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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.

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I asked Prof. Etingof and he answered the following:

  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.

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