Applied Mathematics, 2013, 4, 3540 Published Online November 2013 (http://www.scirp.org/journal/am) http://dx.doi.org/10.4236/am.2013.411A3006 Open Access AM Canonical and Boundary Representations on Rank One ParaHermitian Spaces Anatoli A. Artemov G. R. Derzhavin Tambov State University, Tambov, Russia Email: tria@tsu.tmb.ru Received July 24, 2013; revised August 24, 2013; accepted September 2, 2013 Copyright © 2013 Anatoli A. Artemov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT This work studies the canonical representations (Berezin representations) for paraHermitian symmetric spaces of rank one. These spaces are exhausted up to the covering by spaces GH with SL ,Gn, . For Her mitian symmetric spaces GL 1,Hn GK, canonical representations were introduced by Berezin and VershikGelfandGraev. They are unitary with respect to some invariant nonlocal inner product (the Berezin form). We consider canonical rep resentations in a wider sense: we give up the condition of unitarity and let these representations act on spaces of distri butions. For our spaces GH, the canonical representations turn out to be tensor products of representations of maxi mal degenerate series and contragredient representations. We decompose the canonical representations into irreducible constituents and decompose boundary representations. Keywords: ParaHermitian Symmetric Spaces; Overgroups; Canonical Representations; Boundary Representations; Poisson and Fourier Transforms 1. Introduction Canonical representations on Hermitian symmetric spac es GK were introduced by VershikGelfandGraev [1] (for the Lobachevsky plane) and Berezin [2]. They are unitary with respect to some invariant nonlocal inner product (the Berezin form). Molchanov’s idea is that it is natural to consider canonical representations in a wider sense: to give up the condition of unitarity and let these representations act on sufficiently extensive spaces, in particular, on distributions. Moreover, the notion of ca nonical representation (in this wide sense) can be ex tended to other classes of semisimple symmetric spaces GH, in particular, to paraHermitian symmetric spaces, see [3]. Moreover, sometimes it is natural to consider se veral spaces i GH together, possibly with different i , embedded as open orbits into a compact mani fold , where acts, so that is the closure of these orbits. G G Canonical representations can be constructed as fol lows. Let be a group containing (an overgroup), a series of representations of induced by charac ters of some parabolic subgroup associated with G G P R G GH and acting on functions on . The canonical representations of are restrictions of to G. R GR In this talk we carry out this program for paraHermi tian symmetric spaces of rank one. These spaces are ex hausted up to the covering by spaces GH with SL ,Gn, 1,GLHn. For these spaces GH, an overgroup is the direct product GG and canonical representations turn out to be tensor products of representations of maximal degenerate series and con tragredient representations. These tensor products are studied in [4], see also [5]. So we lean essentially on these papers [4,5]. We decompose canonical representa tions into irreducible constituents and decompose boun dary representations. Notice that in our case the inverse of the Berezin transform , Q can be easily written: precisely it is the Berezin transform ,n Q . Canonical and boundary representations for GH in the case 2n (then GH is the hyperboloid of one sheet in ) were studied in [6]. For the twosheeted hyperboloid in , it was done in [7]. 3 3 In this paper we present only the main results. The de tailed theory of canonical and boundary representations, for example, on a sphere with an action of the generaliz ed Lorentz group, can be seen in [8].
A. A. ARTEMOV 36 Let us introduce some notation and agreements. By we denote . The sign denotes the congruence modulo 2. 0,1,2, For a character of the group we shall use the following notation *0 ,sgn ,tt t where , * t , 0,1 . For a manifold , let denote the Schwartz space of compactly supported infinitely differentiable valued functions on , with a usual topology, and the space of distributions on —of antili near continuous functionals on . 2. The Space GH and the Manifold We consider the symmetric space GH where , , . SL ,Gn GL 1,Hn G3n Mat n The group acts on the space by , 1. gxg Let us write matrices in in block form according to the partition Mat ,n 1nn 1 of n. Let us take the matrix 000 . 01 x The subgroup is just the stabilizer of this point 0 , this subgroup consists of block diagonal matrices: 1 0,GL1,, det 0 hn . Thus, our space GH is the orbit of G0 , it con sists of matrices of rank one and trace one. Equip with the standard inner product n , y, let 12 , xx. Let be the sphere S1s. Let d be the Euclidean measure on . The group acts on by SG S sg sg. Let be a cone in ,nMat consisting of matri ces of rank one. Therefore, the space 0xGH is the section of by the hyperplane tr 1 . Introduce a norm in by Mat ,n 12 tr ,xxx where the prime denotes matrix transposition. Let be the section of by 1x. Define a map by SS ,. tts u It is a twofold covering. The measure d defines a measure on by du 1 ddd, 2SS The action of the group on gives the follow ing action of on G S G : 1 1. ug u ug In particular, the subgroup SO n, a maximal compact subgroup, acts on by translations: 1.ukuk Let us consider on the function ,, .ppu stuts (1) The action on has three orbits: namely, two open orbits (of dimension 22n ): and 0p 0p and one orbit of dimension 2n3 : 0p . The orbit is a Stiefel manifold, it is the boundary of . Denote . Each of orbits can be identified with the space GH. The map is constructed by means of generating lines of the cone . 3. Maximal D e ge n e r at e S er i e s Representations Recall [4] maximal degenerate series representations , , , 0,1 , of the group . Let G S be the subspace of S consisting of functions of parity : 1 s . The representations , act on S by ,, sg ss sg g 1 1 ,1 = '' ' sg gs sg sg . 4. Representations of Associated with G GH Recall [5] a series of representations , T of the group associated with the space GGH. Denote by the space of functions in of parity 0,1 : 1. The representation , T acts on by 1 1 ,1. gg Tg gg gg (2) Let , denote the following sesquilinear form ,d. (3) Define an operator , on by 1,n . uu ftsstuts ,trd .A Open Access AM
A. A. ARTEMOV 37 It intertwines , T and 1,n T . The operator , is a meromorphic function of . Let us normalize this operator (multiplying it by a function of ) such that the normalized operator , is an entire nonvanishing function of . There are three series of unitarizable irreducible representations. The continuous series consists of , T with 12ni , , the inner product is (3). The complementary series consists of , T with 22n 2n, the inner product is ,,A with a factor. The discrete series consists of the repre sentations ,m T where 22mn m , m, 0,1 , which are factor representations of ,m T on the quotient spaces , Ker m A . The representations ,m T with the same and different m are equivalent. It is convenient to take m 1m where for odd and for even . The inner product is induced by the form 0m n n m ,mm ,A . 5. Canonical Representations We define canonical representations , R , , 0,1 , of the group as tensor products: G ,. nn R They can be realized on : let denote the subspace of consisting of functions of parity f : 1 uf u, then the representation , R acts on by a formula similar to (2): 1 1 ,1. n gug Rgfufgug gug The inner product ,fhfuhu u d (4) is invariant with respect to the pair , ,n RR, , i.e. 1 ,, ,, . (5) n Rgfh fRgh Consider an operator , Q on defined by , ,,tr dQfu cuvfvv , It turns out that the composition ,,n QQ is equal to the identity operator up to a factor. We can take E ,c such that ,, , n QQ E namely, 2 121 , coscos . 22 nn n c nn 1 With the form (4) the operator , Q interacts as fol lows: , ,,Qfh fQh , . (6) This operator , Q intertwines the representations , R and ,n R , i.e. ,,,, ,. n RgQQRgg G Let us call it the Berezin transform. Let be the space of distributions on of parity . We extend , R and , Q to by (5) and (6) respectively and retain their names and the notation. Let us introduce the following Hermitian form , ,fh on : , , 1 ,, 2 hQf h Let us call this form the Berezin form. 6. Boundary Representations The canonical representation , R gives rise to two re presentations , L and , M associated with the boun dary of the manifolds (boundary representa tions). The first one acts on distributions concentrated at , the second one acts on jets orthogonal to . We can introduce “polar coordinates” on corre sponding to the foliation of into orbits. The  orbits are level surfaces of the function , see (1). For p 1p1 the orbits are diffeomorphic to . In these coordinates the measure on is du 32 2 d1 dd n up p, where d is the measure on . Let be a function in . Consider it as a function of polar coordinates. Consider its Taylor series 2 01 2 aa app in powers of . Here p aaf mm are functions in . Denote by k k, , 0,1 , the space of distributions in k , having the form 0 , m m m p where mm , is the Dirac delta function on the real line, m its derivatives. Let k . Denote by m af * Taylor coefficients of the function Open Access AM
A. A. ARTEMOV 38 32 2 1n pf u. The distribution acts mp on a function as follows: f * ,1!,. (7) m mm pfma f Denote by , L the restriction of , R to . This representation is written as a upper triangular matrix with the diagonal , . 1,nmm T k m Distributions in can be extended in a natural way to a space wider than . Namely, let k be the space of functions of class C on and of parity and having the Taylor decomposition of order : k 01 , kk k uaapapop where . Then (7) keeps for m a k f with . mk Let denote the column of Taylor coefficients m. The representation af f a, M acts on these co lumns: ,, . gafaRgf It is written as a lower triangular matrix with the di agonal , . ,nmm The boundary representations T m , L and , M are in a duality. 7. Poisson and Fourier Transforms Let us write operators ,; , P and ,; , F intertwining representations , R and , T . We call them Poisson and Fourier transforms associated with canonical representations. The Poisson transform ,; , P is a map given by C , , ,; ,trd . n Pupu It intertwines 1,n T with , R . Here we consider , R as the restriction to C of the representation , R acting on distributions in . For a finite function K and 12n the Poisson transform has the following decomposition in powers of : p , ,; ,,, 0 1, ,, 0 , n k k k k k PupC p pD k p where has polar coordinates u,p . Here ,,k C and ,,k D are certain operators acting on . The factors ,n p and 1, p give poles of the Poisson transform in depending on : ,1nk ,l (8) where ,kl and k , l . If a pole hen the pobelongs one of se tle is simple, and if a pole belongs to both series (8), then only to ries (8), 12n and the pole is of the second or first or Let der. k , k , be simple. The the pole residue ,, ˆk P of ,; , P at this ole is an operator p k . Denote the image of this operator by ,,k V . The Fourier ,; , F transform is a map given by tr , , ,; ,d . fupf uu It intertwines , R with , T . n trThe Fourier Poissoan ea andsforms are conjugate to ch other: ,; ,,; , ,, n Ff fP . Poles in of the Fourier transform are situated at po ,l ints ,1nk (9) where ,kl and k , l (9), then . If a pole belongs the pole is simple, and if a pole belongs to both series (9), then only to one of ries the se 12n and the pole is of the second or first or Let der. the pole nk , k , be simple. The residue ,, ˆ F nk perator of at th ,; F ,is pole is a “boundary” o ,,k B , k : . The operator ,,k B is dyl efficients e terms of Tafined inor co m a is a linear combination of functions f : it ,,nk k Daf . Therefore, we may consider rator , B * mm the following ope acting on columns 012 ,,,aaaa of functio k a : this ope rator to any col ns umn a assigns the column ,,,0,,,,2 ,,,Ba B aBB a of functio 1 ans in the same space —by the same formulas without . This operator , B is given by a lower triangular m trix. 8. D a ecomposition of Boundary Th ure of the Poisson and Fourier Representations e meromorphic struct transforms is a basis for decompositions of boundary re presentations , L and , M . Let the polek the Poisson transform is si of mple, in particupens when lar, it hap12n . Open Access AM
A. A. ARTEMOV 39 Then the boundary representation L, is diagonaliz  able which means that decoses into the di mpo rect sum of ,, , k Vk the restriction of , L , and to ,,k V is eto 1,nkk T (by meansf quivalent o If a pole is of the second order, thee decomposition of ,, k ). ˆ P n th , L contains a finite number of Jordan blocks, this num depends on ber . Let the pole nk rticular, when of turier transform is he Fo simple, in pa 12n . Then the matrix , M is diagonalins that 1 ,,, BMB zablch meae whi a diagonal matrix. Its diagonal is is If a pole is of the second order, thee decomposition of ,nkk T , k. n th , M contains a finite number of Jordan blocks, this numpends on ber de . 9. Decomposition of Canonical Leon ofpresentations. Representations t us write decompositi canonical re We restrict ourselves to a generic case: lies in strips 11nn :Re ,. 22 k Ik kk Case (A): 0 . 1 Let Theorem 0 . The canonical re en th presentation , R decomposes—as the quasiregular representation into irreducible unitary representa tions of continuous and discrete series with multiplicity one. Namely , let us assign to a function [5]— fD the family of its Fourier components ,; , f, 12ni , , 0,1 , and h responde ,;1,nmm F , m. Tis cornce is G equivariant. There is an inversion formula: ,;1, ,;12 ,; ,,;1 0 ,d nni mmm n m fPf PF f , ,mm F (10) and a “Plancherel formula” for the Berezin form: , ,; ,,;1,12 0 ,;1,,; , ,,,;, , ,; , ,. nni m m nmm mm fh FfF h mm FfFh d , (11) Here and m stand for the Plancherel mea sure for GH, ula: see [5 the factoris given by fol lowing form ], ,; , 111cosπ 1sinπ sin2 π1sin 2π co π n n nn . s 21cos 2πnn Case (B): 1, k Ik . Here we continue decomposition (10) analytically in to 1k , k from 0 . Some poles in of the integrating line—the line integrand intersect the Re 12n. They are poles m and 1n m of the give additional summands to the right hand side. So after the continuation we obtain: (12) whn and the ser a Poisson transform with 0mk. They ,, , k m ff 00 mm ere the itegralies mean the sames in (10) and ,,,, ,;1 ˆ ,, mm LmP F ,, nmm ,,Lm are some numbers. Similarly, the continuation of (11) gives ,00 ,, k mm ,; 1,,; , ,, nm m mm ,hM (13) where the integral and the series mean the same as in (11) and m FfFh ,, m he ope are some numbers. Trators ,,m , mk, can be extended from to the space k and therefore to the sum . kk Then these operators ,,m turn out to be operators ontothere are some “or (1 Theo projection ,,m V . Moreover, thogonality relations” for them. Decomposition (13) can also be extended to the space k . This decompo sitionthagorean theorem” is a “Pyfor decomposition 2). rem 2 Let 1k , k. Then the space the represe has to On this space ntation be completed to , R k . splits int s as o the sum of two terms: the first one decompose, R does in Case (A), the second one decomposes into the sum of 1k irreducible representations 1,nmm T , 0,1, ,mk. Namely, let us assign to any k f th e family ,; ,,, ,;1, ,, m nmm fFf f Open Access AM
A. A. ARTEMOV Open Access AM 40 12ni , n , ivariant. where This co equ formula rel form 0,1, ,mk. T rrespondence is Ghere is an inverse , see (12), and a “Plancheula”, see (13). Case (C): 1k Ik, . Now we continue decompion (10) analytically in osit from 0 to 1k . Here poles nm and 1m , m , mk , of the integrand (they are poles of the Fform) give addourier transitional terms. We obtain , k m,, 00mm f (14) where the integral and the series mean the same as in (10 ) an m somebers. Tperat of e op ere are Now we continue decomposition (11) from d ,,,,,; 1,mm m NP B ,,m numhe oors ,,m can be ex tended to the space k , mk. Denote by ,,m the image ,, m N ,,m them th ,, are projection some m. It turn operat “orthogon s ou ors onto ality relatio t that th n erators and for ,,m s”. 0 to 1k . Poles of the integrand which intersect the inte grating line Re 12n and give additionalrms e poles of both Fourier transforms) tu or tely to be of the first order, since at these po te ut fo ints th (they ar tuna functi rn e on ,; , as a function of o has ntinuan we zero of the first order. After the cotiobtain: ,00 ,,, k mm fhK m ,, ,, nm m m AB fBh where the inties mean the same as in (11), ,, egral and the ser (15) m some numbers. It is a “Pythagorean the orem” for decomposition (14). Theore et 1 k Im 3 L , k Then the repre sentation , R. considered on the space k splits into the sum of two termse a space of functions f such that their Taylor coefficients m af are equal to 0 for mk and decomposes as , R does in Case (ce decomposes into the direct sum of 1k A), the onseond irreducible representations ,nm m T , mk acting on the sum of the spaces ,,m . There is annversion formula, see (14), and a herel formula” for the Bn form, see (15). REFERENCES . M. Vershik, I. M. Gelfand and M. I. Graev, “Represen tations of the Group 2,SLR Where R Is a Rin i p “Q ra “Planc [1] A [2] F. A. erezi tematiches Math es Repr g of kikh Nauk, Vol. 28, No. elex Symmetric Spa nematical Society Transla esentations of the Group Functions,” 5, 1973 B Us , pp. 83 rezin, ekhi Ma a te Seri 128. uantization in Comp ces,” Izvestia Akademii nauk SSSR, Vol. 39, No. 2, 1975, pp. 363402. [3] V. F. Molchanov, “Quantization on ParaHermitian Sym metric Spaces,” Americ tions, Vol. 175, Series 2, 1996, pp. 8195. [4] G. van Dijk and V. F. Molchanov, “Tensor Products of Maximal Degene ,SLn ,” Journal de Mathématiques Pures et Appli quées, Vol. 78, No. 1, 1999, pp. 99119. http://dx.doi.org/10.1016/S00217824(99)800117 [5] G. van Dijk and V. F. Molchanov, “The Berezin form for Rank One ParaHermitian Symmetric Spaces,” Journal de Mathématiques Pures et Appliquées, Vol. 77, No. 8, 1998, pp. 747799. http://dx.doi.org/10.1016/S00217824(98)800081 [6] V. F. Molchanov, “Canonical and Boundary Representa tions on a Hyperboloid of One Sheet,” Acta Applicandae Mathematicae, Vol. 81, No. 13, 2004, pp. 191214. [7] V. F. Molchanov, “Ca Sheeted Hyperboloid,” Indagationes Mathematicae, nonical Repr . The first oncts on the sub esentations on the Two Vol. 16, No. 34, 2005, pp. 609630. [8] A. A. Artemov, “Canonical and Boundary Representations on a Sphere with an Action of the Generalized Lorentz Group,” Monography, The Publishing House of TSU, Tambov, 2010, 235 p.
