CC BY
36
MSC 43A85
Invariant boundary distributions for hyperbolic spaces 1
© L. I. Grosheva
Derzhavin Tambov State University, Tambov, Russia
For canonical representations on real and complex hyperbolic spaces, a description of distributions concentrated at the absolute and in particular invariant with respect to a maximal compact subgroup is given
Keywords: real and complex hyperbolic spaces, canonical representations, distributions, boundary representations
In this paper we generalize our results [1] for the Lobachevsky plane SU(1,1)/U(1) to real and complex hyperbolic spaces. The real hyperbolic space (the Lobachevsky space) is the Riemannian symmetric space G/K, where G = SO0(n — 1,1), K = SO(n — 1), and the complex hyperbolic space is the Riemannian symmetric space G/K, where G = SU(n — 1,1), K = U(n — 1).
§ 1. Boundary representations for real hyperbolic spaces
We use the Klein model of the real hyperbolic space (the Lobachevsky space), G/K G = SO0(n — 1,1), K = SO(n — 1). This space is the unit ball B: (u,u) < 1 in Rn-1 with the fractional linear action (from the right):
The boundary S: (u,u) = 1 is the absolute of the Lobachevsky space. Let B = BU S. Canonical representations R\, A E C, of the group G act on the space D(B) bv
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
(Rx(9)f) (u) = f(u ■ 9)(uP + S) X n
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They can be extended to the space D'(B) of distributions on B and, in particular, to the space E(B) of distributions concent rated at S. The restriction La of Ra to E(B) is called a boundary representation.
Introduce on B polar coordinates p, s : u = л/1 — p •ss G S, where p = 1 — (u, u). In these coordinates the Laplaee-Beltrami operator A on B is
д2 д p
A = 4(1 — p)p2^ + 2p(4 — n — 3p)— + ---------------As ,
dp2 dp 1 — p
with the Laplaee-Beltrami operator AS on S. To the Casimir element, the representation Ra assigns a dilferential operator (the Casimir operator)
д
Aa = A + 4(A + n)p(1 — p) dp + (A + n) [A + 2 — (A + n + 1)p].
Its K-radial part (acting on K-invariant functions) is the operator
д2 д
Rad Aa = 4(1 — p)p2d~2 + 2p [2A + n + 4 — (2A + 2n + 3)pj д^~
+ (A + n) [A + 2 — (A + n + 1)p].
The Berezin form (f, h)A on D(B) is a bilinear form defined by:
(f, h)A = c(A) f {1 — (u,v)}A f (u) h(v) dudv,
Jb
где du is the Euclidean measure on B,
c(A) = n-<->/'2 Г ( —A2+i) /Г () .
The Casimir operator and the Berezin form are invariant with respect to Ra,
Representations TCT, a G C, of the group G associated with a cone act on the space D(S) by
(T* (gV)(s) = ^(s • g)(se + .
They are irreducible for all a except a G N and a G 2 — n — N. Here and further
N = {0,1, 2,...}.
For A G — (n — 4)/2 + N, the boundary representation La decomposes into the direct sum of representations T2_ra_A+2k, k G N, as follows. First, we introduce differential operators Wa,k and W^ on D(S), which are polynomials in AS:
WCT,fc = wk(a, As), WCT*fc = w*k(a, As),
where wk, wk are defined by means of a reproducing function (with ^ = /(3 — n — /)
and the Gauss hvpergeometric function F):
= ^ wk(a,^i)pk,
k=0
= ^ wk (a,^i)pk. k=0
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(1 — p)i/2 f(
a + n — 2 + / a + n — 1 + / n ----2-----'-----2----; a +2; p
(1 — p)_i/2 f(
a + 2 — / a + 1 — / n ---2---'---2---; a +2; p
Then we define operators fA,k : D(S) ^ Ek(B):
k!
£« (v) = E^1)'’ w-WbOv) ■ i(k-b,(p),
6=0 ( )!
they intertwine T2_ra_A+2k with LA. The space E(B) decomposes into the direct orthogonal (with respect to the Berezin form) sum of their images VA,k, k G N. These subspaces are eigenspaees of Aa with eigenvalues (A — 2k)(A + n — 2 — 2k), The "old basis" v(s)6(k,(p) is expressed in terms of the "new basis" fA,m(v)
VW^M = '£ (—1)‘_' k1 fv (»2*_„_A+2r.k_r(v)) .
r=0 !
In the subspace E(B)K of E(B) consisting of K-invariant distributions we have two natural bases: the first one consists of derivatives of the delta function 6(p):
6(k) (p), k = 0,1,..., (1.1)
the second one consists of distributions
CA,fc = fA,k (1), k = 0,1,.... (1.2)
We use the notation:
a[m1 = a(a + 1)... (a + m — 1), a(m) = a(a — 1)... (a — m +1).
Basis (1.2) consists of eigenfunctions of Aa and is orthogonal with respect to the
Berezin form:
(CA,fc ,CA,fc )a = P (A,k^ (CA,fc ,ZA,r )a = 0, k = ^
where
P(A, k) = b(A) ■ 2_4k k! (—1)k ( A)[2k](3 n A)[2k]
((4 — n)/2 — A)[2k] ((2 — n)/2 — A + k)[k1 ’
b(A) = 2A+™_3 (n_2,/2 r(A + (n — 2)/2)r((—A + 1)/2)
b(A) n r((n — 1)/2) r((2 — n — A)/2)r(A + n — 2) .
(1.1) (1.2)
triangular matrices with the unit diagonal, namely,
Z = V' ( li* (k \ 2_2‘ + n — 2 — 2:)|261 ?(*_*, (I 3
^<—1) b) 2 (A + n/2 — 2k)[b1 6 (p)- (1-3)
bj (A + n/2 — 2k)lb1
,k, W „2,-2k (3 — n — A + 2s)|2k_2s1
,=0
^ = D— ^ ( J 22,_2k ((4 -n)/2_A^)2s)lk_,1 ZA- (14)
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Notice that formula (1,3) can be written as follows:
/- n/A + n — 2 A + n — 1 n \ (k, , \
CA,fc = F (—2-------------k’ —2--------------k; A + 2 — 2k; pJ 6
Formula (1,4) gives a generating function for 6(k,(p):
exp («d6(p)=g m x
/4 — n — A 3 — n — A 4 — n ,
x FI ------2-----+ m, ---2--+ m; —2-A + 2m; —u ) ■ (a,,
Pairwise inner products of basis (1,1) are given by
= b(A) ■ (—1)^+^ 2_2r_2m (3n)/2— — ^—i'
-2r-2m (3 — n — A)|2m1(3 — n — A)|2r1
2. Boundary representations for complex hyperbolic spaces
The group G consists of matrices g G SL(n, C) preserving the form
[x y] = — x1 y1 — ... — xn_1yn_1 + y„.
Let us write g G G in a block form with respect to the partition n = (n — 1) + 1:
/ a P g =1 Y 6
The group G acts on Cn_1 bv fractional linear transformations (from the right):
za + y
z ^z ■g = Zpn.
The stabilizer K of the point z = 0 consists of block diagonal mat rices. Let (z,w) be the standard inner product in Cn_1: (z, w) = z1w1 + ... + zra_1wn_1. Denote
p =1 — (z, z).
The homogeneous space G/K is the unit ball D : p > 0, Its boundarv S : p = 0 (a sphere of dimension 2n — 3) is also a G-orbit, Set D = D U S.
Canonical representations RA, A G C, of the group G act on D (D) by
(RA(g)f) (z) = f (z ■ g)|zP + 6|_2A_2n.
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They can be extended to the space D' )D) of distributions on Cn_1 with support in D, in particular to the space E (D) of distributions concentrated at S. The restriction LA of RA to E(B) is called a boundary representation.
In the subspace E(D)K of E(D) consisting of K-invariant distributions we have two natural bases: the first one consists of derivatives of the delta function 6(p):
6(k) (p), k = 0,1,..., (2.1)
the second one consists of distributions
<A,k = f A,fc (^o), k = 0,1,..., (2.2)
operators fA,k are defined in [2], the function ^0 on S is equal to 1 identically. Elements of the second basis are eigenfunctions of the Casimir operator Aa:
AaCa,& = (A — k)(A + n — 1 — k)ZA,fc.
This condition determined (A,k up to a factor,
(2.1) (2.2)
by means of triangular matrices with the unit diagonal, namely,
m
ZA,m = £(—1W "I -^ll} 6(-_^ (2*3)
(2.4)
s_0 vs/ (2A + n — 2m)[,]
(m)—n—A+:)i;mf <-
Notice that formula (2,3) can be written as follows:
(A,m = F (A + n — 1 — m, A + n — 1 — m; 2A + n — 2m; p) 6(m) (p),
where F is the Gauss hvpergeometric function. Formula (2,4) gives a generating function for 6(k,(p):
exp (“dp)6(p)=g mx
x F (2 — n — A + m, 2 — n — A + m; 2 — n — 2A + 2m; —u) ■ ZA,m.
Define on D(D) a bilinear form (Berezin form)
BA(f,h) = c(A) / |1 — (z,w)| f (z) h(w) dzdw,
JDxD
where dz is the Euclidean measure in C
n— 1
c( A) = n
1 —n
r(—A)
r(1 — n — A)
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2
This bilinear form is invarant with respect to RA, Denote b(A) = Ba (6(p),6(p))
= (—1)n_V
n- In-1 r(A + n)r(2A + n — 1)
r(n — 1)r(A + 1)r2(A + n — 1) ' (2.2)
BA (CA,m,CA,m) = P(A,:),
Ba ((A,m,(A,r) = 0, k = r,
where
(m,
P (A, m) = b( A)
m! {A(m,(A + n — 2)(m,|
(2A + n — 2)(2m,(2A + n — 1 — m)(m) .
(2.1)
Ba(6(k)(p),6(m)(p))A = b(A) ■ ckm(A),
where
C = ^1)k+m {(2 — n — A)|k1 (2 — n — A)|m1} Ckm (A) ( 1; (2 — n — 2A)[k+m]
For n =2 there is a nice formula:
^ «m vr
V'' Cmr(A) ■ j" ■ j" = F( —A, —A; —2A; —“ — v — “v). m! r!
m,r=0
References
1, L, I, Grosheva, Boundary representations on the Lobachevsky plane, Vestnik Tambov Univ. Ser,: Est, tekhn, nauki, 2005, vol. 10, issue 4, 357-365,
2, L, I, Grosheva, Canonical and boundary representations on complex hyperbolic spaces, Vestnik Tambov Univ. Ser,: Est, tekhn, nauki, 2008, vol. 13, issue 6, 499-555,
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