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PARA-HERMITIAN SYMMETRIC SPACES WITH PSEUDO-ORTHOGONAL GROUP OF TRANSLATIONS1
S. V. Tsykina
G. R. Derzhavin Tambov State University, Russia
Our goal is to construct polynomial quantization on para-Hermitian symmetric spaces G/H with pseudo-orthogonal group G = SOo(p, q)- For all these spaces, the connected component He of the subgroup H containing the identity of G is the direct product SOo(p — 1, q — 1) x SOo(l, 1), so that G/H is covered by G/He (with multiplicity 1, 2 or 4).
A construction of quantization on arbitrary para-Hermitian symmetric spaces was offered in
[1]. Polynomial quantization on rank one para-Hermitian symmetric spaces was constructed in
[2]. Our spaces G/H with G = SOo(p,q) have rank 2. Let n — p + q. Dimensions of G and H are n(n — l)/2 and 1 + (n — 2)(n — 3)/2 respectively, so that dimension of G/H is 2n — 4.
The paper is the first step of our program: we describe various realizations of G/H and write explicit formulas for invariant metric, measure and Laplace operators.
1 The pseudo-orthogonal group and its Lie algebra
Let us equip the space Wn with the following bilinear form
[*£> y\ = • KpVp “I" Xp+lUp+l + • • • + XnUn
n
= ^ ^ \xiVi, i=l
where
Ai = ... = Xp — — 1, = ... = Xn = 1,
and x = (xi,, xn), y = (yi,..., yn) are vectors in Rn.
Let G denote the group SOo ip-, q), the connected component of the identity in the group of linear transformations with determinant 1 of the space Kn, preserving the bilinear form y]. The latter means that g € G satisfies the condition
g'lg = /,
where the prime denotes matrix transposition and I is the matrix of the bilinear form [x,y], i.e.
I = diag {Ai,..., An}.
We assume that G acts linearly on R" from the right: x >->■ xg. In accordance with that, we write vectors in the row form.
We also assume that p > 1, q > 1, it is the general case.
Denote by Ga the subgroup of fixed points of the involution a : g ^ LgL, where L is a diagonal matrix diag{—1,1,..., 1, — 1}. It consists of four connected components. Let us write
Supported by the Russian Foundation for Basic Research (grant No. 05-01-00074a), the Scientific Programs ”Universities of Russia” (grant No. ur.04.01.465) and’’Devel. Sci. Potent. High. School”, (Templan, No. 1.2.02).
matrices g £ G in the block form corresponding to the partition n = l + (n — 2) + l. Denote by H an open subgroup of Ga consisting of matrices
/ a 0 /3 \
h= 0 v 0 , (1.1)
\ /3 0 a J
where a2 — (32 = 1, v € SO(p — l,q — 1). The subgroup H consists of two connected components. Let the subscript e indicate the connected component containing the unit matrix E. We have He = G° and this subgroup consists of matrices (1.1), where a = chi, /3 = sht. Thus, it is SOo(p — 1,9 — 1) x SOo(l, 1)- The second connected component of H (which does not contain E) has as its representative the matrix diag{—1, —1,1,. ..,1,-1,— 1}. Since G% C H C Ga, the homogeneous space G/H is a symmetric space.
The Lie algebra g of G consists of real matrices X of order n satisfying the condition X'l + IX = 0. Dimension of 0 and G is equal to n(n — 1)/2.
The involution a of G induces an involution of 0. which we denote by the same letter. The algebra g splits into the direct sum of +1, — 1-eigenspaces of a: 0 = f) + q. The space f) is the Lie algebra of the subgroup H. This decomposition is orthogonal with respect to the Killing form.
The Lie algebra t) and the space q consist of matrices respectively:
/ 0 0 t \ / 0 <p 0 \
f) : 0 u 0 , q : I <p* 0 -</>* ,
V t 0 0 / \ 0 ^ 0/
written in the block form corresponding to the partition n = 1 + (n — 2) + 1. Here u is a matrix in the Lie algebra of SOo(p — 1, q — 1), i.e. a real matrix of order n — 2 satisfying the condition u'I\ + Ixu — 0 where I\ = diag {A2,..., An_i}, ip, np are rows in Mn~2, (p* denotes Inp' (or (ipli)').
The Lie algebra [) has a centre. It is one-dimensional if n^4 and two-dimensional if n = 4. In the latter case is commutative and coincides with its centre. The matrix
/ 0 0 1 A
Z0 = 0 0 0
\1 0 0/
belongs to this centre and forms a basis for n^4. Thus, the space G/H is para-Hermitian.
The stabilizer of Zo in the adjoint representation is exactly the group H, therefore, the manifold G/H is just the G-orbit of the matrix Zq.
The operator adZo has three eigenvalues: —1,0,+1. The 0-eigenspace is Let q“,q+ be — 1, + 1-eigenspaces respectively. They consist respectively of matrices
/ 0 £ 0 \ / 0 rj 0 \
= C 0 r , = »7* 0 -77* ,
V 0 -e 0 J \ 0 7! 0 )
where £, 77 are rows in R”-2. The sum of q_ and q+ is q, so the whole Lie algebra g decomposes into the direct sum
0 = q~ + f) + q+-
Both spaces q1^1 have dimension n — 2, the dimension of f) is equal to 1 + (n — 2)(n — 3)/2. Here are commutation relations:
[q-,q-] = 0, [q+,q+] = 0,
[f),q+] C q+, [fj,q“] C q~,
[q+,q“] C
Therefore, the subgroup H preserves both subspaces q and q+ under the adjoint action (recall we consider the right actions of groups):
Z ^ h~lZh, h £ H. (1.2)
Let h G H have the form (1.1). Under the action (1.2) vectors ^ £ q“ and r] € q+ are transformed
as follows:
= (a +/3)£w, v ^ ff = [a -(3)rjv. (1.3)
For a matrix h € H, given by (1.1), let us consider the operator (Ad/i) . By (1.3), it
t’+
carries a vector £ 6 q+ to the vector ( E q+ given by
c = (a + (3)C,v~l = (a + P)(hvli.
In the coordinate form we have
71—1
Ci ip1 "t" ft) ^ ' X^XjUijC^j.
3=2
Therefore, the determinant of this operator is
det(Ad h) =(a + (3)n~2. (1.4)
q+
Equip the space Wl~2 with a bilinear form by means of the matrix I\:
{£, V) = -&V2 - • • • - ipr]p + (p+iVp+i + • • • +
71—1
i=2
where £ = (£2, • • • ,£n-i) and 77 = (772, ■ ■ ■ ,7771-2) are vectors in Rn~2.
Consider subgroups Q~ = expq-, Q+ — expq+ of G. They consist of matrices respectively
l + (l/2)<£,£) i (l/2)(e,0
E
-(l/2)<e, 0 l-(l/2)(£,0
( 1 + (1/2) (97, 77) 77 (1/2) (^7, 77)
eYv = 77* E -rj*
\ (1/2)(V,V) V 1 — (1 /2)(??, ??)
where E is the unit matrix of order n — 2. The subgroups P± = HQ± = Q±H are maximal parabolic subgroups of G.
2 Gauss decompositions
The group G = SOo(p, q) preserves manifolds [x,x] = c, c £ 1, in 1". Let C be the cone [a;, x] = 0, x 7^ 0, in M™. Let us fix two points in the cone:
= (1,0,... ,0,1), *- = (1,0,...,0,-1).
Consider the following two sections of the cone:
r+ = {xi + xn = 2} = {[x,s~] =-2}, r~ = {xi-xn = 2} = {[x,s+] =-2).
The points .s,+ . s~ belong to r+, T- respectively. They are eigenvectors of the subgroups P+ = Q+H and P~" = Q~H respectively, with eigenvalues a — /3 and a + /3, where a, ft are parameters of h £ H, see (1.1).
Theorem 2.1 The cone C is connected. The group G acts on it transitively.
The section meets almost every ruling of the cone C, the intersection consists of one
point. Therefore, the linear action of G on the cone gives the following actions of G on F~ and T+ respectively:
2 2
XI—> X = -—---------- —r- • xg = --------T -xg, xer , (2.1)
[xg)i-{xg)n [xg,s+\
^2 2
x i—> x = t;—— • xg = —7-r ■ xg, x G T+. (2.2)
{xg)x + {xg)n y [xg,s~]
These actions are defined almost everywhere on For the subgroups Q~ and Q+ respectively,
these actions turn out to be linear: x xg. Moreover, the subgroups Q± act on T* simply transitively. This allows to define the coordinates £ = (£2, • • •, £n-i) on and 77 = (772, ■ ■ ■, ??n-i) on r+ transferring them from q~ on q+ respectively, namely, for u € F~ and v 6 we set:
U = u(0 = = (1 + (£, 0, 2£, -1 + <£, 0),
V = v(n) = s+eYr> = (1 + (77,77), 2r), 1 - (77,77)).
Actions (2.1) and (2.2) of the group G define its actions
£ £ ri^rj (2.3)
by _ _____________________________
v{v) = v(ri)
In particular, the subgroups Q~ and Q+ act by parallel translations, and the subgroup H acts precisely by formulas (1.3).
The stabilisers in G of the points s~ 6 T- and s+ € r+ under the actions (2.1) and (2.2) are the subgroups P+ = Q+H and P~ = Q~H respectively. Together with Theorem 2.1 it gives the following decompositions.
Theorem 2.2 There are the following decompositions:
G = Q+HQ- = P+Q~, (2.4)
G = Q-HQ+ = P=Q+, (2.5)
where the bar means closure and the sets under the bar are open and dense in G.
Let us call (2.4) the Gauss decomposition and (2.5) the anti-Gauss decomposition (allowing
some slang). The decomposition (2.4) means that almost any element g 6 G can be written in
the form
g = eYr’hexs. (2.6)
In (2.6), parameters £, 77 and parameters a, f3,vofh (see (1.1)) are determined as follows. Let
us introduce a function fj,(a) on the space Mat(71, R), setting for a matrix a = (a^):
/x(a) = — [s a, s~*~] = ~ (flu flin flni "I" ®nn)' (2-7)
Then
6 ~ 211(g){9U 9ni)
“Hi = q j~t(9h — Qin)
2 m)
rv — ft = -------- = n.(n\.
" a + P
where i = 2,... , n — 1. Thus, the decomposition (2.6) is defined for g G G such that fj,(g) ^ 0. Similar formulas are true for the anti-Gauss decomposition:
9
= eXi h e
Y„
The re-decomposition of the anti-Gauss into the Gauss plays an important role. Namely, let £ G q-, rj G q+. The product expXf • exp(—Yv) is the anti-Gauss decomposition. Decompose this matrix according to the Gauss decomposition:
exie Yr, _
where Xy G q~, 7^, G q+ and h G H. Denote this element h by Let us write its
parameters, i.e. vectors and ip and the matrix v (see (1.1)), in terms of £,77. Consider the following polynomial in £,77:
Y* u
Then
N{£,ri) = 1 - 2^,77) + {£,£)(ri,Ti).
<p = ~ (£,On)>
■0 = + favK),
a - /3 = {a + (3)-1 = N,
VZ=E~^ + & £№*1 + ~ 2(^ v))Cri},
(2.8)
(2.9)
where N = N((, rj) and E is the unit matrix of order n — 2.
Denote L(£,rj) = (Ad/1(1,77)) ■ Using (1.3) and (2.8), (2.9), we can write it in the coordi-
t>+
nate form. Let the operator L(£,r]) carries a vector ( G q+ to the vector ( G q+. Then
n—1
> X A t (/*-
i=2
where
Lij(Z,v) _jy | $ij 2Aj
[-Vitj + {v,v)Zi£j + (Z>Z)ViVj + (1 - 2(f,J7))&Tfe] > ,
where N = N((. 77). The inverse operator L(£, 77)_1 is an analog of the Bergman transform for Hermitian symmetric spaces. In virtue of (1.4) and (2.8) the determinant of the operator L(^,rj) is
b(£,ri) =detL(t,r!) = N(H,r,r(n-2l It is an analog of the Bergman kernel for Hermitian symmetric spaces.
3 Realizations of the space G/H
(a) We already said in Section 1 that the space G/H is the G-orbit of the point Zq G g under the adjoint representation. The subgroup H is the stabilizer of Zq. The group G acts on this
orbit by X i-» g lXg, g £ G. A matrix g G G carries the matrix Zq to a matrix X = (Xij) G g whose entries are minors of order two of the matrix g taken with the sign ±, namely,
Xij = -A %Mli
where M%in denotes the minor of the matrix g situated in rows with numbers 1, n and in columns with numbers i,j\
9i i 9ij 9ni 9nj
Ki =
9u9nj 9ni9ij ■
(b) Let us consider the manifold C of rulings of the cone C. This manifold consists of lines (without the origin) {xj- = Mx\ {0}, where x EC. The space G/H can be realized as a manifold in the direct product CxCas follows.
The group G acts on C in the natural way. {x}g — {xg}, g G G. The stabilizers of the
rulings £s~}^and {s+} are P+ = Q+H and P~ = Q~H respectively.
On C x C, the group G acts diagonally: ({a:}, {2/}) '—> ({xg}, {yg}), g & G. The stabilizer of the pair ({s“J, {sj~}) is P+ H P~ = H. Thus, the space G/H is the G-orbit of the pair ({s-},{s+}) inCxC.
Theorem 3.1 The space G/H is the unique open dense G-orbit in C x C. This orbit consists of the pairs ({2:}, {y}), x,y G C such that [x, y\ 7^ 0.
This theorem is a consequence of the following theorem which gives a description of G-orbits in C x C.
Theorem 3.2 If (p,q) / (2,2), then there exist three orbits of G on C x C:
i) the manifold consisting of the pairs ({a;},{y}) with [x,y] 7^ 0; dimension of this orbit is
equal to 2n — 4;
ii) the manifold consisting of the pairs ({2:}, {y}) such that [x,y] = 0 and {x} 7^ {y}, dimension of this orbit is equal to 2n — 5;
in) the manifold of the pairs ({2;}, {2;}), it is the manifold C, dimension is equal to n — 2.
If p = q = 2, then there exist 4 orbits on C x C: two above-mentioned manifolds i) and in) and the manifold ii) splits into two orbits defined by the sign of the product of determinants
CM r-H X3 2:4
y 1 V2 2/3 y±
(c) Let us assign to a pair (x,y) £ C x C the matrix
z = Iy'x = y* x, (3.1)
where, recall, y* = Iy'. All these matrices form the set JA of matrices z of rank 1 satisfying the conditions:
zlz' = 0, z'lz = 0.
The trace of matrix (3.1) is equal to [x,y]. Together with each matrix z, the set M. containes all matrices proportional to z, i.e. the ruling Mz \ {0}. Hereby the direct product C x C becomes the set M of rulings in M.. The linear action (x,y) i—> (xg,yg) of G on C x C becomes the ajoint action on the set M:
z^g~lzg. (3.2)
The condition [a:, y\ ^ 0 giving the G-orbit G/H in C x C, see Theorems 3.1 and 3.2, turns out to be the condition tr z 7^ 0 for matrices z G M. Thus, the space G/H is the subset in M defined by the condition tr z 7^ 0. This subset may be identified with the manifold S7 of matrices z in
M satisfying the condition tv z = 1. The action (3.2) preserves fi. Therefore, we can take the manifold with the action (3.2) as a realization of the space G/H.
The connection between two realizations (b) and (c) of the space G/H is the following: to a pair ({a;}, {y}) in the G-orbit G/H in C x C, we assign the matrix
y*x
Z~[x,y]
in fl.
For matrices 2 given by (3.1), the function /i (see (2.7)) is
1 (xi -xn)(yi +yn)
= _ 2----------M'
Denote by C the set of matrices z £ Q with n(z) ^ 0.
As vectors x,y £ C in (3.3), let us take vectors u = u(£) and v = v(rj) in the sections r~ and
T+ of the cone C respectively. We obtain a map T“ x T+ -> fl given by
vr u
Z = n) = 7--------7, U = u{C), V = v{rj), (3.3)
[u,v j
For this matrix z, we have [u, v] = —2N((,rj) and n(z) = A^(£,?7)_1. Thus, the map (u,v) \—> z given by formula (3.3) is defined for (, rj £ M"-2 such that N(£, r/) ^ 0, and its image is L. Therefore, vectors r) £ Mn~2 with the condition iV(£, if) ^ 0 are the coordinates on C and they can be regarded as local coordinates on fL The action (3.1) of the group G on C is nothing but its action on r~ x F1+ in accordance with (2.3): (£,77) '—> (^, 77). For each g £ G this action is defined on a dense set.
In virtue of [1], a metric ds2, a symplectic form w and a measure du on G/H, all invariant
with respect to G, in coordinates £, 77 are:
ds = 2 y ' r))d^jdrjj,
i,3
ui = 2 ^ ^ Lij(£,rj)d£i A dfjj, du = |A^(£,77)|_("_2)^2---^n-1^2---^77n-l-
(d) Finally, the space G/H can be realized as the set in C x C consisting of pairs (x,y), x,y £ C, for which [x, y] ^ 0 is fixed. Let us take [a;, y] = —2. Then the pair (s~, s+) belongs to this set. Let us assign to a pair(a:,y) the matrix with two rows:
z = f x \ = f x 1 x2 ... xn \
Z v y J \ yi V2 ■■■ Vn J '
Let z° denote the matrix corresponding to the pair (.s“, s+). The group G acts by multiplication from the right: z 1—> zg.
4 A root structure
It will be convenient to us to write matrices in G and g in the block form corresponding to
the partition of number n into the sum of 5 summands: n = 1 + 1 +(n — 4)+ 1 + 1. Any
maximal Abelian subalgebra of q consisting of semisimple elements has dimension 2. It means
that rank of the space G/H is equal to 2. For that subalgebra, we take the following subalgebra ciq consisting of matrices
/ 0 0 0 ti 0 \
0 0 0 0 t2
At = 0 0 0 0 0
tx 0 0 0 0
\ 0 t2 0 0 0 )
where t = (ti, t2) £ ffi2. The dual space a* consists of linear functions of t = (ti,t2). We identify these functions with vectors A = (Ai, A2) in M2 and write them as an inner product:
(A,t) = Aiti + X2t2.
The algebra 0 splits into the direct sum of root subspaces of the pair (0, aq):
0 = 00 + ^2 0a,
a^O
where 0Q consists of X 6 0a such that
[At,X] = (a,t)X.
The set £ of nonzero roots consists of the following 8 vectors in a* :
(±1,±1), (±1,0), (0,±1), (4.1)
where all combinations of the signs ± are taken. It is the root system of type B2. Multiplicities of roots (4.1) are equal to 1, n — 4, n — 4, respectively. In a*, we take the alphabetical ordering with respect to coordinates. The sets £+ and £“ of positive and negative roots consist of vectors
(1,±1), (1,0), (0,1),
(-1,±1), (-1,0), (0,-1),
respectively. The subspace 0o is the centralizer of the subalgebra aq in 0, it is a subalgebra of 0. This subalgebra is the direct sum 0o = Qq + m. The subalgebra m consists of matrices
/ 0 0 0 0 0 \
0 0 0 0 0
0 0 6 0 0
0 0 0 0 0
\ 0 0 0 0 0 /
such that m is isomorphic to the Lie algebra of the group SOo(p — 2, q — 2). Let n and 3 denote the subalgebras of 0 formed by positive and negative root spaces respectively. The dimension of each of them is equal to 2n — 6.
The algebra 0 splits into the direct sum:
n = n4-n_-Um-4-i
v " 1 1 1 0’
The subalgebras n and 3 respectively consist of matrices:
X =
( 0 x + y a 0
—x — y 0 /3 x + y
a* p* 0 -a*
0 x + y a 0
\ —x + y 0 (3 x — y
-x + y \ 0 -P*
-x + y 0 )
f 0 x + y a 0 x-y \
-x-y 0 P -x-y 0
a* p* 0 a* P*
0 -x-y —a 0 -x + y
V x-y 0 -13 x-y 0 /
where x, y € M, a, /3 are vectors (rows) in R"-4, a* = hoi1, /3* = I2P1, h = diag{A3,..., An_2}-Parameters x, y, a, P of the matrix X correspond to the roots (1,1), (1, —1), (1,0), (0,1) respectively, and parameters x, y, a, /3 of the matrix Y correspond to the roots (—1, —1), (—1,1), (—1,0), (0,-1) respectively.
Let N denote the positive root subgroup: N = exp n.
5 Laplace operators
The tangent space of G/H at the initial point {H} can be identified with the space q = q~ + q+, see Section 1. The subgroup H preserves both subspaces q_ and q+ under the adjoint action (1.2). Coordinates £ in q~ and 77 in q+ are transformed by (1.3).
Let S'(q) denote the algebra of polynomials on q. The action (1.2) of the group II on q gives rise to an action of H on S(q). Let S(q)H denote the algebra of polynomials invariant with respect to H.
Theorem 5.1 The algebra S(q)H is generated by two polynomials
Proof. Let Hs denote the subgroup of H consising of matrices (1.1) with a = 1, /3 = 0. It is isomorphic to SOo(p — 1, g— 1). It acts on £ and 77 linearly. It follows from [3] that the algebra of polynomials in £ and 77 invariant with respect to Hs has as generators three polynomials: (£,0,
{77,77), (£,77). Under the action (1.3) these polynomials are multiplied by (a + (3)2, (a — f3)2 and
1 respectively. Since a + (3 = (a — (3)~l, generators for the whole group H are polynomials
(Z,0(v,v) and (^v)-
Let H(G/H) denote the algebra of differential operators on G/H invariant with respect to G.
This algebra is in the one-to-one correspondence with the algebra S(c\)H. Let Ai and A2 denote
operators in 0(G/H) corresponding to generators (£,77) and {£,£){ri,rj) of ^(q)^ respectively.
Let us call these operators Ai and A2 the Laplace operators on G/H. These operators are
differential operators of the second and the fourth order respectively, they are generators in
B>(G/H). Explicit expressions of them are very cumbersome. We write explicit expressions 0 0
for their radial parts Ai and A2 with respect to the root subgroup N. They turn out to be differential operators with constant coefficients.
Let us use the realization (d) of G/H, see Section 3. Consider the set of points 2 = z°an, where a = exp At, t = (^i, ^2), n £ N. It is a neighbourhood U of the point z°. For coordinates in this neighbourhood we obtain t\,t2 and also parameters x,y,a,(3 of the subgroup N (horospherical coordinates). Let/ be a function defined on U such that it does not depend on n.
Then it is a function of ti,t2: f(z°an) = F(t\,t2). Let D be a differential operator in u(G/H).
Then Df also does not depend on n:
Df =D F,
0
where D is a differential operator in t\,t2, the radial part of D with respect to N. Introduce
operators
- (££)+<--«>■■
* - (£-£)’+’(£-£)-*■-<>■
Theorem 5.2 We have
Ai = ^{£>i + D2 - (n- 4)(n - 6)},
A2 = D\D2 + 2(n — 4)3.
The proof of this theorem contains rather long computations, so we omit it.
REFERENCES
1. V. F. Molchanov. Quantization on para-Hermitian symmetric spaces. Amer. Math. Soc. Transl., Ser. 2, 1996, vol. 175, 81-95.
2. V. F. Molchanov, N. B. Volotova. Polynomial quantization on rank one para-Hermitian symmetric spaces. Acta Appl. Math., 2004, vol. 81, Nos. 1-3, 215-232.
3. H. Weyl. The Classical Groups, Their Invariants and Representations. Moscow: IL, 1947 (Russian translation)
