Abstract
There is a close connection between Demazure crystals and tensor products of Kirillov–Reshetikhin crystals. For example, certain Demazure crystals are isomorphic as classical crystals to tensor products of Kirillov–Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to nonsymmetric Macdonald polynomials and $q$-deformed Whittaker functions. Les cristaux de Demazure et les produits tensoriels de cristaux Kirillov–Reshetikhin sont étroitement liés. Par exemple, certains cristaux de Demazure sont isomorphes, en tant que cristaux classiques, à des produits tensoriels de cristaux Kirillov–Reshetikhin via un isomorphisme que l'on peut choisir canoniquement. Ici, nous montrons que cet isomorphisme entremêle la graduation affine naturelle des cristaux de Demazure avec une fonction énergie définie combinatoirement. Comme conséquence, nous obtenons une formule pour le caractère de Demazure exprimée au moyen de la fonction énergie, avec des applications aux polynômes de Macdonald non symétriques et aux fonctions de Whittaker $q$-déformées.
Highlights
Kashiwara’s theory of crystal bases [20] provides a remarkable combinatorial tool for studying highest weight representations of symmetrizable Kac–Moody algebras and their quantizations
Our main tool is an enhancement of the relationship between KR crystals and Demazure crystals due to Fourier, Shimozono, and the first author
In [8, Theorem 4.4] it was shown that, under certain assumptions, there is a unique bijection from the Demazure crystal to the KR crystal respecting the classical crystal structure and such that all zero edges in the Demazure crystal are taken to zero edges in the KR crystal
Summary
Kashiwara’s theory of crystal bases [20] provides a remarkable combinatorial tool for studying highest weight representations of symmetrizable Kac–Moody algebras and their quantizations. We consider finite-dimensional representations of the quantized universal enveloping algebra Uq(g) corresponding to the derived algebra g of an affine Kac–Moody algebra These representations do not extend to representations of Uq(g), but one can define the notion of a crystal basis. In [21], Kashiwara proposed that this relationship is connected to the theory of Demazure crystals [19, 29], by conjecturing that perfect KR crystals are isomorphic as classical crystals to certain Demazure crystals (which are subcrystals of affine highest weight crystals) This was proven in most cases in [4, 5]. We will formulate this precisely, and provide a proof
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