The variety of regular elements g^{reg} carries
a flat group scheme Z, the family of centralizers. Using the Kostant section
we get a flat group scheme over t/W. The vector bundle of Lie algebras is
canonically identified with the cotangent bundle to t/W.

Thus we get a functor Coh^Z(t/W) \to Coh (T(t/W)), coherent sheaves on the
tangent to t/W. Using base change we get a functor
Coh^Z(S) \to Coh (T(t/W)\times_{t/W} S) where S=t\times_{t/W} t is Soergel's
variety (the action of Z everywhere is, of course, trivial).

There is an increasing sequence of closed subschemes Z_n in
T(t/W)\times_{t/W} S  of dimension dim(t) and the above functor lands in
Coh(Z_n) for some n. This is related to affine SBim as follows.

Consider the affine Cartan t' fitting in the s.e.s. t--> t' --> C.

Let's exhaust W_{aff} by finite subsets W_1\subset W_2...
and let Z'_n be the union of the graphs of w\in W_i, thus
Z'_n\subset t'\times_C t'. The action of W_{aff} on t factors
through W, so Z'_n\cap (t\times t) is contained in
S=t\times_{t/W} t.

Let's now blow up S in t'\times_C t' removing the proper preimage
of the zero fiber t\times t. We get a "renormalized" version of
t'\times C t' which maps to C, with nonzero fibers identified with
(t \times t) while the 0 fiber is T(t/W)\times_{t/W} S.
Taking the proper transform of Z_n' we get a subscheme whose fiber
over 0 is our Z_n.

Upshot: Z_n is the renormalized 0 fiber of the union of graphs of
w\in W_n.

By the way, if we don't restrict to the zero fiber, we get a version
of affine SBim which "controls" the category of monodromic
sheaves on the total space of punctured determinant bundle of
\tilde Fl (or, Koszul dually, loop rotation equivariant complexes
on I\Fl). On the Coh^Z(t/W) side this corresponds to deforming
Coh^G(g) to Harish-Chandra bimodules on which one considers
the Whittaker functor, see section 4 of our paper with Finkelberg,
https://arxiv.org/pdf/0707.3799.pdf