Spectral Characters of Finite–dimensional Representations of Affine Algebras

In this paper we study the category C of finite–dimensional representations of affine Lie algebras. The irreducible objects of this category were classified and described explicitly in [2],[4]. It was known however that C was not semisimple. In such a case a natural problem is to describe the blocks of the category. The blocks of an abelian category are themselves abelian subcategories, each of which cannot be written as a proper direct sum of abelian categories and such that their direct sum is equal to the original category. Block decompositions of representations of algebras are often given by a character, usually a central character, namely a homomorphism from the center of the algebra toC, as for instance in the case of modules from the BGG category O for a simple Lie algebra. In our case however, the center of the universal algebra of the affine algebra acts trivially on all representations in the category C and the absence of a suitable notion of character has been an obstacle to determining the blocks of C. In recent years the study of the corresponding category Cq of modules for quantum affine algebras has been of some interest, [5], [6], [10], [11], [13], [14], [15]. In [7] the authors defined the notion of an elliptic character for objects of Cq when |q| 6 = 1 and showed that for |q| < 1, the character could be used to determine the blocks of Cq. The original definition of the elliptic character used convergence properties of the (non–trivial) action of the R–matrix on the tensor product of finite–dimensional representations. Of course in the q = 1 case, the action of the R–matrix on a tensor product is trivial. However, the combinatorial part of the proof given in [7] suggests that an elliptic character can be viewed as a function χ : E → Z with finite support, where E is the elliptic curve C/q and m ∈ N depends on the underlying simple Lie algebra. This then motivated our definition when q = 1 of a spectral character of L(g) as a function C → Γ with finite support, where Γ is the quotient of the weight lattice of g by the root lattice of g. The other ingredient used in [7] to prove that two modules with the same elliptic character belonged to the same block, was a result proved in [2],[13] that a suitable tensor product of irreducible representations was indecomposable but reducible on certain natural vectors. In the classical case however, it was known from the work of [4] that a tensor product of irreducible representations was either irreducible or completely reducible. However, it was shown in [6] that the the tensor product of the irreducible representations of the quantum affine algebra specialized to indecomposable, but usually reducible representations of the classical affine algebra. This led to the definition of the Weyl modules as a family of universal indecomposable modules. The Weyl modules are in general not well–understood, see [6],[8],[9] for several conjectures about them. However in this paper, we are still able to identify a large family of quotients of the Weyl modules, which allow us to effectively use them as a substitute for the methods of [7]. Although, we work with the affine Lie algebra, our results and proofs work for the current algebra, g ⊗C[t], but with the spectral character being defined as functions from C to Γ with finite support. The paper is organized as follows: section 1 is devoted to preliminaries, section 2 to the definition of the spectral character and the statement of the main theorem. In Section 3, we recall the definition of the Weyl modules, give an explicit realization of certain indecomposable but reducible quotients