Symmetric crystals and affine Hecke algebras of type \(B\). (English) Zbl 1130.20008

Summary: The Lascoux-Leclerc-Thibon conjecture [A. Lascoux, B. Leclerc and J.-Y. Thibon, Commun. Math. Phys. 181, 205–263 (1996; Zbl 0874.17009)], reformulated and solved by S. Ariki [J. Math. Kyoto Univ. 36, No. 4, 789–808 (1996; Zbl 0888.20011)], asserts that the \(K\)-group of the representations of the affine Hecke algebras of type \(A\) is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra \(\mathfrak g\) where \(\mathfrak g\) is \(\mathfrak{gl}_\infty\) or the affine Lie algebra \(A^{(1)}_\ell\), and the irreducible representations correspond to the upper global bases. In this note, we formulate analogous conjectures for certain classes of irreducible representations of affine Hecke algebras of type \(B\).


20C08 Hecke algebras and their representations
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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