Irreducible representations of the classical Lie superalgebra of type $$A(0,n)$$ in prime characteristic

Abstract

Let \(\mathfrak{g }=\mathfrak{s }\mathfrak{l }(1|n+1)\) be the classical Lie superalgebra of type \(A(0,n)\) over an algebraically closed field of prime characteristic \(p>2\). A sufficient condition is provided for baby Kac \(\mathfrak{g }\)-modules to be simple. Moreover, simple \(\mathfrak{g }\)-modules with (quasi) regular semisimple characters are classified. In particular, up to isomorphism, all the simple modules for \(\mathfrak{s }\mathfrak{l }(1|2)\) are determined, and representatives and dimensions of simples are precisely given. As an application, simple modules for the general linear Lie superalgebra \(\mathfrak{g }\mathfrak{l }(1|n+1)\) with certain \(p\)-characters are classified. In particular, a complete classification of simple \(\mathfrak{g }\mathfrak{l }(1|2)\)-modules is given.