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Suppose I know the Chern character of an object $F \in D^b(X)$, where $X$ is some smooth complex projective variety with a finite group $G = \mathbb{Z}/m$ acting on it. In $D^b([X/G]) \simeq D^b(X)^G$ where $[X/G]$ is the associated quotient stack, I can consider objects $F \otimes \chi_i$, where $\chi_i$ for $i=0, \dots, m-1$ are the irreducible representations of $G$.

Is there an easy way to compute the Chern character of $F \otimes \chi_i$? I suppose I'd have to work with something like Chen--Ruan cohomology? The paper Stringy K-theory and the Chern character of Jarvis--Kaufmann--Kimra also seems relevant.

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  • $\begingroup$ To make sense of $F \otimes \chi_i$ you need to assume that $F$ itself has equivariant structure. But there may be different equivariant objects which, after forgetting the equivariant structure, become isomorphic. Because of this you cannot expect to be able to compute the Chern character of $F \otimes \chi$ (even for the trivial character $\chi$). $\endgroup$
    – Sasha
    Commented Mar 1, 2023 at 13:56
  • $\begingroup$ @Sasha If $X$ is a threefold with $\mathrm{Pic} X = \mathbb{Z}$ and the basis of its numerical Grothendieck group is $\langle [O_X], [O_H], [O_L], [O_P] \rangle$, is it possible to explicitly write down a basis of the numerical Grothendieck group of $D^b([X/G])$? $\endgroup$
    – alg_et_geom
    Commented Mar 1, 2023 at 14:57
  • $\begingroup$ I am not sure this is true in such generality, and anyway I don't know any reference dealing with this problem. $\endgroup$
    – Sasha
    Commented Mar 2, 2023 at 5:58

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