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Integrals of Products of Eigenfunctions on SL 2(C) 1. Zonal Spherical

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Integrals of Products of Eigenfunctions on SL 2(C) 1. Zonal Spherical (November 24, 2009) Integrals of products of eigenfunctions on SL2(C) Paul Garrett [email protected] http://www.math.umn.edu/˜garrett/ 1. Zonal spherical harmonics on SL2(C) 2. Formula for triple integrals of eigenfunctions 3. Asymptotics of triple integrals 4. Asymptotics of integrals of n-fold products We determine precise asymptotics in spectral parameters for integrals of n-fold products of zonal spherical harmonics on SL2(C). In a variety of situations, integrals of products of eigenfunctions have faster decay than smoothness entails. This phenomenon does not appear for abelian or compact groups, since irreducibles are finite-dimensional, so the decomposition of a tensor product of irreducibles is finite. In contrast, for non-compact, non- abelian groups irreducibles are typically infinite-dimensional, and the decomposition of a tensor product of irreducibles typically contains infinitely-many irreducibles, and the issue is non-trivial. In a few situations integrals of products of eigenfunctions are elementary, and these deserve attention. Zonal spherical functions for complex reductive groups are elementary, and the rank-one case SL2(C) is the simplest. The automorphic version of asymptotics for integrals of products of eigenfunctions is more delicate, and more complicated, as illustrated by [Sarnak 1994]. [Bernstein-Reznikoff 1999] and [Krotz-Stanton 2004] show that the critical mechanism for exponential decay is extendability of matrix coefficient functions to complexifications of Lie groups or symmetric spaces. This phenomenon is well-known for Fourier transforms on Euclidean spaces. The example of SL2(C) is less trivial than the Euclidean case, but still transparent. The references indicate sources for spherical functions more generally. 1. Zonal spherical harmonics on SL2(C) We review some standard facts and set up notation. Let G = SL2(C) and K = SU(2). The standard split component is er/2 0 A+ = {a = : r ≥ 0} r 0 e−r/2 The Cartan decomposition is G = KA+K In Cartan coordinates, the Haar measure on G is ′ 2 ′ 2 ′ Haar measure = d(kark ) = | sinh r| dk dr dk = sinh r dk dr dk The Laplacian ∆ on G/K is the restriction of the Casimir operator Ω, and on left-and-right K-invariant functions is a constant multiple of ′′ ′ ′ f −→ f (r)+2coth r · f (r) (on functions F (kark )= f(r)) A zonal spherical function on G is a smooth K-bi-invariant eigenfunction for ∆, that is, a function F such that ∆F = λ · F 1 Paul Garrett: Integrals of products of eigenfunctions on SL2(C) (November 24, 2009) The standard normalization is that a zonal spherical function takes value 1 at 1 ∈ G, that is, at r = 0. Thus, put sinh(2s − 1)r ϕ (r) = (zonal spherical function, eigenvalue s(s − 1)) s (2s − 1) sinh r 1 On the unitary line s = 2 + it, the spherical function is sin2tr 1 2 ϕ 1 (r) = = ϕ 1 (r) (zonal spherical function, eigenvalue −( + t )) 2 +it 2t sinh r 2 −it 4 For a left K-invariant function f on G/K with sufficient decay, the spherical transform f of f is e f(ξ) = f · ϕ 1 = f · ϕ 1 (with real ξ) 2 +iξ 2 +iξ ZG ZG e Spherical inversion is ∞ 8 1 2 f = f( + iξ) · ϕ 1 · ξ dξ 2 2 +iξ π Z−∞ e The Plancherel theorem for f, F left-and-right K-invariant functions in L2(G) is ∞ 8 1 1 −2 f · F = f( 2 + iξ) · F ( 2 + iξ) · ξ dξ ZG π Z0 e e 2. Formula for triple integrals of eigenfunctions The integral of three (or more) zonal spherical functions for SL2(C) can be expressed in elementary terms. This section carries out the computation for three. In fact, the expressions produced are not as useful as one might have anticipated, because extraction of useful asymptotics is not trivial. Direct determination of asymptotics for arbitrary products is done subsequently. 1 1 2 R Since ϕ 2 +ia ϕ 2 +ib is in L (G) for a,b ∈ , this product has a reasonable spherical transform e ϕ 1 ϕ 1 (c) = ϕ 1 ϕ 1 ϕ 1 2 +ia 2 +ib Z 2 +ia 2 +ib 2 +ic G ∞ ∞ 1 sin2ar sin2br sin2cr 2 1 sin2ar sin2br sin2cr = 3 sinh r dr = dr 8abc Z0 sinh r 16abc Z−∞ sinh r These integrals are expressible in terms of integrals ∞ eitr sin ur dr Z−∞ sinh r The latter is holomorphic in t and u for t,u near the real axis. With t> |u|, by residues, ∞ eitr sin ur ∞ ∞ dr = 2πi · (−1)ℓ eit(πiℓ) sin uπiℓ = π (−1)ℓ eit(πiℓ) eiu(πiℓ) − e−iu(πiℓ) Z sinh r −∞ Xℓ=1 Xℓ=1 −e−π(t+u) −e−π(t−u) e−π(t−u) − e−π(t+u) π sinh πu = π −π(t+u) − −π(t−u) = π −π(t+u) −π(t−u) = π π 1+ e 1+ e (1 + e )(1+ e ) 2 cosh 2 (t + u) cosh 2 (t − u) 2 Paul Garrett: Integrals of products of eigenfunctions on SL2(C) (November 24, 2009) Thus, ∞ sin2ar sin2br sin2cr 1 ∞ ei(±2a±2b)r sin2cr dr = − dr Z sinh r 4 Z sinh r −∞ ±X,± −∞ −π sinh 2πc 1 1 = − 8 cosh π(a + b + c) cosh π(a + b − c) cosh π(a − b + c) cosh π(a − b − c) 1 1 − + cosh π(−a + b + c) cosh π(−a + b − c) cosh π(−a − b + c) cosh π(−a − b − c) −π sinh 2πc 1 1 = − 4 cosh π(a + b + c) cosh π(a + b − c) cosh π(a − b + c) cosh π(a − b − c) Toward putting these fractions over a common denominator, recall cosh(A − B + C) cosh(A − B − C) − cosh(A + B + C) cosh(A + B − C) 1 1 = e2A−2B + e2C + e−2A+2B + e−2C − e2A+2B + e2C + e−2C + e−2A−2B 4 4 = − sinh 2A · sinh 2B Thus, ∞ sin2ar sin2br sin2cr π sinh 2πa sinh 2πb sinh 2πc dr = · Z−∞ sinh r 4 cosh π(a + b + c)cosh π(a + b − c)cosh π(a − b + c)cosh π(a − b − c) With the factor of 1/16abc, π sinh 2πa sinh 2πb sinh 2πc ϕ 1 ϕ 1 ϕ 1 = · 2 +ia 2 +ib 2 +ic ZG 64abc cosh π(a + b + c)cosh π(a + b − c)cosh π(a − b + c)cosh π(a − b − c) [2.0.1] Remark: It is clear that this formula yields asymptotics as one or more of the parameters a,b,c becomes large. For b,c fixed and a → ∞, exponential decay is clear. For c fixed while a = b → ∞, the triple integral only decays like 1/ab. For a = b = c → ∞, again there is no exponential decay. 3. Asymptotics for triple integrals The integral of three zonal spherical functions has different behavior in different parameter regimes, readily seen from the explicit formula above. More economical methods scale better and give cleaner results, as follows. These can be viewed as elementary analogues of [Kr¨otz-Stanton 2004], using analytic continuation to a thickening of the space in its complexification. Without loss of generality, a ≥ b ≥ c ≥ 0. [3.1] The main case: asymptotics for a> |b| + |c| First treat the case that a goes to ∞ faster than b,c. This scenario occurs in the spectral decomposition of a product of two eigenfunctions. Move the contour of integration in ∞ e2iax sin2bx sin2cx I(a,b,c) = dx Z−∞ sinh x upward by an amount π<h< 2π across the first pole πi in the upper half-plane, producing a main term and an error integral: ∞ e2ia(x+ih) sin2b(x + ih) sin2c(x + ih) I(a,b,c)= 2πie−2πa sinh 2πb sinh 2πc + dx Z−∞ sinh(x + ih) 3 Paul Garrett: Integrals of products of eigenfunctions on SL2(C) (November 24, 2009) From the identities sin(x + ih) = sin x · cos ih + sin ih · cos x = sin x · cosh(−h) − i sinh(−x) · cos x and sinh(x + ih) = sinh x · cosh ih + sinh ih · cosh x = sinh x · cos h + i sin x · cosh x for a ≥ 0 the error integral is estimated by ∞ 2ia(x+ih) e sin2b(x + ih) sin2c(x + ih) −2h·(a−|b|−|c|) dx ≪h e (with a ≥ 0) Z−∞ sinh(x + ih) The same idea applies to e−2iax, moving the contour down rather than up, producing another copy of the main term and the same size error term. Thus, for every ε> 0, ∞ sin2ax sin2bx sin2cx −2πa −(4π−ε)(a−|b|−|c|) dx = πe sinh 2πb sinh 2πc + Oε(e ) (for a ≥ 0) Z−∞ sinh x In the regime a > |b| + |c| the error term is smaller than the main term, giving an asymptotic with error term: for every ε> 0, πe−2πa sinh 2πb sinh 2πc e−(4π−ε)(a−|b|−|c|) ϕ 1 ϕ 1 ϕ 1 = + O ( ) (for a> |b| + |c|) 2 +ia 2 +ib 2 +ic ε ZG 16abc 16abc [3.2] Secondary case: asymptotics for 0 ≤ a< |b| + |c| and a ≥ b ≥ c Now consider the secondary case that a does not go to infinity faster than the others, but, rather, a<b + c. Use the identity 1 1 sin2ax · sin2bx = 2 cos2(a + b)x − 2 cos2(a − b) to rewrite ∞ sin2ax sin2bx sin2cx ∞ 1 cos2(a + b)x − 1 cos2(a − b) sin2cx dx = 2 2 dx Z−∞ sinh x Z−∞ sinh x Since a + b>c ≥ 0, shifting contours as above, for all ε> 0, ∞ 1 2 cos2(a + b)x sin2cx 1 −2π(a+b) −(4π−ε)(a+b−c) dx = 2 e sinh 2c + Oε(e Z−∞ sinh x But this will prove to be smaller than the eventual main term, and the error term absorbed entirely.
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