On certain maximal cyclic modules for the quantized special linear algebra at a root of unity ^{∗}

Kaneda Masaharu 558-8585 Osaka, Japan Osaka City University Department of Mathematics kaneda@sci.osaka-cu.ac.jp Nakashima Toshiki 102-8554 Tokyo, Japan Sophia University Department of Mathematics toshiki@mm.sophia.ac.jp

November 28, 2002

Abstract

By properly specializing the parameters irreducible modules of maximal dimension for the De Concini-Kac version of the Drinfeld-Jimbo quantum algebra in type A may be transformed into modules over Lusztig’s infinitesimal quantum algeba. Thus obtained modules have a simple socle and a simple head, and share the same dimension as the infinitesimal Verma modules. Despite these common features we find that they are never isomorphic to infinitesimal Verma modules unless they are irreducible. The same carry over to the modular setup for the special linear groups in positive characteristic.

The finite dimensional irreducible representations of the De Concini-Kac version of the Drinfeld-Jimbo quantized enveloping algebra at a complex ℓ-th root of 1 have their dimensions bounded above, generically attaining a maximal dimension [DK]. In type A Date, Jimbo, Miki and Miwa [DJMM] have given a concrete realization of most of those of maximal dimension. By properly specializing their parametres the second named author of the present paper found in [N] that they afford modules V, rarely irreducible, for Lusztig’s “infinitesimal” quantum algebra 𝔲 [L1] and that each V has a unique, up to scalar, invariant vector u0 relative to a Borel subalgebra 𝔲^{♯} of 𝔲, and hence that V has a simple socle generated by u0. The dimension being right, it is tempting to compare V with Humphreys’ “infinitesimal” Verma modules [H] quantized by Andersen, Polo and Wen [APW], which are the standard objects of study in the representation theory of 𝔲.

It is easy to see that V is isomorphic to an infinitesimal Verma module as 𝔲^{♯}-module, which in turn shows that V has the same simple head as the infinitesimal Verma module. The explicit description of the actions of the standard generators of 𝔲 on V allows us, however, to find that V has also a unique, up to scalar, invariant vector u_{−} with respect to the opposite infinitesimal Borel subalgebra. It follows that V does not lift to an integrable 𝔲U_{ℂ}^{0}-module, and hence V can not be isomorphic to any infinitesimal Verma module as 𝔲-module unless V is simple, where U_{ℂ}^{0} is the Cartan part of Lusztig’s quantum algebra U_{ℂ} at the ℓ-th root of 1.

∗supported in part by JSPS Grants in Aid for Scientific Research

By construction V may be defined over B = ℤ[v, v^{−1}]/(ϕ_{ℓ}) with ϕ_{ℓ} the ℓ-th cyclotomic polynomial in indeterminate v. If ℓ is an odd prime p, B/(v − 1) is a finite field 𝔽_{p} of pelements. Let G be the special linear group scheme over 𝔽_{p} with opposite Borel subgroups B and B^{+}, and let G_{1}, B_{1}, B_{1}^{+} be the Frobenius kernel of G, B, and B_{1}^{+}, respectively. If V_{B} is the B-form of V, then V_{p} = V_{B} ⊗_{B} 𝔽_{p} is naturally a G_{1}-module. We find that u0 ⊗ 1 (resp. u_{−} ⊗ 1) remains a unique, up to 𝔽_{p}^{×}, B_{1}^{+}- (resp. B_{1}-)invariant vector in V_{p}. Hence V_{p} is isomorphic to an infinitesimal Verma module as B_{1}^{+}-module, but not as G_{1}-module unless V_{p} is simple.

If the simple 𝔲-module generated by u0 in V and the simple G_{1}-module generated by u0 ⊗ 1 in V_{p} have the same dimension, Lusztig’s conjecture for the irreducible characters of G-modules will follow from the celebrated theorems of Kazhdan and Lusztig [KL] and Kashiwara and Tanisaki [KT] and Casian [C].

If C is a category, C(A, B) will denote the set of morphisms of C from object X of C to object Y of C. If A is a ring, AMod will denote the category of left A-modules.

We are grateful to H.H. Andersen and J.C. Jantzen for a helpful comment.

1^{∘} Infinitesimal Verma modules

In this section we recollect some facts about infinitesimal Verma modules over an arbitrary quantum algebra of finite type.

(1.1) Let ℚ(v) be the fractional field of the Laurent polynomial ring A = ℤ[v, v^{−1}] in indeterminate v, A = [(A_{ij})] an indecomposable Cartan matrix of finite type and let U be the associated Drinfeld-Jimbo quantum algebra over ℚ(v) with generators E_{i}, F_{i}, and K_{i}^{±1}, i ∈ [1, n]. Let U be Lusztig’s A-subalgebra of U generated by E_{i}^{(r)} = \frac{E_{i}^{r}}{[r]_{i}!}, F_{i}^{(r)} = \frac{F_{i}^{r}}{[r]_{i}!},

K_{i}^{±1}, i ∈ [1, n], r ∈ ℕ, where [r]_{i}! = ∏_{s=1}^{r}[s]_{i} with [s]_{i} = \frac{v_{i}^{s}−v_{i}^{−s}}{v_{i}−v_{i}^{−1}} , v_{i} = v^{d_{i}}, d_{i} ∈ \{1, 2, 3\} minimal such that the matrix [(d_{i}A_{ij})] is symmetric.

Let R (resp. Λ) be the root system (resp. the weight lattice) associated to A and R^{+} a positive subsystem of R with the simple roots α_{i}, i ∈ [1, n]. We equip Λ with a partial order defined by R^{+} as usual. Let Λ^{∨} be the colattice of Λ and denote by ⟨ , ⟩ : Λ×Λ^{∨} → ℤ the perfect pairing. If α ∈ R, let α^{∨} be its coroot. We set λ_{i} = ⟨λ, α_{i}^{∨}⟩ for λ ∈ Λ. Let

U^{0} be the A-subalgebra of U generated by K_{i}^{±1} and [\begin{matrix}K_{i}; c \\ r \end{matrix}] = ∏_{s=1}^{r} \frac{K_{i}v_{i}^{c−s+1}−K_{i}^{−1}v_{i}^{−c+s−1}}{v_{i}^{s}−v_{i}^{−s}},

i ∈ [1, n], c ∈ ℤ, r ∈ ℕ. Each λ ∈ Λ defines an A-algebra homomorphism χ_{λ} : U^{0} → A such that

K_{i} ↦ v_{i}^{λ_{i}}, [\begin{matrix}K_{i}; c \\ r \end{matrix}] ↦ [\begin{matrix}λ_{i} + c \\ r \end{matrix}]_{i} = \frac{∏_{s=0}^{r−1}[λ_{i} + c − s]_{i}}{[r]_{i}!} ∀i ∈ [1, n], c ∈ ℤ, r ∈ ℕ.

(1.2) Let ℓ be a positive integer greater than 2 prime to all entries A_{ij} of the Cartan matrix A, K = ℚ[v]/(ϕ_{ℓ}), and set U_{K} = U ⊗_{A} K. Let 𝔲 (resp. 𝔲^{+}; 𝔲^{−}; 𝔲^{0}) be the K-subalgebra of U_{K} generated by E_{i} ⊗ 1, F_{i} ⊗ 1, K_{i} ⊗ 1 (resp. E_{i} ⊗ 1; F_{i} ⊗ 1; K_{i} ⊗ 1), i ∈ [1, n]. Let also 𝔲^{♯} = 𝔲^{+}𝔲^{0} and 𝔲^{♭} = 𝔲^{0}𝔲^{−}. We will abbreviate x ⊗ 1 of U_{K} as x, and χ_{λ} ⊗_{A} K as χ_{λ}.

Let 𝔲̃ be the K-subalgebra of U_{K} generated by 𝔲 and U_{K}^{0} = U^{0} ⊗_{A} K, and let 𝔲̃^{♯} = 𝔲^{♯}U_{K}^{0}, 𝔲̃^{♭} = U_{K}^{0}𝔲^{♭}. Each λ ∈ Λ defines a 1-dimensional 𝔲̃^{♭}-module by χ_{λ} annihilating all F_{i}, which we will still denote by λ. Let ˜∇(λ) = 𝔲̃^{♭}Mod(𝔲̃, λ). We make ˜∇(λ) into a 𝔲̃-module by setting xf = f(?x) for each x ∈ 𝔲̃ and f ∈ ˜∇(λ).

Let Λ^{res} = {ν ∈ Λ | ν_{i} ∈ [0, ℓ − 1] ∀i}. If we write λ = λ^{0} + ℓλ^{1} with λ^{0} ∈ Λ^{res} and λ^{1} ∈ Λ, one has from [APW, 1.9] an isomorphism of 𝔲̃-modules

(1)  ˜∇(λ) ≃ ˜∇(λ^{0}) ⊗_{K} ℓλ^{1},

where ℓλ^{1} ia a 1-dimensional 𝔲̃-module defined by χ_{ℓλ^{1}} annihilating all E_{i}, F_{i} and K_{i} − 1.

On the other hand, the natural gradation on 𝔲^{+} assigning each E_{i} grade α_{i} equips 𝔲^{+} with a structure of 𝔲̃^{♯}-module such that 𝔲^{+} act by the left multiplication and U_{K}^{0} by χ_{−λ+ν} on the ν-th homogeneous part of 𝔲^{+}, ν ∈ ∑_{i} ℕα_{i}. Recall antiautomorphism Ψ on U such that E_{i} ↦ E_{i}, F_{i} ↦ F_{i} and K_{i} ↦ K_{i}^{−1}, i ∈ [1, n]. If M is a 𝔲̃^{♯}-module of finite type, we will denote by M^{Ψ} the K-linear dual of M made into 𝔲̃^{♯}-module by setting xf = f(Ψ(x)?) for each x ∈ 𝔲̃^{♯}, f ∈ Mod_{K}(M, K). Then we have an isomorphism of 𝔲̃^{♯}-modules

(2)  ˜∇(λ) ≃ (𝔲^{+})^{Ψ} via f ↦ f ∘ (Ψ ⊗_{A} K).

One can likewise define a 𝔲-module ∇(λ) = 𝔲^{♭}Mod(𝔲, λ). By restricting the 𝔲̃-action to 𝔲, ˜∇(λ) yields ∇(λ). Then the isomorphism (2) restricts to an isomorphism of 𝔲^{♯}-modules

(3)  ∇(λ) ≃ (𝔲^{+})^{Ψ}.

(1.3) Recall from Xi [X, 2.5] that

(1)  𝔲^{+} has a simple socle K ∏_{α∈R^{+}} E_{α}^{ℓ−1} as 𝔲^{+}-module,

where E_{α} is a root vector of 𝔲^{+} associated to α ∈ R^{+} [L2] and the product is taken in a certain specific order. It follows that 𝔲^{+} is indecomposable as 𝔲^{+}-module, and hence that

(2)  𝔲^{+} is a projective cover of trivial module K as 𝔲^{+}-module.

By an integrable 𝔲̃- (resp. 𝔲-) module M we will mean a 𝔲̃- (resp. 𝔲-) module M such that

M = ∐_{μ∈Λ} M_{μ}, with M_{μ} = {m ∈ M | tm = χ_{μ}(t)m ∀t ∈ U_{K}^{0} (resp. 𝔲^{0})}.

Each ˜∇(λ) (resp. ∇(λ)) is an integrable 𝔲̃- (resp. 𝔲-) module. Define integrable 𝔲̃^{♯}- and 𝔲^{♯}-modules likewise. One obtains from (2) and (1.2.2, 3)

Proposition. For each λ ∈ Λ the 𝔲^{♯}- (resp. 𝔲̃^{♯}-) module ∇(λ) (resp. ˜∇(λ)) is an injective hull of λ in the category of integrable 𝔲^{♯}- (resp. 𝔲̃^{♯}-) modules.

(1.4) Let ζ be the image of v in K, and let U_{ζ} be the De Concini-Kac algebra [DK] over K associated to the Cartan matrix A with the generators E_{i}, F_{i}, K_{i}^{±1}, i ∈ [1, n], and the same relations for U with v replaced by ζ. For each α ∈ R^{+} let E_{α} (resp. F_{α}) be the root vector of U_{ζ} associated with α (resp. −α), and let \overline{U}_{ζ} = U_{ζ}/(E_{α}^{ℓ}, F_{α}^{ℓ} | α ∈ R^{+}). If \overline{U}_{ζ}^{♭} is the K-subalgebra of \overline{U}_{ζ} generated by all F_{α}, α ∈ R^{+}, and K_{i}, i ∈ [1, n], each λ ∈ Λ

defines a 1-dimensional \overline{U}_{ζ}^{♭}-module by annihilating all F_{α} and letting K_{i} act by ζ^{d_{i}λ_{i}}. Then \overline{U}_{ζ} ⊗\overline{U}{♭ \atop ζ} λ comes equipped with a structure of 𝔲̃-module [AJS, 2.10] such that each

x ∈ U_{K}^{0} acts on ∏_{α∈R^{+}} E_{α}^{c_{α}} ⊗ 1, c_{α} ∈ ℕ, by the scalar χ_{λ+∑_{α} c_{α}α}(x). Put ρ = \frac{1}{2} ∑_{α∈R^{+}} α.

Proposition. For each λ ∈ Λ we have an isomorphism of 𝔲̃-modules

˜∇(λ) ≃ \overline{U}_{ζ} ⊗\overline{U}{♭ \atop ζ} (λ − 2(ℓ − 1)ρ).

Proof: Let ε_{λ} ∈ ˜∇(λ) be the element induced by the counit of 𝔲^{+}. By the universality of ˜∇(λ) [APW, 0.8.1] there is a homomorphism of 𝔲̃-modules

(1)  \overline{U}_{ζ} ⊗\overline{U}{♭ \atop ζ} (λ − 2(ℓ − 1)ρ) → ˜∇(λ) such that E^{+} ⊗ 1 ↦ ε_{λ},

where E^{+} = ∏_{α∈R^{+}} E_{α}^{ℓ−1}. On the other hand, by [AJS, 4.9]

(2)  the 𝔲̃-socle of \overline{U}_{ζ} ⊗\overline{U}{♭ \atop ζ} (λ − 2(ℓ − 1)ρ) is simple of highest weight λ.

It follows that the map (1) is injective, and hence bijective by dimension.

(1.5) Corollary. Each ˜∇(λ) (resp. ∇(λ)), λ ∈ Λ, is the projective cover of λ − 2(ℓ − 1)ρ as integrable 𝔲̃^{♯}- (resp. 𝔲^{♯}-) module.

(1.6) Because of the isomorphism (1.4) we call ˜∇(λ) and also by abuse of language ∇(λ) the infinitesimal Verma module of highest weight λ. By [AJS, 6.3 and 4.10.1]

(1) ˜∇(λ) (resp. ∇(λ)) is simple as 𝔲̃- (resp. 𝔲-) module iff λ ≡ (ℓ − 1)ρ mod ℓΛ .

Let 𝔲^{++} (resp. 𝔲^{−−}) be the augmentation ideal of 𝔲^{+} (resp. 𝔲^{−}). If M is a 𝔲^{±}-module, let M^{𝔲^{±±}} denote the annihilater of 𝔲^{±±} in M. By (1.3)

(2)  ˜∇(λ)^{𝔲^{++}} = ∇(λ)^{𝔲^{++}} = λ.

If λ ≡ (ℓ − 1)ρ mod ℓΛ, then ∇((ℓ − 1)ρ + ℓν) = ∇((ℓ − 1)ρ), ν ∈ Λ, is simple, called the Steinberg module, and hence

(3)  ∇((ℓ − 1)ρ + ℓν)^{𝔲^{−−}} = −(ℓ − 1)ρ + ℓν.

In general, the lowest weight of ˜∇(λ) (resp. the socle of ˜∇(λ)) is λ − 2(ℓ − 1)ρ (resp. w_{0}λ^{0} + ℓλ^{1} if w_{0} is an element of the Weyl group of R such that w_{0}R^{+} = −R^{+} and if one writes λ = λ^{0} + ℓλ^{1} with λ^{0} ∈ Λ^{res} and λ^{1} ∈ Λ [AJS, 4.2.5]). It follows that

(4)  dim ∇(λ)^{𝔲^{−−}} ≥ 2 unless ∇(λ) is 𝔲-simple.

(1.7) Let B = A/(ϕ_{ℓ}) and let 𝔲̃_{B} be the B-subalgebra of U ⊗_{A}B generated by E_{i}⊗1, F_{i}⊗1,

K_{i} ⊗ 1, [\begin{matrix}K_{i}; c \\ r \end{matrix}] ⊗ 1, i ∈ [1, n], c ∈ ℤ, r ∈ ℕ. Define its B-subalgebras 𝔲̃_{B}, 𝔲_{B}, 𝔲̃_{B}^{♭}, and 𝔲_{B}^{♭}

as for 𝔲̃. An infinitesimal Verma module may be defined over B; ˜∇_{B}(λ) = 𝔲̃_{B}^{♭}Mod(𝔲̃_{B}, λ), λ ∈ Λ, admits a structure of 𝔲̃_{B}-module like ˜∇(λ), and we have an isomorphism of 𝔲̃- modules

˜∇_{B}(λ) ⊗_{B} K ≃ ˜∇(λ).

Restricting the 𝔲̃_{B}-action to 𝔲_{B}, one obtains 𝔲_{B}-module ∇_{B}(λ) = 𝔲_{B}^{♭}Mod(𝔲_{B}, λ).

Assume now that ℓ is a prime p. Then B/(v − 1) is a finite filed 𝔽_{p} of p-elements. Let G be a simply connected simple algebraic group over 𝔽_{p} associated to the Cartan matrix A with a Borel subgroup B and a maximal torus T of B both split over 𝔽_{p} such that the roots of B are −R^{+}. Let G_{1} (resp. B_{1}) be the Frobenius kernel of G (resp. B). If Dist(G_{1}) (resp. Dist(B_{1})) is the algebra of distibutions of G_{1} (resp. B_{1}), there are isomorphisms of 𝔽_{p}-algebras [L2]

𝔲_{B}/(K_{i} − 1 | i ∈ [1, n]) ⊗_{B} 𝔽_{p} ≃ Dist(G_{1}), 𝔲_{B}^{♭}/(K_{i} − 1 | i ∈ [1, n]) ⊗_{B} 𝔽_{p} ≃ Dist(B_{1}),

and each ˜∇_{p}(λ) := ˜∇_{B}(λ) ⊗_{B} 𝔽_{p}, λ ∈ Λ, admits a structure of G_{1}T -module (cf. [J, II.9]):

˜∇_{p}(λ) ≃ Dist(G_{1}) ⊗_{Dist(B_{1})} (λ − 2(p − 1)ρ).

Likewise ∇_{B}(λ) ⊗_{B} 𝔽_{p} yields a G_{1}-module, which we will denote by ∇_{p}(λ).

2^{∘} Maximal cyclic modules

In this section we assume that our Cartan matrix is of type A_{n}. Then all d_{i} = 1, and we will suppress i from [ ]_{i}. By properly specializing the parametres a maximal cyclic module V for U_{ζ} of [DJMM] factors through 𝔲, having a unique, up to scalar, 𝔲^{+}-primitive vector. Thus V is of dimension ℓ^{|R^{+}|} and has a simple 𝔲-socle, inviting us to compare V with infinitesimal Verma modules.

(2.1) Fix λ ∈ Λ. We define as we may V to be a K-linear space of basis u_{m}, m ∈ (ℤ/ℓℤ)^{|R^{+}|}. After [N] we reindex R^{+} by the pairs (i, j), 1 ≤ i ≤ j ≤ n, and we will denote the (i, j)- component of m by m_{ij}. Then V admits a structure of integrable 𝔲-module as proved in [N, 5.2] such that for each i ∈ [1, n] and m ∈ (ℤ/ℓℤ)^{|R^{+}|}

(1)  E_{i}u_{m} = ∑_{k=i}^{n}[m_{ik} + m_{i,k−1} − m_{i−1,k−1} − m_{i+1,k}]u_{m+ϵ(i,k)+···+ϵ(i,n)},

(2)  F_{i}u_{m} = ∑_{k=1}^{i}[−λ_{i} + m_{i+1−k,n−k} − m_{i+1−k,n+1−k} + m_{i−k,n+1−k} − m_{i−k,n−k}]
u_{m−ϵ(i+1−k,n+1−k)−ϵ(i+2−k,n+2−k)−···−ϵ(i,n)},
(3)  K_{i}u_{m} = ζ^{λ_{i}+2m_{in}−m_{i−1,n}−m_{i+1,n}}u_{m},

where [r] = \frac{ζ^{r}−ζ^{−r}}{ζ−ζ^{−1}} , ϵ(i, j) ∈ (ℤ/ℓℤ)^{|R^{+}|} such that ϵ(i, j)_{st} = δ_{is}δ_{jt} for each s and t, and any meaningless terms in the sums should be read as 0. As the structure of 𝔲-module on V depends on λ, to be precise, we will denote the 𝔲-module V by V(λ).

A main theorem of [N] is that V(λ) has a unique, up to K^{×}, 𝔲^{+}-primitive element, i.e.,

(4)  V(λ)^{𝔲^{++}} = Ku0 with 0 = (0, …, 0),

and hence by Engel’s theorem

(5)  V(λ) has a simple 𝔲-socle generated by u0.

It also follows from (1.3) by dimension that Proposition. There is an isomorphism of 𝔲^{♯}-modules V(λ) ≃ ∇(λ). (2.2) Recall anitiautomorphism τ of 𝔲 such that

E_{i} ↦ F_{i}, F_{i} ↦ E_{i}, K_{i} ↦ K_{i}, ∀i ∈ [1, n].

If M is a 𝔲-module, let M^{τ} be the K-dual space of M with a 𝔲-action given by

xf = f(τ(x)?), f ∈ M^{∗}, x ∈ 𝔲.

Then the isomorphism of 𝔲^{♯}-modules V(λ) ≃ ∇(λ) from (2.1) yields an isomorphism of 𝔲^{−}-modules

V(λ)^{τ} ≃ ∇(λ)^{τ}
≃ \{\overline{U}_{ζ} ⊗\overline{U}{♭ \atop ζ} (λ − 2(ℓ − 1)ρ)\}^{τ} by (1.4)
≃ \overline{U}_{ζ} ⊗\overline{U}{♯ \atop ζ} λ by [AJS, 4.10]
≃ 𝔲^{−},

where \overline{U}_{ζ}^{♯} is the K-subalgebra of \overline{U}_{ζ} generated by all E_{α}, α ∈ R^{+}, and K_{i}, i ∈ [1, n]. It follows from [X, 2.5] again that V(λ)^{τ} has a unique, up to K^{×}, 𝔲^{−}-primitive vector, and hence Corollary. V(λ) has the same simple 𝔲-socle and the same simple 𝔲-head head as ∇(λ). (2.3) We find, moreover, that Theorem. The 𝔲-module V(λ) has a unique, up to K^{×}, 𝔲^{−}-primitive vector u_{m} with

m_{ij} ≡ − ∑_{s=1}^{i} ∑_{t=i}^{j} λ_{n+s−t} mod ℓ ∀i ≤ j,

which has 𝔲^{0}-weight w_{0}λ. Proof: The argument is the same as for (2.1.4) from [N, 4.2]; let

∑_{m∈(ℤ/ℓℤ)^{|R^{+}|}} c_{m}u_{m} ∈ V(λ)^{𝔲^{−−}}, c_{m} ∈ K.

As F_{i} ∑ c_{m}u_{m} = 0 for all F_{i}, we obtain successively c_{m} = 0 unless

(1)  −λ_{i} + m_{i+1−k,n−k} − m_{i+1−k,n+1−k} + m_{i−k,n+1−k} − m_{i−k,n−k} ≡ 0 mod ℓ

for each i and k ∈ [1, i]. If ∑ c_{m}u_{m} ≠ 0, the system (1) of equations determines m with c_{m} ≠ 0 uniquely as asserted.

(2.4) Corollary. Let λ ∈ X.

(i) There is an isomorphism of 𝔲^{♭}-modules

V(λ) ≃ \overline{U}_{ζ} ⊗\overline{U}{♯ \atop ζ} (w_{0}λ + 2(ℓ − 1)ρ).

(ii) The 𝔲-module V(λ) lifts to an integrable 𝔲̃-module iff λ ≡ (ℓ − 1)ρ mod ℓΛ. In particular, unless λ ≡ (ℓ − 1)ρ mod ℓΛ, V(λ) is not isomorphic as 𝔲-module to any infinitesimal Verma module.

Proof: For (i) argue as in (2.1) and (2.2).

(ii) If λ ≡ (ℓ − 1)ρ mod ℓΛ, the simple 𝔲-socle of V(λ) has dimension ℓ^{|R^{+}|} = dim V(λ) and hence the assertion follows. We may therefore assume that λ ≢ (ℓ − 1)ρ mod ℓΛ.

Just suppose V(λ) lift to an integrable 𝔲̃-module. Then we would have from (1.3) an isomorphism of 𝔲̃^{♯}-modules

V(λ) ≃ ˜∇(λ + ℓν) for some ν ∈ Λ.

Then the unique 𝔲^{−}-primitive in V(λ) should have by (1.4) weight λ + ℓν − 2(ℓ − 1)ρ. That would yield, arguing as in (2.3), an isomorphism of 𝔲̃^{♭}-modules

V(λ) ≃ \overline{U}_{ζ} ⊗\overline{U}{♯ \atop ζ} (λ + ℓν).

Then V(λ) = 𝔲^{−}u0 as u0 is a highest weight vector of V(λ). But 𝔲^{−}u0 = 𝔲u0 is the simple socle of V(λ) and of dimension < dim V(λ) by (1.6.1), absurd.

(2.5) Assume now that ℓ is a prime p and let B = A/(ϕ_{p}) = ℤ[v]/(ϕ_{p}) as in (1.7). By construction V(λ) may be defined over B: let V_{B}(λ) be the free B-module of basis u_{m}, m ∈ (ℤ/pℤ)^{|R^{+}|}, with the 𝔲_{B}-action given by (2.1.1-3). Regarding 𝔽_{p} as the quotient B/(v −1), let V_{p}(λ) = V_{B}(λ) ⊗_{B} 𝔽_{p}. Then V_{p}(λ) is naturally a G_{1}-module for G = SL_{n+1} ⊗_{ℤ} 𝔽_{p} in the setup of (1.7). The proof of (2.1.4) from [N] and the argument of (2.3) carry over to obtain

Theorem. Let λ ∈ Λ

(i) There is an isomorphism of B_{1}^{+}-modules V_{p}(λ) ≃ ∇_{p}(λ).

(ii) V_{p}(λ) has a unique, up to 𝔽_{p}^{×}, B_{1}^{+}- and B_{1}-invariant vector, respectively, and hence has a simple G_{1}-socle and a simple G_{1}-head, where B_{1}^{+} is the Frobenius kernel of the Borel subgroup B^{+} of G opposite to B.

(iii) If λ ≢ (p − 1)ρ mod pΛ, the structure of G_{1}-module on V_{p}(λ) does not lift to G_{1}T - module, and hence V_{p}(λ) is not isomorphic to any infinitesimal Verma module as G_{1}-module.

(2.6) Remark. If the simple 𝔲-module generated by u0 in V(λ) and the simple G_{1}-module generated by u0 ⊗ 1 in V_{B}(λ) ⊗_{B} 𝔽_{p} have the same dimension, then Lusztig’s conjecture for the irreducible SL_{n+1} ⊗_{ℤ}𝔽_{p}-modules will follow from [KL] and [KT], [C]; for p ≥ n + 1 the converse is also expected to hold.

References

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[APW] Andersen, H. H., Polo, P. and Wen, K., Injective modules for quantum algebras, Am. J. Math. 114 (1992), 571-604.

[C] Casian, L., Proof of the Kazhdan-Lusztig conjecture for Kac-Moody algebras (the characters chL_{wρ−ρ}), Adv. Math. 119 (1996), 207–281.

[DJMM] Date, E., Jimbo, M., Miki, K. and Miwa, T., Cyclic representations of U_{q}(𝔰𝔩(n+ 1, ℂ)) at q^{N} = 1, Publ. RIMS 27 (1991), 347–366.

[DK] De Concini, C. and Kac, V., Representations of quantum groups at roots of 1, pp. 471–506 in: A. Conne et al. (eds.), Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Colloque Dixmier), Proc. Paris 1989 (Progress in Mathematics 92), Boston etc. 1990 (Birkhäuser)

[H] Humphreys, J. E., Modular representations of classical Lie algebras and semisimple groups, J. Alg. 19 (1971), 51–79.

[J] Jantzen, J. C., Representations of Algebraic Groups, Pure and Applied Mathematics 131, Academic Press, Boston etc. 1987.

[KT] Kashiwara, M. and Tanisaki, T., Kazhdan-Lusztig conjecture for affine Lie algebras with negative level, Duke Math. J. 77 (1995), 21–62

[KL] Kazhdan, D.A. and Lusztig, G., Tensor structures arising from affine Lie algebras I, II, J. AMS 6 (1993), 905–1011; Tensor structures arising from affine Lie algebras III, IV, J. AMS 7 (1994), 335–453.

[L1] Lusztig, L., Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. AMS. 3 (1990), 257–296.

[L2] Lusztig, L., Quantum groups at roots of 1, Geom. Ded. 35 (1990), 89–114.

[N] Nakashima, T., Irreducible modules of finite dimensional quantum algebras of type A at roots of unity, J. Math. Phys. 43 (2002), 2000–2014.

[X] Xi, N., Irreducible modules of quantized enveloping algebras at roots of 1, Publ. RIMS 32 (1996), 235–276.