From the Hecke Category to the Unipotent Locus

Let $W$ be the Weyl group of a split semisimple group $G$. Its Hecke category $\mathsf{H}_W$ can be built from pure perverse sheaves on the double flag variety of $G$. By developing a formalism of generalized realization functors, we construct a monoidal trace from $\mathsf{H}_W$ to a category of bigraded modules over a certain graded ring: namely, the endomorphisms of the equivariant Springer sheaf over the unipotent locus of $G$. We prove that: (1) On objects attached to positive braids $\beta$, the output is the weight-graded, equivariant Borel-Moore homology of a generalized Steinberg scheme $\mathcal{Z}(\beta)$. (2) Our functor contains, as a summand, one used by Webster-Williamson to construct the Khovanov-Rozansky link invariant. In particular, the Khovanov-Rozansky homology of the link closure of $\beta$ is fully encoded in the Springer theory of $\mathcal{Z}(\beta)$. Decategorifying, we get a trace on the Iwahori-Hecke algebra, valued in graded virtual characters of $W$. We give a formula for it in terms of a pairing on characters of $W$ called Lusztig's exotic Fourier transform, which generalizes it to all finite Coxeter groups. Using this formula, we establish properties of the trace like rationality, symmetry, and compatibility with parabolic induction. We also show that on periodic braids, the trace produces the characters of explicit virtual modules over Cherendik's rational double affine Hecke algebra. For $W = S_n$, we recover an identity of Gorsky-Oblomkov-Rasmussen-Shende relating these modules to the HOMFLY polynomials of torus knots.