# Status of a conjectural definition of H. Nakajima

In his paper '$t$-analogue of $q$-characters of finite dimensional representations of quantum affine algebras' - http://arxiv.org/abs/math/0009231 - H. Nakajima states a conjectural definition of the $t$-analogue of the $q$-character of a standard module $M_{P}$ (with weight $P$) of a quantum affine algebra at level $0$ - this appears as Conjecture 3.1.1 on p. 5 of the arXived preprint above. In summary, Nakajima conjectures that the $q,t$'-character of a standard module $M_{p}$ can be determined using certain filtrations on individual weight spaces.

I was hoping that someone can let me know of the status of this conjecture - is it true that we can describe the '$q,t$'-character as conjectured?

I imagine that this has been resolved since the paper is a bit more mature now. If this is the case, can someone point me in the direction of a resolution?

If this conjecture has not been resolved, does anyone know of any progress towards its resolution/any problems that have arisen in resolving this conjecture?

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Thanks for your response and reference; I will take a look! Cheers, George –  George Melvin Mar 19 '13 at 19:26

The following paper might help get you up to date, by working through its references:

Yoshiyuki Kimura, Fan Qin. Graded quiver varieties, quantum cluster algebras and dual canonical basis.

http://arxiv.org/abs/1205.2066

It contains some material on $q,t$-characters and twisted versions of these, in relation to some problems in the theory of quantum cluster algebras.

In particular it references

David Hernandez. Algebraic approach to q,t-characters. Advances in Mathematics 187 (2004), no. 1, 1–52.

which I imagine (though I've not read it) should be pretty close to what you want.

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Thanks for the references! Cheers –  George Melvin Sep 20 '12 at 20:00