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Is there any character formula for demazure modules in arbitary kac moody settings which does not use demazure operators?

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up vote 2 down vote accepted

Ryom-Hansen, Steen(DK-CPNH) Littelmann's refined Demazure character formula revisited. (English summary) Sém. Lothar. Combin. 49 (2002/04), Art. B49d, 10 pp.

The review:

"The author provides a purely combinatorial proof the Demazure character formula, a generalisation of Weyl's character formula. This is done using only the combinatorial properties of crystals, namely Kashiwara's crystal operators and the ∗-operation. Prior proofs required an appeal to either representation theory or Littelmann's path models."

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Here is an accessible link to the article, which is short but includes comments and references for the tangled history of the original Demazure formula and its generalizations: However, I'm not sure about the interaction of the Kac-Moody setting with the quantum group setting used in Ryom-Hansen's approach. – Jim Humphreys Mar 6 '13 at 1:15

Littelmann's, which gives a positive formula (counting Littelmann paths). His ICM address is here:

He proves its validity using Demazure operators -- I hope that doesn't disqualify it!

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