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On Harish-Chandra's Isomorphism

Eric Opdam, Valerio Toledano-Laredo

TL;DR

This work surveys Harish-Chandra's isomorphism and its far-reaching impact on real and p-adic harmonic analysis, representation theory, and mathematical physics. It develops and contrasts the real (Archimedean) and p-adic pictures through objects like the Harish-Chandra homomorphism, Satake transform, Iwahori-Hecke and Bernstein algebras, Macdonald's explicit spherical formulas, and the graded affine Hecke/Dunkl-Cherednik framework. Central contributions include the real-analytic realization of radial parts via Dunkl operators, the introduction and study of generalized spherical transforms, and the discovery of both symmetric and nonsymmetric shift operators that intertwine parameter shifts in Dunkl-Cherednik theory. These results unify and extend classical spherical function theory, offer robust tools for quantum integrable systems (Calogero-Moser/Yang), and illuminate deep connections between geometric representation theory and special functions, with wide-ranging implications in mathematical physics and algebraic combinatorics.

Abstract

This is the text of a talk given by the first author at the Harish-Chandra centenary meeting held in Allahabad in October 2023. It reviews Harish-Chandra's isomorphism and its many applications to representation theory and mathematical physics. It also announces the existence and uniqueness of nonsymmetric shift operators for an arbitrary root system. These are differential-reflection operators with a transmutation property relative to Dunkl-Cherednik operators: they shift the parameter k of these operators by 1, and restrict on symmetric functions to the hypergeometric shift operators introduced by the first author.

On Harish-Chandra's Isomorphism

TL;DR

This work surveys Harish-Chandra's isomorphism and its far-reaching impact on real and p-adic harmonic analysis, representation theory, and mathematical physics. It develops and contrasts the real (Archimedean) and p-adic pictures through objects like the Harish-Chandra homomorphism, Satake transform, Iwahori-Hecke and Bernstein algebras, Macdonald's explicit spherical formulas, and the graded affine Hecke/Dunkl-Cherednik framework. Central contributions include the real-analytic realization of radial parts via Dunkl operators, the introduction and study of generalized spherical transforms, and the discovery of both symmetric and nonsymmetric shift operators that intertwine parameter shifts in Dunkl-Cherednik theory. These results unify and extend classical spherical function theory, offer robust tools for quantum integrable systems (Calogero-Moser/Yang), and illuminate deep connections between geometric representation theory and special functions, with wide-ranging implications in mathematical physics and algebraic combinatorics.

Abstract

This is the text of a talk given by the first author at the Harish-Chandra centenary meeting held in Allahabad in October 2023. It reviews Harish-Chandra's isomorphism and its many applications to representation theory and mathematical physics. It also announces the existence and uniqueness of nonsymmetric shift operators for an arbitrary root system. These are differential-reflection operators with a transmutation property relative to Dunkl-Cherednik operators: they shift the parameter k of these operators by 1, and restrict on symmetric functions to the hypergeometric shift operators introduced by the first author.

Paper Structure

This paper contains 35 sections, 38 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Harish--Chandra's homomorphism
  3. Zonal spherical functions
  4. An integral formula for zonal spherical functions
  5. The Abel transform
  6. The radial part of an invariant operator
  7. Asymptotic behaviour; Harish--Chandra's $c$-function
  8. Satake's isomorphism
  9. Satake transform
  10. The Iwahori Hecke algebra and Bernstein's isomorphism
  11. Macdonald's explicit formula for spherical functions
  12. Graded Hecke algebras and Dunkl operators
  13. The graded affine Hecke algebra
  14. Dunkl-Cherednik operators on $\mathbb{C}[T]$
  15. Hypergeometric differential operators and $\mathbb{D}(X)$
  16. ...and 20 more sections

Key Result

Theorem 2.1

(Gel) The convolution algebra $C_c(G/\!\!/K)$ of spherical continuous compactly supported functions is a commutative algebra.

Theorems & Definitions (61)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Corollary 2.8
  • Corollary 2.9
  • Theorem 3.1
  • ...and 51 more