Spherical Gaiotto conjecture for cotangent-type quantum supergroups

Establish the braided or categorical equivalence between the category of kappa-twisted (H(F),psi)-equivariant D-modules on the affine Grassmannian Gr_G and the representation category Rep_q^{Lus}(G_s) of the Lusztig quantum supergroup for a basic classical Lie supergroup of cotangent type, including the construction and required root-of-unity properties of the full Lusztig quantum supergroup.

Background

The paper studies geometric realizations of representation categories of quantum supergroups through twisted equivariant sheaves on affine Grassmannians and affine flag varieties. The spherical Gaiotto conjecture is the unramified version of this proposed correspondence, with the quantum parameter related to the twisting by q=exp(pi i kappa).

The paper explicitly notes that the full Lusztig quantum supergroup with the properties required at a root of unity is not constructed there. Consequently, the spherical statements involving this quantum supergroup remain conditional rather than established results of the paper.

References

An important caveat is that the full Lusztig quantum supergroup, with the precise properties needed here at a root of unity, is not constructed in this paper. The spherical statements involving it are understood subject to the hypotheses in Remark~\ref{rem:spherical-gaiotto-conjecture}.

— Iwahori Gaiotto conjecture for cotangent type  (2610.08243 - Travkin et al., 6 Oct 2026) in Section 1.1, subsection “The Gaiotto conjecture,” spherical case

Given a basic classical supergroup G_s, it is known that the odd part \fg_{\bar{1}} of \fg_s is a coisotropic Hamiltonian variety of the even part, cf. . Take \bM\vee to be \fg_{\bar{1}} and {G\vee}=G_{\bar{0}}, and up to a Koszul duality, , the relative Langlands conjecture can be reformulated as

— Iwahori Gaiotto conjecture for cotangent type  (2610.08243 - Travkin et al., 6 Oct 2026) in Section 1.1, subsection “The Gaiotto conjecture,” spherical case