The Laplace-Beltrami operator on rank one semisimple symmetric spaces in polar coordinates Текст научной статьи по специальности «Математика»

THE LAPLACE-BELTRAMI OPERATOR ON RANK ONE SEMISIMPLE SYMMETRIC SPACES IN POLAR COORDINATES 1

A, A. Artemov, V. F. Molchanov

G. R. Derzhavin Tambov State University, Russia

Let X = G/H be a semisimple symmetric space of rank one. The algebra of G-invariant differential operators on X is generated by the Laplace-Beltrami operator A corresponding to a G-invariant metric. It is very important (for many purposes) to know explicit expressions of A in various coordinate systems. For example, polar coordinates associated with the Cartan-

Rprcrpr HprnmnnQilinn Ct — TT ATC (fnr Hpfinifinnc qpp CJppfinn 1^ arp nPPPGcnru frvr ctnrl-u rvf

canonical and boundary representations, for the study of Poisson and Fourier transforms etc.

But, as we know, explicit expressions of A in polar coordinates are written in particular cases only.

For Riemannian (noncompact) symmetric space of rank one X = G/K, the Laplace-Beltrami operator in polar coordinates has the form (see, for example, [4]):

a 1 ® a ® T

Afrr dr+ S'

Here r is the distance between a point x E X and the initial point x°, S the sphere in X with center x° and radius r, the area A of S is given by

Tl (■ T2

A = C ■ | sinh(cr) j | sinh(2cr)|

where c is a number (written explicitly), r\, r2 are multiplicities of roots a, 2a, respectively, and, finally, Ls is the Laplace-Beltrami operator on S.

For real hyperbolic spaces (hyperboloids) X — G/H, where G = SOo(p, q), H = SOo(p, q—1), the Laplace-Beltrami operator in polar coordinates is written as follows (see, for example, [1]). The hyperboloid X is a manifold in Rn, n = p + q, defined by equation

2 2,2, , 2 1

-xx - ..., -xp + xp+l + ... + xn = 1.

Let spheres Si C W and 52 C K9 are defined by equations u\ +... + it2 — 1 and vf +... + w2 = 1,

respectively. Polar coordinates t,u,v (t € K, u £ Si, v £ S2) in X are introduced by

x = (sinh t ■ u, cosh t ■ v).

Then

A = 1 d_v d_ + Ai__________A2

v dt dt sinh2 t cosh2 t ’

where

v = | sinh i|p_1(cosh i)9_1,

Ai, A2 the Laplace-Beltrami operators on Si, S2, respectively.

Supported by the Russian Foundation for Basic Research (grants No. 05-01-00074a and No. 05-01-00001a),

j-i, „ c„:—4-4C„ ” t t„ 4-;„r, o,,™;.,” i\t„ „„ n/i m ark\ C„; D„+„„4-

LUC X X '-'gx Clllio yj 1X1 V Ci Q1U1CO KJ1 ILllOOia y^XClllU 1H-». IXX .un. vx } UiiU IVOtU. kJV^l. X V/UCllU. xxxgxx. UWXUU1 ,

(Templan, No. 1.2.02).

A similar formula for A is true for hyperbolic spaces over complex numbers and octonions, see, for example, [3].

For arbitrary semisimple symmetric spaces X = G/H of rank one, the Laplace-Beltrami operator in polar coordinates t, s, where t 6 R, s G S (for 5, see Section 2 below) was written in [5]:

A = - ttv^- + Vj(t)Dj, v dt dt ^ n ' 3 ’ j

where the function v = v(t) is given by formula (2.14) below, Vj(t) are some functions, Dj are some differential operators on S invariant with respect to K. But [5] does not contain explicit expressions of Vj and Dj. Apparently, such expressions seem to be rather complicated.

We succeeded in the obtaining explicit expressions of A in polar coordinates for all X - at points in a Cartan subset (i.e. points whose angular coordinates are equal to zero), see Theorem 2.1.

Notice that for para-Hermitian symmetric spaces of rank one, an explicit formula of A in horospherical coordinates is written in [2].

§1. Semisimple symmetric spaces of rank one

In this Section we recall some material from [5].

Let X = G/H be a semisimple symmetric space. It means that G is a connected semisimple Lie group, there is an involution cr(^ 1) of G such that H is an open subgroup in the subgroup Ga of all points in G fixed under a. We shall assume that G acts on X from the right and shall denote by R(g) : x xg the translation of X by g. Let us write x° for the initial point {H} of X.

Let g and f) be the Lie algebras of G and H respectively. The involution a of G gives rise to an involution a of g (we use the same symbol). The algebra g can be written as the direct sum:

0 = f) + q

of +1, — i-eigenspaces of a. The commutation relations are:

M c fj, [f),q] C q, [q,q] C fj.

The space q can be identified with the tangent space of X at .t°.

There exists a Cartan involution r of g which commutes with a. The algebra g decomposes into the direct sum of +1, - I-eigenspaces for r:

0 = 6 + P,

here t is a subalgebra of g. The commutation relations are:

[M] ct, [t,p] cp, [p,p] Cl.

There is a joint decomposition:

g = linf)-|-6nq + pnf) + pnq.

All these decompositions are orthogonal with respect to the Killing form Bg of g.

We assume that p ^ 0 and p fl 1} ^ 0, i. e. that g and i} are non-compact (excluding in this way the Riemannian case). Then pflq^O, EDq^O.

From now on we assume that the rank of X is equal to 1. It means that the dimension of any Cartan subspace of a (a maximal Abelian subalgebra of a consisting of semisimple elements) is equal to 1. Fix such a subspace a lying in p fl q.

Let A = exp a. The centralizer of the subgroup A in G is the product AM of two subgroups A and M whose intersection is the identity element e of G. The subgroup M is closed and reductive.

The algebra a is splitted in 5. The corresponding root decomposition is:

0 = 0—2a + 0-a + 00 + 0a + 02aj

where the root a is an element of a*. The subspaces g±2a may sometimes be absent. The involutions a and r give isomorphisms Qja —> Q-ja, j = 0, ±1, ±2. Let us denote:

Tj = dim Qja = dim g_ja, j = 1,2,

r = 1 + ri + r2.

Set

*\j = (1 ~~ bj = 9jai j = 1)2.

Then dimqj = dimfjj = rj, the spaces q and f) decompose into the direct orthogonal (with respect to the Killing form) sums:

q = a + qi + q2, f) = m + f)i + t)2,

where m = go fl b, so that go = a + m. Therefore, dimq = r. The algebra m is the Lie algebra of M.

Fix a basis element Aq G a so that

a(v40) = 1.

Then adylo ■ X = ±jX for X G g±ja, j = 0,1, 2. Let us denote:

at = exp tAo. (1.1)

It is more convenient for us to consider, instead of the Killing form Bs , the proportional form (X, Y) normalized by the condition

(Ao,Ao) = 1.

Recall, that the signature of a non-degenerate quadratic form on a vector space is the pair (p, q) indicating the number of plus and minus signs in a canonical expression of this form.

Denote the signatures of the form {, ) on q and qj by (r+,r~) and {r^rj) respectively (j = 1,2). On a its signature is (1,0). Therefore,

r+ — 1 + rf + T~2 1 r~ = rf +rj.

The signatures of (, ) on fy are (rj”>r’/)-

The operator adylo gives an isomorphism of qj onto fy and conversely and vanishes on go = a + m. The subspaces q^ and f)j are eigenspaces for (ad^lo)2 with eigenvalues j2. In particular, it implies the following.

Lemma 1.1. Let X G qj, j = 1,2. Then the element Y = (l/^ad^o • X belongs to 1jj and the operator ad/l{) transforms the elements X, Y by the matrix:

0 j \

3 0 j’

so that in this basis X, Y the operator Ad at has the matrix

( coshit sinhjt A ^ sinhji coshji )'

The same is true for X £ f)j, then Y £ c\j.

Let P denote the orthogonal projection operator in g onto qi + q2. Lemma 1.1 gives: Lemma 1.2. If X £ qj, then

P(Adat ■ X) = coshjt ■ X,

and if X £ i)j, then

P(Adat ■ X) = - sinhjt ■ ad^lo ■ X.

Let K be the Lie subgroup of G with Lie algebra t. Then K is connected, closed and contains the centre of G. The involution r can be lifted to G so that K = GT. The group K is compact if and only if the centre of G is finite. Then K is a maximal compact subgroup of G.

§ 2. The Laplace-Beltrami operator

The bilinear form (, ) on q gives rise to a G-invariant metric Q on X:

Qxog(dR(g)xoL, dR(g)x0M) = (L,M) (2.1)

where L,M £ q. Let A be the Laplace-Beltrami operator on X generated by this metric. Let us recall that the Laplace-Beltrami operator A on a manifold generated by a metric ^ gij(x)dxidxj is defined as follows:

a = ^E|-E^, m

3 1

where (gLJ) is the inverse matrix of (gtJ) and g = \det(glJ)\.

Let us introduce a system of polar coordinates on X - by means of the Cartan-Berger decomposition G = HAK. This decomposition gives that any x £ X can be written in the form

x = x°atk, (2.3)

where at is given by (1.1) and k £ K. If t ^ 0, then the element k in (2.3) is defined up to the multiplication from the left by elements in K fl H fl M, so that the manifold A x S, where S = K/K n H fl M, is mapped in the natural way onto X. The tangent space to S at the initial point can be identified with the direct sum of spaces

6nt)i, finqi, tnf)2, tnq2. (2.4)

Let us take orthogonal bases Xi, (Xi,Xi) = —1, in these spaces, here 1 ^ i ^ rf, + ri, ri + 1 ^ i ^ r\ + r~2, r\ + rg- + 1 ^ i ^ r\ + r2 = r — 1, respectively. We introduce local coordinates t,u\, ...,ur-\ on X (polar coordinates) by

x — x°g, (2.5)

where

r—1

g = at exp ^ UiXi. (2.6)

l — l

Theorem 2.1. At points x = x°at the Laplace-Beltrami operator A reads:

d2 d A = 7^2 + (r^cothi + rftanhi + 2r2~coth2i + 2rjtanh2t)—+

1 A + 1 A- 1 A+ 1 A-

_]------ ^ + ----K--/\o — -----7)-Ao ,

sinh t cosh t sinh 21 cosh 21

where Af, A}, Ag", Aj are the usual Laplace operators (J2 d2/du2) in spaces (2.4) respectively.

The theorem implies from the following two lemmas.

Lemma 2.2. In the coordinates t, ..., ur_i the metric Qx has the form

Qx = dt2 + ^^bij(t,u)duiduj, (2.7)

with

_8_ dm

u=0

bmm — 0, (2-8)

blm\u-.Q — 0) ^7^ (2-9)

Proof. Let us consider tangent vectors

d_ _d_ d dt’ du\’ ” ’ 9ur_i

at a point x given by (2.5), (2.6). By (2.1), they are the images of some vectors T, U\,..., C/r-i

in q under the map dR(g)xo. Clearly that T = Aq, i.e.

Q

xo Ao.

Let us determine Um G q■ Let 7(/z) be a curve in G such that 7(0) — e and 7/(0) = Um. Then

the following condition has to be satisfied:

*°7 (^9 = a;° at exp {pLXm + ^ .

From here and (2.6) we have

= x°atex.p^fj,Xm + ^2uiX^exp( - y^UjX^aJ1.

Let us apply to the product of these exponents the Taylor expansion of the product in canonical (logarithmic) coordinates up to the order two and differentiate the obtained equality with respect to fj, at jj, = 0. Then we obtain:

Um = -P(Ad at ■ {Xm — - Ui[Xm, Xi] + ■■■}), (2.10)

i^m

where dots mean terms of the order greater than one in Ui.

Since the vectors Xi belong to spaces (2.4), the vector Um lies in qi + q2. Therefore, (Ao, Um) = 0 and Qx(djdt, d/dum) = 0. It proves (2.7).

In order to prove (2.8) we have to show that linear terms (in itj) in (Um, Um) vanish. We use Lemma 1.2. It follows from (2.10) that the linear function m enters {Um,Um) with a coefficient that is the product of

(Xm,[Xm,Xi)) or (ad^o • Xm, [Xm,Xi]) (2-11)

and a function of t. Both inner products in (2.11) are equal to zero: the first one is equal to ([Xm, Xm\, Xi) = 0, the second one vanishes because ad^o-^m belongs to pflq and is orthogonal to [Xm, Xi] G t.

Finally, let us prove (2.9). For that, we have to show that (Ui,Um) = 0 when I ^ m and u = 0. From (2.10) we have for u = 0:

The right hand side is equal to (Xi,Xm) or (AdAo ■ Xi, Xm) with a coefficient depending on t. If / ^ m, then these inner products are equal to zero: the first one because of orthogonality of the bases Xi, the second one because ad^lo ■ Xi G p fl q and Xm G t. □

Qx = dt2 + sinh2t • ^ duj — cosh2t • ^ duf + sinh22t • ^ duf — cosh22t • ^ duf, (2.13) where i ranges sets mentioned above, so that

(Ui, Um) = (P(Adat • Xt), P(Adat ■ Xm)).

(2.12)

Denote

Lemma 2.3. At points x = x°at, t ^ 0, the metric Qx has the form

v = (coshi)r* ■ |sinhi|ri • (cosh2£)r^ • |sinh2i|r2 .

(2.14)

Proof. It follows from (2.9) that the matrix (bij) at x°at is diagonal. By (2.12), a diagonal entry bmm is equal to

(Um, Um) = {P(Adat ■ Xm), P(Adat ■ Xm)).

Let Xm G 6 fl qj, then by Lemma 2.2, P(Adat ■ Xm) = coshjt ■ Xm, so that

(Um, Um) — cosh jt.

Let Xm G £ fl t)j, then P(Adat ■ Xm) = sinhji ■ (l/j)adylo • Xm, so that

(Um, Um) = -^-Sinh2j'i • (ad^o • Xm, ad^40 • Xm) = -sinh2ji • (Xm,Xm) = sinh2j't 3Z

It proves (2.13) and, therefore, (2.14). □

Now we can finish the proof of Theorem 2.1.

Let (bZJ) be the inverse matrix of (bij). It follows from (2.8) and (2.9) that

Therefore, by (2.2), we have at points x = x0af.

It remains to substitute values of v and bn at x°at, see (2.13), (2.14).

REFERENCES

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2. G. van Dijk, V. F. Molchanov. The Berezin form for rank one para-Hermitian symmetric spaces. J. Math. Pures Appl., 1998, tom 77, No. 8, 747-799.

3. J. Faraut. Distributions spheriques sur les espaces hyperboliques. J. Math. Pures Appl., 1979, tom 58, No. 4, 369-444.

4. S. Helgason. Groups and Geometric Analysis. Amer. Math. Soc., 2000.

5. V. F. Molchanov. Harmonic analysis on homogeneous spaces. Encycl. Math. Sci., vol. 59, Springer, 1995, 1-135.