Representation theory of strange Lie superalgebras

In this dissertation, we study the representation theory of strange Lie superalgebras. It is divided into three parts. In the first part, we study categories of finite-dimensional modules over the periplectic Lie superalgebras $mathfrak{p}(n)$ and obtain a BGG type reciprocity. In particular, these categories have only finitely-many blocks. We also compute the characters for irreducible modules over periplectic Lie superalgebras of ranks $2$ and $3$, and obtain explicit description of the blocks for ranks $2$, $3$, and $4$. In the second part, we develop a reduction procedure which provides an equivalence from an arbitrary block of the BGG category for the queer Lie superalgebra $mathfrak{q}(n)$ to a block with weights in $Lambda_{{ell_1},s_{1}}(n_1) imes cdots imes Lambda_{{ell_k},s_{k}}(n_{k})$ (see, Theorem ef{FirstMainThm}) for a BGG category of finite direct sum of queer Lie superalgebras. The descriptions of blocks are given as well. We also establish equivalences between certain maximal parabolic subcategories for $mathfrak{q}(n)$ and blocks of atypicality-one of the category of finite-dimensional modules for $mathfrak{gl}(ell|n-ell)$, where $ell leq n$. In the third part, we establish a maximal parabolic version of the Kazhdan-Lusztig conjecture cite[Conjecture 5.10]{CKW} for the BGG category $mathcal{O}_{k,zeta}$ of $mathfrak{q}(n)$-modules of ``$pm zeta$-weights'', where $kleq n$ and $zetainCsetminushf $. As a consequence, the irreducible characters of these $mathfrak q(n)$-modules in this maximal parabolic category are given by the Kazhdan-Lusztig polynomials of type $A$ Lie algebras. As an application, closed character formulas for a class of $mathfrak q(n)$-modules resembling polynomial and Kostant modules of the general linear Lie superalgebras are obtained.