Blocks of the category of cuspidal sp_2n-modules

Abstract

In this paper we show that every block of the category of cuspidal generalized weight modules with finite dimensional generalized weight spaces over the Lie algebra sp 2n ‫)ރ(‬ is equivalent to the category of finite dimensional ‫[[ރ‬t 1 , t 2 , . . . , t n ]]-modules. Introduction and description of the resultsFix the ground field to be the complex numbers. Fix n ∈ {2, 3, . . . } and consider the symplectic Lie algebra sp 2n =: g with a fixed Cartan subalgebra h and root space decompositionwhere denotes the corresponding root system. For a g-module V and λ ∈ h * setA g-module V is called• a weight module provided that V = λ∈h * V λ ;• a generalized weight module provided that V = λ∈h * V λ ;• a cuspidal module provided that for any α ∈ the action of any nonzero element from g α on V is bijective.If V is a generalized weight module, then the set {λ ∈ h * : V λ = 0} is called the support of V and is denoted by supp(V ).Denote byᏯ the full subcategory in g-mod that consists of all cuspidal generalized weight modules with finite dimensional generalized weight spaces, and by Ꮿ the full subcategory ofᏯ consisting of all weight modules. Understanding the categories Ꮿ andᏯ is a classical problem in the representation theory of Lie algebras. The first major step towards the solution of this problem was made in [Mathieu 2000], where all simple objects inᏯ were classified. Britten et al. [2004] MSC2000: 17B10.

Restriction to Levi subalgebras and generalization of the category \mathcal{O}

“…We will not try to recall all the known results here. We refer the reader to [11,12,23,22] for details.…”

“…The general case for sl n is more complicated. As we will not need it, we do not mention it here (see [12,22]). On the other hand we will need the case of sp 2n .…”

“…p(H α+β 1 +β 2 ⊗ x(k)) = η 1 (k)p(X + ⊗ x(k − 1)), which gives us the following equation: c + d + a 1 + k =(a 2 − k + 1)η 1 (k). (22) We apply then the vector X β 2 to equation (21). We have p(X −α−β 1 ⊗ x(k)) = η 1 (k)p(H 2 X −β 1 ⊗ x(k − 1)) + η 2 (k)p(X −β 1 H 2 ⊗ x(k − 1)), which gives together with lemma 4.15: η(k) =(d + 1)η 1 (k) + dη 2 (k).…”