Imaginary Schur-Weyl duality for quiver Hecke superalgebras

The irreducible modules over quiver Hecke superalgebras $R_\theta$ can be classified in terms of cuspidal modules. To an indivisible positive root $\alpha$ and a non-negative integer $d$, one associates a quotient $\bar R_{d\alpha}$ of $R_{d\alpha}$ called the cuspidal algebra. If the root $\alpha$ is real, the cuspidal algebra is well-understood. But if $\alpha=\delta$, the imaginary null-root, the {\em imaginary cuspidal algebra} $\bar R_{d\delta}$ is rather mysterious. It has been known that the number of the isomorphism classes of the irreducible $\bar R_{d\delta}$-modules equals the number of the $\ell$-multipartitions of $d$, but there has been no way to canonically associate an irreducible $\bar R_{d\delta}$-module to such a multipartiton. The imaginary cuspidal algebra is especially important because of its connections to the RoCK blocks of the double covers of symmetric and alternating groups. We undertake a detailed study of the imaginary cuspidal algebra and its representation theory. We use the so-called Gelfand-Graev idempotents and subtle degree and parity shifts to construct a (graded) Morita (super)equivalent algebra $\mathsf{C}(n,d)$ (for any $n\geq d$). The advantage of the algebra $\mathsf{C}(n,d)$ is that, unlike $\bar R_{d\delta}$, it is non-negatively graded. Moreover, the degree zero component $\mathsf{C}(n,d)^0$ is shown to be isomorphic to the direct sum of tensor products of $\ell$ copies of the classical Schur algebras. This gives the classification (and the description of dimensions/characters, etc.) of the irreducible $\mathsf{C}(n,d)$-modules, and hence of the irreducible $\bar R_{d\delta}$-modules, in terms of the classical Schur algebras. In particular, this allows us to canonically label these by the $\ell$-multipartitions of $d$. The results of this paper will be used in our future work on RoCK blocks of the double covers of symmetric groups.