TITLE: Microlocalizing Hochschild homology
QUESTION [13 upvotes]: A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories.  Without getting too far into details, amongst other things, this formalism allows us to study the associated Hochschild classes of D-modules on stacks, which naturally live in the deRham cohomology of the associated loop stack. So, for example, for equivariant D-modules on a variety $X$ for a group $G$, the Hochschild classes of equivariant D-modules live in the equivariant cohomology of the inertia stack $I(X)=\{(x,g)\in X\times G | g\cdot x=x\}$.
In the case of a smooth variety, it's known how to enrich this picture by microlocalizing it (see, for example, the paper of Kashiwara and Schapira) and in fact to generalize beyond D-modules to modules of sheaves on other deformation quantizations.  This is extremely valuable, since one can define Hochschild classes supported on subvarieties in the cotangent bundle which are non-zero but pushforward to zero in the full cotangent bundle.  Thus, this gives a much richer picture.  In this case however, the loop space makes no interesting appearance, since a smooth variety is its own loop space.  

Is it well understood how to unify these two stories?  How does one think about the Hochschild homology of (say) equivariant D-modules on a smooth variety microlocally?

REPLY [6 votes]: One answer to Ben's question is to calculate the Hochschild homology 
of the category of $D$-modules $D_\Lambda(X)$ with fixed microsupport $\Lambda\subset T^*X$. The functor $D_\Lambda(X)\to D(X)$ has a continuous right adjoint, so defines a map $HH_*(D_\Lambda(X))\to HH_*(D(X))$. Thus given a $D$-module with $\Lambda$ microsupport we have its characteristic class with supports and its image in de Rham cohomology which is its usual characteristic class (but may be zero as you point out). Formally one can apply this idea in the other settings Ben considers, but one needs to calculate the
$HH_*$ of the categories of sheaves with supports geometrically, perhaps along the lines Tom indicates.
I haven't thought through the calculation of the relevant Hochschild homology - one needs to think through the properties of the adjoint functor above (enforcing $\Lambda$-support) which are worked out (in the coherent setting) in Arinkin-Gaitsgory, but I don't think it's easy.
But in applications one often sees microsupport enforced by equivariance - i.e., you might be considering $D(X/H)$ for some group $H$ with finitely many orbits (e.g., the flag variety with $B$ or a a symmetic subgroup, or general spherical varieties), so that $\Lambda$ is the union of the conormals to the orbits. In this case we are asking to calculate $HH_*(D(X/H))$ and the induced map (from pullback under $X\to X/H$) to $HH_*(D(X))$. Note that the category $D(X/H)$ recovers its nonequivariant cousin, the category $D_\Lambda(X)$, from its structure as a category over $BH$ so I think we can ``deequvariantize" this calculation to answer the question for these special $\Lambda$'s.
This is very geometric: by the results Ben quotes, the characters of equivariant $D$-modules are given by Borel-Moore homology of the inertia stack of $X/H$, i.e., of the stabilizers of the $H$-action. These are I believe related by linear Koszul duality to the characteristic classes with support that Ben is talking about (see the papers of Mirkovic-Riche).