Asymptotics for the Radon transform on hyperbolic spaces Текст научной статьи по специальности «Математика»

ISSN 2686-9667. Вестник российских университетов. Математика

Том 24, № 127 2019

© Andersen N.B., Flensted-Jensen M., 2019 DOI 10.20310/2686-9667-2019-24-127-241-251 УДК 517.986.66

Asymptotics for the Radon transform on hyperbolic spaces

Nils Byrial ANDERSEN1, Mogens FLENSTED-JENSEN2

1 Aarhus University 1 Nordre Ringgade, Arhus C DK-8000, Denmark e-mail: [email protected] 2 University of Copenhagen 10 N0rregade, Copenhagen K DK-1017, Denmark e-mail: [email protected]

Асимптотика преобразования Радона на гиперболических пространствах

Нильс Бириал АНДЕРСЕН1, Могенс ФЛЕНСТЕД-ЙЕНСЕН2

1 Орхусский университет DK-8000, Дания, Орхус C, Северная кольцевая улица, 1 e-mail: [email protected] 2 Копенгагенский университет DK-1017, Дания, Копенгаген K, Северная улица, 10 e-mail: [email protected]

Abstract. Let G/H be a hyperbolic space over R, C or H, and let K be a maximal compact subgroup of G. Let D denote a certain explicit invariant differential operator, such that the non-cuspidal discrete series belong to the kernel of D. For any L2 -Schwartz function f on G/H, we prove that the Abel transform A(Df) of Df is a Schwartz function. This is an extension of a result established in [2] for K -finite and KП H -invariant functions.

Keywords: hyperbolic spaces; Radon transform; cuspidal discrete series; Abel transform For citation: Andersen N.B., Flensted-Jensen M. Asymptotics for the Radon transform on hyperbolic spaces. Vestnik rossiyskikh universitetov. Matematika - Russian Universities Reports. Mathematics, 2019, vol. 24, no. 127, pp. 241-251. DOI 10.20310/2686-9667-201924-127-241-251.

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 L2 -Шварца на G/H преобразование Абеля A(Df) функции Df есть функция Шварца. Это — расширение результата, установленного в [2] для K -финитных и K П H -инвариантных функций.

Ключевые слова: гиперболические пространства; преобразование Радона; каспидальные дискретные серии; преобразование Абеля

Для цитирования: Андерсен Н.Б., Фленстед-Йенсен М. Асимптотика преобразования Радона на гиперболических пространствах // Вестник российских университетов. Математика. 2019. Т. 24. № 127. С. 241-251. DOI 10.20310/2686-96672019-24-127-241-251. (In Engl., Abstr. in Russian)

§ 1. Introduction

The Radon transform R on the hyperbolic spaces G/H,

Rf = i f (• nH) dn,

J N *

where N * C 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 Ap = A + p2, where A denotes the Laplace-Beltrami operator on G/H, and consider the G -invariant differential operator D = AP(AP — A2)... (Ap — \2), where Ai,...,Ar 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 fi 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.

§ 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 M 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+1 yp+1+q+1 ,

where x,y e Fp+q+2. Let G = U (p + 1,q + 1;F) denote the group of (p + q + 2)x (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) x 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) x 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 = {z e Fp+q+2 : [z,z] = -1}/

where ~ is the equivalence relation z ~ zu, u e F*.

Let Xt, for t e R, denote the following element in the Lie algebra 0 of G:

X,

V

0 0 . . 0 0

0 0 . . 0 0

1 0 . . 0 0

/

(a matrix of order p + q + 2). Let aq denote the Abelian subalgebra given by Xt, t e R, let at = exp(Xt) denote the exponential of Xt, and also define Aq = exp(aq). Let (considered as row vectors)

u = (u1,...,up) e Fp and v = (vq,...,v1) e Fq,

and let w e ImF (i. e., w = 0 for F = R). Define Nu,v,w e 0 as the matrix given by

Nu

( —w u v w \

-uT 0 0 UT

vT 0 0 —vT

\ —w u v w )

Then

exp(Nu,v,w) = I + Nu,v,w + 2 • Nlvw

and a small calculation yields that

at exp( Nu,v,w) • xo =

= ^sinh t + 1 • et (\u\2 — \v\2) + eJ w, u;

1 x T —v, cosh t + 2 • eJ (\u\2 - \v\2) + et w

for any t E R.

Define the nilpotent subalgebra n* as follows, for p ^ q,

n* = {NUvw : u = (—V^,u'), v E F9, u' E Fp-q}, (2)

and, for p < q,

n* = {Nuvv,w : V = (—u,V), u E Fp, V E Fq-p}, (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 p-factors. Let d = dimRF, and let

Pq =(1/2)(dp + dq + 2(d — 1)) E R, pi = (1/2)(\dp — dq\ + 2(d — 1)) E R.

Let N * = exp(n*) denote the nilpotent subgroup generated by n*. For functions f on G/H, we define, assuming convergence.

Rf (g)= i f (gn*H) dn* (g E G). (4)

Jn *

Let f E 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 0 -equivariant.

We define the associated Abel transform A by Af (a) = aPl Rf (a), for a E 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 E R. Let A denote the Laplace-Beltrami operator on G/H. Then, for f E C2(G/H),

A(Af )=( -dS-2 — P2) Af (s E R). (5)

Finally, for R > 0, let CR?(G/H) denote the subspace of smooth functions on G/H with support in the (K-invariant) 'ball' {kas • x0 \ \s\ ^ R} of radius R. Similarly, let CR? (R) denote the subspace of smooth functions on R with support in [—R,R], and let S(R) denote the Schwartz space on R.

§ 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

{Tx | A = 2 (dq - dp) — 1 + fx > 0, fx e 2Z},

see [1] and [2]. The spherical discrete series are given by the parameters A for which fix ^ 0, including the 'exceptional' discrete series corresponding to A > 0 for which fx < 0.

For q = d =1, the discrete series is parameterized by A e R\{0} such that |A| +pq e 2Z, and there are no spherical discrete series.

The parameters A are, via the formula Af = (A2 — p^)f, related to the eigenvalues of A acting on functions f in the corresponding representation space in L2(G/H).

Let D be the G -invariant differential operator AP(AP—Af)... (Ap—Aj:), where A1 ,...,Ar are the parameters of the non-cuspidal discrete series, and Ap = A + p2.

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, vnth p ^ 0, q > 1.

(i) If d(q — p) ^ 2, then all discrete series are cuspidal.

(ii) If d(q — p) > 2, then non-cuspidal discrete series exists, given by the parameters A > 0 vnth fx < 0. More precisely, if 0 = f e C2(G/H) belongs to Tx, then Af(s) = Cexs, with C = 0.

(iii) Tx is non-cuspidal if and only if Tx is spherical.

(iv) If p ^ q, and f e C£(G/H), for R> 0, then Af e CR?(R).

(v) If d(q — p) ^ 1, and f e C2(G/H), then Af e S(R).

(vi) Assume d(q — p) > 1. Then A(Df) e S(R), for f e 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 fi 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 non-cuspidal non-spherical discrete series corresponding to negative and odd values of f x in the exceptional series, see [1, Section 5].

The conditions in (vi) essentially state that Af is a Schwartz function if f is perpendicular to all non-cuspidal discrete series. The factor Ap, 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 —<x> for A(f ), and thus also for A(Df ). This leaves us to study the Abel transform near

Let f G C2(G/H), and write f [x] = f (gH), where x = g ■ x0. From (1) and (3), we get

Rf (s) = f (asn*H)dn*

In *

f [(sinhs - 1/2es\v'\2 + esw,u;

-dp y

—u, —v', cosh s — 1/2es\v'\2 + esw)] dv' du dw.

Let v' = \v'\v, v = — sinhs + l/2es\v'|2, such that \v'\2 = 1 + 2e sv — e 2s, and w = esw. Then,

(X

Rf (s) = e-ds J dw j f [(w — v,u; —u, — (1 + 2e-sv — e-2s)1/2v,e-s — v + w)] x

— sinh s M

x (1 + 2e—sv — e—2s)(dq—dp)/2—1 dv dv du ,

where M = Sdq—dp—1 x Rdp x 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 = {z E Fp+q+2 : [z,z] < 0}/

and identify a function f on X with a homogeneous function of z of degree zero on {z e Fp+q+2 : [z,z] < 0}.

We now identify Fp+q+2 with Rd(p+q+2) such that the coordinates satisfy Re zj = Xj, for j = 1,.. .,p + q + 2. Consider the real hyperbolic space

X = {z E Fp+q+2 : [z,z] = —1}.

The group G = O(d(p + 1),d(q + 1)) acts transitively on XX. Let K denote the standard maximal compact subgroup O(d(p + 1)) x O(d(q + 1)) of G. Let U(k), respectively U(t), denote the universal enveloping algebra of the Lie algebra t of K, respectively of the Lie algebra t of K.

Lemma 1. Let U e 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 e X can be written as x = ka • x0, where k e K, and a = as, s > 0. Let H = O(d(p + 1),d(q +1) — 1), and let m denote the commutator of Aq in the Lie algebra of

K n H. Then k = k + m.

Let Uk = Ad (k)U, for k e K, then Uf = (Ad(k"1)Ufc)f. By the Campbell-Baker-Hausdorff formula, there exists an element U° e U(k), such that Uk = U° modulo the left ideal generated by m. This implies that

Uf[ka • xo] = (Ad (k-1)U0)f [ka • xo].

The map k M Ad(k-1)U° is continuous into a finite dimensional subspace of U(k), and we can write Uf[ka • x0] = (Ad (k-1)U°) f [ka • x0] = Tliui(k)Uif [ka • x0], for a finite set of elements Ui e U(k) and continuous coefficients Ui(k). It follows that Uf is in C2(G/H). □

Define for t = (t1,t2,t3) e R3, the auxiliary function

Gf (t1,t2,t3)= f [(w + t1,u; —u,t2v,t3 + w)] dvdudw, Jm

and, with the identification z = e-s, define the function F(z) = edsRf (s). Then, since sinh s = —(z — z-1)/2, we get

<x

F(z)= J Gf (—v, — (1 + 2zv — z2)1/2,z — v) (1 + 2zv — z2)(dq-dp)/2-1 dv. (6)

(z-z-l)/2

Lemma 2. The function Gf is homogeneous of degree dp + d — 1 on the cone t2 —

t2 2 - t3

t2 — t2 < 0, it is even in t2, and satisfies Gf(—t1,t2, —13) = Gf(t1,t2,t3).

Let X be the differential operator on R3 given by t3d/dt2 — t2d/dt3. For all f e C2(G/H), and all k,N e N, there exists a constant C, such that

lxkGf(t)| < C(t2 + t3)-d(q-p)/4(1 + iog(t2 + t3))-N, on the hyperboloid t2 — t2 — t3 = — 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

d d

Di f [x] = xd(p+q+2^^ f [x] — xi o-f [x].

dxi dxd(p+q+2)

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.

Let now v = (vd(i+2p)+i, • • • ,vd(1+p+g)) E Sd(q p) 1. The operator

1+p+q

Yv = ^ ViDi, i=2+2p

also maps C2(G/H) into itself, and

\Yvf [x]\ ^ d(q - p) max(IDif [x]\).

i

Applying the operator X to the integrand in the definition of Gf, we get

d d Xf[ti,u; -u,t2V,t3] = t3 V — f [.]vi - t2^-f [.]

— dxi dXd(p+q+2)

d d = hY, dd-f [.]Vi -v2 ^—f [.]

^ dXi ^ dXd(p+q+2)

= Yvf [ti,u; -U,t2v, ¿3]

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 M 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 t = (dq - dp)/2 - ko. Define t = t(z,v) = (-v,-(1 + 2zv - z2)1/2,z - v). Then

(i) For k ^ k0, the function

v m dzk (Gf (¿(z, v))|(1 + 2zv - z2)|(dq"dp)/2"1)

(G Jt(z v))I(1 +2zv - I(dq-dp)/2-lN

dzk

is uniformly integrable over R for z < 1.

(ii) For k ^ k0 odd, this function is an odd function of v for z = 0.

Proof. Notice that tl - ¿2 - ¿2 = -1 and ¿2 + ¿2 = 1 + v2, for t = t(z, v), and that the integral (6) is uniformly convergent for 0 ^ z ^ k < m. The same holds with Gf replaced by Xk Gf.

Repeated use of the formula

d

—Gf (t(z, v))(1 + 2zv - z2)a = - XGf (t(z, v))(1 + 2zv - z2)a-1/2

+ 2aGf (t(z, v))(1 + 2zv - z2)a-1(z - v)

yields (i), and together with the parity properties of XkGf also gives (ii). □

We notice that e =1 if d(q — p) is even, and e =1/2 if d(q — p) is odd, i. e., if d = 1 and q — p is odd.

For k < k0, the derivatives dk/dzk 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 dko/dzkoF(£), for some 0 < £(z) < z,

F(z) = C0 + C1z + C2z2 + ... + Cko-1zk0-1 + Rko (£)zk0,

where 0 < £ < z, and

-I f ™ dL

C = j J-oo dZi

(Gf (t(z, v))(1 + 2zv - z2)(dq-dp)/2-1) dv,

z=0

for j e {0,.. .,k0 — 1}. The remainder term is given by:

Rk0 (£) =7-7

l r™ dk°

koU(í-í-i)/2 dzk0

(Gf (t(z, v))(1 + 2zv - z2)(dq-dp)/2-1) dv.

z=t

Consider Af(s) = ePisRf(s) = z (pi d)F(z), which is equal to

C0z-(pi-d) + C1z-(pi-d-1) + C2z-(pi-d-2) + ... + ck-1z- + z (-e+1)Rko (£).

Here we have used that p1—d = d(q—p)/2—1. For j even, the exponents — d(q—p)/2—1— j, for j e {0,...,k0 — 1}, correspond to the parameters A1,...,Ar 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 e{0,...,k0 — 1}, correspond to parameters A1,...,Ar 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 <x>, due to the term d2/ds2. If e = 1/2, the remainder term e-1/2sRko(£(s)) is clearly rapidly decreasing, and we are thus left to consider the case e =1, in which case k0 = d(q — p)/2 — 1.

Consider the constant term CRk = lims^^ Rko (e-s), which could be non-zero. We want to show that Rko(£) — CRko is rapidly decreasing at where £ = £(s), with 0 < £ < e-s.

We also include the case k0 = 0, where we put £ = e-s.

Define

dk0

H (z,v) = ^ (Gf (t(z,v))(1 + 2zv — z 2)ko).

Then, for £ < z < 1,

Rko(£) - CRko = J (H(£,v) - H(0, v)) dv + J H(0,v) dv = Ii(£) + /2(e).

(?-?-1)/2

For ), there exists ^ = v) < £, such that

d

h (e,v) - h (o, v) = e-

H (z,v).

z = il

and we get:

ш<zl Í

H(z,v)

z=îi

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 E N bounded by

\H(0,v)\ < C(1+ v2)-d(q-p)/4\v\k0 log(1 + v2)-N,

for some positive constant C. Using this, we find that

I2(z) <C v-1(log(v))-N dv = C(N — 1)-1 (log(sinh s))-N+1 ^ Cs-N+1.

J sinh s

It follows that Rko(£) — CRko is rapidly decreasing at whence A(Df) is rapidly

decreasing at which finishes the proof of Theorem 1.

References

[1] N. B. Andersen, M. Flensted-Jensen and H. Schlichtkrull, "Cuspidal discrete series for semisimple symmetric spaces", Journal of Functional Analysis, 263:8 (2012), 2384-2408.

[2] N. B. Andersen, M. Flensted-Jensen, "Cuspidal discrete series for projective hyperbolic spaces", Contemporary Mathematics. V. 598: Geometric Analysis and Integral Geometry, Amer Mathematical Society, Providence, 2013, 59-75.

Список литературы

[1] N. B. Andersen, M. Flensted-Jensen and H. Schlichtkrull, "Cuspidal discrete series for semisimple symmetric spaces", Journal of Functional Analysis, 263:8 (2012), 2384-2408.

[2] N. B. Andersen, M. Flensted-Jensen, "Cuspidal discrete series for projective hyperbolic spaces", Contemporary Mathematics. V. 598: Geometric Analysis and Integral Geometry, Amer Mathematical Society, Providence, 2013, 59-75.

Information about the authors

Nils Byrial Andersen, Candidate of Physics and Mathematics, Associate Professor of the Mathematics Department. Arhus University, Arhus, Denmark. E-mail: [email protected]

Mogens Flensted-Jensen, Professor of Mathematics (Emeritus). University of Copenhagen, Copenhagen, Denmark. E-mail: [email protected]

Информация об авторах

Андерсен Нильс Бириал, кандидат физико-математических наук, доцент кафедры математики. Орхусский университет, г. Орхус, Дания. E-mail: [email protected]

Фленстед-Йенсен Могенс, профессор математики (почетный). Копенгагенский

университет, г. Копенгаген, Дания. E-mail: [email protected]

Конфликт интересов отсутствует.

There is no conflict of interests.

Corresponding author:

Nils Byrial Andersen E-mail: [email protected]

Для контактов:

Андерсен Нильс Бириал E-mail: [email protected]

Received 21 May 2019 Reviewed 19 June 2019 Accepted for press 23 August 2019

Поступила в редакцию 21 мая 2019 г. Поступила после рецензирования 19 июня 2019 г. Принята к публикации 23 августа 2019 г.