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34
Quantization on symplectic symmetric spaces
V.F. Molchanov * Tambov State University, 392622 Tambov, Russia e-mail: molchanov@math-univ.tambov.su
Symplectic symmetric manifolds G/H with G simple are divided into four classes [17]: (a) Hermitian symmetric spaces; (b) semi-Kahlerian irreducible symmetric spaces;
(c) para-Hermitian symmetric spaces of the first category; (d) para-Hermitian symmetric spaces of the second category. The spaces of the three latter classes are not Riemannian, and each has a Riemannian form belonging to the class of Hermitian symmetric spaces.
Berezin constructed quantization on spaces of class (a). We would like to outline a program for a quantization in the spirit of Berezin for other classes of symplectic homogeneous manifolds. In these lectures we restrict ourselves to class (c). The local classification
of spaces of class (c) is given in §3.
There is an inspiring analogy between (a) and (c), which starts at the coordinate level:
? <__> £. ?/, see §3, and continues on the level of formulae and so on. Oil the other hand.
it is well-known, that the passage from the Riemannian case to the non-Riemannian one drastically increases the difficulties. So, in this theory there are still many interesting open problems.
§1. Quantizations on the plane
First we have classical mechanics. Let us consider a particle with oue degree of freedom, say, a linear oscilator with the coordinate q and the impulse p. Then the energy is H = (1/2)p2 + q2. To pass to quantum mechanics we replace functions by operators:
h d „ p-+ p = --г, q -> q = q
г dq
These operators act in L2(®). Then the energy H goes to the Hamiltonian
- p2 ~ h2 d2 , .
* = t + ’ =-¥ lf+r
In a general sense, quantization is the passage from functions to operators. Returning to our particle, we come across the problem : what operator has to be assigned to a function more complicated than p. q, FI. for example, pq,p2q3 etc.? The point is that the operators p and q do not commute: [p. q) = h/i, while the functions p and q do it. This problem can be solved in different ways. First let us consider (^-quantization. In any monomial
‘Partially supported by RFBR (grant 96-15-96249) and Goskomvuz RF (grant 95-0-1.7-41).
we write q from left of p and then we put the huts. We obtain the correspondence A(q,p) -c A for polynomials, and we can extend it on arbitrary functions. The function A(p, q) is called qp-symbol of the operator A. There is a simple way to find A(p, q) for
given A. Denote .
ipq
$p(g) = $(g,p) = exP~Y
Then -
,, >
A(q’p) = W
Therefore the kernel K(q,v) of an operator A is expressed in term of its ^p-symbol as follows:
(all integrals are taken over ffi etc.) The multiplication of operators gives rise to a multiplication of symbols:
{A*B)(q,p) = J A{q,t)B(s,p)B{q,p\s,t)dsdt where «,
P’s, ) 27rh
There is another expression for the multiplication of symbols:
3^
(A * B)(q,p) = exp{-ih-Q^)A(q,t)B(s,p)
t=p,s=q
SOtHat . „ .,dAdB , (ih)2 d2Ad2B t nn
A* B - AB ih dp dq + 2! dp2 dq2
From (1.1) we have
lim A * B = AB (1-2)
0
lim —(A * B — B * A) = {A, B} (1-3)
h-+ o h
where {A, B} is the Poisson bracket. Equalities (1.2) and (1.3) mean that for ^-quantization the correspondence principle holds.
Similarly we consider ^-quantization (in monomials p stands before q). The kernel L(q,v) of an operator A is expressed in terms of its pg-symbol A(q,p) as follows.
L{q’v) = ^hJA{v't]Wj)dt
The multiplication of pg-symbols is given by formulae:
(A*B){q,p) = J A{s,p)B(q,t)B(s,p',q,t)ds dt
r\ 2
= exp (ih-Q^)B(q,t)A{s,p)
t=p,s=q
In this case the correspondence principle (1.2), (1.3) holds too.
* ° A / . There is a connection between pg-symbol A and qp-symbol A of an operator A (given
by A -»• i -» A):
A(q,p) = J B(s,p-,q,t)B(q,t)dsdt
and also a connection between operators A and A± for which A(q,p) is qp- and pg-symbol respectively: L(q,v) = K(v, —q + 2v), i.e. variables q,v are transformed by the matrix
("!)
There are also other quantizations: Wick, anti-Wick, Weyl etc., see, for example [5].
2. Berezin quantization
Recall the concept of quantization proposed by Berezin, see [1-5]. We shall not give it in its full generality, but restrict ourselves to a rather simplified version.
Let M be a symplectic manifold. Then C°°(M) is a Lie algebra with respect to the Poisson bracket {A, B}. A, B £ C°°(M).
Quantization in the sense of Berezin consists of the following two steps:
(I) To construct a family Ah of associative algebras contained in C0C(M) and depending on a parameter h > 0 (called the Planck constant), with a multiplication denoted by * (depending on h also). These algebras must satisfy the conditions (a) through (d):
(a) lim/lx * A2 = A1A2',
' /1—*-0
(b) lim t(Ai * A2 — A2 * .4.1) = {Al,A2} (2-1):
0 a
(c) the function A0 = 1 is the unit element of each algebra Ah]
(d) the complex conjugation A 1—>• A is an anti-involution of any Ah',
where the multiplication on the right-hand side of (a) is the pointwise multiplication and conditions (a) and (b) together are called the correspondence principle (CP).
(II) To construct representations A 1—> A of the algebras Ah by operators in a Hilbert space.
Berezin mainly considered the case when M is Kahlerian, hence has a complex structure. The functions in question are functions A(z, z) analytic on z and z separately. In this case complex conjugation reduces to the permutation of z and z\ f(z,z) = f(z,z).
For our theory we shall slightly change some of the conditions above: namely, the factor i in (2.1) has to be omitted, the anti-involution is the permutation of arguments, and, finally, we give up the Hilbert space structure of the representation space.
§3. Para-Hermitian symmetric spaces of the first category
Let us recall some facts about semisimple symmetric spaces G/H. Here G is a connected semisimple Lie group with an involutive automorphism a ^ 1 Denote by G° the subgroup of fixed points of a. Then H is an open subgroup of Ga. There exists a Cartan involution r of G commuting with a. Let I< = Gr. For Lie groups G,... we denote their Lie algebras by the corresponding small Gothic letters g,.... We assume that the pair (a h) is effective, i.e. f) contains no non-trivial ideals of g. The automorphisms o g induced by a, r are denoted by the same letters <r, r. There is a decomposition of g into
direct sums of +1, — 1-eigenspaces of <7 and r: 0 = f) + q and 0 - ? + p. q
The subspace q can be identified with the tangent space of G/H at the point x - He, it is invariant with respect to the adjoint representations Ad0 of H and ad0 of b-
Now assume, in addition, that G/H is symplectic. Then fj has a non-trivial centre Z(0) For simplicity we assume that G/H is an orbit AdG ■ Z0 of an element Z0 G 9- We can assume that G is simple. Then the statement nG/H is para-Hermitian of the firs category” means that the centre Z(H) of b is one-dimensional: Z(b) - and Z0 can
be normalized so that the operator I = (adZ0)q is an involution There ore o P !>■
A symplectic structure on G/H is defined by the bilinear form lo(X,Y) - B(X,1Y) on
q. where B(X,Y) is the Killing form of g. _ _ _
The ±l-eigenspaces q± C q of I are Lagrangian, H-invariant, and irreducible. T ey
are Abelian subalgebras of 0. So 0 becomes a graded Lie algebra:
0 = q" + f) + q+(= g_i + 9o + 0+i)-
The pair (<Z+,<7~) is a Jordan pair [fl] with multiplication {XYZj — ^[[X, Y"], Z]. Let
r and k be the rank and the genus of this Jordan pair. _ -
Set Q± = expq±. The subgroups P± = HQ± = Q±H are maximal parabolic subgroups of G, with H as a Levi subgroup. One has the following decompositions:
g = q+hQ:s
= Q~HQ+ (3-2)
= Q+HK (3‘3)
= Q~HK, (3-4)
where bar means closure and the sets under the bar are open and dense in G. Let us call (3.1) the Gauss decomposition and (3.3) the Iwasawa-type decomposition. Allowing some slang, let us call (3.2) the anti-Gauss decomposition and (3.4) the anti-Iwasawa-type decomposition. For an element in G all three factors in (3.1), (3.2) and the firs factors in (3.3), (3.4) are defined uniquely, whereas the second and the third factors m (3.3), (3. )
are defined up to an element of I< fl H. _ . ~ „
For g € G we define the transformations £ ^ £ of q and rj i-> V of q taking (an i)
from the Gauss and the anti-Gauss decompositions:
exp£ -g = expy • h -exp|, v3'5)
exp ' g exp X ' h ' exp Tj. (3-6)
These £ and fj are defined on open and dense sets in q and q+ respectively, depending on g. _
Therefore, G acts on q~ X q+ : (£, 77) i—>• (<f, fj). The stabilizer of (0, 0) is P+ fl P~ = H, so that we obtain the embedding (defined on an open dense set)
q- x q+ c—» G/H. (3.7)
We may regard (£,77) as coordinates in G/H.
The connection between the Gauss and the anti-Gauss decompositions gives us an operator and a function, both very important (see (3.9), (3.10) below). Let £ 6 q~, 77 £ q+. Decompose the anti-Gauss product exp£ -exp(—77) according to the Gauss decomposition:
exp£ • exp(—77) = expF • h • expX. (3.8)
Denote this h by h((,rj). On q+ define the operator
K(£, 77) = Ad/i(£, 77)""1 |q+ (3.9)
which is the analogue of the Bergman transform for Hermitian symmetric spaces. In terms of Jordan pairs it becomes:
K(Z,ri)T = T-2{rtT] + {rttnh}-
Under the action of G the operator A'(£, 77) is transformed as follows:
K(£,rj) = (Adr1)5+/^(£,77)(AdA)?+,
where h and h are taken from (3.5) and (3.6).
The function dct K(£,rj) is a polynomial in £,77. Moreover, det K(£, 77) = A (£, 77)''.
where N(£,T]) is an irreducible polynomial in £ and 77 of degree r in £ and 77 separately
[11]. In viewr of (3.7) the function
KtiV) = ldetK(Z’ri)]~1 (3-10)
can be regarded as a function on G/H, becoming an analog of the Bergman kernel. It is invariant with respect to H.
A point x in G/H C 0 with coordinates £,77 is
x = Z0-X + Z + Y
where I £ q',F G q+ are given by (3.8), Z = [£, F] = [tI,X] £ f), and for X,Y we have equations X = £ + AX, Y = -77 - AY, where A = (l/2)ad[£, 77], which allow to find X, Y by means of iterations:
X = (E - A)-1^ Y = -{E + A)_177
Let us write a C-invariant metric ds2, symplectic form u, measure dx on G/H. Take a basis Ei,..., Em of q~ in such a way that B(Ei, tEj) = 8iy Then F{ = tE{ form a basis
of q+. Let £,■ and r]i be the coordinates of £ G q_ and 77 £ q+ in these bases. Then the
desired metric, form and measure are:
ds2 = 2'£ki>(£,T])d£tdT]j ' (3.11)
ш - Adrjj ^ЗЛ2^
dx = |6(£, 7/)|с?£х ... d£mdr)i ... drjm (з ^
^тІгіГ the “trieS °f K((' ,)_1' The fU“Ctbn Xt. v) is the potential of
• d2F 2k,] =
d&drij'
The likeness of (3.11) and (3.12) reflects the fact that G/H has the structure of a manifold over the algebra D = {x + jy, х,уЄШ,р = П. structure of a
The coset spaces S+ = G/P~, S~ = G/P+ Я - K/ tr n u .
diffeomorphic to each other by the following correspondence: ^ C°mpaCt mamf°lds’
s°ki *■ s±k, k є к, (ЗЛ4)
on gl *Pf e,S ~P e,S ~ n H^e are the basic points. The natural action of G
A I і lW°.aCtl0ne °f ° °П 5 : 5 ~ 5 and 3 ~ where ^ = s°k,s = s°k s = s°k and k, k are obtained from the Iwasawa and the anti-Iwasawa decompositions: ’ ’
kg = exp Yi-hi- k, kg = expXj • hx ■ k.
Set
5 = s-g;
(3.15)
(3.16)
(3.17)
(3.18)
then
s = s-T(g). ^ ^
The group G acts on S~ x 5+ in a natural way. The stabilizer of the point (5- s+) is again, so that we obtain the following equivariant embedding G/H <-+ S~ x S+ The identification (3.14) gives rise to the equivariant embedding ’
G/H S x S (3_19)
Denoted n" Tb ^ ^ ‘V- ~ (S’ The image °f (3'19) is a Si”*le °Pe" dense OTbit-
SxS,« IN ^ ^fjS " TPaC‘ifiCati0n °f G/H■ F°r the “"“tore of
s S, see 19]. Note that G/H can be represented as the tangent bnndle of the manifold
The spaces q and q+ can be embedded in S:
£ > 5 -exp£, t) t—> s° ■ exp
where £ e q ,7] e q+, see (3.17), (3.18), with open dense images; thus either £ or 7? can be
™lrrrdmate system on 5,n these ^
ds = yfikiTOdt (3.20)
= /b(TT],Tj)dT}. (3.21)
We now define an important function ||s,£|| on S x S. For s,t E S take ks,kt so that 5 = s°ks,t = s°kt, and apply to ksk^1 the Gauss decomposition:
kskt_1 = expF • h ■ expX (3.22)
It turns out that det(Ad/i)q+ depends only on s,t, but not on the choice of ks, kt. We set
\\s, t\\ = | det(Ad/i)q+ (3.23)
where h is taken from (3.22). Formula (3.23) defines \\s,t\\ on an open dense subset of S x S. This function is continuous, symmetric and invariant with respect to the diagonal action of K. It can be expanded on the whole S x S, keeping all these properties.
In terms of this function, we can rewrite (3.17) as follows:
dx = dx(s,t) = \\s,t\\~Kds dt,
where x i—(s^t) by (3.19). The orbit fi is characterized by the condition ||s,t|| / 0.
The following table contains the list of simple symmetric Lie algebras g/fj that correspond to para-Hermitian symmetric spaces GjH with G simple, see [10]. Here Gpq(l) denotes the Grassmann manifold of p-planes in IF", where IF = E or H; 5m_1 is the sphere in !m; P2(0) denotes the octonion projective plane; n — p + q. For aesthetic reasons we denote Lie algebras by capital Latin letters instead of small Gothic ones.
0 f) 5
SL(n, ®) SL(p,m) + SL{q,R) + E Gp,(e)
SUm(2n) SU*{2p) + SU*{2q) + R Gpq{m)
SU(n,n) SL{n,C) + E U(n)
SO* (4n) SU*{2n) + R U (2n) j Spin)
SO(n, n) SL(n, E) + R SO(n)
SO(p,q) SO(p — 1, <7 — 1) + E (5f,_1 x Sq~x)/Z2
Sp(n, e) SL(n,T8) + E U (n)/0(n)
Sp(n, n) SU*(2n) + E Sp{n)
Eq(&) 50(5,5) +ffi G22 (iHl) /^2
E6(-26) 50(1,9) + E P2( 0)
£V(7) Em + ® 5t/(8)/5p(4) • Z2
-E'7(-25) ^6(-26) + ® S1 ■ E6/F:4
§4. Representations induced from P±
For fiEC, let Li>n be the character of H\
u^(h) = | det(Ad/i)q+ \~^K
We restrict ourselves to such characters of H, for simplicity. Extend to the character of P±, setting it equal to 1 on Q±. Consider the representations of G acting on C°°(S)'.
TJ - IndpitJ^
Ч
In more detail,
= wm(^i)^(5), {T+(g)ip)(s) = шДА"1)^)
we use (3.15), (3.16) and put s = s°k,s = s°k,s = s°k; note that u^hi) and u^^1) are well defined because и^(1) = 1 for I £ К П H. For the same ц, the representations are connected by r: T~ = T+ о r, so that if r is an inner automorphism, then Г+ and T~ are equivalent.
Let (ip,ip) be the inner product in L2(S):
(^,Ф) = J <p(s)if>(s)ds,
for ds, see (3.20), (3.21). This Hermitian form is G-invariant for the pairs (TM+,TVJ and (Г-,ГГд.й). Therefore, for Re/i = —к/2, the representations Tare unitarizable, and we obtain two continuous series of unitary representations.
In a generic case, Г* are irreducible: the reducibility is possible only for real ц satisfying some integrality conditions. Therefore the representations of the continuous series are irreducible for Im/i ф 0.
On C°°(S) define the operator A
(Лму?)(з) = f ||М|Гм~>(0Л;
J s
the integral converges absolutely for Reft < -к + 1 and is extended on /х-plane as a
meromorphic function. This operator intertwines T* with T^_K:
Д Гр± _ A
Moreover,
= c(fi)~1E (4.1)
where E is the identity operator and c(/i) is a meromorphic function.
The representations (degenerate series representations) were studied for separate spaces and with different degree of completeness (for references, see, for example, [16]).
§5. Supercomplete systems and symbols
Let us construct a quantization on G/H (a symbol calculus). The main role belongs
to the kernel of the intertwining operator from §4, i.e., to the function
Ф((з) = Ф (s,t) = ||МГ;
this function is an analog of Berezin’s supercomplete system. The function Ф* has the reproducing property (which is formula (4.1) written in another form):
Let A be an operator acting on functions on S. Define the covariant symbol A(s, t) of A as follows:
We can regard it as a function A(x) on G/H, using (3.19). The operator is recovered by its covariant symbol:
The identity operator has 1 as its symbol. Themultiplication of operators gives rise to the multiplication of the symbols: AXA2 - {Ai * A2), where
the contravariant symbol of the operator A. Thus, we have a chain of correspondences: A -+ A -* A. Their composition B, called the Berezin transform, links the contra- and covariant symbols by means of the same Berezin kernel.
Thus, we have a method for constructing a family of algebras Ah,'- they consist of the covariant symbols A{s,t) = A(x) of operators from some class, the multiplication * in Ah is given by (5.2), the representations are A A. For the Planck constant we take
fi = __d/ ft, where d depends on normalizations of measures, metrics, etc.
Define the bilinear form FJ^,^) on C°°(S) by setting
form: F (Atp,ip) = F^,A'^). Then their symbols are connected by the transposition
of the arguments: A\s, t) = A{t, s). The map A A' changes the order of the factors in the product (5.2): (Aj * A2)' = A'2 * A[, so it is an anti-involution of any A- In order that CP be in agreement with this anti-involution, we must omit the factor i = >/—1 in
formula (2.1). _
By (3.19) the Berezin kernel can be regarded as a function B(x,x) on G/H x G/H.
In coordinates £,77, it can be written in terms of the function (3.10):
(5.1)
Let us call this kernel the Berezin kernel.
U5 bino -------- a
On the other hand, let A(s,t) be a function on S x S. It givs rise to an operator A
by the formula
(iy>)(s) = c(/i) js^A(5,t)^S^(s)dx(s,t)
(it differs from (5.1) in the first argument of A only). Let us call the function A(s,t)
F^,^) = {A^,4>) = js Ik, <11 M K?(s)4;(t)dsdt.
Let A' be the operator conjugated to an operator A with respect to this
B(x,x) = c(fi) I
K£>W£,q) I-»*/* K£>J?)K£^)
where (£,??) i-> i-> X accordingly to (3.7). In particular (recall that x° = He is
the basic point of G/H):
B(x,z°) = cM IfcK,,)!"7".
The kernel of an intertwining operator depends on a realization of a representation. If we use the coordinates £,r/, then we must take the function $(£,??) = |in direct analogy with the Hermitian case.
§6. Tensor products
For //(El the tensor product
R, = ® r_V„
acting on C°°(S x S) has the following invariant Hermitian form:
= c(fi) J ^(MMM) (||s,f|| • ||Mir dsdtd~sdl
The representation together with E'^ of G in C00(S' x S) can be considered as an analog of the canonical representation from [18].
Let us restrict this representation to the space X>(fl) (the space of C°°-functions on Q with compact support). An operator <p / on V(Q) defined by
f(s,t) = <p(s,t)\\s,tr+K
takes the representation T.TM_K ® T^_K of G in the representation U of G in V{Vt) by translations (see (3.17), (*3.18)):
U{g)f{s,t) = f{s-g,t- r(g)),
and the Hermitian form E' in the Hermitian form EM with the Berezin kernel (let us call
H-
E^ the Berezin form):
Enifh /2) — J f-iis,t)f2(s}t)B(s,t',s,t)dx(s,t)dx(s,t), (6-1)
or, in terms of G/H:
(U{g)f) (x) = f{xg),
^(/1, /2) = j fi{x)f2{x)B(x,x) dx dx. (6.2)
Thus, we obtain a densely defined G-invariant Hermitian form on L2(G/H) (with V(G/H) as the domain). The integral (6.1), or (6.2), converges absolutely for Re// > -1 and is understood as the analytic continuation for other fi s .
We can regard £>(x,x°) as a //-invariant distribution on G/H. Suppose that we succeed expanding B(x,x°) in terms of spherical functions (distributions) on G/H. This is equivalent to writing a Plancherel formula for E^. Then we can write expressions of EM
in terms of Laplace operators Ai,..., Ar. This gives us information about the behaviour
of E^ as // —> -00, and we can say whether CP is true.
The representation RM on C°°(5 x 5) is equivalent to the representation U for fi sufficiently near to fi = -k/2. For other fi the decomposition of contains additional terms so that the space C°°(S x 5) needs some "completion” to contain an orthogonal decomposition with respect to E'^.
§7. Examples
(a) The hyperboloid of one sheet (the imaginary Lobachevsky plane) G/H, where G = 51(2, E), H = GL( 1,1), see [13]. The Lie algebra g consists of real 2x2 matrices with the zero trace. Let Z0 = diag{l/2,-1/2}. Then H consists of diagonal matrices, d = Z{ fj) = mZ0,
(7-i;
The space G/H consists of matrices
_ 1 ( X3 XX-X2\
x ~ 2 \ -Xi - x2 -x3 )
satisfying the condition det x = -1/4. In ffi3, define a bilinear form [x,y\ = -xxyx + x2y2 + x3y3. Then the condition detx = -1/4 is [x,x] = 1, i.e., exactly the equation of
the hyperboloid of one sheet.
The group G acts on G/H by x g^xg and on q" and q+ by fractional linear
transformations:
„ a£ + 7 Srj + /3 ( a /3\
t i—>-------ri i—>----------, g = r 6 (j.
? № + S 777 + 0; V 7 S J
The embedding (3.7) is
£ + »/ 1 + f»?
xi — i----7~i X2 ~ i----IT' X3 ~ 1 cZ'
1 -(rt 1 - £77 I-???
The manifold 5 is the unit circle |u| = 1 in C. For this example it is convenient to take the embedding (3.19) as follows: x 1—> (u,v), |it| = |u| = 1, where
in X3 + ix2 iff X3 + ix2
U = e = ---------------V = e p-------------------
X\ "h 1 2-1 ^
(now a,j3 are not the entries of g). The action of an element of G on u,v is a fractional
linear function from 5f7(l, 1), the same for both u,v.
Let us take the measure dx and the Laplace-Beltrami operator A on G/H as follows
dx\dx2 2 d£ dr] da d/3
dx ~ |i3| _ (1 - tvY 1 - cos(« - PY
d2 92
A = (1 - <^/)2-^-^- = -2(1 - cos (a - /3))
0^- ^ r))dadp
Let U be the unitary representation of G on L2(G/H) by translations (the quasiregulai representation). It decomposes into irreducible unitary representations of three series: the continuous series representations Ta, a = —| + iu, u > 0, with multiplicity 2, and the
discrete series representations T„, n = 0,1,2,..., with multiplicity 1, see, for instance,
[12]. Correspondingly, L2(G/H) decomposes into the direct sum of four subspaces:
L\G/H) = L<°' + I.M + Lt + L;.
Let us write out the expressions for the Berezin transform on these subspaces in terms of A:
- a - 1) sin//7r + (-l)esin(77r r(— //)r( — fJ, — 1) sin//7T C ’
r(—// + cr)r(—/i — a — I) ±
B = ------T?---VT?-------------- °n Ld ’
where the right-hand sides should be considered as functions of A = <t(<t + 1). For LJ and L2 we obtain:
B ~ E-------A (// —> — oo).
№
Thus, CP holds for the discrete spectrum and does not hold for the continuous spectrum. As to algebras with the multiplication (5.2), we can take as such the subspaces of L\ or L~l consisting of -finite vectors. They have no identity element.
(6) The space G/H, where G = SL(n,M), H = GL(n — 1,1), n > 3. Here it is more convenient to consider G/H as the orbit of the matrix x° = diag{0,... ,0,1} under the action x i—» g~lxg of G. Then G/H consists of matrices x of rank one and trace one. This space has rank r = 1 and genus k = n. The spaces of examples (a) and (b) exhaust all para-Hermitian symmetric spaces of rank one up to the covering.
The stabilizer H of x° consists of matrices diag{a,fr} where a £ GL(n — 1,1), b = (det
i-i
The subalgebras q and q+ consist of matrices of the form (7.1), where £ is the row (£i,..., £n-i) and tj is the column (r/j,..., //„_]) from I"-1. The embedding (3.7) is
1 ( -vt -v \
1 - tv V f 1 /
In these coordinates on GjH, the Laplace-Beltrami operator is:
d2
A = (i -tn)YL(hi - km)
d&drii
For x,y £ ffin we write [x,y) = X\y\ + ••• + xnyn and |x| = y(x,x). The manifold S is the unit sphere 5"-1 : |s[ = 1 in 1” with the identification of points s and —5, i.e., S is the (n — l)-dimensional real, projective space. We have \\s,t\\ = |(5,^)|. Any matrix x £ G/H can be written as
t's
x —
where t,s £ 5’n_1, (t, s) 7^ 0, the prime denotes matrix transposition. Let ds be the Euclidean measure on S'"-1. The following measure dx on G/H is G-invariant:
dx = |(i, s)\~ndt ds.
-1
The supercomplete system is = |(s, ^) |In terms of G/H the Berezin kernel is:
B(x,x) = c(/i)|tr(xx)|M,
where
( Tl T17T ^ "
c(fi) = |2"+17rn_2r(—// - n + l)T(fi + l)[cos(// + -)ir - cos —]|
The quasiregular representation U of G on G/H decomposes into irreducible unitary representations of two series: the continuous series representations Ta £, a = (1 /2)(1 — n) -f iu,u > 0,e = 0.1, and the discrete series representations TCT(m), <r(m) = (l/2)(2 — n) + m, m = 0,1, 2,...; all with multiplicity 1, see [14], [15], [8]. Let us write the expressions of the Berezin form (fi < (1 — n)/2) in terms of A:
n T(-fi + cr)r(-/i - a - n + 1) COS//7T + (-1)£ COSCT7T
T (—fj.)T (—fi — n + 1) COS//7T + 1 (n°
T( — a + <r)r( — a — a — n + 1) sin//7T + ( —1)£ sin <J7T
B = —------;----—-------------:---- • --------;---------------- (Tl even).
T(—m)T(—fi — n + 1) sm/£7r
The right-hand sides should be regarded as functions of A = a (a + n — 1). In both formulae the first fraction behaves as 1 — /z-1 A when fi —> — oo. It is just what we need for CP. In the second fractions, the term with ( —1)£ disappears on the discrete spectrum for n even. So we have CP on the discrete spectrum for n even.
We can unite both formulae above
r_, r(^)r(^)
' ' y-cr + 1+e fi+a+n+e r( M )T( )
For the decomposition of tensor product R^ acting on C00(S' x S), see [6,7]. For ^ > (—7i + 1)/2 additional representations act on distributions on S x S concentrated on the boundary T of fi. This action is diagonalizable (the corresponding representation decomposes into the direct sum of irreducible representations). In general, the appearance of representations acting on distributions concentrated on manifolds of lower dimension is one of the intriguing phenomena in harmonic analysis.
References
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