We give a short elementary proof of the fact that connected semisimple real
Lie groups satisfy property RD. The proof is based on a process of
linearization

Boyer, Adrien

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 335–338Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comGroup Theory/Harmonic AnalysisSemisimple Lie groups satisfy property RD, a short proofLes groupes de Lie semi-simples ont la propriété RD, une preuve courteAdrien BoyerLATP, Centre de mathématiques et informatique (CMI), Aix–Marseille Université, Technopôle de Château-Gombert,39, rue Frédéric-Joliot-Curie, 13453 Marseille cedex 13, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 6 December 2012Accepted after revision 21 May 2013Available online 30 May 2013Presented by the Editorial BoardWe give a short elementary proof of the fact that connected semisimple real Lie groupssatisfy property RD. The proof is based on a process of linearisation.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous donnons dans cette note une preuve courte et élémentaire du fait que les groupesde Lie semi-simples réels connexes satisfont la propriété RD. La preuve est basée sur unprocédé de linéarisation.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionA length function L : G →R∗+ on a locally compact group G is a measurable function satisfying:(i) L(e) = 0 where e is the neutral element of G ,(ii) L(g−1) = L(g),(iii) L(gh) L(g) + L(h).A unitary representation π : G → U (H) on a complex Hilbert space has property RD with respect to L if there existsC > 0 and d  1 such that for each pair of unit vectors ξ and η in H , we have:∫G|〈π(g)ξ,η〉|2(1 + L(g))d dg  Cwhere dg is a (left) Haar measure on G , see [10]. We say that G has property RD if its regular representation has propertyRD with respect to L. First established for free groups by U. Haagerup in [6], property RD has been introduced and studied assuch by P. Jolissaint in [8], who notably established it for groups of polynomial growth, and for classical hyperbolic groups.See [12, Chap. 8, p. 69] for more details.If π denotes a unitary representation on a Hilbert space H , then π denotes its conjugate representation on the conjugateHilbert space H . The process of linearisation consists in working with σ : G → U (H ⊗ H) the unitary representation σ =π ⊗ π , see [3, Section 2.2].E-mail address: aboyer@cmi.univ-mrs.fr.1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2013.05.007336 A. Boyer / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 335–338A connected semisimple real Lie group with finite center can be written G = K P , where K is a compact connectedsubgroup, and P a closed amenable subgroup. We denote by P the right-modular function of P . Extend to G the map Pof P as  : G → R∗+ with (g) = (kp) := P (p). It is well defined because K ∩ P is compact (observe that P |K∩P = 1).The quotient G/P carries a unique quasi-invariant measure μ, such that the Radon–Nikodym derivative at (g, x) ∈ G × G/P ,denoted by c(g, x) = dg∗μdμ (x), with g∗μ(A) = μ(g−1 A), satisfies dg∗μdμ (x) = (gx)(x) for all g ∈ G and x ∈ G/P (notice that for allg ∈ G , the function x ∈ G/P 	→ (gx)(x) ∈R∗+ is well defined). We refer to [2, Appendix B, Lemma B.1.3, pp. 344–345] for moredetails. Consider the quasi-regular representation λG/P : G → U (L2(G/P )) associated with P , defined by (λG/P (g)ξ)(x) =c(g−1, x) 12 ξ(g−1x). Denote by dk the Haar measure on K , and under the identification G/P = K/(K ∩ P ), denote by d[k] themeasure μ on G/P .The well-known Harish-Chandra function is defined by Ξ(g) := 〈λG/P (g)1G/P ,1G/P 〉 where 1G/P denotes the character-istic function of the space G/P .In the rest of the paper, we set σ = λG/P ⊗ λG/P . Observe that L2(G/P ) ⊗ L2(G/P ) ∼= L2(G/P × G/P ), via: ξ ⊗ η 	→((x, y) 	→ ξ(x)η(y)). Notice that σ preserves the cone of positive functions on L2(G/P × G/P ).Let G be a (non-compact) connected semisimple real Lie group. Let g be its Lie algebra. Let θ be a Cartan involution.Define the bilinear form denoted by (X, Y ) such that for all X, Y ∈ g, (X, Y ) = −B(X, θ(Y )), where B is the Killing form.Set |X | = √(X, X). Write g = l ⊕ p the eigenvector space decomposition associated with θ (l for the eigenvalue 1). Let Kbe the compact subgroup defined as the connected subgroup whose Lie algebra l is the set of fixed points of θ . Fix a ⊂ pa maximal Abelian subalgebra of p. Consider the roots system Σ associated with a and let Σ+ be the set of positive roots,and define the corresponding positive Weyl chamber as:a+ := {H ∈ a, α(H) > 0, ∀α ∈ Σ+}.Let A+ = Cl(exp(a+)), where Cl denotes the closure of exp(a+). Consider the corresponding polar decomposition K A+K .Then define the length function:L(g) = L(k1eHk2) := |H|where g = k1eHk2 with eH ∈ A+ . Notice that L is K bi-invariant. The desintegration of the Haar measure on G according tothe polar decomposition is:dg = dk J (H)dH dkwhere dk is the Haar measure on K , dH the Lebesgue measure on a+ , andJ (H) =∏α∈Σ+(eα(H) − e−α(H)2)nαwhere nα denotes the dimension of the root space associated with α. See [9, Chap. V, Section 5, Proposition 5.28,pp. 141–142], [5, Chap. 2, §2.2, p. 65] and [5, Chap. 2, Proposition 2.4.6, p. 73] for more details.The aim of this note is to give a short proof of the following known result [4,7].Theorem 1 (C. Herz). Let G be a connected real semisimple Lie group with finite center. Then G has property RD with respect to L.See [4, Proposition 5.5 and Lemma 6.3] for the case G has infinite center.2. ProofProof. We shall prove that the quasi-regular representation has property RD with respect to L defined above. This impliesthat the regular representation has property RD with respect to L by Lemma 2.3 in [11]. Write G = K P , where K is acompact subgroup and P is a closed amenable subgroup of G . It is sufficient to prove that there exists d0  1 and C0  0such that∫G〈λG/P (g)ξ,ξ〉2(1+L(g))d0 dg < C0, for positive functions ξ , with ‖ξ‖ = 1.Take ξ ∈ L2(G/P ) such that ξ  0, and ‖ξ‖ = 1. Define the function:F : G/P × G/P →R+,(x, y) 	→∫σ(k)(ξ ⊗ ξ)(x, y)dk.KA. Boyer / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 335–338 337For all (x, y) ∈ G/P × G/P , we have by the Cauchy–Schwarz inequality:∫Kσ(k)(ξ ⊗ ξ)(x, y)dk =∫Kξ(k−1x)ξ(k−1 y)dk(∫Kξ2(k−1x)dk) 12(∫Kξ2(k−1 y)dk) 12.Observe that the function f : x ∈ G/P 	→ ∫K ξ2(k−1x)dk ∈ R+ is constant. Indeed, fix x ∈ G/P and let y in G/P . Writey = hx for some h ∈ K (as K acts transitively on G/P ). By invariance of the Haar measure, we have f (y) = ∫K ξ2(k−1 y)dk =∫K ξ2(k−1hx)dk = ∫K ξ2(k−1x)dk = f (x). If e is the neutral element in G , we write [e] ∈ G/P . We have, for all x ∈ G/P ,f (x) = f ([e]).Hence, for all x ∈ G/P we have:∫Kξ2(k−1x)dk =∫Kξ2(k−1[e]) dk=∫K/K∩Pξ2([k−1])d[k]= ‖ξ‖2 = 1.Therefore ‖F‖∞ := sup{F (x, y), (x, y) ∈ G/P × G/P }  1. Hence 0  F  1G/P×G/P , where 1G/P×G/P denotes the character-istic function of G/P × G/P .Let r be the number of indivisible positive roots in a. We know that there exists C > 0 such that, for all H ∈ a whereeH ∈ A+ , we have:Ξ(eH) Ce−ρ(H)(1 + L(eH))rwith ρ = 12∑α∈Σ+ nαα ∈ a+ , see [5, Chap. 4, Theorem 4.6.4, p. 161]. Hence for d0 > dim(a) + 2r, we have:∫a+Ξ2(eH )(1 + L(eH ))d0 J (H)dH < ∞.We obtain for all d  0 and for all positive functions ξ , with ‖ξ‖ = 1:∫G〈λG/P (g)ξ, ξ〉2(1 + L(g))d dg =∫G〈λG/P (g)ξ, ξ〉〈λG/P (g)ξ, ξ〉(1 + L(g))d dg=∫G〈σ(g)ξ ⊗ ξ, ξ ⊗ ξ〉(1 + L(g))d dg=∫K∫a+∫K〈σ(k1eHk2)ξ ⊗ ξ, ξ ⊗ ξ〉(1 + L(k1eHk2))d J (H)dk1 dH dk2=∫K∫a+∫K〈σ(eH )σ (k2)(ξ ⊗ ξ),σ (k−11 )(ξ ⊗ ξ)〉(1 + L(eH ))d J (H)dk1 dH dk2=∫a+〈σ(eH )(∫K σ(k2)(ξ ⊗ ξ)dk2), (∫K σ(k−11 )(ξ ⊗ ξ)dk1)〉(1 + L(eH ))d J (H)dH=∫a+〈σ(eH )F , F 〉(1 + L(eH ))d J (H)dH∫+〈σ(eH )1G/P×G/P ,1G/P×G/P 〉(1 + L(eH ))d J (H)dHa338 A. Boyer / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 335–338=∫a+〈λG/P (eH )1G/P ,1G/P 〉2(1 + L(eH ))d J (H)dH=∫a+Ξ2(eH )(1 + L(eH ))d J (H)dH .Take d0 > dim(a) + 2r and C0 =∫a+Ξ2(eH )(1+L(eH ))d0 J (H)dH . We have found d0  1 and C0 > 0 such that for all positivefunctions ξ in L2(G/P ) with ‖ξ‖ = 1, we have:∫G〈λG/P (g)ξ, ξ〉2(1 + L(g))d0 dg  C0. Remark 1. The same approach applies to algebraic semisimple Lie groups over local fields. See [1, Section 1, (1.3)] and[13, Lemma II.1.5.].Remark 2. It’s not hard to see that this approach shows that the representations of the principal series of G (of class one,see [5, (3.1.12), p. 103]) satisfy also property RD.AcknowledgementsWe would like to thank C. Pittet for helpful criticisms, and P. Delorme to point out the papers [1] and [13] for the caseof semisimple Lie groups over local fields.References[1] J. Arthur, A local trace formula, Publ. Math. Inst. Hautes Études Sci. 73 (1991) 5–96.[2] B. Bekka, P. de la Harpe, A. Valette, Kazhdan’s Property (T), New Math. Monogr., vol. 11, Cambridge University Press, Cambridge, 2008.[3] A. Boyer, Quasi-regular representations and property RD, preprint, arXiv:1305.0480, 2013.[4] I. Chatterji, C. Pittet, L. Saloff-Coste, Connected Lie groups and property RD, Duke Math. J. 137 (3) (2007) 511–536.[5] R. Gangolli, V.S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Springer-Verlag, New York, 1988.[6] U. Haagerup, An example of a nonnuclear C*-algebra which has the metric approximation property, Invent. Math. 50 (3) (1978/1979) 279–293.[7] C. Herz, Sur le phénomène de Kunze–Stein, C. R. Acad. Sci. Paris, Sér. A–B 271 (1970) A491–A493.[8] P. Jolissaint, Rapidly decreasing functions in reduced C*-algebras of groups, Trans. Amer. Math. Soc. 317 (1) (1990) 167–196.[9] A.-W. Knapp, Representation Theory of Semisimple Groups, Princeton Landmarks Math., 2001.[10] M. Perrone, Rapid decay and weak containment of unitary representations, 2009, unpublished notes.[11] Y. Shalom, Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann.Math. (2) 152 (1) (2000) 113–182.[12] A. Valette, Introduction to the Baum–Connes Conjecture, Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 2002.[13] J.-L. Walspurger, La formule de Plancherel pour les groupes p-adiques. D’après Harish-Chandra, J. Inst. Math. Jussieu 2 (2) (April 2003) 235–333.

Semisimple Lie groups satisfy property RD, a short proof

Adrien Boyer

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Semisimple Lie groups satisfy property RD, a short proof

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