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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 flagvariety of $G$. By developing a formalism of generalized realization functors,we construct a monoidal trace from $\mathsf{H}_W$ to a category of bigradedmodules over a certain graded ring: namely, the endomorphisms of theequivariant Springer sheaf over the unipotent locus of $G$. We prove that: (1)On objects attached to positive braids $\beta$, the output is theweight-graded, equivariant Borel-Moore homology of a generalized Steinbergscheme $\mathcal{Z}(\beta)$. (2) Our functor contains, as a summand, one usedby Webster-Williamson to construct the Khovanov-Rozansky link invariant. Inparticular, the Khovanov-Rozansky homology of the link closure of $\beta$ isfully encoded in the Springer theory of $\mathcal{Z}(\beta)$. Decategorifying,we get a trace on the Iwahori-Hecke algebra, valued in graded virtualcharacters of $W$. We give a formula for it in terms of a pairing on charactersof $W$ called Lusztig's exotic Fourier transform, which generalizes it to allfinite Coxeter groups. Using this formula, we establish properties of the tracelike rationality, symmetry, and compatibility with parabolic induction. We alsoshow that on periodic braids, the trace produces the characters of explicitvirtual modules over Cherendik's rational double affine Hecke algebra. For $W =S_n$, we recover an identity of Gorsky-Oblomkov-Rasmussen-Shende relating thesemodules to the HOMFLY polynomials of torus knots.

From the Hecke Category to the Unipotent Locus