Is there any character formula for demazure modules in arbitary kac moody settings which does not use demazure operators?
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 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."
Littelmann's, which gives a positive formula (counting Littelmann paths). His ICM address is here: http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0298.0308.ocr.pdf
He proves its validity using Demazure operators -- I hope that doesn't disqualify it!