Two-dimensional irreducible representations of finite W-superalgebras

Establish that the finite W-superalgebra U(g,e) associated with an even nilpotent element e in a basic classical Lie superalgebra g always admits a two-dimensional irreducible representation.

Background

The paper discusses a conjecture from \cite[Conjecture 1.3 (2)]{ZS17} asserting the existence of a two-dimensional irreducible representation for every finite W-superalgebra U(g,e). The results proved in the paper establish the existence of finite-dimensional representations and relate the representation theories of U(g,e) and the associated algebra W, but they do not establish this uniform two-dimensionality assertion in general. The remark explains how the conjecture is connected to a separate conjecture concerning one-dimensional representations of W.

References

It was conjectured in Conjecture 1.3 (2) that the finite $W$-superalgebra $U(g, e)$ always admits a two-dimensional irreducible representation.

— Categorical Equivalences of Finite W-Superalgebras and Clifford Twists  (2608.22749 - Chen et al., 24 Aug 2026) in Remark \ref{res::fdim2}, Subsection “Finite-dimensional representations of W”