×

Unique functionals and representations of Hecke algebras. (English) Zbl 1261.22011

Authors’ abstract: J. D. Rogawski [Invent. Math. 79, 443–465 (1985; Zbl 0579.20037)] used the affine Hecke algebra to model the intertwining operators of unramified principal series representations of \(p\)-adic groups. On the other hand, a representation of this Hecke algebra in which the standard generators act by Demazure-Lusztig operators was introduced by G. Lusztig [J. Am. Math. Soc. 2, No. 3, 599–635 (1989; Zbl 0715.22020)] and applied by D. Kazhdan and G. Lusztig [Invent. Math. 87, 153–215 (1987; Zbl 0613.22004)] to prove the Deligne-Langlands conjecture. These operators appear in various other contexts. B. Ion [Adv. Math. 201, No. 1, 36–42 (2006; Zbl 1103.33010)] used them to express matrix coefficients of principal series representations in terms of nonsymmetric Macdonald polynomials, while B. Brubaker, D. Bump and A. Licata [“Whittaker functions and Demazure operators”, Preprint (2011), arXiv:1111.4230] found essentially the same operators underlying recursive relationships for Whittaker functions.
Here, we explain the role of unique functionals and Hecke algebras in these contexts and revisit the results of Ion from the point of view of Brubaker et al. [loc. cit.].

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
PDFBibTeX XMLCite
Full Text: DOI Link