# Hirzebruch-Riemann-Roch for quiver varieties?

These days, I attended a workshop at North Carolina State University. The key lecturer is Professor Nakajima. He introduced two types of quiver variety. One of them is affine, another one is quasi-projective variety.

The reference is the following: Nakajima's quiver variety

I wonder whether is there anyone developing the Riemann-Roch theorem for this quasi projective scheme. It is known that the H-R-R for flag variety of finite dimensional Lie algebra is Weyl character formula

Is there any representation theoretical interpretation of H-R-R for quiver variety? Of course, one can say that some quiver variety correspondence to flag variety of some Lie algebra. However, I think quiver variety is more general.

The further question: Is there BGG category on quiver representation settings. I know one can construct Hall algebra associated to some quiver. It seems that one can define analogue of Verma module in this setting comparing to the Verma module for Lie algebra. Therefore, is there a Character formula here, if there exists, can it be re-interpretated as H-R-R for quiver variety?

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It's definitely not my area of expertise but character formulae for quivers are related to Kac's conjectures [V.G. Kac, in Invariant Theory, pp. 74--108, Lecture Notes in Math. 996 (1983)] concerning the number of isomorphism classes of absolutely indecomposable representations of quivers over finite fields. For finite and tame quivers (i.e., classical and affine ADE) Kac's conjectures are known to be true, and there are explicit combinatorial identities for these cases (see [J. Hua, J. Algebra 226 (2000) 1011--1033] and [J. Fulman, Bull. London Math. Soc. 33 (2001) 397--407]). – Wadim Zudilin May 29 '10 at 8:17
I have edited the question to say Hirzebruch-Riemann-Roch rather than Grothendieck-Riemann-Roch, as it is really about HRR, not GRR. But feel free to change it back if you think that I am wrong. – Kevin H. Lin Oct 27 '10 at 6:26