Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic with Borel subgroup $B$. Let $\Lambda$ denote the weight lattice of $G$; we write elements of the group ring $\mathbb Z[\Lambda]$ of $\Lambda$ as linear combinations of elements of the form $e^\lambda$, $\lambda \in \Lambda$. In particular, characters of finite-dimensional $B$-modules are elements of $\mathbb Z[\Lambda]$.

For dominant $\lambda \in \Lambda$ let $ V(\lambda) $ denote the Weyl module for $G$ with highest weight $\lambda$ and for any element $w$ in the Weyl group $W$ of $G$ let $V_w(\lambda)$ denote the Demazure submodule of $V(\lambda)$ associated to $w$; this is the $B$-submodule of $V(\lambda)$ generated by an extremal vector of weight $w\lambda$. (Remark in particular that $V_{w_0}(\lambda) = V(\lambda)$).

For any simple root $\alpha_i$ of $G$ with associated simple reflection $s_i \in W$ define the Demazure operator $D_{s_i} : \mathbb Z[\Lambda] \to \mathbb Z[\Lambda]$ by $$ D_{s_i}(e^\lambda) = \frac{ e^\lambda - e^{s_i \lambda - \alpha_i} }{ 1-e^{-\alpha_i} } . $$ It is easy to see that this is well-defined. For any word $\mathfrak w = (s_{i_1}, \ldots, s_{i_k})$ of simple reflections in $W$ we have a Demazure operator $D_{\mathfrak w}$ defined by the obvious composition.

We now have the following theorem: Choose $w \in W$ and let $ \mathfrak w $ be any (not necessarily reduced!) word of simple reflections representing $w$. Then the character of $V_w(\lambda)$ is $D_{\mathfrak w}(e^\lambda)$. [A reference for this is, say, section 3.3 of Brion-Kumar's Frobenius splitting book].


Has anyone studied $q$-analogues of these Demazure operators? In light of recent work on $q$-character formulas and Kazhdan-Lusztig polynomials, this seems like a natural combinatorial thing to consider. For example, I would (perhaps naively) expect that an appropriate $q$-analogue of the Demazure operators computes, say, the $q$-analog of weight multiplicity considered by Kazhdan-Lusztig, R. Brylinski, Joseph, and others. (Also, I don't know much about the path model or crystal bases, but it seems as though there may be a connection to those as well).

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    $\begingroup$ You can get Demazure operators by specializing $q=0$ in (an appropriate normalization/integral form of) the Iwahori-Hecke algebra. I believe this is pursued in some form or another in a recent preprint of Dan Bump and collaborators. $\endgroup$ – David Hill Oct 9 '12 at 15:21
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    $\begingroup$ As far as I can tell, you're being fairly abstract about what you mean about q-analog here. Perhaps you mean quantum Schubert calculus and what Demazure operators correspond to this. Perhaps you mean quantum K-theory and those Demazure operators. Perhaps you even mean Demazure operators associated to quantum groups. As far as I understand, there has been lots of work in the first two directions, and I am interested in the third direction. $\endgroup$ – B. Bischof Oct 9 '12 at 15:51
  • $\begingroup$ If you want these operators to compute KL multiplicities in the quantum group setting you need a lot more framework, if you just want to "q-ify" the formulas from the classical case, maybe this is already true from quantum Schubert polynomials? $\endgroup$ – B. Bischof Oct 9 '12 at 15:51
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    $\begingroup$ After looking at your profile, I worry that you knew already everything I said. $\endgroup$ – B. Bischof Oct 9 '12 at 15:55
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    $\begingroup$ @Chuck: An added note in proof in Kumar's 1996 Invent. Math. paper "The nil Hecke ring ..." asks for q-characters leading to a q-analogue of Demazure's character formula. I'm not sure exactly how far the later literature goes in that direction, but see for instance the paper by S. Ryom-Hansen (preprint on arXiv 0905.0236) influenced especially by Andersen-Polo-Wen, along with his references to Kashiwara and others. Quantum group methods have at least provided new approaches to the classical Demazure formula. $\endgroup$ – Jim Humphreys Oct 9 '12 at 17:47

Dear Chuck Hague,

It may not answer your question. But let me suggest you to take a look at sections 4.5, 5.3 and 5.4 of:

A. Joseph, Modules with a Demazure flag. Studies in Lie theory, 131–169, Progr. Math., 243, MR2214249 (2007b:17023).


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