Irreducible Generalized Restricted Representations of g∆2

Generalized restricted representations were introduced to facilitate the study of representations of modular Lie algebras that lack a restricted structure. At present, generalized restricted representations are defined for Lie algebras of the Cartan type and for contragredient Lie algebras over fields of positive characteristic. Classifications of irreducible generalized restricted representations have been obtained for both classes of Lie algebras. However, irreducible generalized restricted representations with non-restricted highest weights, as well as their dimensions, remain largely unexplored. For contragredient Lie algebras, the dimensions of such representations are known only in the case of a three-dimensional simple algebra over a field of characteristic 2. In this paper, we study irreducible generalized restricted representations of the ten-dimensional simple contragredient Lie algebra g∆2 with a Cartan matrix of ∆2=0111 over an algebraically closed field of characteristic p=2. We provide a complete classification of the irreducible generalized restricted g∆2 modules and their dimensions (Theorem 1). These modules are parameterized by the fundamental weights ω1 and ω2 and by elements of the finite field F2s.