Demazure Modules, Chari-Venkatesh Modules and Fusion Products

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $\theta$. Given two non-negative integers $m$, $n$, we prove that the fusion product of $m$ copies of the level one Demazure module $D(1,\theta)$ with $n$ copies of the adjoint representation ${\rm ev}_0 V(\theta)$ is independent of the parameters and we give explicit defining relations. As a consequence, for $\mathfrak{g}$ simply laced, we show that the fusion product of a special family of Chari-Venkatesh modules is again a Chari-Venkatesh module. We also get a description of the truncated Weyl module associated to a multiple of $\theta$.


Introduction
Let g be a finite-dimensional complex simple Lie algebra with highest root θ. The current algebra g[t] associated to g as a vector space is equal to g ⊗ C[t], where C[t] is the polynomial ring in one variable. The degree grading on C[t] gives a natural Z ≥0 -grading on g[t] and the Lie bracket is given in the obvious way such that the zeroth grade piece g ⊗ 1 is isomorphic to g. Let g be the untwisted affine Lie algebra corresponding to g. In this paper, we shall be concerned with the g[t]-stable Demazure modules of integrable highest weight representations of g. The Demazure modules are actually modules for a Borel subalgebra b of g. The g[t]-stable Demazure modules are known to be indexed by a pair (l, λ), where l is a positive integer and λ is a dominant integral weight of g and we denote the corresponding module by D(l, λ) (see [7], [9]). We call D(l, λ) the level l Demazure module with weight λ; it is in fact a finite-dimensional graded g[t]-module.
The study of the category of finite-dimensional graded g[t]-modules has been of interest in recent years for variety of reasons. An important construction in this category is that of the fusion product. The fusion product of finite-dimensional graded g[t]-modules [5] is by definition, dependent on the given parameters. Many people have been working in recent years, to prove the independence of parameters for the fusion product of certain g[t]-modules, see for instance [3], [4], [7], [9], [10]. These works mostly considered the fusion product of Demazure modules of the same level and gave explicit defining relations for them. We ask the most natural question: what about the fusion product of different level Demazure modules? In this paper, we answer this question for some important cases; namely we prove (Corollary 3) that the fusion product of many copies of the level one Demazure module D(1, θ) with many copies of the adjoint representation ev 0 V (θ) is independent of parameters, and we give explicit defining relations. We note that ev 0 V (θ) may be thought of as a Demazure module D(l, θ) of level l ≥ 2.
More generally, the following is the statement of our main theorem (see §3 for notation).
We obtain the following two important corollaries: Corollary 3. Given m, n ≥ 0, we have the following isomorphism of g[t]-modules, In [4], Chari and Venkatesh introduced a large collection of indecomposable graded g[t]-modules (which we call Chari-Venkatesh or CV modules) such that all Demazure modules D(l, λ) belong to this collection. In the case that g is simply laced, Theorem 1 enables us to obtain (see Theorem 19) interesting exact sequences between CV modules and to show that the fusion product of a special family of CV modules is again a CV module. Theorem 19 generalizes results of Chari and Venkatesh (see [4, §6]), where they only consider the case g = sl 2 .
Let n ≥ 1, the truncated algebra A n = C[t]/(t n ). We consider local Weyl modules W An (kθ), k ≥ 1 for the truncated current algebra g ⊗ A n . These modules are known to be finite-dimensional, but they are still far from being well understood; even their dimensions are not known. As a corollary to Theorem 1, we are able to obtain the following nice description about these truncated Weyl modules in terms of the local Weyl modules W (kθ), k ≥ 1 of the current algebra g[t]. These later modules are very well understood.
Corollary 4. Assume that g is simply laced. Given k, n ≥ 1, we have the following isomorphism of g[t]-modules, The Corollary 4 is proved in §5.
Acknowledgements. The author thanks Vyjayanthi Chari, K.N. Raghavan and S. Viswanath for many helpful discussions and encouragement. Part of this work was done when the author was visiting the Centre de Recherche mathematique (CRM), Montreal, Canada, as a part of thematic semester on New Directions in Lie Theory. The author acknowledges the hospitality and financial support extended to him by CRM.

Preliminaries
Throughout the paper, C denote the field of complex numbers, Z the set of integers, Z ≥0 the set of non-negative integers, N the set of positive integers and C[t] the polynomial ring in the indeterminate t.

2.2.
Let g be a finite-dimensional complex simple Lie algebra, with cartan subalgebra h. Let R (resp. R + ) be the set of roots (resp. positive roots) of g with respect to h and θ ∈ R + be the highest root in R. There is a non-degenerate, symmetric, Weyl group invariant bilinear form (.|.) on h * , which we assume to be normalized so that the square length of a long root is two. For α ∈ R, α ∨ ∈ h denotes the corresponding co-root and we set d α = 2/(α|α). For α ∈ R, g α be the corresponding root space of g and fix non-zero elements Let P + be the set of dominant integral weights g. For λ ∈ P + , V (λ) be the corresponding finite-dimensional irreducible g-module generated by an element v λ with the following defining relations: We define a morphism between two graded g[t]-modules as a degree zero morphism of g[t]modules. For r ∈ Z, let τ r be the grade shift operator: if V is a graded g[t]-module then τ r V is the graded g[t]-module with the graded pieces shifted uniformly by r and the action of g[t] unchanged. For any graded g[t]-module V and a subset S of V , < S > denotes the submodule of V generated by S. For λ ∈ P + , ev 0 V (λ) be the irreducible graded For α ∈ R + and r, s ∈ Z ≥0 , we define an element where for any non-negative integer b and any The following was proved in [8] (see also [4,Lemma 2.3]).

Weyl, Demazure modules and fusion product
In this section, we recall the definitions of local Weyl modules, level one Demazure modules and fusion products. We also understand them for multiples of θ.
3.1. Weyl module. The definition of the local Weyl module was given originally in [2], later in [1] and [5].
Definition 6. Given λ ∈ P + , the local Weyl module W (λ) is the cyclic g[t]-module generated by an element w λ , with following defining relations: We note that the relation (3.2) implies which is easy to see from Lemma 5. We set the grade of w λ to be zero; then W (λ) becomes a Z ≥0 -graded module with We now specialize to the case λ ∈ Nθ, and obtain some further useful relations that hold in such W (λ).
The following relations hold in the local Weyl module W ((k + 1)θ): , completes the proof of part (1). Part (2) follows easily by putting r = 2, s = 2m + 1 and α = θ in Lemma 5, and using the fact that 3.2. Level one Demazure module. Let λ ∈ P + and α ∈ R + with λ, α ∨ > 0. Let s α , m α ∈ N be the unique positive integers such that If λ, α ∨ = 0, set s α = 0 = m α . We take the following as a definition of the level one Demazure module.
The following proposition gives explicit defining relations for D(1, kθ).
We record below a well-known fact, for later use: In particular, dim D(1, θ) = dim V (θ) + 1. (3.6) The following is a crucial lemma, which we use in proving Theorem 1.

Fusion product.
In this subsection, we recall the definition of the fusion product of finitedimensional graded cyclic g[t]-modules given in [5] and give some elementary properties.
For a cyclic g[t]-module V generated by v, we define a filtration F r V, r ∈ Z ≥0 by We say F −1 V is the zero space. The associated graded space gr V = r≥0 F r V /F r−1 V naturally becomes a cyclic g[t]-module generated by v + F −1 V , with action given by Observe that, gr V ∼ = V as g-modules.
The following lemma is trivial but useful.
Lemma 11. Let V be a cyclic g[t]-module. For r, s ∈ Z ≥0 , the following equality holds in the for all a 1 , · · · , a s ∈ C, x ∈ g, w ∈ F r V.

Given a g[t]
-module V and z ∈ C, we define an another g[t]-module action on V as follows: We denote this new module by V z . Let V i be a finite-dimensional cyclic graded g[t]-module generated by v i , for 1 ≤ i ≤ m, and let z 1 , · · · , z m be distinct complex numbers. We denote the corresponding tensor product of g[t]-modules. It is easily checked (see [5,Proposition 1.4]) that V is cyclic g[t]-module generated by v 1 ⊗ · · · ⊗ v m . The associated graded space grV is called the fusion product of V 1 , · · · , V m w.r.t. parameters z 1 , · · · , z m , and is denoted by V 1 z 1 * · · · * V m zm . We denote v 1 * · · · * v m = (v 1 ⊗ · · · ⊗ v m ) + F −1 V, a generator of grV. For ease of notation we mostly, just write V 1 * · · · * V m for V 1 z 1 * · · · * V m zm . But unless explicitly stated, it is assumed that the fusion product does depend on these parameters.
The following lemma is very useful in showing some elements in fusion products are zero.
Proof. Let z 1 , · · · , z m be distinct complex numbers and let V = V 1 z 1 ⊗· · ·⊗V m zm . By using Lemma 11, we get the following equality in grV, Now proof follows by the definition of fusion product.

Proof of the main theorem
In this section, we prove the existence of maps φ + and φ − and then prove our main theorem (Theorem 1). 4.1. Given k ≥ 1 and 0 ≤ i ≤ k, we denote Using Proposition 9, V i,k is the cyclic graded g[t]-module generated by the element v i,k , with the following defining relations: (4.5) Existence of φ + is trivial, which we record below.
>. Now we prove the existence of φ − in the following proposition. 4.2. The existence of the surjective maps φ + and φ − , give the following:

Proposition 14. There exist a surjective morphism of g[t]-modules such that
(4.6) The following proposition helps in proving the reverse inequality.
Proposition 15. The map ψ : Proof. We only need to show that, ψ(v i,k+1 ) satisfies the defining relations of V i,k+1 . But they follow easily from the relations ( Further from Lemma 12, by using the defining relations of D(1, θ) and ev 0 V (θ).
We record below a result from [6] in our notation, and use this in proving our main theorem.

CV modules and truncated Weyl modules
We start this section by recalling the definition of CV modules given in [4]. For g simply laced, we shall restate Theorem 1 in terms of these modules. At the end, we also discuss truncated Weyl modules.
The following lemma (implicit in the proof of Theorem 1 of [4]) useful in understanding CV modules.
The CV modules corresponding to these two, have nice descriptions, which we record below for later use.
5.2. Given k ≥ 1 and 0 ≤ i ≤ k, we define the following |R + |-tuple of partitions: For g simply laced, we can restate Theorem 1 in terms of CV modules as follows: Theorem 19. Assume that g is simply laced. Given k ≥ 1 and 0 ≤ i ≤ k, we have the following: (1) A short exact sequence of g[t]-modules, Proof. This follows from Theorem 1, by using Lemma 18 and (5.1). 5.3. For n ≥ 1, we define A n = C[t]/(t n ). The truncated current algebra g ⊗ A n , can be thought of as the graded quotient of the current algebra g[t]: g ⊗ A n ∼ = g[t]/(g ⊗ t n C[t]).
Let k ≥ 1. The local Weyl module W An (kθ) for the truncated current algebra g ⊗ A n is defined in [1], and we call as the truncated Weyl module. It is easy to see that W An (kθ) naturally becomes a g[t]-module and the following is an isomorphism of g[t]-modules, W An (kθ) ∼ = W (kθ)/ < (x − θ ⊗ t n ) w kθ > .