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The too long, didn't read form of the question would simply be, has someone completed A. Joseph's proof of the Demazure character formula? Is Joseph's proof considered complete?

In more detail, Joseph's paper "On the Demazure character formula" only proves the formula for finite $\mathfrak{g}$-modules of the form $L(\lambda)$ where $\lambda$ is 'sufficiently large'. The Demazure operators make their appearance in the lemma 2.5, where for a finite $\mathfrak{b}$-module $F$, one has the character formula, $$ ch \mathscr{D}_\alpha F = \Delta_\alpha (ch(Im(F\to \mathscr{D}_\alpha F))). $$ So the difficulty in the proof of the full formula for $L(\lambda)$ is in showing that at each stage the natural $\mathfrak{b}$-module map, $$ \mathscr{D}_{\alpha_{i}}\cdots \mathscr{D}_{\alpha_{1}} (\mathbb{C}_\lambda) \to \mathscr{D}_{\alpha_{i+1}} \mathscr{D}_{\alpha_{i}}\cdots \mathscr{D}_{\alpha_{1}} (\mathbb{C}_\lambda) $$ is injective, where $w = s_{\alpha_n}\cdots s_{\alpha_1}$ is a reduced decomposition of the longest element of the Weyl group, and the $\mathscr{D}_{\alpha_{i}}$ are related the Zuckerman functors.

It isn't expressly mentioned in Kashiwara's paper, "The crystal base and Littelmann's refined Demazure character formula", but is it known if injectivity follows from Kashiwara's 'string property'? This seems immediate, but the review of Joseph's paper on mathscinet makes it sound as though Joseph's proof was never finished.

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  • $\begingroup$ It seems I need to have read the paper a bit closer. The injectivity of those maps is actually shown in an earlier part of Joseph's paper, in lemma 2.8. Along with proposition 2.13 he actually has proven the character formula for finite dimensional $\mathfrak{g}$-modules, just not all Demazure modules. See for instance see theorem 2.21 in the same paper, where the formula is proved for all Demazure submodules of $E(\lambda)$, but only for sufficiently large $\lambda$. Sorry for the uninformed question :-/ $\endgroup$
    – denomme
    Aug 14, 2013 at 17:41

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Rather than attempt my own answer to this complicated question, I'll add some explicit links and literature references to help the discussion along. The classical Demazure character formula and its generalization to symmetrizable Kac-Moody Lie algebras have a long and difficult history, starting with Demazure's own influential but somewhat flawed paper. It would be helpful to get feedback from some of the still-active participants, of course, including Joseph, Littelmann, Andersen, Kashiwara. Meanwhile, here are some details about the papers mentioned (though the one by Kashiwara probably needs special permission to access online):

Joseph, part I here

Joseph, part II here

Kashiwara: The crystal base and Littelmann’s refined Demazure character formula. Duke Math. J. 71 (1993), no. 3, 839–858.

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