Quasi-Whittaker modules for the affine Lie superalgebra sl(2,1)̂

This paper investigates the representation theory of the affine Lie superalgebra sl(2,1)̂ by introducing and studying a new class of modules termed quasi-Whittaker modules. While Whittaker modules have been extensively studied for various Lie algebras and quantum groups, their exploration for affine Lie superalgebras remains largely undeveloped. Drawing inspiration from methods used for the quasihighest weight sl(2,1)̂-modules and the structure of irreducible Whittaker modules on the affine Lie algebra sl2̂ and Heisenberg Lie algebra, we construct these quasi-Whittaker modules Ŵψ,k,l for sl(2,1)̂ by inducing from irreducible Whittaker modules of its even subalgebra. Our main result establishes the irreducibility of this quasi-Whittaker module Ŵψ,k,l. This work represents an initial effort to investigate Whittaker-type representations for affine Lie superalgebras.