TY - JOUR

T1 - Harmonic analysis on the Iwahori–Hecke algebra

AU - Flicker, Yuval Z.

N1 - Publisher Copyright:
© 2014, The Mathematical Society of Japan and Springer Japan.

PY - 2014/9/1

Y1 - 2014/9/1

N2 - These are purely expository notes of Opdam’s analysis [O1] of the trace form τ(f) = f(e) on the Hecke algebra H = Cc(I\G/I) of compactly supported functions f on a connected reductive split p-adic group G which are biinvariant under an Iwahori subgroup I, extending Macdonald’s work. We attempt to give details of the proofs, and choose notations which seem to us more standard. Many objects of harmonic analysis are met: principal series, Macdonald’s spherical forms, trace forms, Bernstein forms. The latter were introduced by Opdam under the name Eisenstein series for H. The idea of the proof is that the last two linear forms are proportional, and the proportionality constant is computed by projection to Macdonald’s spherical forms. Crucial use is made of Bernstein’s presentation of the Iwahori–Hecke algebra by means of generators and relations, as an extension of a finite dimensional algebra by a large commutative subalgebra. We give a complete proof of this using the universal unramified principal series right H-module M = Cc(A(O)N\G/I) to develop a theory of intertwining operators algebraically.

AB - These are purely expository notes of Opdam’s analysis [O1] of the trace form τ(f) = f(e) on the Hecke algebra H = Cc(I\G/I) of compactly supported functions f on a connected reductive split p-adic group G which are biinvariant under an Iwahori subgroup I, extending Macdonald’s work. We attempt to give details of the proofs, and choose notations which seem to us more standard. Many objects of harmonic analysis are met: principal series, Macdonald’s spherical forms, trace forms, Bernstein forms. The latter were introduced by Opdam under the name Eisenstein series for H. The idea of the proof is that the last two linear forms are proportional, and the proportionality constant is computed by projection to Macdonald’s spherical forms. Crucial use is made of Bernstein’s presentation of the Iwahori–Hecke algebra by means of generators and relations, as an extension of a finite dimensional algebra by a large commutative subalgebra. We give a complete proof of this using the universal unramified principal series right H-module M = Cc(A(O)N\G/I) to develop a theory of intertwining operators algebraically.

KW - Bernstein forms

KW - Bernstein presentation

KW - Iwahori–Hecke algebra

KW - Macdonald’s spherical forms

KW - generating function

KW - intertwining operators

KW - trace

KW - trace forms

UR - http://www.scopus.com/inward/record.url?scp=84907615911&partnerID=8YFLogxK

U2 - 10.1007/s11537-014-1365-9

DO - 10.1007/s11537-014-1365-9

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AN - SCOPUS:84907615911

SN - 0289-2316

VL - 9

SP - 171

EP - 216

JO - Japanese Journal of Mathematics

JF - Japanese Journal of Mathematics

IS - 2

ER -