  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.

Representation Theory;Algebraic Geometry;Quantum Algebra