Mirabolic Satake equivalence and supergroups

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup to the category of $\operatorname{GL}(N-1,{\mathbb {C}}[\![t]\!])$ -equivariant perverse sheaves on the affine Grassmannian of $\operatorname{GL}_N$ . We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.

WHEN ARE THERE ENOUGH PROJECTIVE PERVERSE SHEAVES?

Let X be a topologically stratified space, p be any perversity on X and k be a field. We show that the category of p-perverse sheaves on X, constructible with respect to the stratification and with coefficients in k, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if X has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.

Irregular perverse sheaves

We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse $t$-structure, which is a straightforward generalization of usual perverse $t$-structure, and we prove that its heart is equivalent to the abelian category of holonomic ${\mathcal {D}}$-modules. We also develop the algebraic version of the theory.