Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let G be a finite group,let Rep(G) denote the category of finite dimensional representations over C. Let V,W be representations of G in Rep(G). One can define a bilinear form on Rep(G) or inner product in K0(Rep(G)) (in Teleman’s notes) as dimCHom(V,W)G which is G invariant of Hom(V,W).Then there is a homomorphism:χ:K0(Rep(G))Fun(G,C),such that χ(V,ρ)=Tr(ρ(g)).Notice that the inner product on Fun(G,C) is 1|G|ΣgG¯χ(W,g)χ(V,g).I would like to consider the geometrization of this set up as following:

I identify the category of Rep(G) with category of quasi coherent sheaves on quotient stack [spec(C)/G] and equip with a bilinear form on Qcoh([spec(C)/G]) as the Euler characteristic: χ(V,W):=Σi(1)idimExti(V,W),V,W are representation of G which can be regarded as quasi coherent sheaves. Since the category Rep(G) is semisimple,the higher extension are vanishing,hence χ(V,W) is just dimHomQcoh(V,W)=h0([spec(C)/G],VW)dimHom(V,W)G

My question is what is the inner product on the right handside? How can I geometrize the 1|G|ΣgG¯χ(W,g)χ(V,g).Notice that if we consider the case of infinite group G,say Lie group. The summation over gG should become the intergration over G. I have the guess that the inner product on the right hand side should be Hizebruch-Riemann-Roch for computing χ(V,W) which is [spec(C)/G].ch(V)ch(W)Td([spec(C)/G]),it looks like similar to the formular 1|G|ΣgG¯χ(W,g)χ(V,g)

But I dont know how to make it very precise and how to prove or disprove my guessing. I found a paper online

It talks about the Grothendieck Riemann Roch for quotient stack. I am trying to understand his work and probably solve my question.

Thanks for your help.
Any comments and suggestions are welcomed


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