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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?

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This isn't really an answer to Ben's question, more like an attempt to isolate one possible way of thinking about how to make $HH_*$ very explicit.

Let’s first look at a variety $X$ (assumed affine for the discussion so I can avoid too many irrelevant technicalities). The $HH_*$ of $D$-modules is typically computed in two ways, which are related but not identical. Both are computing the self-Tor over $D \otimes D$ of the diagonal $D$-bimodule. One of them first computes the derived monoidal product of $D$-modules on $X \times X$—that is, the derived $\mathcal{O}$-module tensor product—and then computes the de Rham cohomology of that. Standard $D$-module identities (projection formula basically) imply that this is computing $HH_*$. This calculation immediately generalizes to stacks (it only takes a few lines, even) to identify the $HH_*$ with de Rham cohomology of the inertia stack.

There's also another approach. Again, first for varieties, that’s to resolve $D$ as a $D$-bimodule “all at once” and then compute the tensor product as $D\otimes D$-modules directly, i.e., tensor over $D\otimes D$).

For special cases, like when $X$ is a vector space, one can use the (noncommutative version of the) Koszul resolution of the diagonal. This is implicitly realizing $D(X)$ as a quadratic algebra with inhomogeneous relations; the method is pretty standard (I like Kapustin-Kuznetsov-Orlov or Polishchuk-Positselski as readable references, but there are many good ones).

More generally, there’s a standard trick that seems to appear in basically all the literature in the 80s on $HH_*$ of deformation quantizations (Brylinski, Brylinski-Getzler, Wodzicki, etc.; actually I don't think I've checked Feigin-Tsygan but I would guess it's similar?). That’s to filter the algebra $D$ (again, a choice here) and then use the spectral sequence to compute $HH_*$ of the filtered algebra. The associated graded will be some commutative thing (in the case of $D$-modules on $X$, functions on $T^*X$) and you get its $HH_*$ via HKR; then you just have to compute the $E_1$-differential, which you find (using the conical structure of the commutative algebra and some halfway explicit calculation) is the de Rham differential. This is nicely explained in Brylinski-Getzler in an explicitly conical setting. Finally then you need that the spectral sequence degenerates at $E_2$, and a standard way is to use local coordinates (say in the $C^\infty$ setting) or a formal geometry argument (in algebraic geometry; see Bezrukavnikov-Kaledin for the standard formalism in symplectic algebraic geometry) to reduce to knowing this for the Weyl algebra, where you just crank it out using the previous paragraph. Now this line of attack “evidently” works to identify $HH_*$ of a filtered deformation with de Rham cohomology of the underlying conical symplectic variety.

Note the difference: when applied to the category of $D$-modules, the first attack naturally gives de Rham cohomology of the base $X$, the second de Rham cohomology of the cotangent bundle itself $T^*X$. Of course that’s not an important difference in the result, but the point is that in calculating sort of “base then fiber” (or maybe “fiber then base”) vs. “all at once” the fiber contribution in the first method gets naturally integrated out so you're not explicitly seeing it in the answer.

So, this second approach should microlocalize well. The trick is to resolve the kernel of the identity endofunctor of $D$-modules on the stack in microlocal terms, in a reasonably explicit way. In $D$-module terms, this endofunctor is the $D$-module push forward along the diagonal $X/G \rightarrow X/G \times X/G$ of the “constant sheaf,” which lifts under the flat covering $X \rightarrow X/G$ to the push forward of $\mathcal{O}$ along $G \times X \rightarrow X \times X$ (via action, projection). So that, at least, is not mysterious. The tensor product of $D$-modules in either sense (i.e. over O, which gives a monoidal structure on the category of D-modules, or over D, which lands in vector spaces and is exactly what we want to compute) is computed (or even better, defined!) by descent from $X \times X$, so in principle we can calculate it “upstairs” on the flat cover.

OK, now the trick is to formulate that calculation in a nice invariant way! Which unfortunately I haven't done...

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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).

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To follow up on your comment about Mirkovic-Riche: it seems you're proposing that we realize the preimage of the moment map for $X$ in $T^*X \times \mathfrak{g}^*$ as the intersection of $T^*X$ and the graph of the moment map. Mirkovic-Riche tells me this is dual to the intersection in $TX\times \mathfrak{g}$ of $X\times \mathfrak{g}$ and the graph of the action map; this is a sort of infinitesimal isotropy stack. The differential operators on $X/G$ correspond to deforming $T^*X$, which will somehow be Koszul dual to adding the Chevalley-Eilenberg differential to the isotropy stack... – Ben Webster Mar 11 '14 at 11:43
Am I getting warmer? – Ben Webster Mar 11 '14 at 11:43
Yes that sounds exactly right. If you want you can say "deforming $T^*X$ is Koszul dual to working $S^1$-equivariantly on the inertia/loop stack"... – David Ben-Zvi Mar 16 '14 at 0:38

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