Affine Hecke Algebras via DAHA
Tóm tắt
A method is suggested for obtaining the Plancherel measure for Affine Hecke Algebras as a limit of integral-type formulas for inner products in the polynomial and related modules of Double Affine Hecke Algebras. The analytic continuation necessary here is a generalization of “picking up residues” due to Arthur, Heckman, Opdam and others, which can be traced back to Hermann Weyl. Generally, it is a finite sum of integrals over double affine residual subtori; a complete formula is presented for $$A_1$$ in the spherical case.
Tài liệu tham khảo
citation_journal_title=Ser. Math. et Phys.; citation_title=A finite analog of the reciprocal of a theta function, Pubblications de la Faculté D’électrotechnique De L’Université À Belgrade; citation_author=L Carlitz; citation_volume=412–460; citation_publication_date=1973; citation_pages=97-99; citation_id=CR1
citation_title=Double Affine Hecke Algebras, London Mathematical Society Lecture Note Series; citation_publication_date=2006; citation_id=CR2; citation_author=I Cherednik; citation_publisher=Cambridge University Press
citation_journal_title=IMRN; citation_title=Difference Macdonald–Mehta conjecture; citation_author=I Cherednik; citation_volume=10; citation_publication_date=1997; citation_pages=449-467; citation_doi=10.1155/S1073792897000317; citation_id=CR3
citation_journal_title=Selecta Math.; citation_title=Nonsemisimple Macdonald polynomials; citation_author=I Cherednik; citation_volume=14; citation_issue=3–4; citation_publication_date=2009; citation_pages=427-569; citation_doi=10.1007/s00029-009-0493-1; citation_id=CR4
citation_journal_title=IMRN; citation_title=Whittaker limits of difference spherical functions; citation_author=I Cherednik; citation_volume=20; citation_publication_date=2009; citation_pages=3793-3842; citation_id=CR5
Cherednik, I.: Integration of quantum many-body problems by ffine Knizhnik-Zamolodchikov equations. Preprint RIMS 776 (1991) [Advances in Math. 106, 65–95 (1994)]
Cherednik, I.: On Harish-Chandra theory of global nonsymmetric functions.
arXiv:1407.5260
(2014)
Cherednik, I., Ma, X.: Spherical and Whittaker functions via DAHA I, II. Selecta Mathematica (N.S.) 19(3), 737–817, 819–864 (2013)
Cherednik, I., Orr, D.: One-dimensional nil-DAHA and Whittaker functions I. Transform. Groups 17(4), 953–987 (2012).
arXiv:math/0111130v1
(2011)
citation_journal_title=Mathematische Zeitschrift; citation_title=Nonsymmetric difference Whittaker functions; citation_author=I Cherednik, D Orr; citation_volume=279; citation_issue=3; citation_publication_date=2015; citation_pages=879-938; citation_doi=10.1007/s00209-014-1397-0; citation_id=CR10
Cherednik, I., Ostrik, V.: From double Hecke algebras to Fourier transform. Selecta Math. New Ser. 8, 1–89 (2003).
arXiv:math/0111130
citation_journal_title=Inventiones mathematicae; citation_title=On characters and formal degrees of discrete series of affine Hecke algebras of classical types; citation_author=D Ciubotaru, M Kato, S Kato; citation_volume=187; citation_issue=3; citation_publication_date=2012; citation_pages=589-635; citation_doi=10.1007/s00222-011-0338-3; citation_id=CR12
citation_journal_title=J. Math. Kyoto Univ.; citation_title=Composition factors of polynomial representation of DAHA and crystallized decomposition numbers; citation_author=N Enomoto; citation_volume=49; citation_issue=3; citation_publication_date=2009; citation_pages=441-473; citation_doi=10.1215/kjm/1260975035; citation_id=CR13
Etingof, P., Stoica, E., with an appendix by Griffeth, S.: Unitary representations of rational Cherednik algebras. Represent. Theory 13, 349–370 (2009)
citation_title=Harmonic analysis for affine Hecke algebras; citation_inbook_title=Current Developments in Mathematics; citation_publication_date=1996; citation_id=CR15; citation_author=GJ Heckman; citation_author=EM Opdam; citation_publisher=Intern. Press
citation_journal_title=Adv. Math.; citation_title=Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series; citation_author=B Ion; citation_volume=201; citation_publication_date=2006; citation_pages=36-62; citation_doi=10.1016/j.aim.2004.10.020; citation_id=CR16
citation_journal_title=Inventiones Math.; citation_title=Proof of the Deligne–Langlands conjecture for Hecke algebras; citation_author=D Kazhdan, G Lusztig; citation_volume=87; citation_publication_date=1987; citation_pages=153-215; citation_doi=10.1007/BF01389157; citation_id=CR17
citation_journal_title=Ann. Math.; citation_title=Green functions and character sheaves; citation_author=G Lusztig; citation_volume=131; citation_publication_date=1990; citation_pages=355-408; citation_doi=10.2307/1971496; citation_id=CR18
citation_journal_title=Ann. Math.; citation_title=The c-function expansion of a basic hypergeometric function associated to root systems; citation_author=J Stokman; citation_volume=179; citation_issue=1; citation_publication_date=2014; citation_pages=253-299; citation_doi=10.4007/annals.2014.179.1.4; citation_id=CR19
citation_journal_title=Acta Math.; citation_title=Harmonic analysis for certain representations of graded Hecke algebras; citation_author=E Opdam; citation_volume=175; citation_publication_date=1995; citation_pages=75-121; citation_doi=10.1007/BF02392487; citation_id=CR20
Opdam, E.: Hecke algebras and harmonic analysis. In: Proceedings of the International Congress of Mathematicians -Madrid, vol. II, pp. 1227–1259. EMS Publ. House (2006)
Opdam, E.: A generating formula for the trace of the Iwahori–Hecke algebra. Prog. Math. 210, 301–323 (2003).
arXiv:math/0101006
citation_journal_title=Acta Math.; citation_title=Discrete series characters for affine Hecke algebras and their formal degrees; citation_author=E Opdam, M Solleveld; citation_volume=205; citation_issue=1; citation_publication_date=2010; citation_pages=105-187; citation_doi=10.1007/s11511-010-0052-9; citation_id=CR23