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27
MSC 43A85
Poisson transforms for the complex hyperboloid 1
© O. V. Betina
Derzhavin Tambov State University, Tambov, Russia
The representation U of the Lorentz group G = SL(2, C) bv translations in polynomials on the complex hyperboloid X in C3 is the multiplicity free direct sum of finite dimensional representations. We write operators (Poisson transforms) intertwining finite dimensional
U
G
transforms. Finally, we write explicitly spherical functions on X.
Keywords: the Lorentz group, representations, distributions, hyperboloid, Poisson transforms, spherical functions
1. Principal series and intertwining operators
The group G = SL(2, C) consists of complex 2 x 2 matrices:
g = ( a p ) , aS - = L (L1)
For such a matrix g, let j denote the matrix obtained by permutations a <—> S and
p <—> rS 7
j ' P a
The map g j is an involutive ^^^^morphism of the group G. For A G C, k E Z a S C\{0}, we denote
- = ^ (±
Observe that the following differentiation formulae hold:
d A,k = A + k \-l,k-1 zX,k = A —
dzz = 2 z , dzz = 2
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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We also use the notation ("generalized" powers):
a(n) = a(a — 1)(a — 2)... (a — n +1).
Define principal series representations of the group G, Let a G C, 2m G Z. Denote by Da,m the space of functions f (z) in C~(C) such that "inverse" functions f(z) = z2a,2mf (—1/z) also belong C^(C), The representation Ta,m of the principal series acts on Da,m bv
(T„,(g)f) (z) = f (z ■ g) (Pz + S)2a,2m , z ■ g = az+2- .
The eontragredient representation Ta,m is obtained from Ta,m by means of the involution g M j:
T<T,m(g) = Ta,m(g).
Representations Ta,m and Ta,m are equivalent. A bilinear form
(F,f )c = F(z) f (z) dxdy, z = x + iy,
J C
is invariant with respect to the pair (Ta,m,T-a-2—m).
Introduce an operator Aa,m:
(Aa,mf) (z) = f (1 — zw)-2a-4,-2mf (w) dudv , w = u + iv.
C
This operator intertwines T,,m with T-a-2-m as well Ta,m with T-a-2—m:
Aa,mTa,m(g') T—/J—2,—m(g')A/J,m,
Aa,mTa,m(g) T—,—2 ,—m(g) Aa,m.
(z —
w)—2a—'4,—2m. It intertwines Ta,m with T—a—2,—m. But for our purposes (we want to construct polynomial quantization and to study finite dimensional analyse on a complex hyperboloid), just introduced operator A,,m is much more convenient.] The composition of operators A—,—2,—m and Aa,m is a scalar operator, i, e, an operator of multipleation by a number:
A—a—2,—mAa,m ^0,0 (a,'m)Ei
where
2 n2
uo,o(a,m) = —(—1)2
(a + 1)2 — m2
Let 0 ^e Lie algebra of the group G. It acts on Da,m bv means of some first order differential operators. These operators give rise to a representation of the Lie algebra 0 and its universal enveloping algebra not only on Va,m, but also on other
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spaces, for example, on the space C~(C), on the space Pol (C) of polynomials on C, on the space D'(C) of distributions on C, on the space D0 (C) distributions on C concentrated at zero.
The space D0(C) consists of linear combinations of the delta function 8(z,z) and its derivatives
d k+i
8{k’l)(z,z) = k 8(z,z).
dzk dz
The spaces Pol (C) и D0 (C) are Verma modules with respeet to Ta,m.
The intertwining operator carries out the basis zkzl in Pol (C) to the basis (z,z) in D0(C) and back (with factors):
Aa,m (zkzl) = Uk,i(a,m) ■ 5(k’l\z,z), (1.2)
A-a-2—m (8(k'l)(z,z)) = (a + m)(k)(a - m)(l) ■ zkzl,
where
n2
^(a,m) = -(-1)2m(a +1 + m)(k+.)(a +1 - m)(l+i) ' <L3>
§ 2. Finite dimensional representations
Let 2k, 2l e N = {0,1, 2,...}. Let Vkyl be the space of polynomials p(z,z) in two variables ^d z of degrees ^ 2^d ^ 2l in variables z and z respectively. On this space the representation of G acts bv
(nk,l(g)p)(z,z) = p(z ■ g,z^g)(pz + 6)2k(pz + 6)21.
A basis in Vk^ consists of monomials zpz\ where 0 ^ p ^ 2k., 0 ^ q ^ 2l, so that dimVki = (2k + 1)(2l + 1). All representations nk,l are irreducible. Conversely, any irreducible finite dimensional representation is equivalent to one of nk,l. In particular, the eontragredient representation ^defined by nk,l (g) = (g) is equivalen t to nk,l.
The representation nk,l preserves the following bilinear form Bk>l(^,p) on Vk,f. at basis elements it is given by
Bk>l (zrzs, zpzq) = (-1)p+^^ ^^ 6p,2k-rSq,2l-s,
Sij being the Kroneeker delta. Such a form is unique up to a factor. Parallel with it we consider on Vkl the bilinear form B'kl (^, p) = Bk,l (^, p), so that
Bh(zrzS,z’z ) = (-1)P+^2pk) (2q() Sp,r Sq,s. (2.1)
The form B'k l is invariant with respect to the pair (jrk,l,nktl), i. e,
Bk ,l (^,nk,l (g)p) = B'k l (n'k,l(g-1)^,p).
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For a = k + l, m = k — l, the representation Ta>m = Tk+l>k-l is reducible, it has an invariant finite dimensional irreducible subspace Vk>l. The module D0 (C) with respect to T-k-l-2--k+l has a submodule invariant with respect to g, The quotient module is equivalent to Vkyl.
We can write the form B'k l by means of the intertwining operator Ak+l,k-l (which is defined just on Vk,l):
Bk l P) =-------2 (2k + 1) (2l + 1) iAk+l,k-l^, p)c , (2.2)
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it follows from (1.2), (1.3) and (2.1).
§ 3. Hyperboloid
Introduce in C3 a bilinear form
[x,y] = —Xiyi + X2V2 + X3y3.
Denote bv X hyperboloid [x,x] = 1. The manifold X can be realized as a set of matrices
1 1 — X3 X2 — X1 x = — (
2 \ X2 + Xi 1 + X3
with det x = 1. The group G acts on the space Mat(2, C) as follows: x ^ g-1Xg. On X
x0 = (0, 0,1) ' 0 0
01
is the diagonal subgroup H of G, it consists of matrices
h = ( 0 S ) ■ =1.
An action of G on functions / on X by translations is denoted by U:
(U(g)/)(x) = /(g-1xg), g e G.
Introduce on X horospherical coordinates £, n
X = N-1(^ + — v, 1+ Cn), N = 1 — &,
in the matrix form we have:
_ 1 f —£n —n X = N (, £ 1
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These coordinates are defined on X except x3 = — 1. The action x g 1 xg of G is
given bv a linear-fraetion transformaton in each variable £ and n separately:
£ ^ £ = £ ■ g, n ^ V = n ■ 9\
so that (U(g)/)(£,n) = /(£, V)- The initial point x0 has eoordinates £ = 0 n = 0.
An element g, see (1.1), moves x0 to a point x with coordinates
£ = t, n =~ , (3.1)
o a
so that N = 1/aO,
There are two Laplace operators A h A on X:
a = n*JI , a = n2~d
d£dn ’ d£dn ’
GX
Let us denote by A and A spaces of analytic and antianalvtie polynomials in C3 respectively. A polvnomial / in A is called harmonic (with respect to the form
[x,y]) if
_d!_ \
8x1 + dx2 + dx2 / .
Let us denote by % the space of harmonic polynomials and by H the corresponding subspace in A. Let H^ be the subspace in H ® H consisting of homogeneous polynomials of degree k in x1, x2, x3 and degree l in x1,x2,x3. We denote restrictions of spaces H ® H and Hkyl to X bv (H ® H)(X^d Hk,l(X) respectively. This restricting map is one-to-one. Notice that (H ® H)(X) coincides with the space of restrictions to X of all polynomials on C3, The space Hk,l(X) is invariant and irreducible with respect to U, the corresponding representation is equivalent to nk,l, see below. Polynomials in this space are eigenfunctions for Laplace operators:
A/ = k(k + 1)/, A/ = l(l + 1)/.
§ 4. Poisson transforms, spherical functions
For representations nkj and nrk,^ an invariant with respeet to H in the space Vkyl kl
9k,l (z,z) = zk zl.
It is unique up to a factor. It gives rise to an intertwining operator mapping Vkyl to functions on X, we call it the Poison transform Pk,l. This transform assigns to a polynomial <p(z,z) in Vkyl the following function (Vkj<p)(x) on X:
(Vk,l^)(x) = Bk,l(n(g-1)0k,l,<p)
= Bk,l(0k,l, n(g)tp), (4.1)
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where g is an element in G such that g 1x0g = x. It intertwines and U:
Pk,l nk,l (g) = U(g) Pk,l.
The forms Bk>l (0, ^^d Bkl(^, ^coincide for ^ = 9k,f.
Bk,l (9k,h = Bk, l (9k,l, ^).
Therefore, we can rewrite (4,1) as
(Vk,l^)(x) = Bk ,l (V(g-1 )0k,uv)
= Bk ,l (°k,l ,п(g)v),
The latter formula, together with (2,2), (1.2), (1.3), gives a differential form for the Poisson transform:
(Pk,l<p)(x) = c(k,l) J 0(k,l)(z,z){Tk+l,k-l(g)<p(z,z)j dxdy
Qk+l
c(k,l) ■ (—1)k+l
dzkdz1
<P (z ■ g,z ■ g) (pz + o)2k (pz + 8)21
z=0
where c(k,l) = k! l!/(2k)l (2l)L We get (Pk,l<^)(x) as a function of horospherieal coordinates £, n, see (3.1). This function is just a polynomial on X lying in Hk,l(X), Therefore, the Poisson transform Pk,l maps nkj isomorphically onto Hk,l(X),
In particular, H-invariant 9k,l goes to a H-invariant ^k,l in Hk,l (X) (a .spherical function):
2k)-1(2l)-\. k A (k)2t
kj
•k'lxi - (-i f) -(a -N -k
■ N -lS C)'®"'
= < — 1)k+i(2k) 1(2/) lpk<X3) Pl<X3).
where Pm(t) is the Legendre polynomial.
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