1⊗e)∘Δ = 1, where m is multiplication, the canonical map Hom (A, C^{ϱ}) → Hom (A ⊗ C^{ρ_{1}}, K) is a U-module isomorphism. Thus Hom_{U}(A, C^{ϱ}) ≅ Hom_{U}(A ⊗ C^{ρ_{1}}, K). Second, it is straightforward to check Hom_{U}(A⊗C^{ρ_{1}}, F) ≅ ℙ(A, C, F), for any U-module F. One then combines these facts to obtain isomorphisms;

ℙ(A, B⊗F^{ϱ}) ≅ Hom_{U}(A⊗(B⊗F^{ϱ})^{ρ_{1}}, K) ≅ Hom_{U}(A⊗B^{ρ_{1}}, F) ≅ ℙ(A, B, F)

and

ℙ(A ⊗ (F^{ϱ})^{ρ_{1}}, B) ≅ Hom_{U}(A ⊗ (F^{ϱ} ⊗ B)^{ρ_{1}}, K)

≅ Hom_{U}(A ⊗ B^{ρ_{1}} ⊗ (F^{ϱ})^{ρ_{1}}, K) ≅ Hom_{U}(A ⊗ B^{ρ_{1}}, F) ≅ ℙ(A, B, F) ,

where the second isomorphism is implemented by _{f}R. The reader can check that these isomorphisms are the inverses to θ_{r} and θ_{l} respectively. □

(6.3) Now define the right and left liftings ϕ_{rF_{i}} and ϕ_{lF_{i}} ∈ ℙ_{ϱ|_{_{R}U_{i}}} (A_{F_{i}}, B_{F_{i}},_{R}F) by the identities:

(ϕ_{rF_{i}})^{r} = (ϕ^{r})_{F_{i}} , (ϕ_{lF_{i}})^{l} = (ϕ^{l})_{F_{i}}.  (6.3.1)

THEOREM. For ϕ ∈ ℙ_{ϱ}(A, B, F) the right and left liftings of ϕ are equal; i.e. ϕ_{rF_{i}} = ϕ_{lF_{i}}.

Proof. By (4.2) it is sufficient to prove the identity for _{K}ϕ in place of ϕ. Let ν denote the natural pairing of_{K}F and_{K}F^{ϱ}. Choose a basis {x_{i}} and dual basis {x_{i}^{ϱ}} for _{K}F and _{K}F^{ϱ} and let i denote the inclusion _{K}A →_{K} A ⊗_{K}F^{ϱ} ⊗_{K} F given by a ↦ ∑ a ⊗ x_{i}^{ϱ} ⊗ x_{i}. Now let ϕ_{c}^{l} = ϕ^{l} ⊗ ν ∘ (i ⊗ 1_{_{K}B⊗_{K}F^{ϱ}}) denote the contraction as a K-valued pairing on _{K}A ×_{K} B ⊗_{K} F^{ϱ}. We claim: for all y ∈ _{K}F, a ∈ _{K}A and b ∈ _{K}B

ϕ^{r}(a, b ⊗ y) = yϕ(a, b) = ∑ ϕ^{l}(a ⊗ y_{i}, b_{i}) = ϕ_{c}^{l}(a,_{f}R(b ⊗ y)).  (6.3.2)

Fix an index j. Then

ϕ^{l}(a ⊗ x_{j}^{ϱ}, b) = ϕ^{l} ⊗ ν(∑_{i} a ⊗ x_{i}^{ϱ} ⊗ x_{i}, b ⊗ x_{j}^{ϱ}) = ϕ_{c}^{l}(a,_{f}R(∑ x_{k}^{ρ} ⊗ b_{k})) ,

where b ⊗ x_{j}^{ϱ} =_{f} R(∑ x_{k}^{ϱ} ⊗ b_{k}). Extending this linearly over the basis {x_{j}^{ϱ}} proves (6.3.2).

By imbedding B into B ⊗ _{K}F^{ϱ} ⊗ _{K}F we can define in a similar way the contraction ϕ_{c}^{r}. Then from (6.3.2) we find that knowing any one of the maps ϕ, ϕ^{r}, ϕ^{l}, ϕ_{c}^{r} or ϕ_{c}^{l} determines all the others. Also by (6.3.2) for any _{K}U_{i}- invariant _{K}F-valued pairings ψ and ξ ; ψ = ξ ⇔ ψ_{c}^{l} ∘ 1 ⊗_{f}R = ξ^{r}. Thus to prove the theorem it is sufficient to establish the identity: (ϕ_{lF_{i}})_{c}^{l} ∘1⊗_{f}R = (ϕ_{rF_{i}})^{r}.

By (5.2) we have (ν ⊗ ϕ^{l})_{F_{i}} ≅ ν ⊗ ϕ_{F_{i}}^{l}. Now applying localization to the definition of ϕ_{c}^{l}, its functoriality and (6.3.1) we obtain: (ϕ_{c}^{l})_{F_{i}} = ν ⊗ (ϕ^{l})_{F_{i}} ∘ (i ⊗ 1_{_{K}B_{F_{i}}} ⊗_{K}F^{ϱ}) = (ϕ_{lF_{i}})_{c}^{l}. Combining this with the first identity in (6.3.1) and the result of localizing identity (6.3.2) we obtain; (ϕ_{rF_{i}})^{r} = (ϕ^{r})_{F_{i}} = (ϕ_{lF_{i}})_{c}^{l} ∘ 1 ⊗_{f}R which completes the proof of the theorem.

(6.4) If M is a U-module, then Hom_{K}(_{K}M, K), becomes a U-module by defining: u.ϕ(x) = ϕ(S′(u)) for u ∈ U, where S′ : U → U is the skew-antipode given on generators by: S′(E_{i}) = −E_{i}K̃_{−i}, S′(F_{i}) = −K̃_{i}F_{i}, S′(K_{μ}) = K_{−μ} (cf. [L, §3.3]). Let this U-module be denoted by M′.

Suppose F is a finite dimensional U_{i}-stable subspace of U under the adjoint action ad u(b) = ∑ u_{(1)}bS(u_{(2)}) and ϕ ∈ ℙ_{ϱ}(A, B). Define_{K}F′-valued pairings ϕ_{l} and ϕ_{r} by the formula: for X ∈ F, a ∈ A and b ∈ B

ϕ_{l}(a, b)(X) = ϕ(X · a, b) , ϕ_{r}(a, b)(X) = ϕ(a, ϱ(X) · b).  (6.4.1)

Since Δ ∘ S′ = τ ∘ S′ ⊗ S′ ∘ Δ where τ(a ⊗ b) = b ⊗ a,

∑ ϕ(x_{(1)}a, ρ_{1}(x_{(2)})b)(f) = ∑ ϕ(fx_{(1)}a, ρ_{1}(x_{(2)})b) = ∑ ϕ(S′(x_{(2)})fx_{(1)}a, b)

= ∑ ϕ((S′(x).f)a, b) = (x.ϕ_{l}(a, b))(f) as SS′ = S′S = 1,

for x ∈ U_{i}, f ∈ F, a ∈ A and b ∈ B . This shows ϕ_{l} ∈ ℙ_{ϱ|_{_{R}U_{i}}} (A, B, F′). A similar calculation shows ϕ_{r} ∈ ℙ_{ϱ|_{_{R}U_{i}}} (A, B, F′).

LEMMA. For ϕ as above, (ϕ_{F_{i}})_{l}^{l} = (ϕ_{l}^{l})_{F_{i}} , (ϕ_{F_{i}})_{r}^{r} = (ϕ_{r}^{r})_{F_{i}} .

Proof. Let π be the U_{i}-module map , π : F ⊗ A → A given by f ⊗ a ↦ f · a. Then ϕ_{l}^{l} ∘ 1_{A} ⊗ _{f}R = ϕ ∘ (π × 1_{B}). By functoriality of localization , (ϕ_{l}^{l})_{F_{i}} ∘ 1_{A_{F_{i}}} ⊗_{f}R_{F_{i}} = ϕ_{F_{i}} ∘ (π_{F_{i}} × 1_{B_{F_{i}}}). The isomorphism (5.1) implies π_{F_{i}} has the same form as π ; i.e. π_{F_{i}}(f ⊗ a) = f · a, for f ∈ F and a ∈ A_{F_{i}}. As we remarked earlier _{f}R_{F_{i}} = _{f}R, which proves the first identity. The second identity is proved in a similar fashion. □

(6.5) Proof of Theorem (6.1) From [JL1,§ 6] U can be expressed as a localization of its ad-integrable part and is equal to T_{<}^{−1}I(U)[T/T_{0}] where I(U) is a sum of integrable U_{i}-stable subspaces, T_{<} is an Ore subset of I(U) contained in the abelian group {K_{μ}| μ ∈ Y } and [T/T_{0}] is a finite set of elements in {K_{μ}| μ ∈ Y }. To prove (6.1) we must first show that ϕ_{F_{i}} is invariant on the locally finite part of U; i.e. (ϕ_{F_{i}})_{l} = (ϕ_{F_{i}})_{r} . Combining the definition (6.3.1) and the preceding lemma gives ((ϕ_{l})_{lF_{i}})^{l} = (ϕ_{l}^{l})_{F_{i}} = (ϕ_{F_{i}})_{l}^{l}. This implies (ϕ_{l})_{lF_{i}} = (ϕ_{F_{i}})_{l} and by a similar argument (ϕ_{r})_{rF_{i}} = (ϕ_{F_{i}})_{r} . Finally since ϕ is ϱ-invariant, ϕ_{l} = ϕ_{r} and by (6.3) the left and right liftings agree giving (ϕ_{l})_{lF_{i}} = (ϕ_{r})_{rF_{i}}.

To complete the proof we must show ϕ_{F_{i}} is ϱ-invariant by elements from U^{0}. By Lemma 2.7 and 4.2 one may reduce to case that _{K}A and _{K}B are direct sums of Verma modules _{K}M_{m+λ} for some λ ∈ X. Since _{K}ϕ_{F_{i}} is _{K}ϕ localized at F_{i} distinct weight spaces are orthogonal. This is the desired invariance. □

(6.6) The significance of the invariance theorem for us comes from its implications regarding the Shapovalov forms on Verma modules: By [L, 3.2.5] we have a direct sum decomposition: U^{0} ⊕ (U_{+}^{−}U + UU_{+}^{+}), where U_{+}^{±} = U^{±} ∩ ker e. For a ∈ U^{0} and λ ∈ X let a(λ) denote the value obtained when each K_{μ} is replaced by v^{⟨λ,μ⟩}. If we let ψ : U → U^{0} denote the projection then the bilinear form ψ_{λ} on U given by ψ_{λ}(a, b) = ψ(ϱ(a)b)(λ) factors to the Shapovalov form on M_{λ} which we also denote by ψ_{λ}.

COROLLARY. Let A = _{R}M_{m+λ} and let ϕ equal the Shapovalov form on A. Then A_{π} ≅ ^{T} M_{m+λ} and ϕ_{F_{i}} is a scalar multiple of the pullback to A_{F_{i}} of the Shapovalov form on A_{π}.

Proof. The space of _{K}U-invariant forms on A_{π} is a one dimensional K-space.

§7 Induced Forms and Pairings

In this section we assume U is of finite type.

(7.1) Let M and N be _{R}U-modules and ϕ = ϕ_{M,N} an R-valued invariant pairing of M and N. Let E and F be finite dimensional U-module. We now consider a natural family of pairings of M ⊗_{R}E and N ⊗_{R}F. Recalling the isomorphisms of Lemma (6.2) we obtain an isomorphism:

Θ : ℙ(M ⊗_{R} E ⊗_{R}F^{ρ_{1}}, N) → ℙ(M ⊗_{R}E, N ⊗_{R}F),

defined by Θ(χ) = θ_{r} ∘ θ_{l}^{−1}(χ), for χ ∈ ℙ(M ⊗_{R} E ⊗_{R}F^{ρ_{1}}, N).

(7.2) Let τ denote a U-module homomorphism τ :_{R} E ⊗_{R} F^{ρ_{1}} → U. For each pairing ϕ of M and N and τ, we define invariant pairings ψ′_{τ,ϕ} and ψ_{τ,ϕ} in ℙ(M ⊗_{R}E ⊗_{R}F^{ρ_{1}}, N) and ℙ(M ⊗_{R}E, N ⊗_{R}F) respectively by the formulas:

ψ′_{τ,ϕ}(m ⊗ e ⊗ f, n) = ∑ ϕ(τ(e_{i} ⊗ f_{i}) · m_{i}, n), and ψ_{τ,ϕ} = Θ ψ′_{τ,ϕ} , (7.2.1)

where _{f}R(m ⊗ e ⊗ f) = ∑ e_{i} ⊗ f_{i} ⊗ m_{i} . We call the pairing ψ_{τ,ϕ} the pairing induced by τ and ϕ. In the cases when M, N and ϕ are fixed we write ψ_{τ} in place of ψ_{τ,ϕ} and say this pairing is induced by τ. Let Ind_{ϕ}(M ⊗_{R} E, N ⊗_{R} F) denote the subspace of ℙ(M ⊗_{R}E, N ⊗_{R}F) of those pairings which are induced for fixed ϕ and varying τ. In the cases where M = N and E = F we write Ind_{ϕ}(M ⊗_{R}E) for this set and refer to elements as induced forms. When M, N and ϕ are fixed we shall frequently suppress the subscript ϕ in the notation for both pairings and forms.

(7.3) As above let S denote the set of positive integral powers of F = F_{i}. There are times when the subscript notation for localization of U with respect to S is clumsy and easily confused with other required subscripts. When this occurs we write loc_{F} M in place of M_{F} and loc_{F} for the localization functor in general.

PROPOSITION. With Θ defined as in (7.1), the following diagram is commutative.

ℙ(M ⊗_{R} E ⊗_{R} F^{ρ_{1}}, N) ⟶^{Θ} ℙ(M ⊗_{R} E, N ⊗_{R} F)
loc_{F} ↓  ^{loc_{F}} ↓
ℙ(M_{F} ⊗_{R} E ⊗_{R} F^{ρ_{1}}, N_{F}) ⟶^{Θ} ℙ(M_{F} ⊗_{R} E, N_{F} ⊗_{R} F)

Proof. Let C be a finite dimensional U-module and A and B objects in C_{R}. We have isomorphisms by (6.2): θ_{r} : ℙ_{ϱ}(A, B, C) → ℙ_{ϱ}(A, B ⊗ C^{ϱ}), and θ_{l} : ℙ_{ϱ}(A, B, C) → ℙ_{ϱ}(A ⊗ (C^{ϱ})^{ρ_{1}}, B) . The map Θ is θ_{r}θ_{l}^{−1} with B = N, C = F^{ϱ} and A = M ⊗E. The theorem on the equality of the left and right liftings (6.6) can be rephrased : for ϕ ∈ ℙ_{ϱ}(A, B, C), θ_{r}^{−1}loc_{F_{i}} θ_{r}(ϕ) = θ_{l}^{−1}loc_{F_{i}} θ_{l}(ϕ) . Setting ϕ = θ_{l}^{−1}(γ) and multiplying by θ_{r} we obtain: Θ loc_{F_{i}} (γ) = loc_{F_{i}} Θ(γ). □

(7.4) LEMMA. Let A, B, C and D be objects in C_{R} and δ and γ homomorphisms δ : A → C and γ : B → D. Set F = F_{i} for i ∈ I fixed. The following diagram is commutative.

ℙ(C, D) ⟶^{δ⊗γ} ℙ(A, B)
loc_{F} ↓  ^{loc_{F}} ↓
ℙ(C_{F} , D_{F}) ⟶^{δ_{F} ⊗γ_{F}} ℙ(A_{F} , B_{F})

Proof. Recall from (4.3) the definition of lifting and the surrounding notation. For all maps S and T of M_{m+ϵλ} into A and B respectively and ϕ ∈ ℙ(C, D), we have:

((δ ⊗ γ)ϕ)(Sw_{ϵλ,−n}, T w_{ϵλ,−n}) = ϕ(δSw_{ϵλ,−n}, γT w_{ϵλ,−n})
= [n − 1]_{i}^{!}[ϵιK̃_{i}; n − 1]_{(n−1)}ϕ_{F} ((δS)_{F} w_{ϵλ,n}, (γT )_{F} w_{ϵλ,n})

= [n − 1]_{i}^{!}[ϵιK̃_{i}; n − 1]_{(n−1)}((δ_{F} ⊗ γ_{F})ϕ_{F})(S_{F} w_{ϵλ,n}, T_{F} w_{ϵλ,n}) .

But this is precisely the desired identity: ((δ ⊗ γ)ϕ)_{F} = (δ_{F} ⊗ γ_{F})ϕ_{F}. □

(7.5) PROPOSITION. Localization and induction of pairings commute ; loc_{F} ψ_{τ,ϕ} = ψ_{τ,loc_{F} ϕ} .

Proof. By (7.3) it is sufficient to prove the proposition for the case where F equals the trivial representation and τ is a homomorphism of E into U ;i.e. ψ′_{τ,ϕ} equals ψ_{τ,ϕ}. The induced pairing is determined by the maps,

M ⊗_{R} E →^{_{f} R}_{R} E ⊗ M →^{τ⊗1} U ⊗ M →^{a} M , where a denotes the action of U on M. Since localization of a gives the action on M_{F}, we have:

M_{F} ⊗_{R} E →^{_{f} R}_{R} E ⊗ M_{F} →^{τ⊗1} U ⊗ M_{F} →^{a} M_{F} .

Let τ′ denote the composition of these maps. Then ψ_{τ,ϕ} = ϕ ∘ (τ′ × 1) and so we conclude: loc_{F} ψ_{τ,ϕ} = ϕ_{F} ∘ (τ′ × 1) = ψ_{τ,loc_{F} ϕ} . □

(7.6) We end this section by pointing out one important instance where all pairings are induced .

PROPOSITION. Suppose E and F are finite dimensional U-modules. Let λ ∈ X and M be the U-Verma module M_{m+λ} and ϕ the Shapovalov form on M. Then every invariant pairing of M ⊗_{R} E and M ⊗_{R} F is induced by ϕ.

Proof. From (4.2) it is sufficient to prove the theorem with R replaced by K. Since ℙ(_{K}M ⊗ _{K}E,_{K}M ⊗ _{K}F) ≅ ℙ(_{K}M ⊗ _{K}E ⊗ _{K}F^{ρ_{1}},_{K}M), the dimension of this space equals d_{0} the dimension of the weight space for the trivial U^{0} module in E ⊗ F^{ρ_{1}}.

Set 𝔽 equal to the ad-finite submodule of End (_{K}M_{m+λ}). The work of Joseph and Letzter [JL2,§7] and [JL3,§4] proves that the ad-finite part of U maps surjectively onto 𝔽 and 𝔽 = ∑_{δ,1≤i≤m_{δ}} 𝔽_{δ,i} with m_{δ} equal to the dimension of the weight space for the trivial U^{0} submodule. Let E ⊗ F^{ρ_{1}} = ∑_{δ,1≤j≤n_{δ}} F_{δ,j} denote the corresponding isotypic decomposition E ⊗ F^{ρ_{1}}. Since the Shapovalov form is nondegenerate on _{K}M_{m+λ}, for any subspace L ≠ 0 of 𝔽, ϕ(L·M, M) ≠ 0 . For 1 ≤ i ≤ n_{δ} and 1 ≤ j ≤ m_{δ}, let β_{δ,j,i} be any map of E ⊗F^{ρ_{1}} to U inducing a map to 𝔽 which is an isomorphism of F_{δ,j} onto 𝔽_{δ,i} and is zero on the other summands of E ⊗F^{ρ_{1}}. Now by the nondegeneracy of ϕ, the induced pairings χ_{β_{δ,j,i},ϕ} are linearly independent. So the dimension of the space Ind(_{K}M ⊗_{K}E,_{K}M ⊗_{K}F) is greater than or equal to ∑ n_{δ} m_{δ} where the sum is taken over δ. However we know m_{δ} equals the dimension of the weight subspace for the trivial U^{0} submodule. So d_{0} = ∑ n_{δ} m_{δ} and thus the dimension of the space of induced pairings is greater than or equal to the dimension of the space of all invariant pairings. □

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   E-mail address: ma blc@selway.umt.edu ; tenright@euclid.ucsd.edu