Originally asked on Math SE but it was suggested I move it here.

Suppose one has a distinguished cocycle in the group-equivariant sheaf cohomology $\Phi \in H^n(X, G, \mathcal{F})$ for a "nice" -- at least locally compact -- scheme $X$, a group $G$, and a possibly coherent sheaf $\mathcal{F}$. I would like to write down a natural "homology" space $H_r(X, G, \mathcal{F})$ together with a pairing $$H^n(X, G, \mathcal{F}) \times H_r(X, G, \mathcal{F}) \to H_{r-n}(X, G, \mathcal{F})$$ so that I can compute $\Phi \times \alpha$ for suitably chosen $\alpha \in H_r(X, G, \mathcal{F})$. Is there a natural such space $H_r(X, G, \mathcal{F})$? Note that it need not be a homology theory despite the suggestive notation.

In the derived category $D(R)$ of $R$-modules, there is a natural pairing

$$Ext^n(A,B) \times Tor_r(A,B) \to Tor_{r-n}(A,B)$$

using the fact that $\operatorname{Ext}^n(A,B) = \operatorname{Hom}_{D(R)}(A, T^n B)$. So I thought maybe one could adopt this pairing to the sheaf cohomological context using $$\operatorname{Ext}^n(\mathcal{O}_X, \mathcal{F}) = H^n(X, \mathcal{F}),$$ where $\mathcal{O}_X$ denotes the structure sheaf of $X$. Then the desired $H_r(X, G, \mathcal{F})$ could be an appropriate formulation of a $G$-equivariant $``\operatorname{Tor}''(X, \mathcal{F})$, but is not clear to me what that might be.

Alternatively, I read a bit about the theory of Borel-Moore homology. The existence of a cap product in the sheaf-cohomological framework here seems useful, but the derived perspective strikes me as more natural for incorporating group equivariance.

Is anyone aware of what might function as a sort of $G$-equivariant sheaf $\operatorname{Tor}$, or of an existing use/application of cap products with group equivariant sheaf cohomology?