(November 4, 009) Integrals of products of eigenfunctions on SL (C) Paul arrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/. Zonal spherical harmonics on SL (C). 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 SL (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, nonabelian 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 SL (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 994]. [Bernstein-Reznikoff 999] and [Krotz-Stanton 004] 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 SL (C) is less trivial than the Euclidean case, but still transparent. The references indicate sources for spherical functions more generally.. Zonal spherical harmonics on SL (C) We review some standard facts and set up notation. Let = SL (C) and K = SU(). The standard split component is ( ) A + e r/ 0 = {a r = 0 e r/ : r 0} The Cartan decomposition is = KA + K In Cartan coordinates, the Haar measure on is Haar measure = d(ka r k ) = sinhr dk dr dk = sinh r dk dr dk The Laplacian on /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) + cothr f (r) (on functions F(ka r k ) = f(r)) A zonal spherical function on is a smooth K-bi-invariant eigenfunction for, that is, a function F such that F = λ F
Paul arrett: Integrals of products of eigenfunctions on SL (C) (November 4, 009) The standard normalization is that a zonal spherical function takes value at, that is, at r = 0. Thus, put sinh(s )r ϕ s (r) = (zonal spherical function, eigenvalue s(s )) (s ) sinhr On the unitary line s = + it, the spherical function is sin tr +it(r) = t sinhr = it(r) (zonal spherical function, eigenvalue ( 4 + t )) For a left K-invariant function f on /K with sufficient decay, the spherical transform f of f is f(ξ) = f +iξ = f +iξ (with real ξ) Spherical inversion is f = 8 π f( + iξ) +iξ ξ dξ The Plancherel theorem for f, F left-and-right K-invariant functions in L () is f F = 8 π 0 f( + iξ) F( + iξ) ξ dξ. Formula for triple integrals of eigenfunctions The integral of three (or more) zonal spherical functions for SL (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. Since +ia +ib is in L () for a, b R, this product has a reasonable spherical transform ( ) e(c) +ia = +ib +ia +ib +ic = sin ar sin br sin cr 8abc 0 sinh 3 r These integrals are expressible in terms of integrals sinh r dr = sin ar sin br sincr dr 6abc e itr sin ur sinhr The latter is holomorphic in t and u for t, u near the real axis. With t > u, by residues, dr e itr sin ur dr = πi ( ) l e it(πil) sinuπil = π ( ) l e it(πil) ( e iu(πil) e iu(πil)) l= l= ( e π(t+u) ) e π(t u) = π + e π(t+u) + e π(t u) = π e π(t u) e π(t+u) ( + e π(t+u) )( + e π(t u) ) = π sinh πu cosh π (t + u) cosh π (t u)
Thus, = = Paul arrett: Integrals of products of eigenfunctions on SL (C) (November 4, 009) sin ar sinbr sin cr dr = 4 ±,± e i(±a±b)r sin cr π sinhπc ( 8 coshπ(a + b + c) coshπ(a + b c) cosh π(a b + c) coshπ(a b c) coshπ( a + b + c) coshπ( a + b c) + ) coshπ( a b + c) coshπ( a b c) π sinh πc ( 4 coshπ(a + b + c) coshπ(a + b c) ) coshπ(a b + c) coshπ(a b c) Toward putting these fractions over a common denominator, recall Thus, cosh(a B + C) cosh(a B C) cosh(a + B + C) cosh(a + B C) = ( e A B + e C + e A+B + e C) e 4 4( A+B + e C + e C + e A B) sin ar sin br sin cr With the factor of /6abc, dr = π 4 +ia +ib +ic = π 64abc = sinha sinh B sinhπa sinh πb sinh πc coshπ(a + b + c)cosh π(a + b c)coshπ(a b + c)coshπ(a b c) sinh πa sinh πb sinh πc coshπ(a + b + c)coshπ(a + b c)coshπ(a b + c)cosh π(a b c) dr [.0.] 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 /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 [Krötz-Stanton 004], using analytic continuation to a thickening of the space in its complexification. Without loss of generality, a b c 0. [3.] 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 I(a, b, c) = e iax sin bx sin cx upward by an amount π < h < π across the first pole πi in the upper half-plane, producing a main term and an error integral: I(a, b, c) = πi e πa sinh πb sinh πc + e ia(x+ih) sinb(x + ih) sin c(x + ih) sinh(x + ih) 3
From the identities and Paul arrett: Integrals of products of eigenfunctions on SL (C) (November 4, 009) sin(x + ih) = sin x cosih + sin ih cosx = sin x cosh( h) i sinh( x) cosx sinh(x + ih) = coshih + sinhih coshx = cosh + i sinx coshx for a 0 the error integral is estimated by e ia(x+ih) sin b(x + ih) sin c(x + ih) sinh(x + ih) h e h (a b c ) (with a 0) The same idea applies to e iax, 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, sin ax sin bx sin cx = π e πa sinh πb sinh πc + O ε (e (4π ε)(a b c ) ) (for a 0) 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, +ia +ib +ic = πe πa sinh πb sinh πc 6abc + O ε ( e (4π ε)(a b c ) ) (for a > b + c ) 6abc [3.] 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 sinax sin bx = cos(a + b)x cos(a b) to rewrite sin ax sin bx sin cx = Since a + b > c 0, shifting contours as above, for all ε > 0, ( cos(a + b)x cos(a b)) sincx cos(a + b)x sin cx = e π(a+b) sinh c + O ε (e (4π ε)(a+b c) But this will prove to be smaller than the eventual main term, and the error term absorbed entirely. Indeed, the assumption a < b + c gives a b < c, and the above argument, with c and a b now in the former roles of a + b and c, gives cos(a b)x sincx = e πc coshπ(a b) + O ε (e (4π ε)(c a b ) (for c > a b ) Thus, sin ax sin bx sin cx = e πc coshπ(a b) + O ε (e (4π ε)(c (a b)) + e π(a+b) sinh c + O ε (e (4π ε)(a+b c) Since a b c 0, always a + b c c (a b). Thus, one of the error terms is absorbed by the other, and sin ax sin bx sin cx = e πc cosh π(a b) + e π(a+b) sinh c + O ε (e (4π ε)(c (a b)) 4
Paul arrett: Integrals of products of eigenfunctions on SL (C) (November 4, 009) The error term absorbs the term e π(a+b) sinh c when the obvious comparison of exponents holds: a + b c (c (a b)) which is a 3 b + c Indeed, even where a < b + c, possibly a 3b + c. In fact, the remaining case a < b + c (negating a b + c) a < 3 b + c (negating a 3 b + c) a b c 0 is atypical, in the sense that dehomogenizing the inequalities by dividing through by c yields a bounded region. The other two regions are unbounded even after dehomogenizing. [3.3] Summary Simply because ϕ +ib +ic is smooth and has L derivatives of all orders, the triple integral of +ia +ib +ic is rapidly decreasing in a for fixed values of b, c. But this a weaker assertion than what is true, since this coefficient is exponentially decreasing. In fact, It has an asymptotic expansion with a strictly smaller exponential error term: for every ε > 0, +ia +ib +ic = πe πa sinh πb sinh πc 6abc + O ε ( e (4π ε)(a b c ) ) (for a b + c ) 6abc On the other hand, scenarios in which the two largest parameters increase at about the same rate produce exponential decay in the smallest parameter. In particular, when the smallest parameter is bounded and the difference between the larger two parameters is bounded, there is definitely not decay. 4. Asymptotics of integrals of n-fold products An n-fold integral of zonal spherical functions.. = +ic. +icn n+ c...c n sin c x...sinc n x sinh n x is the c th spectral decomposition coefficient of the product of n zonal spherical functions. We are first interested in its asymptotics as a function of c, with c,..., c n fixed. Other parameter configurations are subtler. [4.] Main case: c > c +... + c n Without loss of generality, take c c... c 0 Moving the contour upward by an amount π < h < π, e icx sin c x...sinc n x sinh n x = πires x=πi e icx sinc x...sin c n x sinh n x + 5 e ic(x+ih) sinc (x + ih)...sin c n (x + ih) sinh n (x + ih)
Paul arrett: Integrals of products of eigenfunctions on SL (C) (November 4, 009) πires x=πi e icx sin c x...sin c n x sinh n x + O ε (c n 3 e (4π ε)(c c... cn ) ) where h = 4π ε. The indicated residue is a finite sum of exponentials of the form e π(c±c±c3±...±cn) with polynomial coefficients of degree at most n 3. Thus, the indicated error is much smaller than the main terms, and a precise asymptotic follows. In particular, with c,...,c n fixed,... +ic c n 4 e πc (with c +icn,..., c n fixed) [4.] An opposite case: two largest parameters close together Continue to take c... c n 0, without loss of generality. Now assume c c is small. In particular, suppose that c 3 c 4 +... + c n + c c This requires something of the parameters c 4, c 5,... when n 4. Rewrite the n-fold integral as a difference of two similar integrals using sinc x sinc x = cos(c + c )x cos(c c )x Since certainly c +c c 3 +...+c n, the contour-shifting argument shows that the integral with cos(c +c )x is dominated by (c + c ) n 3 e (4π ε)(c+c c3... cn ) This error will be absorbed easily by the error term in estimating the integral with cos(c c )x. For the latter, use the fact that c 3 > c 4 +... + c n + c c without concern for the size of c c relative to c 3,..., c n. The contour-shifting argument proves that the corresponding integral is a finite sum of exponentials of the form e π(c3± c4 ±...± cn ± c c ) with polynomial coefficients of degree at most n 3, with an error on the order of c 3 n 3 e (4π ε)(c3 c4... cn c c ) Thus, the first error term is absorbed by this error term. Then it is clear that the term (up to a non-zero constant) (c 3 c 4... c n c c ) n 3 e π(c3 c4... cn c c ) is largest. Thus, for c c small in this sense, under various mild hypotheses the n-fold integral is bounded away from 0, or even asymptotic to a non-zero constant. [4..] Remark: An explicit elementary expression for the n-fold integral of zonal spherical harmonics can be obtained by the same devices as for the three-fold integral, but such an expressions seems sub-optimal for understanding the asymptotics of the decay. [Berezin 956a] F.A. Berezin, Laplace operators on semisimple Lie groups, Dokl. Akad. Nauk SSSR 07 (956), 9-. 6
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