O-freeness of imaginary cuspidal algebras and their Gelfand–Graev truncations

Determine whether the imaginary cuspidal algebra \hat C_{d,\mathcal O} and the Gelfand–Graev idempotent truncation C_{d,\mathcal O}=f_d\,\hat C_{d,\mathcal O}\,f_d are \mathcal O-free (torsion-free) as \mathcal O-modules for a principal ideal domain \mathcal O. Equivalently, ascertain whether all graded components of \hat C_{d,\mathcal O} and C_{d,\mathcal O} are free \mathcal O-modules so that graded dimensions do not depend on the characteristic after base change to a field.

Background

Section 6.4 studies integral forms and base change from a principal ideal domain \mathcal O to a field F. While base change is straightforward for many objects (e.g., F\otimes_{\mathcal O} R_{d,\mathcal O} \cong R_{d,F} and F\otimes_{\mathcal O} C_{d,\mathcal O} \cong C_{d,F}), establishing \mathcal O-freeness is subtler and essential for characteristic-independent graded dimensions.

The algebra \hat C_{d} is the imaginary cuspidal algebra (a quotient of the quiver Hecke superalgebra), and C_{d}=f_d\,\hat C_{d}\,f_d is its Gelfand–Graev idempotent truncation. The authors note that R_{d,\mathcal O} is known to be \mathcal O-free, and they do establish freeness for the quotient _{d,j,\mathcal O}, but they do not prove freeness for \hat C_{d,\mathcal O} or C_{d,\mathcal O}. Clarifying the \mathcal O-freeness of these two algebras would strengthen integral results and ensure uniform behavior under base change.