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Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux. (English) Zbl 1127.20005
Author’s summary: The main part of this paper is a survey on crystal bases, the Littelmann path model, the affine Weyl group and the affine Hecke algebra. The new results of this paper are in Section 3 and 4, where a tight connection of affine Hecke algebras and path models is established. The key new idea is due to C. Schwer [Int. Math. Res. Not. 2006, No. 21, Article ID 75395 (2006; Zbl 1121.05121)], who noticed that the periodic Hecke module defined by G. Lusztig encodes the combinatorics of the positively folded galleries used by S. Gaussent and P. Littelmann [Duke Math. J. 127, No. 1, 35-88 (2005; Zbl 1078.22007)]. H. Pittie and the author [Electron. Res. Announc. Am. Math. Soc. 5, No. 14, 102-107 (1999; Zbl 0947.14025), and Prog. Math. 210, 155-176 (2003; Zbl 1100.14526)] discovered the \(q^{-1}=0\) version of this connection when they established a relation between the Littelmann path model and the K-theory of flag varieties. The generalized path model studied in this paper provides a \(q\)-version of the theory of crystals which is a combinatorics for the spherical functions for a \(p\)-adic group. This combinatorics specializes, at \(q^{-1}=0\), to the path model combinatorics for working with Weyl characters.

20C08 Hecke algebras and their representations
43A90 Harmonic analysis and spherical functions
05E10 Combinatorial aspects of representation theory
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20G05 Representation theory for linear algebraic groups
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
22E46 Semisimple Lie groups and their representations
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