Pursuing Coxeter theory for Kac-Moody affine Hecke algebras

• Dinakar Muthiah ^[1] ; Anna Puskás ^[1]
  1. [1] University of Glasgow

     University of Glasgow

     Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 32, Nº. 1, 2026
• Idioma: inglés
• DOI: 10.1007/s00029-025-01117-7
• Enlaces
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• Resumen

  • The Kac-Moody affine Hecke algebra H was first constructed as the Iwahori-Hecke algebra of a p-adic Kac-Moody group by work of Braverman, Kazhdan, and Patnaik, and by work of Bardy-Panse, Gaussent, and Rousseau. Since H has a Bernstein presentation, for affine types it is a positive-level variation of Cherednik’s double affine Hecke algebra. Moreover, as H is realized as a convolution algebra, it has an additional “T -basis” corresponding to indicator functions of double cosets. For classical affine Hecke algebras, this T -basis reflects the Coxeter group structure of the affine Weyl group. In the Kac-Moody affine context, the indexing set WT for the T -basis is no longer a Coxeter group. Nonetheless, WT carries some Coxeter-like structures: a Bruhat order, a length function, and a notion of inversion sets. This paper contains the first steps toward a Coxeter theory for Kac-Moody affine Hecke algebras. We prove three results. The first is a construction of the length function via a representation of H. The second concerns the support of products in classical affine Hecke algebras.

    The third is a characterization of length deficits in the Kac-Moody affine setting via inversion sets. Using this characterization, we phrase our support theorem as a precise conjecture for Kac-Moody affine Hecke algebras. Lastly, we give a conjectural definition of a Kac-Moody affine Demazure product via the q = 0 specialization of H.

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