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How much is known about the Weyl character formula for quantum groups? More specifically, has the formula been generalized to the general setting of deformed coordinate algebras $\mathbb{C}[G_q]$ of semi-simple Lie groups and their associated flag varieties? I am most interested in the non-root of unity case.

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The label "quantum group" covers several different constructions having some common connection to a given root system or complex semisimple Lie algebra/group. So the question needs careful formulation. Apparently the "Weyl character formula" is supposed to result from something like global sections of a line bundle? Work from around 1990 by Lusztig, Andersen-Polo-Wen, and others showed how much carries over from the classical theory for quantum groups attached to a universal enveloping algebra. "Weyl modules" occur naturally here, with the usual character formula, but may not be simple. –  Jim Humphreys Jun 28 '10 at 18:25
I agree with Jim. Your question is rather pointless in the generality you ask. Simple modules for $U_q (g)$ when $q$ is not root of unity will be deformations of simple $g$-modules and satisfy usual Weyl character formula. Is this what you are asking or are you after roots of unity? –  Bugs Bunny Jun 28 '10 at 18:32
I'm asking about the deformed coordinate rings $\mathbb{C}[G_q]$ of semi-simple Lie groups for q not a root of unity. – John McCarthy 0 secs ago –  John McCarthy Jun 28 '10 at 18:46
@John: Maybe you should edit this qualification into your question? The representation theory in that case is less familiar to me, but has been studied a lot by DeConcini, Kac, Procesi and others. For quantized enveloping algebras (including those at a root of unity), the work I mentioned covers the connection with classical Weyl theory. –  Jim Humphreys Jun 28 '10 at 19:40
not quite an answer. Grothendieck-Riemann-Roch of usual flag variety of Lie algebra is Weyl character formula. Therefore, "quantum Weyl character formula" should be G-R-R for quantized flag variety(as a noncommutative projective scheme). The definition of quantized flag variety is given by Lunts-Rosenberg in their paper:"localization for quantum group". I am now trying to calculate G-R-R for quantized flag variety of $sl_2$ –  Shizhuo Zhang Jun 29 '10 at 1:08

1 Answer 1

The question still feels a bit vague to me, but this started to get too long to be a comment. There are a number of issues:

There's the question of the definition of $\mathbb C[G_q]$, and say an analogue of the Peter-Weyl theorem, and there is also the issue of doing this at a root of unity case, or better studying things integrally. For this say, a recent paper of Lusztig gives a definition of a quantum coordinate ring for any (finite type) root datum, which specializes to the Kostant-Chevalley form.

Andersen-Polo-Wen and others have studied studied quantum induction functors which correspond to taking global sections on the classical flag variety, and these might be what you want (Ryom-Hansen also proved a version of Kempf vanishing in this context for example). This was also more recently taken up by Kumar and Littelmann in the context of studying Frobenius splitting. Finally there's the issue of understand quantum flag varieties as noncommutative spaces as in the previous comment, for which along with the Lunts-Rosenberg paper, there is also more recent work of Backelin and Kremnizer.

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