Abstract
We classify a "dense open" subset of categories with an action of a reductive
group, which we call nondegenerate categories, entirely in terms of the root
datum of the group. As an application of our methods, we also:
(1) Upgrade an equivalence of Ginzburg and Lonergan, which identifies the
category of bi-Whittaker $D$-modules on a reductive group with the
category of $W$-equivariant sheaves on a dual Cartan subalgebra
$t^*$ which descend to the coarse quotient
$t^*//W$, to a monoidal equivalence (where $W$
denotes the extended affine Weyl group) and
(2) Show the parabolic restriction of a very central sheaf acquires a Weyl
group equivariant structure such that the associated equivariant sheaf descends
to the coarse quotient $t^*//W$, providing evidence for a
conjecture of Ben-Zvi-Gunningham on parabolic restriction.
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