Whittaker Modules for W type Cartan Lie superalgebras

Abstract

We consider the category of Whittaker modules for the Lie superalgebra $W_{m,n}$ of vector fields on ${\mathbb{C}}^{(m|n)}$. For any $\mathbf{a}\in {\mathbb{C}}^m$ we show the equivalence between the blocks $\hat{I}{\text{c◯}}_{\mathbf{a}}^{{\tilde{W}}_{m,n}}$ of the category of $(AW{)}_{m,n}$-Whittaker modules with finite-dimensional Whittaker vector spaces and the category of finite-dimensional modules over certain Lie subsuperalgebra $T_{m,n}$ of $(AW{)}_{m,n}$ (and also of $\mathfrak{gl}(m,n))$. Then we apply the covering technique to study Whittaker $W_{m,n}$-modules and describe simple modules in the category $\hat{I}{\text{c◯}}_{\mathbf{a}}^{W_{m,n}}$ of such modules with finite-dimensional Whittaker vector spaces and with non-singular $\mathbf{a}$.

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