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The Heat Kernel on Noncompact Symmetric Spaces Jean-Philippe Anker, Patrick Ostellari

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The Heat Kernel on Noncompact Symmetric Spaces Jean-Philippe Anker, Patrick Ostellari The heat kernel on noncompact symmetric spaces Jean-Philippe Anker, Patrick Ostellari To cite this version: Jean-Philippe Anker, Patrick Ostellari. The heat kernel on noncompact symmetric spaces. S. G. Gindikin. Lie groups and symmetric spaces, Amer. Math. Soc., pp.27-46, 2003, Amer. Math. Soc. Transl. Ser. 2, vol. 210. hal-00002509 HAL Id: hal-00002509 https://hal.archives-ouvertes.fr/hal-00002509 Submitted on 9 Aug 2004 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THE HEAT KERNEL ON NONCOMPACT SYMMETRIC SPACES Jean{Philippe Anker & Patrick Ostellari In memory of F. I. Karpeleviˇc (1927{2000) The heat kernel plays a central role in mathematics. It occurs in several fields : analysis, geometry and { last but not least { probability theory. In this survey, we shall focus on its analytic aspects, specifically sharp bounds, in the particular setting of Riemannian symmetric spaces of noncompact type. It is a natural tribute to Karpeleviˇc, whose pioneer work [Ka] inspired further study of the geometry of theses spaces and of the analysis of the Laplacian thereon. This survey is based on lectures delivered by the first author in May 2002 at IHP in Paris during the Special Quarter Heat kernels, random walks & analysis on manifolds & graphs. Both authors would like to thank the organizers for their great job, as well as Martine Babillot, Gilles Carron, Sasha Grigor'yan and Jean{Pierre Otal for stimulating discussions. 1. Preliminaries We shall briefly review some basics about noncompact Riemannian symmetric spaces X = G=K and we shall otherwise refer to standard texbooks ([GV]; [H1], [H2], [H3]; [Kn]) for their structure and harmonic analysis thereon. Thus G is a semisimple Lie group (real, connected, noncompact, with finite center) or more generally a reductive Lie group in the Harish-Chandra class and K is a maximal compact subgroup. Let θ be the Cartan involution and let g = k p be the Cartan decomposition at the Lie algebra level. g is equipped with the inner⊕product (1:1) X; Y = B(X; θY ) ; h i − where B is the Killing form, appropriately modified if g has a central component. (1.1) enables us to identify g with its dual g∗ , and likewise for subspaces of g . (1.1) induces the Riemannian structure on X = G=K , whose tangent space at the origin 0 = eK is identified with p. Let a be a Cartan subspace of p, let m be the centralizer of a in k and let g = a m α Σ gα be the root space decomposition of g⊕with⊕respf ⊕ect2 to a. Select in a a positive Weyl + + + chamber a , in Σ the corresponding sets Σ of positive roots, Σ0 of positive indivisible roots, Π of simple roots, and in g the corresponding nilpotent subalgebra n = α Σ+ gα. 1 ⊕ 2 Let % = 2 α Σ+ mα α be the half sum of positive roots, counted with multiplicities 2 2000 MathematicsP Subject Classification. Primary 22E30, 35B50, 43A85, 58J35; Secondary 22E46, 43A80, 43A90. Key words and phrases. Abel transform, heat kernel, maximum principle, semisimple Lie group, symmetric space, subLaplacian. Both authors partially supported by the European Commission (IHP Network HARP ) Typeset by AMS-TEX 1 2 JEAN{PHILIPPE ANKER & PATRICK OSTELLARI mα = dim gα , let ` = dim a be the rank of X and let n = ` + α Σ+ mα be the dimension of X. Finally A = exp a and N = exp n are closed subgroups2 of G and M denotes the centralizer of A in K. P We shall need the following classical decompositions of G G = N A K (Iwasawa) G = K (exp a+ ) K (Cartan) and the resulting decompositions of X = G=K. Let us write x = n(x) a(x) k(x) in the Iwasawa decomposition, which is unique, and A(x) = log a(x) a . The a+{ component in the Cartan decomposition is uniquely determined, contrarily2 to the K{ components. We shall write x = H if x = k eH k . j j j j 1 2 jk Let (Xj) be a basis of g, let bjk = B(Xj; Xk) and let (b ) be the inverse matrix jk of (bjk). Then Ω = j;k b XjXk is the Casimir element in the universal enveloping algebra (g). Consider in particular orthonormal bases (Y ) and (Z ) of p and k with U P j k respect to (1.1). Then 2 Ωp = j Yj ; Ω = Ωp Ωk where − 2 Ωk = Pk Zk : The Laplace{Beltrami operator on X = G=K can be recovered from the action of the P Casimir element : ∆f (xK) = f (x : Ω) = f (x : Ωp) x G : 8 2 Here we consider { as we shall always do { functions on X as right{K{invariant functions on G and we use the following notation for the right{action of (g) on C 1(G) : U @ @ t1X1 tr Xr f (x : X1 Xr ) = : : : f x e e · · · @t1 t1=0 @tr tr =0 · · · for all x G and X ; : : : ; X g . 2 1 r 2 L2 harmonic analysis on X = G=K may be summarized by the Plancherel formula ⊕ (1:2) ( L ; L2(X)) dλ (π ; ) X ∼= c(λ) 2 λ λ a+ j j H which expresses the decomposition of the leftZ regular representation of G on L2(X) into spherical principal series. Recall that 2 (i) πλ is realized on λ L (K=M) by H ≡ %+iλ;A(k−1 x) 1 π (x) ξ (kM) = eh i ξ(x− :kM) ; f λ g (ii) the Harish{Chandra c {function, which enters the Plancherel measure, was com- puted explicitly by Gindikin & Karpeleviˇc [GK] ; (iii) the decomposition (1.2) is realized by the Fourier transform %+iλ;A(k−1x) (1:3) f (λ, kM) = dx f(x) π (x)1 (kM) = dx f(x) eh i H f λ g and its inverse ZG ZG (1:4) f(x) = const. dλ f(λ); π (x)1 = c(λ) 2 H λ L2(K=M) Za j j dλ % iλ;A(k−1 x) = const. 2 dk f(λ, kM) eh − i : c(λ) H Za j j ZK In the bi{K{invariant case, (1.2) boils down to 2 2 dλ W L (K G=K) = L a; 2 n ∼ c(λ) and the integral transforms (1.3) and (1.4) to j j dλ f(λ) = dx f(x) 'λ(x) and f(x) = const. c(λ) 2 f(λ) ' λ(x) : H G a j j H − These formulas inZvolve the spherical functions Z THE HEAT KERNEL ON NONCOMPACT SYMMETRIC SPACES 3 %+iλ;A(kx) 'λ(x) = πλ(x)1; 1 L2(K=M) = dk eh i : ZK The heat equation is the following parabolic evolution equation @ (1:5) @t u(x; t) = ∆x u(x; t) with Cauchy data u(x; 0) = f (x). It is solved u(x; t) = ht(x; y) f(y) dy ZX by the heat kernel ht(x; y), whose fundamental properties are recalled next. (i) ht is symmetric and positive : ht(x; y) = ht(y; x) > 0 ; (ii) ht(x; y) is a smooth function of x X, y X and t (0; + ) ; @ 2 2 2 1 (iii) It satisfies the heat equation : @t ht(x; y) = ∆x ht(x; y) = ∆y ht(x; y) ; (iv) For every x X, h (x; y) dy is a probability measure on X, which converges to 2 t the Dirac measure δ (y) as t > 0 . x −! These properties hold for quite general manifolds. In our setting, the G{invariance implies moreover that 1 (v) ht(xK; yK) = ht(y− x) is a right convolution kernel; (vi) x ht(x) is a bi{K{invariant function on G, which is thus determined by its restriction7−! to the positive Weyl chamber. An alternative consists in solving the heat equation (1.5) via the Fourier transform (1.3) : @ u(λ, t) = ( λ 2 + % 2) u(λ, t) @t H − j j j j H u(λ, 0) = f(λ) H H ( λ 2+ % 2)t yields u(λ, t) = f(λ) e− j j j j , hence u(x; t) = (f h )(x), where H H ∗ t dλ t( λ 2+ % 2) (1:6) ht(x) = const. c 2 e− j j j j ' λ(x) : (λ) Za j j 2. Explicit expressions The expression (1.6) for the heat kernel, as inverse spherical Fourier transform, is neither explicit nor easy to handle in general. For instance, it is not obvious from (1.6) that ht(x) > 0 . We shall now list some particular cases, where (more or less) explicit and manageable expressions are available. Case 1 : G complex This is the most elementary case, as far as spherical Fourier analysis is concerned, which boils down to Euclidean Fourier analysis on a or on p . The following formula was written down first by Gangolli [Ga] (and must have been known to the Russian School led by Dynkin, Gelfand, Karpeleviˇc, : : : ) : 2 n 2 α,H jHj H 2 % t 4 t ht(e ) = (4πt)− e−| j α Σ+ sinhh α,Hi e− t > 0; H a : 2 h i 8 8 2 Notice that the expression betweenn Qbraces coincideso here both with the Jacobian J(H) 1 of the exponential map exp0 : p X , raised to the power 2 , and with the basic H ! − spherical function '0(e ) .
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