Equivariant Riemann-Roch on DM stacks?

Does an equivariant version of (Toen)-Riemann-Roch theorem hold say over a smooth Deligne-Mumford stack $X$ over the complex numbers?

Any references that state this explicitely?

Are there formulas for the equivariant Chern character and Todd genus? (as for the Chern character, I am content with the case of a coherent sheaf supported at a point and with $X$ of dimension $2$)

• What are you asking, precisely? Where are the Chern character and Todd class supposed to take values? The Riemann-Roch formula is usually only defined up to $\mathbb{Q}$-coefficients. If you are using $\mathbb{Q}$-coefficients, the Chow groups of the stack agree with the Chow groups of its coarse moduli space. Thus, some of the applications of Riemann-Roch for the stack follow from the corresponding statements for the coarse moduli space. – Jason Starr Apr 3 '16 at 13:25
• Have you had a look at Edidin's paper: arxiv.org/pdf/1205.4742.pdf ? – Ariyan Javanpeykar Apr 3 '16 at 19:12
• You can also consider e-mailing A. Krishna tifr.res.in/People_Finder/compcode.php?param1=39 I think he's working on generalizing work of Edidin; see his abstract here maths-people.anu.edu.au/~alperj/kioloa-abstracts.html#Krishna – Ariyan Javanpeykar Apr 3 '16 at 19:13
• @Jason Starr : the point of Toen's construction is precisely to use a cohomology theory such that the cohomology of the stack does not coincide rationally with the cohomology of its moduli space (Toen uses, morally, "ordinary" cohomology of the inertia stack). For instance, the cohomology ring of $BG$ is (morally at least) the rational character ring of $G$, and not $\mathbb Q$. – Niels Apr 4 '16 at 7:46
• This paper of mine spells it out in Appendix A and has a few examples: arxiv.org/pdf/1008.4205 The results is due to Toen: The ́ore`mes de Riemann-Roch pour les champs de Deligne-Mumford. K-Theory, 18(1):33–76, 1999. And is also spelled out in Hsian-Hua Tseng. Orbifold quantum Riemann-Roch, Lefschetz and Serre. Geom. Topol., 14(1):1–81, 2010. – Jim Bryan Apr 6 '16 at 22:34