# Two identities involving Ext functors in the context of D-modules

I have several questions regarding proposition 2.3 in "Cherednik and Hecke algebras of varieties with a finite group action", by Pavel Etingof. Let $$X$$ be a complex affine algebraic variety and $$g$$ an automorphism of finite order. We define the $$D(X)$$-bimodule $$D(X)g$$ by $$a(bg)c=abg(c)g$$.

1. In the aforementioned proposition, it is stated that the standard equivalence of left and right $$D$$-modules on the second factor induces two left $$D$$-modules $$\tilde{D}(X)$$ and $$\tilde{D}(X)g$$ on $$X\times X$$ coming from $$D(X)$$ and $$D(X)g$$ respectively. I would like to understand how this equivalence can be used to induce $$D$$-modules on $$X\times X$$.
2. Let $$i:X\rightarrow X\times X$$ be the morphism given by $$x \mapsto (x, g^{-1})$$ and $$\Delta$$ be the diagonal. The proposition states that $$\tilde{D}(X)=\Delta_{*}O_{X}$$ and $$\tilde{D}(X)g=i_{*}O_{X}$$. Then, the following chain of equalities is stated:

$$Ext_{D(X)\otimes D(X)^{op}}^{m}(D(X),D(X)g)=Ext_{D(X\times X)}^{m}(\tilde{D}(X),\tilde{D}(X)g)= Ext_{D(X\times X)}^{m}(\Delta_{*}O_{X},i_{*}O_{X})=Ext_{D(X)}^{m}(O_{X},\Delta^{!}i_{*}O_{X})$$ I would like to understand the first and third equalities.

1. A $$D$$-module on $$X\times X$$ is a module over $$D(X)\otimes D(X)$$, while initially, these are bimodules over $$D(X)$$, which are modules over $$D(X)\otimes D(X)^{op}$$. The equivalence I mentioned is a Morita equivalence between $$D(X)$$ and $$D(X)^{op}$$ which defines a Morita equivalence between $$D(X)\otimes D(X)$$ and $$D(X)\otimes D(X)^{op}$$.
2. The first equality follows from my answer to (1). The third equality follows because the functors $$\Delta_{ * }$$ and $$\Delta^{ * }$$ are adjoint.