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Affine Hecke algebras for Langlands parameters

211   0   0.0 ( 0 )
 نشر من قبل Maarten Solleveld
 تاريخ النشر 2017
  مجال البحث
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118 - Weideng Cui 2015
We establish an explicit algebra isomorphism between the affine Yokonuma-Hecke algebra $widehat{Y}_{r,n}(q)$ and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type $A.$ As an application of this resu lt, we show that $widehat{Y}_{r,n}(q)$ is affine cellular in the sense of Koenig and Xi, and further prove that it has finite global dimension when the parameter $q$ is not a root of the Poincare polynomial. As another application, we also recover the modular representation theory of $widehat{Y}_{r,n}(q)$ previously obtained in [CW].
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Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the centers. W e further develop connections of these algebras with the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras.
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We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W a nd two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
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The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl ope rator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras.
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Inspired by the work [Ra1], we directly give a complete classification of irreducible calibrated representations of affine Yokonuma-Hecke algebras $widehat{Y}_{r,n}(q)$ over $mathbb{C},$ which are indexed by $r$-tuples of placed skew shapes. We then develop several applications of this result. In the appendix, inspired by [Ru], we classify and construct irreducible completely splittable representations of degenerate affine Yokonuma-Hecke algebras $D_{r,n}$ and the wreath product $(mathbb{Z}/rmathbb{Z})wr mathfrak{S}_{n}$ over an algebraically closed field of characteristic $p> 0$ such that $p$ does not divide $r$.
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