Vanishing of Shapovalov determinants in class (IND)

Determine whether Shapovalov determinants det S_\nu vanish identically for some weight \nu for Kac-Moody superalgebras of class (IND), including the corresponding question for Kac-Moody algebras of class (IND).

Background

The paper establishes irreducibility criteria for Verma modules over symmetrizable Kac-Moody superalgebras using factorizations of Shapovalov determinants. In the non-symmetrizable setting, the authors explain that the behavior of these determinants is not generally understood for class (IND), which consists of infinite-dimensional Kac-Moody superalgebras of infinite Gelfand-Kirillov dimension.

References

For Kac-Moody superalgebras of class (IND), it is not known whether $\det S_\nu$ vanishes identically for some $\nu$. To the best of our knowledge, the corresponding question is open even for Kac-Moody algebras of class (IND).

— Irreducibility of Verma modules for symmetrizable Kac-Moody superalgebras  (2610.06109 - Gorelik, 5 Oct 2026) in Section 5, subsection “Non-symmetrizable cases”