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22
Representations on distributions on the Lobachevsky plane concentrated at the boundary
L. I. Grosheva Tambov State University, 392622 Tambov, Russia grosheva@main.tsu.tambov.ru
In [4], the so-called canonical representations on the Lobachevsky plane were introduced - for the case when they are unitary. They are used for the construction of quantization etc. We consider them in a more general aspect and study their action on distributions concentrated at the boundary. It turns out that this action is diagonalizable. We give explicit expressions for distributions in irreducible constituents. The diagonalizability in question was discovered in [1] for para-Hermitian symmetric spaces.
1 Canonical representations on the Lobachevsky plane
Let us realize the Lobachevsky plane as the unit disk D : zz < 1 on 2-plane. The group G = SU(1,1) acts on D by fractional linear transformations
az + b fab
z z = z ■ g = ---------. a= r
y fcz + a V b a
(the group of motions of the Lobachevsky plane). Let T be the unit circle zz — 1 and D = D U T.
For A E C, let R\ be the representation of G acting on V(D) by the formula:
(R\(9)f)(z) = f{z • 9)\bz + a|-2A-4
It preserves the following sesqui-linear (Hermitian for A £ M) form
(/i,/2)a = c(A) [ fi(z)f2(w)\l - zw\2Xdxdydudv (1.1)
JDxD
where z = x-\-iy,w = u + iv, and
m -A”1 c(A) = —-—
7T
Integral (1.1) converges absolutely for ReA > — 1 and can be extended on the A-plane by analyticity to a meromorphic function.
For A > —2 the representation R\ can be regarded as the tensor product of the analytic series
representation T+ of the universal covering group G of G and its conjugate representation T~ where
0- = (—A — 2)/2.
Let be the inner product in L2(D) with respect to the Euclidean measure:
(rp,<p)= / xl>(z)<p(z)dxdy Jd
It is invariant with respect to the pair R\, R_x_2-
(Rx(g)ip,<fi) = (L2)
Extend R\ to distributions on D by formula (1.2) where (xJj, ip) is understood as the value of xp £ 'D'(D) at ip £ V(D).
Denote by Y the multiplication by
p— 1 — zz (1-3)
This operator intertwines R\ and R\-i, so that Ya intertwines R\ and R\-a-
PQRx(g)f = Rx-a(g)(Paf)
2 Poisson transform
Elementary representations Ta, <J G C, (see, for example, [5]) of G/TL^ act on V(Y) by the formula:
(Ta(g)ip)(u) = <p(u ■ g)\bu + a|2<T, it G T
The inner product from £2(T):
r2*
(V>> ¥>)r = / xj>{v)<p{u)da, u = ela
Jo
is invariant with respect to the pair (Tff,T_^_i). The operator Ba on X>(r), defined by the formula
(Bff(p)(u) = [ |1 - uv\~2a~2<p(v)d/3, v = elp
Jo
intertwines Ta and T-a-The basis ipm(u) = eima, m G Z, consists of eigenfunctions of Ba\
B(j Ipm — brn((T^'lprn
where
bm(<r) = 2tt(—l)m r(-2ff-1)
r(—<7 -f m)r(—cr — m)
For a not integer Ta and T_a_i are equivalent (for a integer there is a ”partial equivalence”). For a G C, define the Poisson transform Va by the formula
(Vaip)(z) = I |1 - zu\2oip(u)da, u = eta Jo
It carries V{Y) to C°°(D) and intertwines T_a_i and the restriction R-a-1 to C°°(D)\
R_a-2(g)Vff = VaT-ff-i(g)
It has the following asymptotics at the infinity, i.e. when p —> 0 (for p, see (1.3)):
oo oo
(TM(z) ~ £<c,lM,)(«)V+P2°+' (21)
jfc=0 k=0
where 2 = ru,r = \z\,u = e,Qf, and Ca,k, Da,k are some operators on X^T). There are connections between them:
(2.2)
(2.3)
where
j(a) = 40(<0 = 2flT(—2<r - l)/r2(-<r)
The basis functions ipm are transformed by Va to functions
(»>,*.)(,) = ^(«)(-l)m2 (£5) (24)
where P™ is the Legendre function, see [2], Ch 3. Applying [2] 3.2 (18), we express (2.4) in terms of the Gauss hypergeometric function:
(Varpm)(z) = rl>m(u)(l-p)-ml2[bm{-(T-l)F(-<r,-<r-m,-2<r,p)
+p2a+lj(<r)F(a + 1,0-4- 1 - m; 2<r + 2; p)] (2.5)
Comparing (2.5) with (2.1), we observe (because (d2/da2)ipm = —m2ipm) that Daik are differential
operators on T (and, by (2.2), (2.3), C0}k are integral operators). Namely, consider the following power
series in p:
(1 - p)-m'2F{(T + 1, <x + 1 - m; 2a + 2; p) (2.6)
Its coefficients are polynomials in m with coefficients rational in a. In virtue of [2] 2.1(23), the function
(2.6) is invariant with respect tomn — m so that it depends on m2 only. Therefore, we have
OO
(1 - p)~m/2F(a + 1, a + 1 - m; 2a + 2;p) = ^ Wk(a, -m2)pk (2.7)
k=0
where Wk{a, t) are polynomials in t with coefficients rational in a. Therefore, operators Da<k are:
D',k=j(<r)Wk(a,-^) (2.8)
Write a recurrence relation for Wk and several polynomials Wk'
(k + l)(2a + k + 2)Wk+\{a, t) - [{a + 2k + l)2 - 2k(k + 1 )]Wk(a, t) + [{a + k)2 + t/4]Wk-i(a, t) = 0
W0(a,t) = l Wi(<M) = \{<r+ !)
W2(a t) =_________-_____t + (<T+1)(^ + 2)2
; 8(2o+ 3) 4(2o+ 3)
W3(a t) =_________<T + 3 t + (°~+ 1)((T + 2)(<T + :3)">
31 ’ ' 16(2(7 + 3) 24(2(7 + 3)
\/y4(a t) = __________^___________t2 - 0-3 + ^a2 + 33cr + 35^ ^ (a + 1 )((7 + 2)((7 + 3)2((7 + 4)J
128(2(7 + 3)(2(7 + 5) 32(2(7 + 3)(2o + 5) 96(2(7 + 3)(2<r + 5)
3 The diagonalization of representations on distributions concentrated at the boundary
Let us denote by Am, m £ N = {0,1,2,...} the space of distributions on D having the form
V(u)6{m\p)
where ip £ X>(r),<5(m) the m-th derivative of the Dirac delta function; for p, see (1.3). The space
= Ao + Ai + ... + Am
is invariant under R\ (but each of Ai, A2, • • • not)
Theorem 3.1 Let A ^ 1/2 + 7L. Then for any m £ N there exists a unique subspace Vm C invariant and irreducible with respect to R\ such that its projection to Am is the whole Am. The space £m decomposes into the direct sum of irreducible invariant subspaces:
£m = Vb + Vi + ... + (3.1)
The restriction of R\ to Vm is equivalent to T_,\-i+m(~ ^A-m). The decomposition (3.1) is orthogonal with respect to (, )\. A distribution £ in Vm is characterized by its highest term ■ <$(m)(p), namely,
m | *2
f = Ef-1)' T^TTv 551 )* • *""«« (3.2)
j= 0 '
where Wj are polynomials defined by (2.7).
Proof. First let m — 1/2 < A < m + 1/2, m £ N. Take an arbitrary function <p £ V(T) and denote
$ = V\-mV
The map ip t-* pm<$ intertwines T-x-i+m and R-x-2 and its image is irreducible subspace - where R-a-2 acts as T_A-i+m(~ T\-m). The function pm<b has the following asymptotic when p —*• 0 (see (2.1)),
oo oo
(pm<t>)(*) ~vmY. ' P* + pu-m+1 £ Dx-m,.<P ■ p* (3.3)
5=0 3=0
Apply to pm$ the operator (A — p.)Y~2^~2 where [i is a complex parameter. This operator is the multiplication by (A — /i)p-2/i-2. The function (A — p)p-2/i-2+m<l> is regarded as a distribution on D. Let fi —► A. Then (see [3])
(A - rip»-2"—'> - ^(-lyiWfp)
Therefore, by (3.3) our distribution tends to
1 m (—] \m~5 4=0 V 7
The map y? i—>- C intertwines T_A-i+m and R\, its image is exactly the desired subspace Vm. By analyticity we can free ourselves from the conditions m — 1/2 < A < m + 1/2, m 6 N, so that A ^ 1/2 + 7L remains only.
Substituting (2.8) in (3.4), we obtain (3.2). □
Write down distributions from Vm for m = 0,1, 2, 3,4:
<p6,
<p6' -
VS" - (A - \)vb' + -^i_^[2A2(A - 1)*> - p'% <pS'" - |(A - 2)Vi" + 4(2/_ 3)'[2(A - 1)2(A - 2)<p - f"W+ + 8(2a‘_ 3) [~2A2(A - 1)(A - 2)<p + 3Av"]6, v6W _ 2(A - 3M'" + 2(2A3_5)[2(A - 2)2(A - 3)*> - *>"]«"+ + 2(2a1-5)[—2(A - 1)2(A - 2)(A - 3V + 3(A - %"]«'+
+ 16(2A-3)(2A-~5j[4A2(A - i)2(A - 2)(A “ 3)V “ 12(A3 - 2A2 + A - W + V4’]*-
References
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