ANNALES DE LA FACULTÉ DES SCIENCES Mathématiques SIMON GINDIKIN Harish-Chandra’s c-function; 50 years later Tome XXV, no 2-3 (2016), p. 385-402. <http://afst.cedram.org/item?id=AFST_2016_6_25_2-3_385_0> © Université Paul Sabatier, Toulouse, 2016, tous droits réservés. L’accès aux articles de la revue « Annales de la faculté des sci- ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Annales de la Facult´e des Sciences de Toulouse Vol. XXV, n◦ 2-3, 2016 pp. 385-402 Harish-Chandra’s c-function; 50 years later Simon Gindikin(1) RESUM´ E.´ — Nous discutons diff´erents aspects de la fonction c de Harish- Chandra, en soulignant ses interactions avec la transform´ee horosph´erique. ABSTRACT. — We discuss different aspects of the c-function of Harish- Chandra with focus on its connection with the horospherical transform. I have known Vadim Schechtman for many years, but it was only a few years ago that I found out that we share two strong interests: sur- prises which it is possible yet to mine in works of great mathematicians and product-formulas for some special functions on Lie groups. Vadim has col- lected a very impressive exhibition of such formulas (see unpublished notes on his webpage). It starts with the product-formula for the c-function which Karpelevich and I found more than 50 years ago [19] in the beginning of my mathematical life. Vadim’s anniversary for me is a pleasant reason to talk about this old subject. Starting at least with Gauss (who continuously pro- duced new proofs of the Fundamental Theorem of Algebra), old mathemati- cians like to return to subjects of their first mathematical love. Perhaps it has a similar nature to criminals, returning to the scene of the crime? What about the c-function, for the past several years, there were quite a few pop- ular generalizations of the product-formula on arbitrary fields, but in these notes we will stay inside old fashioned real or complex considerations. (1) Departm. of Math., Hill Center, Rutgers University, 110 Frelinghysen Road, Pis- cataway, NJ 08854 gindikinath.rutgers.edu – 385 – Simon Gindikin Origin of the c-function The c-function appeared in the publication of Harish-Chandra of 1958 [22] about the asymptotic of zonal spherical functions on Riemannian sym- metric spaces of non compact type and its applications to the Plancherel formula on these spaces. So it appeared as an important, but intermediate object. Let us start from a discussion if it has an independent conceptual nature. Often such kind of questions are artificial, but I believe that here it is a perfectly legitimate. Let us remind of the definitions. Let G be a simple connected semisimple Lie group; K be its maxi- mal compact subgroup; A, N be complimentary Cartanian and maximal unipotent subgroups; G = KAN be the Iwasawa decomposition; N¯ be the unipotent subgroup opposite to N and X = G/K be the Riemannian symmetric space. Let a(g) be the projection on the factor A at the Iwasawa decomposition, a be the Lie algebra Lie of A, a∗ be the dual space, Σ, Π be the system of positive and simple roots, ρ be the half-sum of positive roots; aξ = exp(ξ,log a),ξ a, a A. ∈ ∈ We can extend this definition on ξ Ca. Let μ(dn¯) be the invariant measure on N¯. Now we can define the c-function∈ ρ iλ c(λ)= a(¯n)− − μ(dn¯). ¯ N It is possible to prove that this integral is absolutely convergent if Re(iλ) lies in the positive Weyl chamber W+. Let M be the centralizer of A at K and we call Ξ=G/MN¯ the horospherical space. Points of Ξ parameterize non degenerate orbits E(ξ) of unipotent subgroups conjugated to N which are called the horospheres. On Ξ we have a “left” action of A commutating with the action G, since A normalizes MN¯. Then the horospherical space Ξ fibers on A-fibers over the flag manifold F =Ξ/A = G/AMN.¯ So the isotropy subgroup of F is parabolic. – 386 – Harish-Chandra’s c-function; 50 years later In E.Cartan’s conception the irreducible finite dimensional spherical rep- resentations on X are characterized by unique K-invariant elements - zonal spherical functions. Another possibility is to connect with them N-invariant elements - highest weight vectors. This duality for spherical representations extends from finite dimensional representations on some infinite dimensional representations. It requires some analytical justifications which we will not discuss here. At Borel-Weil modification of highest weight method the irreducible rep- resentations are realized at sections of line bundles on the flag spaces F . Gelfand-Graev’s conception of integral geometry suggests realizing of ir- reducible representations at functions on the horospherical space Ξ. If we decompose the representation at functions on Ξ relative to the “left” ac- tion of A we will obtain Borel-Weil realizations of irreducible components on F , which are parameterized by characters of A. This looks as a small modification, but since points of Ξ admit a geometrical realization at X, it opens a possibility for a new construction of geometric analysis. Gelfand and Graev considered the integral geometry corresponding to infinite di- mensional representations [6]. We will discuss later a similar approach to finite dimensional representations. So irreducible spherical representations can be realized either on X or on Ξ. The principal moment of Gelfand-Graev’s approach was that since spectrums on X and Ξ coincide it must be an invariant intertwining oper- ator between functions on X and Ξ. This is Radon’s type operator - the integration along horospheres - the horospherical transform: f(ξ)= f(x)μ(dn),ξ Ξ, H ∈ E(ξ) where μ(dn) is the invariant measure on N¯ translated on the horospheres. The first area of applications was the Plancherel formula on complex semisim- ple Lie groups and some homogeneous spaces: since the decomposition on irreducible ones is reduced to commutative Fourier transform on A, we will have the decomposition as soon as we can invert the horospherical trans- form. Later we will return to this possibility. Heuristically, the horospherical transform is scalar on irreducible con- stituents if we identify them on X and Ξ. Modulo all complications con- nected with the continuous spectrum and non L2 eigenfunctions the c- function represents the eigenvalues of the horospherical transform under a natural normalization. I believe that this heuristic view is crucial for the understanding of the nature of the c-function. Its appearance at the asymp- totic of the zonal spherical function is already a secondary event. – 387 – Simon Gindikin As test functions for the definition of eigenvalues let us consider the “highest weight vectors” on X and Ξ: ρ iλ ρ iλ χ(x λ)=a(x)− − ,x X; χ(ξ λ)=a(ξ)− − ,ξ Ξ; | ∈ | ∈ Here a(ξ) is the projection on A in the decomposition NMAN¯ at the open part of G and, as a consequence, of Ξ. Then χ(ξ λ) can be interpreted as boundary values of χ(x λ). | | The direct computation shows that χ(x λ)=c( λ)χ(ξ λ) H | − | if Re( iλ) is in the positive Weyl chamber W+ and we take boundary values for real− λ. There is a natural intertwining operator acting from Ξ to X - the dual horospherical transform F (x)= F (ξ)μ(dξ). P S(x) Here S(x) is the set of parameters of horospheres passing through x X. We have S(x)=K/M and μ(dξ) is the invariant measure. So in the∈ dual horospherical transform we have the integration on the compact manifold K/M. Let us remark that χ(ξ λ) is defined only on an open set of Ξ and | the integral converges only for Re(iλ) W+. On the highest weight vectors it is scalar: ∈ χ(ξ λ)=d(λ)χ(x λ). P | | The direct computation shows that d(λ)=c(λ),Re(iλ) W . ∈ + We need just to apply the Harish-Chandra’s trick of the replacement of integrations along compact orbits by integrations along unipotent ones [23]. He applied it for the integral representation of zonal spherical functions on the unipotent subgroup. It corresponds to the appearance of the factor c(λ) at the Poisson formula for reconstructing eigenfunctions of invariant differential operators on X through the boundary values on the boundary F ([23, 24] . The integrals defining the direct and dual horospherical transforms , have no joint convergence area, but we can in both cases consider the bound-P H ary values for real λ and then their composition will have the eigenvalues c(λ)c( λ)= c(λ) 2. − | | – 388 – Harish-Chandra’s c-function; 50 years later So if we add the operator with the eigenvalues L p(λ)=1/ c(λ) 2 | | we have the identical operator and the operator inverts . It illus- trates the principal Harish-Chandra’sLPH observation thatLPp(λ) is theH density in the Plancherel formula and is the reason why in the Plancherel density appears the square of modulus of the c-function. It would be interesting to transform these heuristic discussions into rig- orous statements.