# Littelmann path operators for an arbitrary positive root

Looking at a few of Littelmann's papers, he seems to only apply root operators $f_\alpha$ for $\alpha$ a simple root. However, the definition seems to make perfect sense for $\alpha$ any positive root. (Indeed one can change the hyperplane that divides positive and negative roots and get a new set of simple roots corresponding to that choice.)

My question is: how much have the $e_\alpha$ and $f_\alpha$ been studied or used for nonsimple $\alpha$?
(This question also applies to other path models, like LS paths.)

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The question may be somewhat open-ended, but perhaps I can focus some aspects of it. First, I'm not sure which papers by Littelmann you've looked at, but probably the most definitive treatment of his operators occurs in his often-cited paper Paths and root operators in representation theory. Ann. of Math. (2) 142 (1995), no. 3, 499–525. Here and in related papers the objects of study are the finite dimensional irreducible representations of a semisimple Lie algebra over $\mathbb{C}$ or more generally the irreducible integrable representations of a symmetrizable Kac-Moody Lie algebra (for which there is a Weyl-Kac character formula).