One-dimensional representations of the finite W-algebra W

Determine whether the finite W-algebra W associated with an even nilpotent element e in a basic classical Lie superalgebra g always admits a one-dimensional representation.

Background

The paper identifies a conjecture from \cite[Conjecture 4.2]{ZS19} asserting that the finite W-algebra W always has a one-dimensional representation. A sufficient condition for the two-dimensional-representation conjecture for U(g,e) was previously obtained under the assumption that this one-dimensional-representation conjecture holds. The paper instead derives equivalences between representation dimensions and types, without proving the universal one-dimensional-representation assertion for W.

References

A sufficient condition for this conjecture was subsequently established in Proposition~4.3, which presumes the validity of the conjecture Conjecture 4.2 that ${ W}$ always admits a one-dimensional 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”