SIGMA 4 (2008), 079, 12 pages      arXiv:0809.0534      https://doi.org/10.3842/SIGMA.2008.079
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Non-Gatherable Triples for Non-Affine Root Systems

Ivan Cherednik and Keith Schneider
Department of Mathematics, UNC Chapel Hill, North Carolina 27599, USA

Received September 03, 2008, in final form November 08, 2008; Published online November 14, 2008

Abstract
This paper contains a complete description of minimal non-gatherable triangle triples in the lambda-sequences for the classical root systems, F₄ and E₆. Such sequences are associated with reduced decompositions (words) in affine and non-affine Weyl groups. The existence of the non-gatherable triples is a combinatorial obstacle for using the technique of intertwiners for an explicit description of the irreducible representations of the (double) affine Hecke algebras, complementary to their algebraic-geometric theory.

Key words: root systems; Weyl groups; reduced decompositions.

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References

1. Bourbaki N., Groupes et algèbres de Lie, Ch. 4-6, Hermann, Paris, 1969.
2. Cherednik I., Non-semisimple Macdonald polynomials. I, Selecta Math., to appear, arXiv:0709.1742.
3. Cherednik I., Factorizable particles on a half-line, and root systems, Teoret. Mat. Fiz. 61 (1984), 35-44 (in Russian).
4. Cherednik I., Quantum Knizhnik-Zamolodchikov equations and affine root systems, Comm. Math. Phys. 150 (1992), 109-136.
5. Cherednik I., Special bases of irreducible representations of a degenerate affine Hecke algebra, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 87-88 (in Russian).
6. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
7. Kazhdan D., Lusztig G., Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153-215.