CC BY
8
UDK 517.98
DOI: 10.20310/1810-0198-2017-22-6-1218-1228
CANONICAL AND BOUNDARY REPRESENTATIONS ON THE LOBACHEVSKY PLANE ASSOCIATED WITH LINEAR BUNDLES
© L.I. Grosheva
Tambov State University named after G.R. Derzhavin, 33 Internatsionalnaya st., Tambov, Russian Federation, 392000 E-mail: [email protected]
We describe canonical representations on the Lobachevsky plane, associated with sections of linear bundles, corresponding boundary representations and Poisson and Fourier transforms. Keywords: Lobachevsky plane; canonical representations; distributions; boundary representations; Poisson and Fourier transforms
We give a generalization of the work [1] where we studied canonical and boundary representations of the group G = SU(1,1) on the Lobachevsky plane D. Canonical representations in [1] are deformations of the quasi-regular representation U of G on D . Now we study similar deformations of representations of G in the space sections of linear bundles on D, or, what is the same, deformations of representations of G induced by characters of a maximal compact subgroup K. See also our note [2].
The Lobachevsky plane is the unit disk D : zz< 1 on the complex plane with the linear-fractional action of G :
The boundary S of D is the circle zz = 1, it consists of points s = exp ia, the measure ds on S is da. Let D be the closure of D : D = D U S. The stabilizer of the point z = 0 is the maximal compact subgroup K = U(1) consisting of diagonal matrices:
so that D = G/K. Recall principal non-unitary series representations of G trivial on the center. Let a € C . The representation Ta acts on the space D(S) by
§ 1. Canonical representations associated with a character of K
(1.1)
(Ta(g)<p)(s) = <p(s ■ g)\bs + â\2°.
If a€ Z , then Ta is irreducible and equivalent to T-a-1 (for a € Z there is a "partial equivalence"). The following operator Aa acts on D(S) and intertwines Ta and T-a-1:
A<p)(s) = f |1 - su\-2a-2 ip(u)du, (1.2)
Js
exponents sn are eigenfunctions for Aa with eigenvalues an(a):
°"<a) = 2' r^j-Tr- - n) • <L3)
We shall use denotation
a[q = a(a + 1) ... (a + q - 1), z^ 'n = \z\»( Z)n, fi € C, n € Z, q € N.
V \z\ J
Let A € C, m € Z. We define the canonical representation Rx,m of the group G as follows:
(Rx, m(g)f) (z) = f (z ■ g) (bz + a)-2X-4 • 2m,
it acts on the space D(D). This space consists of functions f (z) = f (z,z) such that for any f € D(D) there is a neighbourhood U of D such that f belong^ to D(U). The representation Rx,m is the restriction to G of a representation of the overgroup G = SL(2, C). Introduce the inner product
(f, h )d = f (z) h(z) dxdy, z = x + iy. (1.4)
Jd
It is invariant with respect to the pair (Rx,m, R_x_2 m):
(RX,m(g)f, h )d = (f, R-j-2,m(g-1)h )d, (1.5)
where g €G . Let us define the operator Qx,m - first on D(D):
(Qx,mf) (z) = c(A, m) i (1 - zw)2X,2m f (w)dudv, Jd
-A + m - 1
id
where
c(A, m) =
n
It intertwines Rx,m and R-\-2,m :
Qx Rx (g) = R — X—2,m (g) Qx g G, and interacts with the form (1.4) as follows:
(Qxmf,h ) D = (f, Qx, m h) D. (16)
The formulae (1.5) and (1.6) allow to extend the representation Rx,m and the operator Qx,m to the space D'(D) of distributions on D .
§ 2. Boundary representations
Any canonical representation of the group G generates 2 representations related to the boundary S of the disk D (see L\m and M\,m below).
Introduce on C polar coordinates (r,s): z = rs, r ^ 0, s € S . Let
p = 1 — zz = 1 — r2,
so that D = {p> 0} and S = {p = 0} . The Euclidean measure dxdy on D is (1/2) dpds .Consider the Taylor series of f € D(D) in powers of p :
f (z) ~ ao + ai p + a2p2 +----, (2.1)
where ak = ak(s) are functions in D(S):
ak(s) = k {dp) l=of(z)
Denote by Sk(D) the space of distributions on C concentrated at S and of the form
c = Ms) m + Ms) S'(p) + ••• + pk (s) 5(k)(p), (2.2)
where 5(p) is the Dirac delta function on the real line (being a continuous linear functional on D(R)) and 5(j)(p) its j -th derivative. Set
S(D) = U=o ^(D).
There is a natural filtration
So(D) C Si(D) C MD) C-^ (2.3)
A distribution p(s) 5(l\p) acts on a function f €D(D) as follows:
(p(s) S{l)(p),f) = 1(—1)11. (P, ai)s.
The canonical representation R\,m acting on D'(D), preserves the space S (D) and the filtration (2.3). Denote by L\, m the restriction of R\,m to S (D). Let us assign to the distribution (2.2) the column (p0, pi,..., pk, 0, 0,...) .
Lemma 2.1. On these columns the representation L\,m is a upper triangular matrix. It is equivalent to a upper triangular matrix with diagonal T-\-i, T-\, T-\+i,... . The equivalence is given by multiplication of the functions pk(s) by s-m .
Proof. Set pk(s) s-m = ik(s). We have to trace how the operator L\ m(g) (g € G) acts on the distribution ik(s)sm • 5(k\p). This distribution is mapped on
ik(s) s™ 5(k\p) (bz + a)-2X-4'2m. (2.4)
Since ps = p •\bz + a\-2 and 5(k\p) is homogeneous of degree —k — 1, the distribution (2.4) is equal to
ik(s) sm (bz + a)-2X-2-2k,2m 6(k)(p) = ik (s) sm (bs + a)-2X-2-2k,2m S(k\p) + •••, (2.5)
where the dots means a distribution in Sk-i(D). Since s = s-1, we have
_ as + b bs + a ,0 2 . .
s = --= s •--= s • (bs + a)0,-2, (2.6)
bs + a bs + a ' y '
so that the distribution (2.5) is equal to
4>k (S) lbs + a-2X-2+2k ■ sm ô(k) (p) + ■■■.
The factor in front of sm 5(k'l(p) is precisely (T-\-\+k(g)^k) (s). □
For f eV(D), let a(f) denote the column (a0,a\,...) of the Taylor coefficients of f , see (2.1). The representation M\,m acts on these columns by:
Mx,m(g) a(f ) = a(Rx,m(g)f )■
Lemma 2.2. The representation M\,m is a lower triangular matrix. It is equivalent to a lower triangular matrix with diagonal T-\-2, T-\-3,... . The equivalence is given by multiplication of the Taylor coefficients ak(s) by s-m .
Proof. By expanding functions in a Taylor series, we find that the k -th Taylor coefficient of the function fg(z) = (Rx,m(g)f)(z) is
ak (s) (bs + â)-2X-4-2k '2m + ■■■
= ak (s) SS-m lbs + âl-2X-4-2k ■ sm + ■■■, (2.7)
where the dots means a linear combination of a0(s),..., ak-1(sS) whose coefficients are some functions of s . Here we used again that p = p ■ lbz + al-2 and formula (2.6). Now setting ak(s) = dk(s) sm , we see from (2.7) that the coefficient dgk (s) for fg (s) is (T-\-2-k (g) dk ) (s) + ■■■. □
§ 3. Poisson transform
Let X, a € C and m € Z . We define the Poisson transform associated with the canonical representation R\,m as the map P? : D(S) — Crx(D) by the following formula
(P\m) v) (z) = P-X-a-2 J (1 - sz)2a'-2m sm v(s) ds. (3.1)
Theorem 3.1. The Poisson transform P^ intertwines the representations T-a-\ and the canonical representation R\,m :
R\,m(g) p? = Pff T-a-i(g) (geG).
Theorem 3.2. With the intertwining operators A* and Q\,m the Poisson transform P^? interacts as follows:
P? A* = a-m(a) P("-i-V (3.2)
Qx,m P? =A(m\X, a) P™ , (3.3)
where
A (m)(A a) = r(-A + a)r(-A - a - 1) A (A'a) T(-A - m)T(-A + m - 1) ^
Proof. Formula (3.2) follows immediately from (3.1). Let us prove (3.3). Applying the operator QX,m P^rnm? to a function p € D(S) , we get the multiple integral:
5 (m) ^ (z) = c(\ m) I (1 _ zW)2X,2m (1 _ W'W)-X-a-2dudV
(Q x,m P? v) (z) = c(X, m) j(1 - zw)2x'2m (1 - ww)
x j (1 - sw)2a'-2m sm v(s) ds, w = u + iv (3.4)
JS
By (3.1), the function (p^ pj (z) behaves as Ci p { a 2 + C2 p x+a 1 when p ^ 0 .Therefore, the integral (3.4) converges absolutely for Re a> —1/2 , Re (A + a) < —1, Re (—A + a) > 0 , and we can then change the order of integration. We obtain
Q,m P{m p) (z) = c(A, m) J K(z, s) sm p(s) ds, (3.5)
where the kernel K(z, s) is given by
K(z, s)= f (1 — zw)2{,2m (1 — ww)-{-a-2 (1 — sw)2a'-2m dudv. Jd
Let us compute it. Using the formula
1 — zw
1 — zw =
(bz + a)(bw + a)
and similar formulae with replacing z by s and by w, we find that the kernel K (z, s) has the following invariance property:
K (I, I) (bz + a)2X,2m (bs + a)2a,-2m = k (z, s).
Take here z = 0, s = 1 and write z and s instead of I and I respectively. Then we have
K (z, s) = K (0,1) a-2X'2m (b + a)-2a'2m (3.6)
and _ _
b a + b
z = -, s = -—=.
a b + a
For these z and s we find
1 _ 1 1 _ 1 a + b b 1
1 — zz = —, 1 — sz = 1 —
aa b + a a a(b + a)'
so that
a-2X,2m (b + a)-2a'2m = (1 - zz)X-a (1 - sz)2a'-2m. (3.7)
It remains to compute K(0,1) :
K(0,1)= j (1 - ww)-X-a-2 (1 - w)2a'2m dudv. (3.8)
Jd
Expand (1 — w)2a'2m in a binomial series:
(1 - w)2a'2m = (1 - w)a+m (1 - w)a-m
= £ ( *+m )( *) (-1)«*wqw.
A non-zero contribution to (3.8) is given by the terms with q = j only, so that
K(0,1) = * + m ) (* -qm^ D (1 - ww)-X-°-2 (ww)q dudv.
The latter integral is equal to B(q +1, —X - a - 1), so that
(a + m)[q1 (a - m)[q1 T(-X - a - 1)
oo
K (0,1) =
= n ■
q=0 q! r(-X - a + q)
^ (-a - m)[q1 (-a + m)[q1 ^ (-X - a - 1)lq+1]q\
n A (-a - m)[q1 (-a + m)[q1 -X - a - 1 ^ (-X - a)[q1 q!
q=0 v '
IT
—--- F (-a - m, -a + m; -X - a; 1)
-X - a - 1
T(-X - a - 1)T(-X + a) T(-X + m)T(-X - m)
1 A(m)(X,a).
c(A, m)
Thus, collecting (3.5), (3.6) and (3.7), we obtain
(Qx,m pm p) (z) = A(m)(A,a) px- S (1 - sz)2°,-2m sm p(s)ds,
which is just (3.3). □
Theorem 3.3. Introduce on D polar coordinates r, s : z = rs, 0 ^ r ^ 1, s€S . Let 2a € Z . For any K -finite function p € D(S), the Poisson transform P^ P of p has the following expansion in powers of p = 1 - r2 :
(P? p) (z) = p-A-a-2 s^ (C? p)) (s) ■ pk
<x
+ pA+a-1 smYl (D? p) (s) ■ pk. (3.9)
k=0
Let us the factors px a 2 and p x+a 1 in (3.9) leading factors. The factors yield that P^ is
„A - a - 2
A,a
meromorphic in a , and has poles at the points
a = A - k, a = -A - 1 + l (k,l € N). (3.10)
All poles are simple except in the case when the two sequences (3.10) have a non-empty intersection and the pole belongs to this intersection. This happens when 2A + 1 €N and 0 ^ k, l ^ 2A + 1, k +1 = 2A + 1. In this case the pole i is of the second order. Let us write down the principal part
of the Laurent series of Pm at the poles ¡i of the first and the second order respectively:
p(m)
P? = --- + ■■■ (3.11)
iM
P? = f_ A'^)2 + + ■ ■■ (3.12)
(m)
(a - fx)2 a - f
The first Laurent coefficient ( p(m and PA ^ respectively) intertwines T-^-\ with RA
Let us write down the Laurent coefficients in (3.11) and (3.12) explicitly.
For that we introduce the following differential operators WO"k on S . Let us set
Va,m,n(p) = (1 - p)(m+n)/2 F (* + 1 + m,* + 1 + n; 2* + 2; p), where F is the Gauss hypergeometric function. Expand V in powers of p :
Vamn (p) = ^ w^mk (n) Pk, k=0
here wO^k are polynomials in n of degree k. The coefficients of these polynomials are rational
functions of a with simple poles at a = —1, —3/2,..., (—k — 1)/2 . Now we set
(1 d).
\i da J
If a pole i belongs only to one of the sequences (3.10), then it is simple and
W™ = w(mA- -
I , da,
Pitk = (—1)k+m 1 a-m(A — k) (3.13)
P{%-i+l = (—1)l+m 1 & ◦ A{-i, (3.14)
where {k is the following operator D(S) — Sk(D):
p = s™ h—1)n (Witn p) (s) 5{k-n)(p). (3.15)
n=0 ( )!
The operator {{m is meromorphic in A . For fixed k = 1,2 ... it has poles (simple) at the points
k1
A = k — 1, k — 3/2, k — 2,...,-
(k poles in total). It intertwines T-{-i+k with L{,m (restricted to Sk(D)). A number A0 € N/2 is a pole for {k for those k that satisfy
Ao + 1 ^ k ^ 2Ao + 1.
In particular, let A0 € N. Denote by the residue of ({lk at A = A0 and denote by W^
the residue of W^ at the pole a = t . The contribution to the residue of {j} is given by the summands in (3.15) for which n ^ 2k — 2A0 — 1. So we have (we omit the index 0) for A € N and A + 1 ^ k ^ 2A + 1:
p = sm £ (—1)n ^^ (Wtinp) (s) • s(k-n)(p). n=2k-2{-i (k — n)!
Let the pole i belong to both sequences (3.10). This happens when 2A + 1 € N . Then i = A — k = = —A — 1 — l, where k,l € N , so that k +l = 2A + 1 and l — k = 2i + 1.
Let first A € N . Then the pole i is of the second order (m = 0). Here we have a difference with the case m = 0 : in that case the pole ¡ was of the first order.
We shall write down, for X € N , only the first Laurent coefficients P . The expressions for the residues are rather complicated and not interesting for us, even more because they turn out to be not concentrated at S .
If X + 1 ^ k ^ 2X + 1 (so that k>l and f ^ -1 ), then
(m) I w \
PAA-k = (-1)k+m ^ a-m(X - k) pm, and if X + 1 ^ l ^ 2k + 1 (so that k < l and f ^ 0 ), then
(m)
Px,-x-i+l=(-1)i+m 1 vxm> o Ax-i.
Therefore, the operator i^k intertwines T-x-1+k with Lxm restricted to Y*k(D).
Let now A € -1/2 + N. This case is similar to such a case for m = 0. The pole i is of the second order.
If k ^ l, then
p(m) (-1)k+m
PA= 2 k! P-m(f) £A,k ,
pA? v = -smE -nn C(mn)v ■ *l-n(p),
n=0
and if k ^ l, then
■^(m) (_-\)l+m , , ^
V o ( 1) c(m) A
px,t = 2-ji-Zx,i o Ax-i,
pxm p = smE -l- d^p ■ s(k-n)(p),
where a-m(i) is the residue of a-m(a) at a = i (Air is the residue of Aa at a = t ) and
^(m) f Ci"n\ n< 2I + 1,
",n I Ctn - D0(m!-2,-i, n > 2i + 1,
D(m) f Dm, n< -2I - 1
{ D°t,n - C°t,n+2t+1, n ^ -2l - 1. Theorem 3.4. Up to a factor, the composition of the operators Qx,m and ^x^ is the Poisson transform P(mx^_2 x-k '■
Q. e(m) = q(m) . P(m) (3i6)
Qx,m ?x,k = qx,k P-x-2,x~k,
k
where
q? = 1 (-1)k+m k! a-m(-X - 1 + k) A(?)(X), (3.17)
A(m)(A)= 1 (2A 2k + 1)nA±m±HiA-m±21
Ak (A) = -(2A - 2k + 1) kW(2A + 2 - k) .
Proof. Taking the residue of both sides of (3.3) at a = A - k and using (3.13), we obtain (3.16), where
q{m = (-1)k+m ki aJx - k) Resa=x-k A(m)(A,a).
The latter residue is equal to
v—kTT *
Finally, computing the product a-m(a) a-m(—a — 1), we obtain expression (3.17). □
Remark. Formula (3.3) seems to contain a contradiction: indeed, the Poisson transform Pm_2 , in the right hand side has poles at the points a = A + 1 +1 and a = —A — 2 — k (k,l € N), but the left hand side seems to have no poles at these points. In fact, the left hand side does have poles at these points; the poles in question are poles of distributions; the left hand side, regarded as a distribution, assigns to a function f € D(D) the scalar
(Q{m P{m\ f )d = (P{m\ QXmf )d,
but the function Q{,mf has asymptotics Ci + C2 p2{+2 when p — 0, and the function p2{+2
m _
§ 4. Fourier transform
together with the leading terms p { « 2 and p {+tJ 1 of pa™ gives the desired poles.
Let A, a € C and m € Z . We define the Fourier transform associated with the canonical representation R{,m as the map F{m«_ : D(D) —D(S) by the following formula
(F^f) (s) = s-m j (1 — zz)2a,2m p{-« f (z)dxdy.
The integral converges absolutely for Re (A — a) > —1, Re (A + a) > —2 and can be meromorphically continued in a and A .
Theorem 4.1. The Poisson and the Fourier transform are conjugate to each other:
(F{m_f, p)s = (f,pm_2wp)d.
This allows to transfer statements about the Poisson transform to the Fourier transform. The Fourier transform interacts with the intertwining operators as follows:
A, Fm = a - m(a) F{mJa - v
F-mU« Q{m = A((A, a) F^.
It has poles in a at the points
a = —A — 2 — k, a = A + 1 + l (k,l € N). (4.1)
All poles are simple, except the case —2A — 3 € N and the pole i belongs to both sequences (4.1), i.e 0 ^ k,l ^ —2A — 3 and k +1 = —2A — 3 .In this case i is of the second order.
For the Laurent coefficients of the Fourier transform we use a similar notation as in case of the Poisson transform.
m_ m_
The first Laurent coefficient (i.e. F{m_ if I is of the first order and F{ ^ if i is of the second order) intertwines R{ m with T
№
m_ m_ Let us write down F{ ^ and F{,№ explicitly.
If the pole i belongs to one of the sequences (4.1), then it is simple and
t-{-2-k = 2(—1)m a-m(—A — 2 — k) b<$,
FS+i+l = — 2(—1)m A-{-2-i b
where bm is a "boundary" operator D(D) — D(S) which is defined in terms of the Taylor coefficients cn of f as follows:
bti_(f) = E W{-m---k*-n (s-mcn).
n=o
theorem 4.1 now gives:
Theorem 4.2. The operators and b(m) are conjugate to each other (up to a factor):
(f, ({m)-2k p)D = 1( — 1)k k. (b(3(f), p)s.
The operator bmk intertwines R{,m with T-{-2-k. It is meromorphic in A. For fixed k = 1,2,... it has poles (simple) at the points:
k3
A = -k - 1, -k - 1/2,
2
( k poles in total). A scalar Ao G -2 - N/2 is a pole for b^k if
-Aq - 1 < k < -2Ao - 3.
In particular, let A0 G-2 - N. Denote by ^mk the residue of bm at A = A0 . Then (cf. § 3) (we omit the index 0) for A G-2 - N and -A - 1 ^ k ^ -2A - 3 we have
-2X-3-k
t$(f) = - £ w-mU-kk-n (s-mcn).
Let the pole i belong th both sequences (4.1). This happens when —2A — 3 € N . Then i = —A — — 2 — k = k + 1 +1, where k,l € N , so that k +1 = —2A — 3, l — k = 2i + 1. This pole is of the second order.
Let i €—2 — N . As in § 3 we write only down the first Laurent coefficient: if —A — 1 ^ k (then k>l and i ^ —1), then
Sm) 1
sFm)
FX,-X-2-k = 2 (-1)m a-m(-A - 2 - k) bXkk
and if -A - 1 ^ l (then k < l and f ^ 0 ), then
2
Let f G -5/2 - N .If k < l, then
(m)
F(m) 1 m F.m)
FX,X+1+l = -ô (-1)m F-X-2-l bX,l .
Fx , n = (-1)m F-m(f) b™ 1l
FXmjf = - 1e cm (s-mci-n),
2 n=o
and if k ^ l, then
(m)
fy. = (-1)m f-x-2-i b
(m)
X= (-1) F-X-2-l bx,l k
I(m) (s
n=o
FXm)f = 1e ds (s-m ck-n).
REFERENCES
1. Molchanov V.F., Grosheva L.I. Canonical and boundary representations on the Lobachevsky plane // Acta Applied Mathematics, 2002. V. 73. P. 59-77.
2. Grosheva L.I. Canonical representations on sections of linear bundles on the Lobachevsky plane // Tambov University Reports. Series: Natural and Technical Sciences. Tambov, 2007. V. 12. Iss. 4. P. 436-438.
Received 4 September 2017
Grosheva Larisa Igorevna, Tambov State University named after G. R. Derzhavin, Tambov, the Russian Federation, Candidate of Physics and Mathematics, Associate Professor of the Functional Analysis Department, e-mail: [email protected]
УДК 517.98
DOI: 10.20310/1810-0198-2017-22-6-1218-1228
КАНОНИЧЕСКИЕ ПРЕДСТАВЛЕНИЯ НА ПЛОСКОСТИ ЛОБАЧЕВСКОГО В СЕЧЕНИЯХ ЛИНЕЙНЫХ РАССЛОЕНИЙ
© Л. И. Грошева
Тамбовский государственный университет им. Г.Р. Державина 392000, Российская Федерация, г. Тамбов, ул. Интернациональная, 33 E-mail: [email protected]
Мы описываем канонические представления, связанные с сечениями линейных расслоений, соответствующие граничные представления и преобразования Пуассона и Фурье. Ключевые слова: плоскость Лобачевского; канонические представления; обобщенные функции; граничные представления; преобразования Пуассона и Фурье
СПИСОК ЛИТЕРАТУРЫ
1. Molchanov V.F., Grosheva L.I. Canonical and boundary representations on the Lobachevsky plane // Acta Applied Mathematics, 2002. V. 73. P. 59-77.
2. Грошева Л.И. Канонические представления в сечениях линейных расслоений на плоскости Лобачевского // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2007. Т. 12. Вып. 4. С. 436-438.
Поступила в редакцию 4 сентября 2017 г.
Грошева Лариса Игоревна, Тамбовский государственный университет им. Г.Р. Державина, г. Тамбов, Российская Федерация, кандидат физико-математических наук, доцент кафедры функционального анализа, e-mail: [email protected]
For citation: Grosheva L.I. Canonical and boundary representations on the Lobachevsky plane associated with linear bundles. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Reports. Series: Natural and Technical Sciences, 2017, vol. 22, no. 6, pp. 1218-1228. DOI: 10.20310/1810-0198-2017-22-6-1218-1228 (In Engl., Abstr. in Russian).
Для цитирования: Грошева Л.И. Канонические представления на плоскости Лобачевского в сечениях линейных расслоений // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2017. Т. 22. Вып. 6. С. 1218-1228. DOI: 10.20310/1810-0198-2017-22-6-1218-1228.
