Razlozhenie granichnykh predstavleniy na ploskosti Lobachevskogo v secheniyakh lineynykh rassloeniy

Abstract

Earlier we described canonical (labelled by λ ∈C ) and accompanying boundary representations of the group G = SU (1,1) on the Lobachevsky plane D in sections of linear bundles and decomposed canonical representations into irreducible ones. Now we decompose representations acting on distributions concentrated at the boundary of D . In the generic case 2λ ∉N they are diagonalizable, in the exceptional case Jordan blocks appear.

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1. The Lobachevsky plane The Lobachevsky plane is the unit disk D : zz < 1 on the complex plane with the linear-fractional action of G: z 7! z g = az + b bz + a ; g = a b b a ; aa bb = 1: The boundary S of D is the circle zz = 1; it consists of points s = exp i ; the measure ds on S is d : Let D be the closure of D: D = D [ S: Let p = 1 zz; so that D = fp > 0g and S = fp = 0g: The stabilizer of the point z = 0 is the maximal compact subgroup K = U(1) consisting of diagonal matrices: k = a 0 0 a ; aa = 1; so that D = G=K: The Euclidean measure dxdy on D is (1=2) dp ds; a G-invariant measure d (z) on D is d (z) = p 2dxdy: If M is a manifold, then D(M) denotes the space of compactly supported infinitely differentiable C-valued functions on M; with a usual topology, and D0(M) denotes the space of distributions on M - of antilinear continuous functionals on D(M): We use the notation N = f0; 1; 2; : : :g: Recall principal non-unitary series representations of G trivial on the center, see also [4]. Let 2 C: The representation T acts on the space D(S) by (T (g)')(s) = '(s g)jbs + aj2 : The inner product from L2(S; ds) : h ; 'iS = Z S (s) '(s) ds is invariant with respect to the pair (T ; T 1): If =2 Z; then T is irreducible and equivalent to T 1 (for 2 Z there is a “partial equivalence”). For = v 2 N the representation Tv has an invariant irreducible subspace Ev spanned by exp ir ; r = v; v + 1; : : : ; v: 370 L. I. Grosheva A basis of the Lie algebra g of G is L0 = i=2 0 0 i=2 ; L1 = 0 i=2 i=2 0 ; L2 = 0 1=2 1=2 0 : We also use their linear combinations (they belong to the complexification of g ): L+ = L2 + iL1 = 0 1 0 0 ; L = L2 iL1 = 0 0 1 0 : Denote by g the twice Casimir element of the universal enveloping algebra Env(g) of g : g = L20 + L21 + L22 : The representation T of g assigns to L0; L+; L the following operators: T (L0) = d d ; T (L+) = ei + i d d ; T (L ) = e i i d d ; T ( g) = ( + 1): 2. Canonical representations Let D(D) be the space of restrictions to D of functions from D(C) with the induced topology, and by D0(D) the space of distributions on C with supports in D: Consider the inner product with respect to the Lebesgue measure on D: hF; fiD = Z D F(z)f(z)dxdy; z = x + iy: (2.1) The space D(D) can be embedded into D0(D) by assigning to h 2 D(D) the functional f 7! hh; fiD; f 2 D(D): We shall use denotation: z ;m = jzj z jzj m ; 2 C; m 2 Z: Let 2 C: We define the canonical representation R ;m of the group G associated with a character of K as follows: (R ;m(g)f) (z) = f(z g) (bz + a) 2 4;2m; it acts on the space D(D): DECOMPOSITION OF BOUNDARY REPRESENTATIONS 371 The inner product (2.1) is invariant with respect to the pair (R ;m; R 2;m) : hR ;m(g)f; h iD = hf; R 2;m(g 1)h iD; g 2 G: (2.2) The formula (2.2) allows to extend the representation R ;m to the space D0(D) of distributions on D: Here are formulae for basic elements of g in variables p and : R ;m(L0) = @ @ im; R ;m(L ) = e i rp @ @p 1 2 (r + r 1)i @ @ ( + 2 m)r : (2.3) Let us also write the operator corresponding to g : R ;m( g) =(p3 p2) @2 @p2 + - (2 + 4)p (2 + 5)p2 @ @p + +imp @ @ + 1 4 p2 1 p @2 @ 2 + + - ( + 2)( + 1) ( + 2)2 m2 p : (2.4) In (2.4) one has to use the binomial expansions ( r = (1 p)1=2 ): r = X1 n=0 1=2 n ( 1)n pn; (2.5) r 1 = X1 n=0 1=2 n ( 1)n pn; (2.6) 1 2 (r + r 1) = X1 n=0 1=2 n (1 n)( 1)n pn: (2.7) Applying these formulae to distributions ; we have to keep in mind the following: pn (k)(p) = ( 1)n k! (k n)! (k n)(p): 3. Boundary representations Canonical representations R ;m generate two boundary representations L ;m and M ;m: For simplicity, in this paper we restrict ourselves to the first one. It acts on distributions in D0(D) concentrated at S: Consider distributions of the following form: = '(s) (k)(p); 372 L. I. Grosheva where ' 2 D(S) and (p) is the Dirac delta function on the real line (being a continuous linear functional on D(R) ) and (k)(p) its k -th derivative. The space of these distributions will be denoted by k(D): Define also k(D) = 0(D) + 1(D) + + k(D); so that a distribution in k(D) is a linear combination = '0(s) (p) + '1(s) 0(p) + + 'k(s) (k)(p): We get a filtration: 0(D) = 0(D) 1(D) 2(D) : : : Let (D) denote the union of all k(D): The canonical representation R ;m acting on D0(D); preserves the space (D) and the filtration (2.3). The boundary representation L ;m is the restriction of R ;m to (D): 4. Poisson transform Let ; 2 C and m 2 Z: We define the Poisson transform associated with the canonical representation R ;m as the map P(m) ; : D(S) ! C1(D) by the following formula P(m) ; ' - (z) = p 2 Z S (1 sz)2 ; 2m sm '(s) ds: The Poisson transform P(m) ; intertwines the representations T 1 and the canonical representation R ;m : R ;m(g) P(m) ; = P(m) ; T 1(g) (g 2 G): The Poisson transform P(m) ; is meromorphic in ; and has poles at the points = k; = 1 + l (k; l 2 N): (4.1) All poles are simple except in the case when the two sequences (4.1) have a non-empty intersection and the pole belongs to this intersection. This happens when 2 + 1 2 N and 0 6 k; l 6 2 +1; k +l = 2 +1: In this case the pole is of the second order. Let us write down the principal part of the Laurent series of P(m) ; at the poles of the first order: P(m) ; = b P(m) ; + : The residue intertwines T 1 with R ;m: Let us write it explicitly. We set V ;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 : V ;m;n(p) = X1 k=0 w(m) ;k (n) pk; DECOMPOSITION OF BOUNDARY REPRESENTATIONS 373 here w(m) ;k are polynomials in n of degree k: The coefficients of these polynomials are rational functions of with simple poles. We set W(m) ;k = w(m) ;k 1 i d d : If a pole belongs only to one of the sequences (4.1), then it is simple. In particular, b P(m) ; k = ( 1)k+m 1 k! a m( k) (m) ;k ; where an( ) = 2 ( 1)n ( 2 1) ( + n) ( n) and (m) ;k is the following operator D(S) ! k(D) : (m) ;k ' = sm Xk n=0 ( 1)n k! (k n)! W(m) k;n ' - (s) (k n)(p): The operator (m) ;k is meromorphic in : For fixed k = 1; 2 : : : it has k poles (simple) at the points = k 1; k 3=2; k 2; : : : ; (k 1)=2: It intertwines T 1+k with L ;m (restricted to k(D) ). Let us write three first operators: (m) ;0 ' = sm' (p); (m) ;1 ' = sm n ' (p) 1 2 ( 2' m i'0) 0(p) o ; (m) ;2 ' = sm ' (p) 1 1 ( 1)2' m i'0 0(p) + 1 4( 1)(2 1) n- ( 1)2 2 + m2 ' 2( 1)(2 1)m i'0 - ( 1)2 + 2m2 '00 o 00(p) : 5. Decomposition of boundary representations Theorem 5.1. The representation L ;m is equivalent to a upper triangular matrix with diagonal T 1; T ; T +1; : : : : P r o o f. The formulae (2.3) and (2.5)-(2.7) show that operators R ;m(L ) move subspaces k(D) to k(D): Also these formulae show that the operator R ;m(X) where X 2 g moves a distribution sm'(s) (k)(p) in k(D) to the distribution sm(T 1+k(X)')(s) (k)(p) + : : : in k(D): Let V (m) ;k be the image of (m) ;k : This space is contained in k(D) and its projection to k(D) is the whole k(D): It gives: 374 L. I. Grosheva Theorem 5.2. In the generic case 2 =2 N the boundary representation L ;m is diagonalizable which means that (D) decomposes into the direct sum of subspaces V (m) ;k ; k 2 N; the restriction of L ;m to V (m) ;k is equivalent to T 1+k (by ;k ), so that L ;m is the direct sum of the T 1+k (k 2 N): Now let 2 (1=2)N: This number is a pole (of the first order) of (m) ;k in for k 2 N such that +1 6 k 6 2 +1: For example, if = 0; then k = 1 ; if = 1=2; then k = 2 ; if = 1; then k = 2; 3 ; if = 3=2; then k = 3; 4: For these the spaces V (m) ;k are defined for all k 2 N such that k < + 1 and 2 + 1 < k; for the others these spaces are absent. Let us write down the Laurent expansion of (m) ;k at = : (m) ;k = b (m) ;k + (m) ;k + : : : For the indicated k we define the spaces bV (m) ;k and V (m) ;k as the images of the operators b (m) ;k and (m) ;k respectively. The space bV (m) ;k is isomorphic to V (m) ;l where l + k = 2 + 1; namely there is a relation (m) ;k (') = (m) ;l ( ) where is obtained from ' by means of some operator. Therefore the operator b (m) ;k intertwines T 1+l with L ;m; notice that it vanishes on El: The space V (m) ;k has the full projection to k(D): On the pair bV (m) ;k ; V (m) ;k the representation L ;m is the block T 1+l 0 T 1+k : Since 1+l = ( 1+k) 1; representations T 1+l and T 1+k are isomorphic, so that this block is a genuine Jordan block. Here is the matrix corresponding to the Casimir operator R ;m( g) : ( + 1) 0 ( + 1) ; where = 1 + l or = 1 + k: Thus, me obtain the following theorem (we use the notation [a] for the integral part of a number a ). Theorem 5.3. Let 2 (1=2)N: Then the space (D) is the direct sum of the subspaces V (m) ;k with k > 2 + 2 and k 6 and subspaces V (m) ;k with + 1 6 k 6 2 + 1: The representation L ;m is equivalent to the direct sum of [ +1] Jordan blocks with the diagonal (T 1+j ; T j); j = 0; 1; : : : ; [ ]; acting on subspaces V (m) ;l + V (m) ;k ; k+l = 2 +1; the representation T1=2 for half-integer ; and the representations T +1; T +2; : : : : Let us write b (m) ;k and (m) ;k for some ; k: Let = 0; k = 1: Then b (m) 0;1 (') = im 2 sm'0 (p); (m) 0;1 (') = sm' 0(p): Let = 1; k = 2: Then b (m) 1;2 (') = imsm n '0 0(p) 1 2 ('0 im'00) (p) o ; (m) 1;2 (') = sm n ' 00(p) + 1 4 m2 2im'0 + (4m2 1)'00 (p) o : DECOMPOSITION OF BOUNDARY REPRESENTATIONS 375 Let = 1=2; k = 2: Then b (m) 1=2;2(') = 1 32 (4m2 1)sm (' + 4'00) (p); (m) 1=2;2(') = sm n ' 00(p) + 1 2 (' 4im'0) 0(p) + 1 16 4' + im'0 + 16m2'00 (p) o :