Гармонический анализ на плоскости Лобачевского-Галилея Текст научной статьи по специальности «Физика»

MSC 43A85

Harmonic analysis on the Lobaehevsky-Galilei

plane 1

© Yu. V. Dunin

Derzhavin Tambov State University, Tambov, Russia

Harmonic analysis on the Lobachevskv-Galilei plane is constructed

Keywords: Lobachevsky plane, dual numbers, Poisson and Fourier transforms, spherical functions, Plancherel formula

1. Lobaehevsky-Galilei plane

Let A be an algebra over R of dimension 2 consisting of elements z = x + iy, x,y E R with relation i2 = 0 (the algebra of dual numbers). It is not a field: pure imaginary numbers iy are zero divisors. For z = x + iy, the conjugate number is z = x — iy.

The Lobachevskv-Galilei plane L is a domain on the plane A, defined by zz < 1, It is a vertical strip bounded by lines x = ±1, Denote the line x = 1 by T, The group G of translations of the Lobaehevskv-Galilei plane L consists of linear-fractional transformations

It preserves the measure

Matrices

g

az + b bz + a

da(z)

ab

b a

aa — bb =1, a, be A.

dx dy

(1.1)

(1 — x2)2

aa — bb = 1,

occuring in (1.1), form the group SU(1,1; A). Denote a = a + ip, b = ft + iq. The condition aa — bb = 1 is equivalent to a2 — ft2 = 1, so that a2 ^ 1, therefore a ^ 1 or a ^ — 1. Thus, the group SU(1,1; A) consists of two connected parts. The connected

G

1 Supported by the Russian Foundation for Basic Research (RFBR): grant 09-01-00325-a, Sci. Progr. "Development of Scientific Potential of Higher School": project 1.1.2/9191, Fed. Object Progr. 14.740.11.0349 and Templan 1.5.07

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of matrices g with a ^ 1. For this component we retain the notation G, We can

write parameters a and ft of the matrix g E G as a = ch t h ft = sh t, where t E R.

Therefore, any matrix g E G can be written as follows

g = g(t) + ic(p,q), (1.2)

where

g(t)^cht sh0- —pq —,)■

The stabilizer of the point z = 0 is the subgroup K consisting of diagonal matrices:

k =(1+0iP ! —0p)- <1.3)

so that

L = G/K.

Let (/, h)L be the inner product in the space L2(L, da):

(/,h)L = J f (z) h(z) da(z).

The quasiregular representation U of G acts on this space: (U(g)/)(z) = /(z ■ g).

§ 2. Representations of the group G

G

characters (one-dimensional representations) of "parabolic" subgroups P0 h P», The first one P0 is the stabilizer of the point 70 = 1 in r, the second one P» is obtained bv a limit passage from the stabilizer of the point Yy = 1 + iy when y M ro.

The subgroup P0 consists of matrices (1.2) with p = q, i, e. matrices h(t,q) = g(t) + ic(q, q), Its character is defined by a complex number A:

wA(h) = eAt = (a + ft )A.

The set G/P0 can be identified with the line r to a point Yy one assigns the diagonal matrix (1,3) with p = y/2. The representation TA of G induced by wA acts in functions ^(7) in D(r) by

TA(g)^(7) = P (Y ■ g)(a + ft)A, A E C

The stabilizer Py of the point Yy consists of matrices (1.2) such that p = q + y sht. The subgroups Py and P0 are isomoephic, so representations induced by characters these subgroups are equivalent.

Let us find the limit P» of Py when y M ro. We consider parameters t and q depending on y: t = ty, q = qy, in such way that the following limits exist: limty = t,

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lim qy = q, lim(qy + y shty) = p. It gives t = 0. Therefore, the sub group P» consists of matrices

h = h(u,v) = E + ic(u,v), (2.1)

where E is the identity matrix of the second order. This subgroup is a commutative

G R2

uxM = eXue^v,

where A,^ E C and h is the matrix (2,1), Any matrix (1,2) can be written as

g = h(u,v) g(t) (2.2)

where

h(u, v) = E + ic(p, q)g(t)-1,

so that

u = p ch t — q sh t, v = —p sh t + q ch t.

So we can identify G/P» with the subgroup of matriees g(t) and hence with R, The

representation T\;jU of G induced bv acts in functions <^(s) in D(R) bv:

TA,M(gMs) = ^(S)

where s^d h are obtained if we decompose g(s) g in accordance with (2,2): g(s) g = hg(S). Let us write g also as (2,2) then

g(s) g = {E + v)g(s)-1} ■ g(s + t),

so that S = s + t, h(u, v) = E + i c(S, S), where

c(s,s) = g(s)c(u,v)g(s)-1

= g(s)c(p,q)g(s +1)-\

therefore,

S = p ch(2s + t) — q sh(2s + t), (2,3)

S = —p sh(2s + t) + q ch(2s + t). (2,4)

Finally for g = g(t) + ic(p, q) we get

PUg) ?)« = + t) eAii+"s, (2.5)

with S and S given by (2,3) and (2,4),

A Hermitian form (the inner product from L2(R,ds))

(^) = ^(sMs) ds

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is invariant with respect to the pair (TA;jU, T_^ _-), i, e,

(TA, M(g)^,^) = (^,T_A, _?(g_1)^), (2.6)

so that T\ is unitarizable for pure imaginary A,

Using (2,6) we can extend TA to the space D'(R) of distributions ^ on R,

3. Poisson and Fourier transforms, spherical functions

We need the second series from § 2,

Theorem 3.1 A non-trivial .space of K-invariants in D'(R) under TAjU exists provided that A = r^, where —1 < r < 1; if so let us set r = th2r, t E R, so that

A = th 2t ■ ^, t E R. (3.1)

This space is one-dimensional, a basis function 9 is the delta function:

9(s) = <S(s — t).

Proof. Let a function 9(s) is K-invariant under TA;jU, Bv (1,3) and (2,5), (2,3), (2,4) it means:

ep(Ach2s_Msh2s) 9(s) = 9(s)

p E R

with respect to p at zero: (A ch 2s — ^ sh 2s) 9(s) = 0, or (A — ^ th 2s) 9(s) = 0, The factor in front of 9(s) has to vanish at some point s = t. It gives (3.1). □

The representation TA;M with condition (3,1) is equivalent to that with A = 0, Indeed, the translation operator C: (C^)(s) = <^(s + t), intertwines TA;jU with T0)V, where v = ^/ch2T. So we can take A = 0 from the beginning. Then Theorem 3,1

claims that the representation T0)M, ^ E C, has a K-invariant 9(s) = £(s) unique up

to a factor.

The K-invariant 9(s) = £(s) gives rise to a Poisson kernel Pu(z, s) = (T0;M(g_1)9) (s), z E L, s E R, where g is an element in G moving 0 to z, for example, the matrix

= 7r=^(1 0 ■ z =x +iy.

We get

y

PM(z,s) = £(£ — s)exp| ^ 1 — x2

= (1 — x2) £(x — c) exp <j ^

1 x2

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where x = th£, c = th s. This kernel gives rise to two transforms: the Poisson transform : D(R) ^ and the Fourier transform : D(L) ^ D(R)

defined respectively by:

- . y

(P. ^)(z) = / P.(z,s) ^(s) ds = ) exp<J ^---------— [> , (3.2)

1 — x2

(F. /)(s) = / P.(z,s) /(z) da(z)

1 f y

/(c + iy) exp < --------- }> dy, c = th s.

They intertwine T0,-M with U and U with T0,M, respectively. These transforms are conjugate to each other:

(PM )L = (^,P/).

The spherical function is defined as the Poisson image of the K-invariant 9:

It follows from (3,2) that

ФДг) = ад e.y.

4. Decomposition of the quasiregular representation

Theorem 4.1 The quasiregular representation U of the group G in L2(L,da) decomposes in the direct integral of representations T0)ip, p E R, with, 'multiplicity one as follows (here i = \f—1 E C, the complex number). Let us assign to a function / E D(L) the family of its Fourier components Фурье Fip/, p E R. This G

1

/ = 2П J P-ip Fip /^ (4.1)

and Йе Plancherel formula:

1 f ™

(/,h)L = 2П J ^ /,Fiph dP. (4.2)

Therefore the 'map / M {Fip/} can be extended to the whole space L2(L,da).

Formulas (4,1), (4,2) are obtained from corresponding formulas for the classical Fourier transform bv the change p = (1 — x2)n

These formulas can be united by the formula decomposing the delta function £(z) concentrated at z = 0 into spherical functions:

1 Г ™

^(z) = 2П J ^(z) dP.

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