Harmonic analysis on the Iwahori–Hecke algebra

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)171-216
Number of pages46
JournalJapanese Journal of Mathematics
Issue number2
StatePublished - 1 Sep 2014


  • Bernstein forms
  • Bernstein presentation
  • Iwahori–Hecke algebra
  • Macdonald’s spherical forms
  • generating function
  • intertwining operators
  • trace
  • trace forms


Dive into the research topics of 'Harmonic analysis on the Iwahori–Hecke algebra'. Together they form a unique fingerprint.

Cite this