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ISSN 1810-0198. Вестник Тамбовского университета. Серия Естественные и технические науки
Том 23, № 122
2018
DOI: 10.20310/1810-0198-2018-23-122-113-124
DECOMPOSITION OF CANONICAL REPRESENTATIONS ON THE LOBACHEVSKY PLANE ASSOCIATED WITH LINEAR BUNDLES
oc L. I. Grosheva
Tambov State University named after G. R. Derzhavin 33 Internatsionalnaya St., Tambov 392000, Russian Federation E-mail: gligli@mail.ru
Abstract. We decompose canonical representations on the Lobachevsky plane, associated with sections of linear bundles
Keywords: Lobachevsky plane; canonical representations; distributions; boundary representations; Poisson and Fourier transforms
Introduction
In our work [1] we described canonical and boundary representations of the group G R U[ )2,2+on the Lobachevsky plane D in sections of linear bundles on D. Now we decompose these representations into irreducible ones. We lean on works [2], [3].
The Lobachevsky plane is the unit disk D ; zz < 2 on the complex plane with the linear-fractional action of G :
The boundary S of D is the circle zz R 2, it consists of points s R o t ia. the measure ds on S is da. Let D be the closure of D \ D R D { S. Let
so that D R }p > 1{ and S R }p R 1{. The stabilizer of the point z R 1 is the maximal compact subgroup K R [ )2+ consisting of diagonal matrices:
1. Representations of U[ )2,2+ induced by characters of [ )2+
z z àg R — ^ , g R bz 0 a
J b a
da bb R 2.
p R 2 zz,
so that D R G/K. The Euclidean measure dxdy on D is )2/3-jdpds, a G-invariant measure dp)z+ on D is
dfi)z+R p~2dxdy.
If M is a manifold, then V)M + denotes the Schwartz space of compactly supported infinitely differentiable C-valued functions on M, with a usual topology, and P') Af+denotes the spacc of distributions on M - of antilincar continuous functionals on 'P)M-\r
Recall principal non-unitary series representations of G trivial on the center. Let a / C. The representation Ta acts on the spacc V)S-\-hy
)T„)g-tp$a-№<p)alg-&sQ af.
The inner product from L2)S, ds~H
)i>,(p\s R t}j)u^p)uSs)u+ )2.2+
s
is invariant with respect to the pair T_^_!-)r
If a / Z, then Ta is irreducible and equivalent to T-a-i (for a / Z there is a "partial equivalence").
The following operator Aa acts on V)S-\- and intertwines Ta and T_ir_1 :
s
exponents ^„)s+R sn are eigenfunctions for Aa with eigenvalues an)a-H
an)a+R3n) 2-P —-± 2+
) <J U 71+ ) <J The composition A^A ^ x is a scalar operator:
AaA_a_ i R -—-—- ±E
67TU!)a+
where £*j)cr+ is a "Plancherel measure" (see Theorem 1.1):
2 \ 2 /
0 3 epxtTTT,
The operator Aa is mcromorphic in a with simple poles at u / )2/3-K) N.
There are four series of unitarizable irreducible representations: the continuous series: Ta, a R )2/3+0 ip, p / M, an inner product is (1.1); the complementary series: Ta, 2 < a < 1, an inner product is the form )A^i/j,(p\s with a suitable factor; the holomorphic and antiholomorphic series consisting of subfactors T„t± of a / Z.
We shall use denotation:
Let us take characters (one dimensional representations) of the group K that are trivial on the center QG, namely,
ojm)fc+Ra2mRa"2m, k / K, m / Z.
Denote by U^ the representation of the group G induced by the character uim. It acts by translations on the space of functions ip / V) G + satisfying the condition
tfj)kg+R ujm)k-\^p)g-\r It can be realized on functions on the disk D :
U{m))g~tf \ )z+R f)z^g+)bzQ a4'2m.
The representation U^ moves the Casimir element of the Lie algebra g to the Casimir operator (a differential operator on D). Its radial part is the following differential operator on (2, e +:
The representation [/!m> preserves the inner product
)/, h-fa R f)z-th)z-tfii)z-It
D
We denote the unitary completion of £Am) acting on L2)D1dp,+ by the same symbol.
Let V)D+bc the space of restrictions to D of functions from 7?)C+ with the induced topology, and by V)D-|-the space of distributions on C with supports in D. Consider the inner product with respect to the Lcbesgue measure on D :
)F,f\DR F)z-tf)z-fady, zRxOiy. )2.4+
D
The space V)D-\- can be embedded into V)D-\- by assigning toh/ V)DJr the functional / U )h,f\D, f / V)D± So we shall write the value of F / V')D+at f / V)D+ in the same form: )F1f\s-
We define the Poisson transform Pa ; V)S-|-oo C°°)D-\- and the Fourier transform F& ; V)D+oc associated to the character U)m, as integral operators
PimV' )2 sz-?a'~2m Sm ip)s^ds. -s
)s+Rs"m )2 sz-F'^p-" f)z-tifi)z+ D
The Poisson and Fourier transforms Pa and intertwine representations 1 —o—\
with and fn-m' with Ta respectively. The Poisson and the Fourier transform are
conjugate to each other:
)Fim)f, ^|sR)/,4mVv
Using the spectral resolution of the radial part of the Casimir operator (1.2), we obtain the following Plancherel theorem for
Theorem 1.1. Let us assign to a function f / V)D+ the family }Fa f( where o R 2/3 0 ip1 p / M, of its Fourier components of the continuous series and the family , /{ where k R 1, 2,..., 2, of its Fourier components of the analytic (if m < 1) or the anti-analytic (if m > 1) series. This correspondence is G -equivariant. One has the inversion formula:
oo
f)z+ R u^P™, FW Hz-^L=-i/2+ipdP
—oo H-1
0 / ¿)3fc0
1
1 o
1=0
and the Plancherel formula for functions f,h / V)D-\~:
CO
L '
0 f (1-4)
Therefore, the previous correspondence can be extended from the space V)S-\- to L2 )D, dp+ and gives then the decomposition of the unitary representation on L2)D, dp-\- into
the direct integral of the representations Ta, a R 2/3 0 ip of the continuous series, and the direct sum of §f?i§ representations 7jt + or Tk R 1,2,..., §m,§ 2, of the analytic (m > 1) or anti-analytic (m. > 1) series. This decomposition is multiplicity free.
2. Canonical representations
Let A / C. We define the canonical representation R\,m of the group G associated with a character of K as follows:
)Rx,m)g^f+)z+R f)z^g+)bz 0 a-F2A_4,2m,
it acts on the space V)D-\r
The inner product (1.3) is invariant with respcct to the pair )Rx,m, R-x-2 m+*
)Rx,m)g^, h\DR)f, R_^_2tJg-'4i\D, g/G. )3.2+
Let us define the operator Q.\,TO - first on V)D~H
)Q\,mf+)z+R c)A,m+ )2 zw-?'2™ f)w-&udv,
D
where
„ AO m 2 cJA, m+R-.
7T
It intertwines and -fl_.\-2,m :
Q\,mR\,m)g+R- R \ 2,m)g^Q\,mi £7 / G,
and interacts with the form (1.3) as follows:
)Qx^f,h\DR)f,QXtmh\D. )3.3+
The formulae (2.1) and (2.2) allow to extend the representation R\,m and the operator Qx,m to the space V')D-|-of distributions on D.
Canonical representations Rx,m generate boundary representations L\.m and M\m. Consider the Taylor series of / / V)D+ in powers of p:
f)z+C cio 0 ciipO a2p2 0 =t±±^
where aj. R afc).s+are functions in V)S-H
Let a)/+denote the column )ao,«1, ■ ■ . + of the Taylor coefficients.
Denote by ! ^D+the space of distributions on C concentrated at S and of the form
where ¿)p+is the Dirac delta function on the real line (being a continuous linear functional on 'P)M+) and its j - th derivative. Set
! )D+R{%L0! k)D+
There is a natural filtration
! i2)D-h^±S± )3.4+
A distribution yj)s4^®)jH-acts on a function / / V)D-\-as follows:
|dr|) 2il{)ip1al\s. )3.5+
Distributions from ! fc)JD+can be extended to a wider space than V)D-\r Namely, let 14)Z)+ be the space of functions / on D of class C°° on D and on S and having a Taylor decomposition of order k :
f)z+R a0 0 aip 0 a2p2 0 ... 0 akpk 0
uniformly with respect to u / 51, where am R am)f+ belong to V)S-\r Then (2.4) is well preserved for / / UjjD-lr
The canonical representation R\,m acting on V')D-\^ preserves the spacc ! )D+and the filtration (2.3). The first boundary representation L\ m is the restriction of Rx,m to ! )D-\r The second boundary representation acts on columns a)/+by:
M^m)g^a)f+R a)Rx,m)gtf+
Theorem 2.1. The representation L\m is equivalent to a upper triangular matrix with diagonal T_\_i, .... The equivalence is given by multiplication of the
functions tpk)s+ by s~m. The representation M\jTn is equivalent to a lower triangular matrix
with diagonal T-x-2, T-\s,____ The equivalence is given by multiplication of the Taylor
coefficients a^)s-\-by s~m.
Let N R }1,2,3,...{. In the generic case: 3A / N, the representation L\m is diagonalizable, which means that the space ! )D+ is the direct sum of the spaces Vff )k / N+ so that L\ m is the direct sum of the T^x—i+k )k / N+
3. Poisson transform
Let A, a / C and m / Z. We define the Poisson transform associated with the canonical representation Rx,m the map Pj^ ; V)S+oo C'°°)Di-by the following formula
) Pff <p p-x~a-2 )2 sz-^'"2™ sm ip)s^ds.
The Poisson transform P-J™ intertwines the representations T_fT_1 and the canonical representation Rx,m '■
Rx,m)gW^ R PT „1)ff+ )g / G+
With the intertwining operators Aa and Qx,m the Poisson transform interacts as follows:
P^A, R
where
7 ' ) A m+ ) A 0 m 2+ The Poisson transform is mcromorphic in <r, and has poles at the points
a R A k, a R A 2 0 I )k,l / N+ )4.2+
All poles are simple except in the case when the two sequences (3.1) have a non-empty intersection and the pole belongs to this intersection. This happens when 3A 0 2 / N and
1 ^ A;, / ^ 3A 0 2, At 0 I R 3A 0 2. In this case the pole fi is of the second order. Let us write down the principal part of the Laurent series of P^1 at the poles fi of the first order:
p(m)
p(m) R Q ^
' <J ¡1
The residue intertwines Tpi with Rx,m- Let us write it explicitly. We set
Va,m,n)p+R )2 p-fm+n}/2 F)u 0 2 0 m, a 0 2 0 7i=3tr 0 where F is the Gauss hypergeometric function. Expand V in powers of p:
00
va,m,n)p+R f j <i
here w^j? are polynomials in n of degree k. The coefficients of these polynomials are rational functions of a with simple poles. Now we set
W(m) R ™>(m) V d ^
If a pole fi belongs only to one of the sequences (3.1), then it is simple and
Pt\1+i R )
where ™ is the following operator V)S-foo ! jt)Z?-H
k
& V R ^ j ) 2-F ^J^- ) )4-3+
The operator is mcromorphic in A. For fixed k R 2,3. . . it has k poles (simple) at the points A R k 2, k 4/3, k 3,..., )k 2-1/3. It intertwines T-x-i-i-t with L\ m (restricted to ! *)£)+).
Theorem 3.1. Up to a factor, the composition of the operators Qx,m and is the Poisson transform P^_2 \-k
n R i ptm)
where
?S>R§) A 20 Mr'JH
n<™))A+R A)3A 3t0 24 )A0'"02+)A "'03+ k ' 3tx2> k( )3A 0 3 k+
4. Fourier transform
Let A, a / C and m / Z. We define the Fourier transform associated with the canonical representation Rx,m the map Fj^ ; V)D-\-oc V)S+ by the following formula
jfy/^s-"1 )2 zs^2mpX-° f)zMxdy.
The integral converges absolutely for So)A <j-\- > 2, So)A0 <7+> 3 and can be mcromorphically continued in a and A. The Poisson and the Fourier transform are conjugate to each other:
This allows to transfer statements about the Poisson transform to the Fourier transform. The Fourier transform interacts with the intertwining operators as follows:
A. iff R a ^aW^l v F^U„Qx,m R <Hi£?.
It has poles in a at the points
a R A 3 k, a R A 0 2 0 I )k,l / N+ )5.3+
All poles are simple, except the case 3A 4 / N and the pole fi belongs to both sequences (4.2), i. e. 1 ^ k,l ^ 3A 4 and k 0 / R 3A 4. In this case fi is of the second order. For the Laurent cocfficicnts of the Fourier transform we use a similar notation as in case of the Poisson transform. The first Laurent coefficient Fj™/ for the first order fi intertwines R\,m with Tfi. Let us write it explicitly:
Ft-x-2-fc H 3) A 3
i-M R 3)
where h ™2 is a "boundary" operator V)D-\-oc V)S+ which is defined in terms of the Taylor coefficients c„ of / as follows:
d=o
The operators £<m> and
are conjugate to each other (up to a factor): 2 "i *(>©/+Pis-
The operator intertwines R\,m with '/' \ 2 t- It is ineromorphic in A. It has k poles (simple) at the points A R k 2, k 2/3,...,) k 4-|/3.
5. Decomposition of canonical representations
For simplicity we restrict ourselves to generic A lying in the strips )k / Z+:
4/30 k < SoA < 2/30 k.
Case A: A / /q. Let /, h / V)D-1 Consider the functions
f0)z+Rpx+2f)z^ h0)z+Rp~Jh)z+
Since A / Jo, both functions /o)z+and /10)2 + belong to L2)D,dp-\r Let us apply to this pair of functions /0, h0 the Planchercl formula (1 19). We obtain:
00 ,
)/o, KM, R f0, hQ\st dp
-00 b=-l/2+ip
H-1 9 ^
0 £ ^)3n0 2+)F^f0:F(^_1h0\s.
Then we return to / and h :
—00
H-i 9 v
¿¿H2-»Fg f, fQ^^s. (5-1)
Lr dP
W= l/2+ip
0
Now usimg the conjugacy (4.1), we transfer the Fourier transform of h to the Poisson transform of F^J f. We obtain a formula that gives an expansion of / regarded as a distribution in V)D-H
,(m) p(m)
L/JU -pi
H-1
/ R ^..C/1 , dp
0 f ¿)3n0 2^„_iil?/- (5.2)
d 0
Theorem 5.1. Lei A / I0. Then the canonical representation R\,m decomposes, in a similar way as U M, see 2, into the direct integral of the representations Ta, a R 2/3 0 ip, of the continuous series and the direct sum of fyre§ representations T,1 + or , 71 R 1, 2,... j 2, of the analytic (771 < 1) or the anti-analytic series (m > 1) with multiplicity one. Namely, if we assign to f / V)D+ the family of Fourier components
is
}F^f{ where a R 2/3 0 ip and a / }1,2, ...,ffre§ 2{, then this correspondence i G -equivariant. There is an inversion formula ): .3+ and a decomposition ): .2+ of the form
)/, Md-
Case B: A / , k / N. Wc perform analytic continuation of (5.2) from the strip I0 to the right, to the strip Here the poles of the Poisson transform intersect the line of
integration Soct R 2/3 and give additional terms. Wc obtain
oo M"1 k
»
— CO
/ 0 f
j/=0 J=0
f R 0 0 ):.4+
where the integral and the first sum mean the same as in (5.2) and
_M T> 9J+m 2 _2_ ¿.(m) p(m)
A,ti ti, o J ¿ + — ------„,J>X,v A,—A—l-pu"
a_mJ A z (J
The operators , v ^ k, can be extended to ! because the Fourier transforms
occuring in are already extended. Thus, the operators , v ^ k, are defined on the spacc
Vk)D+RV)D^O ! k)D+ ):.5+
The operators , v ^ i:, acting on the space are projection operators onto
the spaccs see § 2 for them, i.e. the following relations hold:
(m) (m) R (m) A,?; X,v ^ X,v i
4'? R 1, vRs.
Thus, in Case B we have
Theorem 5.2. Let A / Ik+i, k / N. Then the space V)D+ has to be completed to the space Vk)D-\^ see ): .5+ On this space the canonical representation R\tTn splits into the sum of two terms: the first term decomposes as does in Case A, the second term decomposes
into the sum of the irreducible representations c T\_v with v R 1,2,..., k. Namely,
let us assign to any f / Vk)D+ the family { where a R 2/3 0 ip, a R n,
n R 1,2,. ..,jj7i§ 2, and a R A 2 0 v, v R 1,2,..., k. This correspondence is G -equivariant. The function f is recovered by the inversion formula ): .4+
Case C: A / I_k _i, k / N. Now we perform analytic continuation of (5.2) to the left, to the strip I-k-1- Here the poles
<j R A 3 v, ctRA0 20 ?j, t> / N, v ^ k,
of the integrand (they are poles of the Fourier transform) intersect the line of integration Soct R 2/3 and give additional terms. We obtain
oo I™!"1 k
f R 0 f 0 f ):.: +
J=o J= o
—oo
where the integral and the first sum have the same meaming as in (5.2) and
-R \ 9jn _2_ p(m)
hx>* K } a m)XQ 2 0 ^ ■
Denote by „ the image of the operator . The operators 4"' with v ^ k can
be extended to the space Lfe) D+ since the operators b with v ^ k are defined on this space. In particular, 4™ can be applied to , s ^ k, and we can consider the products 474? with v,s^k.
Theorem 5.3. The operators 4w , v ^ A;, are projection operators on , namely, the following relations hold:
y,(m) v(m) R
^A.u ^A.u IV ^A.U >
R i, sRv.
Thus, in Case C we have
Theorem 5.4. Let X / A; / N. Then the canonical representation Rx,m
considered on the space 14)D+ splits into the sum of two terms. The first term acts on the subspace of functions f such that their Taylor coefficients e„)/+ are equal to zero for v ^ A', and decomposes as R\,m in Case A, the second term decomposes into the direct sum of the A; 0 2 irreducible representations T-\-2~v (C Ta+i+» ), w R 1,2,..., fc, acting on the sum of the spaces „ . One has an inversion formula, see ): .: -fc-
REFERENCES
1. 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.
2. Molchanov V.F., Grosheva L.I. Canonical and boundary representations on the Lobachevsky plane. Acta Appl. Math., 2002, vol. 73, pp. 59-77.
3. Grosheva L.I. Kanonicheskie predstavleniya v secheniyakh lineynykh rassloeniy na ploskosti Lobachevskogo [Canonical representations on sections of linear bundles on the Lobachevsky plane]. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Reports. Series: Natural and Technical Sciences, 2007, vol. 12, no. 4, pp. 436-438. (In Russian).
Received 23 March 2018 Reviewed 25 April 2018 Accepted for press 5 June 2018
Grosheva Larisa Igorevna, Tambov State University named after G. R. Derzhavin. Tambov, the Russian Federation. Associate Professor of Physics and Mathematics, e-mail: gligli@mail.ru
For citation: Grosheva L.I. Decomposition of canonical representation on the Lobachevsky plane associated, with linear bundless. Vestnik. Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Reports. Series; Natural and Technical Sciences, 2018, vol. 23, no. 122, pp. 113-124. DOI: 10.20310/1810-0198-2018-23-122-113-124 (In Engl., Abstr. in Russian).
DOI: 10.20310/1810-0198-2018-23-122-113-124 УДК 517.98
РАЗЛОЖЕНИЕ КАНОНИЧЕСКИХ ПРЕДСТАВЛЕНИЙ НА ПЛОСКОСТИ ЛОБАЧЕВСКОГО В СЕЧЕНИЯХ ЛИНЕЙНЫХ РАССЛОЕНИЙ
Л. И. Грошева
ФГБОУ ВО «Тамбовский государственный университет им. Г.Р. Державина» 392000, Российская Федерация, г. Тамбов, ул. Интернациональная, 33
E-mail: gligli@mail.ru
Аннотация. Мы разлагаем канонические представления, действующие в сечениях линейных расслоений на плоскости Лобачевского
Ключевые слова: плоскость Лобачевского; канонические представления; обобщенные функции; граничные представления; преобразования Пуассона и Фурье
СПИСОК ЛИТЕРАТУРЫ
1. Grosheva L.I. Canonical and boundary representations on the Lobachevsky plane associated with linear bundles // 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.
2. Molchanov V.F., Grosheva L.I. Canonical and boundary representations on the Lobachevsky plane // Acta Appl. Math. 2002. Vol. 73. P. 59-77.
3. 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 23 марта 2018
Прошла рецензирование 25 апреля 2018 г.
Принята в печать 5 июня 2018 г.
Грошева Лариса Игоревна, Тамбовский государственный университет им. Г.Р. Державина, Тамбов, Российская Федерация, кандидат физико-математических наук, доцент, e-mail: gligli@mail.ru
Для цитирования: Грошева Л.И. Разложение канонических представлений в сечениях линейных расслоений на плоскости Лобачевского // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2018. Т. 23. № 122. С. 113-124. БОТ: 10.20310/1810-0198-2018-23-122-113-124
