Whittaker Modules for W type Cartan Lie superalgebras

TL;DR

This paper advances the representation theory of Witt-type Lie superalgebras by classifying simple non-singular Whittaker modules for $W_{m,n}$. It builds an AW-extension framework and identifies a finite-dimensional subsuperalgebra $T_{m,n}$ (and the ambient $rak{gl}(m,n)$-structure) to describe Whittaker modules via tensor-field constructions $T(A^{oldsymbol{a}},V)$. A covering technique transfers results from the extended Witt algebra to $W_{m,n}$, showing that every simple non-singular Whittaker $W_{m,n}$-module embeds as a subquotient of such tensor-field modules; together with prior work, this yields a complete classification of simple non-singular Whittaker $W_{m,n}$-modules. The work connects Whittaker theory with tensor modules, weightings, and finite-dimensional $rak{gl}(m,n)$-representations, offering a precise structural description with potential implications for vector field superalgebras and their representations.

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 $Ω_{\mathbf a}^{\widetilde{W}_{m,n}}$ 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 $Ω_{\mathbf a}^{{W}_{m,n}}$ of such modules with finite-dimensional Whittaker vector spaces and with non-singular ${\mathbf a}$.

Theorems & Definitions (24)

• Theorem 1.1
• Proposition 2.1
• Lemma 3.1
• Lemma 3.2: LX, Lemma 3.5
• Lemma 3.3: LX, Lemma 3.6
• Lemma 3.4: LX, Lemma 3.7
• Lemma 4.1
• proof
• Lemma 4.2
• Lemma 4.3