Асимптотика преобразования Радона на гиперболических пространствах

Аннотация

Пусть G/H - гиперболическое пространство над R ; C или H ; пусть K - максимальная компактная подгруппа группы G. Пусть D обозначает некоторый явно выписываемый дифференциальный оператор - такой, что некаспидальные дискретные серии принадлежат ядру оператора D . Мы доказываем, что для всякой функции f из пространства L 2 -Шварца на G/H преобразование Абеля A(Df) функции Df есть функция Шварца. Это - расширение результата, установленного в [2] для -финитных и K∩H -инвариантных функций.

Полный текст

§ 1. Introduction The Radon transform R on the hyperbolic spaces G=H; Rf = Z N f( nH) dn; where N G is a certain unipotent subgroup, and the associated Abel transform A; were introduced and studied in [1] and [2]. Generalizing Harish-Chandra’s notion of cusp forms for real semisimple Lie groups, a discrete series is said to be cuspidal if it is annihilated by the Radon transform. In contrast with the Lie group case, however, non-cuspidal discrete series exist. For the projective hyperbolic spaces, these are precisely the spherical discrete series, but for some real non-projective hyperbolic spaces, there also exist non-spherical non-cuspidal discrete series. Let C2(G=H) denote the space of L2 -Schwartz functions on G=H: Except for some boundary cases, A maps C2(G=H) into Schwartz functions in the absence of non-cuspidal discrete series. On the other hand, Af can be explicitly calculated for functions f belonging to the non-cuspidal discrete series. To complete the picture, we prove below that A essentially maps the orthocomplement in C2(G=H) of the non-cuspidal discrete series into Schwartz functions. To be more precise, let = + 2q ; where denotes the Laplace-Beltrami operator on G=H; and consider the G-invariant differential operator D = ( 21 ) : : : ( 2r ); where 1; : : : ; r are the parameters of the non-cuspidal discrete series. Then A(Df) is a Schwartz function. This extends our previous result, [2, Theorem 6.1], valid only for the dense G-invariant subspace of C2(G=H) generated by the K -irreducible (K\\H) -invariant functions, to all Schwartz functions. In [2] we also considered the exceptional case corresponding to the Cayley numbers O: We expect our new result to hold for this case as well, but we have not been through the rather cumbersome details. The second author wants to thank Professor Vladimir Molchanov for the invitation to visit Tambov University, where our results were first reported, in October 2012. We would also like to thank Henrik Schlichtkrull and Job Kuit for helpful discussions and comments. ASYMPTOTICS FOR THE RADON TRANSFORM ON HYPERBOLIC SPACES 243 § 2. The Radon transform In this section, we define the Radon transform and the Abel transform for the projective hyperbolic spaces over the classical fields F = R;C and H: We have tried to keep the presentation and notation to a minimum, see [1] and [2] for further details (including results and proofs). Let x 7! x be the standard (anti-) involution of F: Let p > 0; q > 1 be two integers, and consider the Hermitian form [ ; ] on Fp+q+2 given by [x; y] = x1y1 + : : : + xp+1yp+1 xp+2yp+2 : : : xp+1+q+1yp+1+q+1 ; where x; y 2 Fp+q+2: Let G = U(p+1; q+1; F) denote the group of (p+q+2)-(p+q+2) matrices over F preserving [ ; ]: Thus U(p+1; q+1;R) = O(p+1; q+1); U(p+1; q+1;C) = U(p + 1; q + 1) and U(p + 1; q + 1;H) = Sp (p + 1; q + 1) in standard notation. Put U(p ; F) = U(p; 0; F); and let K = U(p + 1; F) - U(q + 1; F) be the maximal compact subgroup of G fixed by the Cartan involution on G: Let x0 = (0; : : : ; 0; 1)T ; where superscript T indicates transpose. Let H be the subgroup U(p+1; q; F)-U(1; F) of G stabilizing the line F x0 in Fp+q+2: The reductive symmetric space G=H can be identified with the projective hyperbolic space X = X(p + 1; q + 1; F); X = fz 2 Fp+q+2 : [z; z] = 1g= ; where is the equivalence relation z zu; u 2 F : Let Xt; for t 2 R; denote the following element in the Lie algebra g of G: Xt = 0 BBBBB@ 0 0 : : : 0 1 0 0 : : : 0 0 : : : : : : : : : : : : : : : 0 0 : : : 0 0 1 0 : : : 0 0 1 CCCCCA (a matrix of order p + q + 2 ). Let aq denote the Abelian subalgebra given by Xt; t 2 R; let at = exp(Xt) denote the exponential of Xt; and also define Aq = exp(aq): Let (considered as row vectors) u = (u1; : : : ; up) 2 Fp and v = (vq; : : : ; v1) 2 Fq; and let w 2 Im F (i. e., w = 0 for F = R). Define Nu;v;w 2 g as the matrix given by Nu;v;w = 0 BB@ w u v w uT 0 0 uT vT 0 0 vT w u v w 1 CCA : 244 N. B. Andersen, M. Flensted-Jensen Then exp(Nu;v;w) = I + Nu;v;w + 1 2 N2 u;v;w ; and a small calculation yields that at exp(Nu;v;w) x0 = = sinh t + 1 2 et (juj2 jvj2) + et w; u; v; cosh t + 1 2 et (juj2 jvj2) + et w T ; (1) for any t 2 R: Define the nilpotent subalgebra n as follows, for p > q; n = fNu;v;w : u = ( vr; u0); v 2 Fq; u0 2 Fp qg; (2) and, for p < q; n = fNu;v;w : v = ( ur; v0); u 2 Fp; v0 2 Fq pg; (3) where ur; vr means that the order of the indices is reversed. By abuse of notation, we leave out the superscript r in what follows. We finally also define the following -factors. Let d= dimR F; and let q =(1=2)(dp + dq + 2(d 1)) 2 R; 1 = (1=2)(jdp dqj + 2(d 1)) 2 R: Let N = exp(n ) denote the nilpotent subgroup generated by n : For functions f on G=H; we define, assuming convergence, Rf(g) = Z N f(gn H) dn (g 2 G): (4) Let f 2 C2(G=H); the space of L2 -Schwartz functions on G=H: From [1] and [2], we know that the Radon transform Rf is a smooth function. Also, the integral defining R converges uniformly on compact sets, and R is G- and g -equivariant. We define the associated Abel transform A by Af(a) = a 1Rf(a); for a 2 Aq: We are mainly interested in the values of Rf and Af on the elements as; and thus define Rf(s) = Rf(as); and, similarly, Af(s) = Af(as); for s 2 R: Let denote the Laplace- Beltrami operator on G=H: Then, for f 2 C2(G=H); A( f) = d2 ds2 2q Af (s 2 R): (5) Finally, for R > 0; let C1 R (G=H) denote the subspace of smooth functions on G=H with support in the (K -invariant) ‘ball’ fkas x0 j jsj 6 Rg of radius R: Similarly, let C1 R (R) denote the subspace of smooth functions on R with support in [ R;R]; and let S(R) denote the Schwartz space on R: ASYMPTOTICS FOR THE RADON TRANSFORM ON HYPERBOLIC SPACES 245 § 3. The discrete series and the Abel transform Let q > 1; or d > 1: The discrete series for the projective hyperbolic spaces can then be parametrized as fT j = 1 2 (dq dp) 1 + > 0; 2 2Zg; see [1] and [2]. The spherical discrete series are given by the parameters for which 6 0; including the ’exceptional’ discrete series corresponding to > 0 for which < 0: For q = d = 1; the discrete series is parameterized by 2 Rnf0g such that j j+ q 2 2Z; and there are no spherical discrete series. The parameters are, via the formula f = ( 2 2q )f; related to the eigenvalues of acting on functions f in the corresponding representation space in L2(G=H): Let D be the G-invariant differential operator ( 21 ) : : : ( 2r ); where 1; : : : ; r are the parameters of the non-cuspidal discrete series, and = + 2q : We have a complete classification of the cuspidal and non-cuspidal discrete series for the projective hyperbolic spaces, also including information about the asymptotics of the Radon and Abel transforms: Theorem 1. Let G=H be a projective hyperbolic space over R; C; H; with p > 0; q > 1: (i) If d(q p) 6 2; then all discrete series are cuspidal. (ii) If d(q p) > 2; then non-cuspidal discrete series exists, given by the parameters > 0 with 0: More precisely, if 0 6= f 2 C2(G=H) belongs to T ; then Af(s) = Ce s; with C 6= 0: (iii) T is non-cuspidal if and only if T is spherical. (iv) If p > q; and f 2 C1 R (G=H); for R > 0; then Af 2 C1 R (R): (v) If d(q p) 6 1; and f 2 C2(G=H); then Af 2 S(R): (vi) Assume d(q p) > 1: Then A(Df) 2 S(R); for f 2 C2(G=H): The above theorem is almost identical to [2, Theorem 6.1], except for item (vi), which was only proved for functions in the (dense) G-invariant subspace V of C2(G=H) generated by the K -irreducible (K \\H) -invariant functions. Additionally, [2, Theorem 6.1] furthermore included the exceptional case corresponding to the Cayley numbers O: Theorem 1 (including the reformulation of (vi)) also holds for the real non-projective spaces SO (p + 1; q + 1)e=SO (p + 1; q)e; except for item (iii), due to the existence of noncuspidal non-spherical discrete series corresponding to negative and odd values of in the exceptional series, see [1, Section 5]. 246 N. B. Andersen, M. Flensted-Jensen The conditions in (vi) essentially state that Af is a Schwartz function if f is perpendicular to all non-cuspidal discrete series. The factor ; however, seems to be necessary (except in the real case with q p odd), even for the case d(q p) = 2; where there are no non-cuspidal discrete series. In the next section, we prove Theorem 1(vi). § 4. Proof of Theorem 1(vi) First we note, following [2, Section 10], that the Schwartz decay conditions are satisfied near 1 for A(f); and thus also for A(Df): This leaves us to study the Abel transform near +1: Let f 2 C2(G=H); and write f[x] = f(gH); where x = g x0: From (1) and (3), we get Rf(s) = Z N f(asn H)dn = Z Rdq dp-Rdp-Rd 1 f - (sinh s 1=2esjv0j2 + esw; u; u; v0; cosh s 1=2esjv0j2 + esw) dv0 du dw: Let v0 = jv0jv; v = sinh s + 1=2esjv0j2; such that jv0j2 = 1 + 2e sv e 2s; and w = esw: Then, Rf(s) = e ds Z1 sinh s dw Z M f - (w v; u; u; (1 + 2e sv e 2s)1=2v; e s v + w) - - (1 + 2e sv e 2s)(dq dp)=2 1 dv dv du ; where M = Sdq dp 1 - Rdp - Rd 1 and Sr is the unit sphere in Rr: We will use the identification of X = X(p + 1; q + 1; F) with X = fz 2 Fp+q+2 : [z; z] < 0g= ; and identify a function f on X with a homogeneous function of z of degree zero on fz 2 Fp+q+2 : [z; z] < 0g: We now identify Fp+q+2 with Rd(p+q+2) such that the coordinates satisfy Re zj = xdj ; for j = 1; : : : ; p + q + 2: Consider the real hyperbolic space eX = fz 2 Fp+q+2 : [z; z] = 1g: The group eG = O(d(p + 1); d(q + 1)) acts transitively on e X: Let ~K denote the standard maximal compact subgroup O(d(p + 1)) - O(d(q + 1)) of e G: Let U(ek ); respectively U(k); denote the universal enveloping algebra of the Lie algebra ek of eK ; respectively of the Lie algebra k of K: ASYMPTOTICS FOR THE RADON TRANSFORM ON HYPERBOLIC SPACES 247 Lemma 1. Let U 2 U(~k); then U maps C2(G=H) into itself. Proof. The lemma is obvious for d = 1: So assume d > 1: We note that any element x 2 eX can be written as x = ka x0; where k 2 K; and a = as; s 0: Let eH = O(d(p + 1); d(q + 1) 1); and let ~m denote the commutator of Aq in the Lie algebra of eK \\ eH : Then ~k = k + ~m: Let Uk = Ad (k)U; for k 2 K; then Uf = (Ad (k 1)Uk)f: By the Campbell-Baker- Hausdorff formula, there exists an element U0 k 2 U(k); such that Uk = U0 k modulo the left ideal generated by ~m: This implies that Uf[ka x0] = (Ad (k 1)U0 k )f[ka x0]: The map k 7! Ad (k 1)U0 k is continuous into a finite dimensional subspace of U(k); and we can write Uf[ka x0] = (Ad (k 1)U0 k ) f[ka x0] = iui(k)Uif[ka x0]; for a finite set of elements Ui 2 U(k) and continuous coefficients ui(k): It follows that Uf is in C2(G=H): Define for t = (t1; t2; t3) 2 R3; the auxiliary function Gf (t1; t2; t3) = Z M f [(w + t1; u; u; t2v; t3 + w)] dv du dw; and, with the identification z = e s; define the function F(z) = edsRf(s): Then, since sinh s = (z z 1)=2; we get F(z) = Z1 (z z 1)=2 Gf v; (1 + 2zv z2)1=2; z v (1 + 2zv z2)(dq dp)=2 1 dv: (6) Lemma 2. The function Gf is homogeneous of degree dp + d 1 on the cone t21 t22 t23 < 0; it is even in t2; and satisfies Gf ( t1; t2; t3) = Gf (t1; t2; t3): Let X be the differential operator on R3 given by t3@=@t2 t2@=@t3: For all f 2 C2(G=H); and all k;N 2 N; there exists a constant C; such that jXkGf (t)j C(t22 + t23 ) d(q p)=4(1 + log(t22 + t23 )) N; on the hyperboloid t21 t22 t23= 1: Proof. The first statement follows from the homogeneity of f and the definition of Gf : As before we identify Fp+q+2 with Rd(p+q+2): For i = d(1+2p)+1; : : : ; d(1+p+q); we define the differential operator Dif[x] = xd(p+q+2) @ @xi f[x] xi @ @xd(p+q+2) f[x]: This operator is defined by the left action of an element Ti in O(d(q + 1)) (with value 1 in the last entry of the i’th row, value 1 in the last entry of the i’th column, and 0 otherwise), and Lemma 1 thus gives that Di maps C2(G=H) into itself. 248 N. B. Andersen, M. Flensted-Jensen Let now v = (vd(1+2p)+1; : : : ; vd(1+p+q)) 2 Sd(q p) 1: The operator Yv = 1X+p+q i=2+2p viDi; also maps C2(G=H) into itself, and Yvf[x] 6 d(q p) max i ( Dif[x] ): Applying the operator X to the integrand in the definition of Gf ; we get Xf[t1; u; u; t2v; t3] = t3 X @ @xi f[:]vi t2 @ @xd(p+q+2) f[:] = t3 X @ @xi f[:]vi t2 X v2 i @ @xd(p+q+2) f[:] = Yvf[t1; u; u; t2v; t3] the summations are taken over i = d(1 + 2p) + 1; : : : ; d(1 + p + q): The inequality for Xkf follows from repeated use of this formula and from the asymptotic estimates of functions in C2(G=H): In particular, it follows that the function v 7! XkGf ( v; 1; v) has the same parity as k: Lemma 3. Let k0 be the largest integer such that k0 < (dq dp)=2; and let = (dq dp)=2 k0: Define t = t(z; v) = ( v; (1 + 2zv z2)1=2; z v): Then (i) For k 6 k0; the function v 7! @k @zk Gf (t(z; v)) (1 + 2zv z2) (dq dp)=2 1 - is uniformly integrable over R for z < 1: (ii) For k 6 k0 odd, this function is an odd function of v for z = 0: Proof. Notice that t21 t22 t23 = 1 and t22 +t23 = 1+v2; for t = t(z; v); and that the integral (6) is uniformly convergent for 0 6 z 6 k < 1: The same holds with Gf replaced by XkGf : Repeated use of the formula @ @z Gf (t(z; v))(1 + 2zv z2) = XGf (t(z; v))(1 + 2zv z2) 1=2 + 2 Gf (t(z; v))(1 + 2zv z2) 1(z v) yields (i), and together with the parity properties of XkGf also gives (ii). ASYMPTOTICS FOR THE RADON TRANSFORM ON HYPERBOLIC SPACES 249 We notice that = 1 if d(q p) is even, and = 1=2 if d(q p) is odd, i. e., if d = 1 and q p is odd. For k < k0; the derivatives @k=@zk of Gf (t(z; v))(1 + 2zv z2)(dq dp)=2 1 are zero at v = sinh s = (z z 1)=2; whence the integrand is at least k0 times differentiable near z = 0; and we can compute the derivatives dk=dzkF(z) by differentiating under the integral sign in (6). If k0 > 0; we can use Taylors formula to express F(z) as a polynomial of degree k0 1; plus a remainder term involving dk0=dzk0F( ); for some 0 < (z) < z; F(z) = c0 + c1z + c2z2 + ::: + ck0 1zk0 1 + Rk0( )zk0 ; where 0 < < z; and cj = 1 j! Z 1 1 dj dzj z=0 (Gf (t(z; v))(1 + 2zv z2)(dq dp)=2 1) dv; for j 2 f0; : : : ; k0 1g: The remainder term is given by: Rk0( ) = 1 k0! Z 1 ( 1)=2 dk0 dzk0 z= (Gf (t(z; v))(1 + 2zv z2)(dq dp)=2 1) dv: Consider Af(s) = e 1sRf(s) = z ( 1 d)F(z); which is equal to c0z ( 1 d) + c1z ( 1 d 1) + c2z ( 1 d 2) + ::: + ck0 1z + z( +1)Rk0( ): Here we have used that 1 d = d(q p)=2 1: For j even, the exponents d(q p)=2 1 j; for j 2 f0; : : : ; k0 1g; correspond to the parameters 1; : : : ; r for the non-cuspidal discrete series, and cj = 0 for j odd, since the integrand is an odd function. For the real non-projective hyperbolic spaces the condition concerning the parity j does not hold, but in that case all the exponents d(q p)=2 1 j; for j 2 f0; : : : ; k0 1g; correspond to parameters 1; : : : ; r for the non-cuspidal discrete series, see [1, Section 3]. From the definition of the differential operator D and (5), we see that A(Df) at most has a contribution from the remainder term, and further that A(Df) does not have a constant term at 1; due to the term d2=ds2: If = 1=2; the remainder term e 1=2sRk0( (s)) is clearly rapidly decreasing, and we are thus left to consider the case = 1; in which case k0 = d(q p)=2 1: Consider the constant term CRk0 = lims!1 Rk0(e s); which could be non-zero. We want to show that Rk0( ) CRk0 is rapidly decreasing at +1; where = (s); with 0 < < e s: We also include the case k0 = 0; where we put = e s: Define H(z; v) = dk0 dzk0 (Gf (t(z; v))(1 + 2zv z2)k0): 250 N. B. Andersen, M. Flensted-Jensen Then, for < z < 1; Rk0( ) CRk0 = Z1 ( 1)=2 (H( ; v) H(0; v)) dv + ( Z 1)=2 1 H(0; v) dv = I1( ) + I2( ): For I1( ); there exists 1 = 1( ; v) < ; such that H( ; v) H(0; v) = d dz z= 1 H(z; v); and we get: I1( ) < z Z 1 1 d dz z= 1 H(z; v) dv: By Lemma 3, the integrand is uniformly integrable for z < 1; and we conclude that I1( ) is bounded by Ce s: For s large, the function H(0; v) is for every N 2 N bounded by jH(0; v)j C(1 + v2) d(q p)=4jvjk0 log(1 + v2) N; for some positive constant C: Using this, we find that I2(z) < C Z 1 sinh s v 1(log(v)) N dv = C(N 1) 1(log(sinh s)) N+1 6 Cs N+1: It follows that Rk0( ) CRk0 is rapidly decreasing at +1; whence A(Df) is rapidly decreasing at +1; which finishes the proof of Theorem 1.