Blocks and characters of $D(2|1;ζ)$-modules of non-integral weights

Abstract

We classify blocks in the BGG category O of modules of non-integral weights for the exceptional Lie superalgebra D(2|1; ζ). We establish various reduction methods, which connect some types of non-integral blocks of D(2|1; ζ) with integral blocks of general linear Lie superalgebras gl(1|1) and gl(2|1). We compute the characters for irreducible D(2|1; ζ)-modules of non-integral weights in O.

Blocks and characters of $G(3)$-modules of non-integral weights

“…Finite-dimensional modules have been studied in [Ger00,Ma14,SZ16]. Subsequently, the authors of the present article solved the irreducible character problem of D(2|1; ζ)-modules in the non-integral weight case in [CCL20], thus completing the computation of irreducible characters for D(2|1, ζ) in O.…”

“…1.3. Our strategy of computing characters of tilting modules is similar to [CCL20] and is divided into two methods: reduction methods and the application of translation functors.…”

“…The method of translation functor follows that of [CW19,CW18,CCL20]. Namely, for a given tilting module T whose character we wish to compute, we make a reasonable choice of an initial tilting module.…”

“…The main tools used there are twisting, odd reflection, and parabolic induction functors. Along this line of reduction, there have been some applications to other types of Lie superalgebras in [Ch16,CCL20]. In the present paper, we establish equivalences of categories connecting the blocks in O e , O Z 2 with blocks of the special linear Lie algebra sl(2), the exceptional Lie algebra G 2 , the general linear Lie superalgebras gl(1|1), gl(2|1) and the ortho-symplectic Lie superalgebra osp(3|2) in Sections 4 and 5.…”

A Schrödinger model, Fock model and intertwining Segal-Bargmann transform for the exceptional Lie superalgebra $D(2,1;α)$

“…This makes the study of these Lie superalgebras more challenging but also intriguing since we do not know what to expect. Representation theory of the Lie superalgebra D(2, 1; α) has already been studied extensively, see for example [18,19,20].…”