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Invariant algebras of functions on spheres.
I. A. Latypov Omsk State University, 644077 Omsk, Russia e-mail: analysis@univer.omsk.su
Let X = G/H be a compact homogeneous space; an invariant algebra on X is a closed subalgebra of C(X) which is invariant with respect to the action of G on X. For example, the restiction to the skeleton (Shilov boundary) X of the algebra of analytic and continuous up to the boundary functions in a symmetric domain is an invariant algebra on X. There are many examples of this kind, and the question is if any nontrivial invariant algebra may be realized as an algebra of analytic (somewhere) functions.
The case of bi-invariant algebras on compact Lie groups is the most investigated one. Algebraic and topological properties of a group restricts the structure of invariant algebras on it. For example, Rider [15] proved that if a compact Lie group G admits an invariant Dirichlet algebra (this means that real parts of functions in A are dense in the space of all real-vaued continuous functions) then G is abelian and connected. Wolf [18] and Gangolli [2] showed that every uniformly closed bi-invariant algebra on a semisimple group is self-adjoint (a function algebra A is called self-adjoint if A = A, where the bar denotes the complex conjugation; the algebra A is called antisymmetric if A fl A contains only constant functions). Gichev [4] proved that a bi-invariant algebra A on a group G is antisymmetric if and only if the Haar measure of G is a multiplicative functional on A; Rosenberg [16] gave a characterization of bi-invariant antisymmetric algebras on compact groups in terms of harmonic analysis.
The case of invariant algebras on general homogeneous spaces G/H is not well-understood even for compact G. The hypothesis is that, for every nontrivial G-invariant algebra on G/H, there is a G-invariant foliation with leaves of the type G/H such that the restriction of A to each leaf is some algebra of boundary values of holomorphic functions on a domain in Gc /Hc whose skeleton is G/H. Such domains appears in a paper of Gel’fand and Gindikin [3]. They considered a real semisimple Lie group G as a boundary of a certain complex domain in Gc . Ol’shanskii [13] proved that these complex domains are interiors of subsemigroups of Gc of the form Gexp(zC), where C is any Ad(G)-invariant cone in the Lie algebra of G.
In this article we consider invariant algebras on spheres Sn; they are the simplest examples of compact homogeneous spaces. Montgomery, Samelson and Borel (see [14]) found all realizations of Sn as a homogeneous space G/H, where G is a compact group, H is its closed subgroup (the isotropy group of some point). Their results are summarized in the following table:
G H G/H
(1) S0(n+1) SO (n) Sn
(2) U(n) U(n — 1) S2n-l
(3) SU(n+ 1) SU(n) S2 n+1
(4) Sp(n) Sp(n - •1) S4n-1
(5) Sp(n) x U(l) Sp(n - l)xU(l) S4n-1
(6) Sp(n) x SP(1) Sp(n - ■ 1) x Sp(l) 54n-l
(7) Spin9 Spin7 s15
(8) Spin7 g2 s7
(9) G2 SU(3) s6
Invariant algebras on S1 = S0(2) = U(l) are in one-to-one correspondence with subsemigroups of the group 7L of integers [17]. K. de Leeuw and H. Mirkil [11] showed that there are only three SO(n + 1)-invariant algebras on 5n (n > 1): the algebra C of constant functions, the algebra of even functions and C(Sn). Note that they are self-adjoint. W. Rudin and A. Nagel characterized U(n)-invariant algebras on S2n~l ([17], [12]); J. Kane [9] described their maximal ideal spaces and realized almost all antisymmetric
U(n)-invariant algebras as algebras of holomorphic functions. We study invariant algebras on spheres in the remaining cases. Main results are stated in Theorems 1-3.
Theorem 1. All invariant algebras in the cases (6)-(9) are self-adjoint. If n > 2 then SU(n)-
invariant algebras on S'2”-1 are U(n)-invariant. Every Sp(n) x U(I)-invariant antisymmetric algebra on 54n-l
is a subalgebra of some \J(2n)-invariant antisymmetric algebra.
This theorem is proved in Propositions 2-7 (Propositions 2-4 are proved in Section 1, Proposition 5 — in Section 2, and Propositions 6-7 — in Section 3).
As a corollary of Propositions 5-7 we have a classification of invariant algebras on PnC and PnIHI (groups acting transitively on projective spaces were classified by Onishchik [14]). All these algebras are self-adjoint.
The case (4) is most complicated because the decomposition of the quasi-regular representation contains irreducible representations with a multiplicity > 1 ([10]). So in Section 4 we consider only the case n = 1 and only a family Aa of antisymmetric Sp(l)-invariant algebras of even functions on the three-dimensional sphere depending on a continuous parameter. These algebras are characterized by
Theorem 2. There are invariant CR-conditions on S3 such that Aa consists of all CK-functions. For every nonstandard invariant CR-conditions on S3 there exists or > 0 such that Aa is isomorphic to the algebra of all CR-functions; in particular, each CR-function f is even.
These algebras differ from the invariant algebras on U(n)/U(n — 1). Every antisymmetric containing constant functions invariant algebra A on U(n)/U(n — 1) has the following properties:
a) A admits an invariant Z+-grading, i. e. A = (Bj2kez+ suc^ that. AiAj C Ai+j where 2£+ is the set of non-negative integers and Ai is an invariant (non-trivial) subspace for each
b) a linear functional corresponding to the normalized invariant measure is a multiplicative functional on A\
c) the group U(n) have a fixed point in the maximal ideal space of A.
An invariant Z+-grading is given by numbers {p — q : H(p,q) C -A}. The properties b) and c) follow from [5]—[7]: every invariant algebra on U(n)/U(n — 1) is an averaging of a bi-invariant algebra on U(n). By the Theorem 1 every antisymmetric containing constant functions invariant algebra A on Sp(n) x U(l)/Sp(n — 1) x U(l) also has these properties.
Theorem 3. Aa is an antisymmetric algebra isomorphic to the algebra of all analytic in the relative interior and continuous up to the boundary functions on the set
{(zi> z2> 23) £ C3 : 2|2i|2 + \zi\2 + 2|z3|2 < 1 + 2a2, z\ — ^.z\z% — 1}
which coincides with its maximal ideal space Ma. For Aa each of the properties a), b), and c) doesn’t occur.
In this paper invariant algebras on the spheres are studied by the following way. Since G is compact every invariant subspace X of C(Sn)..is uniquely determined by minimal ones included to X. They are finite dimensional complex linear spaces of polynomials. The action of G extends to the action of the complexification Gc in these spaces (representations of G, Gc and tangent representations of the corresponding Lie algebras will be denoted by the same letter). We find minimal invariant spaces and the corresponding highest weights of gc . Since any group G in the table above is naturally embedded to SO(n) or U(n) and the problem for these two groups is solved it is sufficient to find the decompositions of minimal SO(n)- or U(n)-invariant spaces. We find the highest vectors of the irreducible representations of G, i. e. the polynomials which are annihilated by n+ (g07 = t © n_|_ © n_, n+ is the nilpotent
subalgebra of g corresponding to positive roots, t is the Cartan subalgebra). To prove coincidence of minimal invariant spaces with SO(n)-invariant spaces we use the Weyl formula for the dimension d\ of the irreducible representation of the Lie algebra g with the highest weight A:
p>o x 7
where 6 is the half-sum of positive roots of g. Finally, we have to describe invariant subspaces which are closed under the multiplication; the complete solution to this problem is given in the cases (6)-(9), in the case (3) for n > 1 and a partial one in the case (5). In the cases (5)-(7) we use the Peter-Weyl theorem, the usual scalar product will be denoted by ( , ).
We finish this introductionary part with the following remark. The problem of the description of all self-adjoint algebras has a geometrical interpretation.
Proposition 1. A function algebra A is a self-adjoint G-invariant algebra on a homogeneous space M = G/H, G is compact, H is its closed subgroup, if and only if there exists a homogeneous space M' and a continuous equivariant mapping ir : M —*■ M' such that A = C(M') o n.
Proof. Set x ~ y if f(x) = f(y) for all / E A. Clearly, this is a G-invariant equivalence and A separates its classes. Hence M' = M/~ is a homogeneous space of G and A may be identified with C(M') by the Stone-Weierstrass theorem.
Corollary. Self-adjoint algebras on M are in one-to-one correspondence with closed subgroups of G which include H.
We don’t consider the geometrical problem of a description of all closed subgroups of a compact Lie group G which include a closed subgroup H but we receive a solution of this problem in the cases (5)-(9) as a consequence of the description of spectrums of self-adjoint invariant algebras.
1 Exceptional spheres
Let Ok be the space of homogeneous polynomials of degree k on Sn 1 and Hk be its subspace of harmonic polynomials. The dimension of the space Ok is equal to (n — 1 + k)\/(n — 1)!A:!. Since O,t = Ok-2 © Hk ([17]), the dimension of the space Hk is (n — 2 + 2k)\/(n — 2)!£!. Let p be the representation of GL(n, IR) in Ojfc, p(g)p(x) = p(xg). The tangent representation of gl(n,M) is defined by p(etj) = xid/dxj where etJ-is a matrix which element (i,j) is 1 and other elements are 0. The restriction of this representation to SO(n) is irreducible in Hk- The spaces Hk are invariant because SO(n) commutes with the Laplacian. The Lie algebra so(n,C) consists of all skew-symmetric matrices.
/ 0 -U1 -V* \
It is convenient to realize so(2m + 1,C) as the set of matrices of the form K X Y I ,
\U Z -X* )
where X is arbitrary, Y and Z are skew-symmetric m x m matrices, U and V are m x 1 vector-columns. This realization is obtained by the reduction (via the change of variables) of the usual quadratic form in
/ i ° ° \
C2m+1 to the form defined by the matrix JO 0 Im I , where Im is the identity m x m matrix. In
\ 0 lm 0 /
the new coordinates a function / is harmonic if and only if (d/dx\ + 2 YHiLi d/&xi+idxm+i+i)f — 0. There is an embedding of g2 to so(7,C) as the set of matrices
/ 0 —wiy/2 -W3y/2 — Z4>/2 N 1 — Z3\fi — W4\/2 \
ZIV2 hi W2 Zs 0 Z\ -w3
z3y/2 2-1 , h2 ?6 -Z4 0 Wi
Wqy/Ô, w5 w6 —hi — h2 w3 — Wi 0
W1VÔ, 0 W4 -z3 -hi -Zl —w5
w3y/2 —W4 0 Zl -w2 -h2 -We
V Z4\/2 Z3 -Zl 0 -Z5 -Z6 hi + h2 /
9 =
(see [19]), hi corresponds to t, Z{ corresponds to n+ and Wj corresponds to n_. Note that hi and /*2 are short roots of the Lie algebra g2.
Proposition 2. All G2-invariant spaces on S6 are SO(7)-invariant.
Proof. The harmonic polynomial £7 (xy is a coordinate function) is annihilated by p(n+). It is the highest vectors of the irreducible representation with the highest weight k(h 1 + h2). By the Weyl formula
(1) the dimension of the invariant space, generated by £7, is equal to (A: -|- 4)(2Ar -h 5)/5!Ar! and is equal to the dimension of //*. It means that G2-invariant spaces are SO(7)-invariant.
It is convenient to realize so(2m, C) as the set of matrices ^ ^ ^ , where X is arbitrary, Y
and Z are skew-symmetric m x m matrices. This realization is obtained by the reduction of the usual quadratic form in C2m to the form f ^
by a linear change of variables. In the new coordinates
a function / is harmonic if (X^i d/dx{dxm+i)f = 0, where x7 = xm+«- To describe invariant algebras in the cases (8) and (9) we consider an embedding of so(2/ -f 1, C) to so(2/, C) corresponding to the spinor representation, details see in [8].
Let C+ be a subalgebra of even elements of the Clifford algebra. Choose generating elements v\ wi ... ,wi of C+ such that V{Vj + VjV{ = 0, W{Wj + WjWi = 0, V{Wj + wjV{ = — 26,j. A basis of C+ is the set of elements v,-, ...VirWj1 ...Wj, where *‘i < ... < ir, j\ < ... < js. The space generated by vi ... viwjl ... wj,, ji < ... < js, is 2,-dimensional right ideal in C+.
Let (/i,-, e,-, fi) be sl(2, (C)-triple corresponding to the simple root a,- of the Lie algebra so(2/ + 1,C). The elements h{ generate t, e* generate n+. The spinor representation r of so(2/ + 1,C) is defined by formulas
r(ei)z = ±zviWi+1, i= 1,...,/- 1; r(ei)z = \zvr,
r{hi)z = \z{viWi - vi+lwi+i), i= 1......./ — 1; r(hi)z = 1 + vtwi.
Let xi = v\ .. .vi be a coordinate function. The harmonic polynomial x^ is the highest vector of the irreducible representation with the highest weight khi. The harmonic polynomial + i ^owes^
vector of this representation, x2i-i+i = v\ .. .v\w \ .. .wi.
Proposition 3. All Spin7-invariant spaces on S7 are SO(8)-invariant.
Proof. The dimension of the irreducible representation of so(7, C) with the highest weight kh3 is equal to (k + 5)!(2fc 4- 6)/6!fc! by the Weyl formula (1) and is equal to the dimension of Hk■ It means that Hk is irrducible.
Proposition 4. There are only five Spin ^-invariant algebras and they are self-adjoint: the three SO(l 6)-invariant algebras, the algebra of functions which are constant on the fibres of the Hopf fibration S15 —>• S8 and its subalgebra of functions which are even on the base of the jibration.
Proof. Set
y = (v)(vW2W3W4) - (vwi)(vw3w4) + — (VW^){vW2W3),
yi = Xq(vWi) — (vW2W2)(vWi W3WA) + (vWiW3)(vWiW2W4) — {vWiWA){vW\W2W3),
where the elements of the Clifford algebra are contained in the brackets and v = V\V2V3V4, we multiply the brackets as coordinate functions. The harmonic polynomial Si = x\yl is annihilated by p(n+), it is the highest vector of the irreducible representation with the highest weight kh4 + lh\. The lowest vector of this representation is = x9Vi — x\yl ■ Denote the corresponding invariant subspace of Hk+21 by 14,/-It could be shown that Hk C C^15) is a direct sum of Vk-2i,i, * = 0,.. -, [A:/2] (see [19, pp. 304-305] with another notation).
Proposition 1 implies that the algebra B of functions which are constant on the fibres of the Hopf fibration Spin9/Spin7 = S15 —► S8 = Spin9/Spin8 and its subalgebra of functions which are even on the base of the fibration are invariant algebras, namely the closures of ]T) Vo,/ and ^2 Vo,2/• There are no other nontrivial SO(9)-invariant subalgebras of B ~ C(S8).
Suppose that an invariant algebra. A contains the space Vk,i, k > I. Since s3 = r(e4)s2 = (vwiw2w3)xl~1yl{ we have ((vw\ w2w3)xi, s\s3) ^ 0, i. e. s4 = (vw\w2w3)x\ G A. Projections of s4 on the spaces V^.o and Vo,i are non-zero, so all even functions are contained in A. If k is odd then A coincides with C(S15).
2 Complex spheres
Let 0(p, q) be the space of homogeneous polynomials of degree p on z and q on ~z, H(p, q) be its subspace of harmonic polynomials. Let 7r be the representation of U(n) in Ok, 7r(fl,)s(-2r) = s(<7-l20- The tangent representation of u(n) is defined by
7r(X>(z) = 4kexPH-’0*)]l<=o = {V2s(z),-A-2} + {Vr«W,-X7},
at
where Vys = (d/dyi,. ..d/dyn), {a, 6} =
Choose a basis of u(n) in the form Ujk = ejk — ekj,j < k; vjk = i(ejk + ejfc;); tj = iejj. Then
, v d d __ d _ d x
TC(Vjk) = — l(Zk -r-h 2,- —---Zjfc —---------Zj —-IS,
V jk, \ k dz_ t d_ j d_> ,
a \ ■( 9 — 9 \
*(*;) =->(^-^5=)*.
Choose a basis of u(n)(C = gl(n,C) as Ijj = —itj = ejj, Ijk = (Ujk — ivjk)/2 — e.jk■ We have 7r(ei*) = -zkd/dzj +zjd/dzk-
It was shown in [17] that the spaces H(p, q) are irreducible components of the representation of U(n). The Lie algebra su(n)c = sl(n, C) consists of all matrices with zero trace, its subalgebra n+ is generated
by e,j, i < j. The polynomial z£ziq £ H(p,q) is annihilated by 7r(n+), so it is the highest vector of
the irreducible representation. The operator Yl]b=i ekk acts on H(p,q) as the multiplication on (q — p) hence the spaces H(p, <7) are irreducible components of representation of sl(n, C). Moreover, if n > 2 then the highest vectors of different H(p,q) have different eigenvalues under the action of 7r(en — 622), i- e. SU(n)-invariant spaces are U(n)-invariant and we have
Proposition 5. If n > 2 then S\](n)-invariant algebras on S2n~x are U(n)-invariant.
Since J2H(p,p) coincides with the set of all polynomials which are constant on all complex lines we have
Corollary. All invariant algebras on PnC = SU(n+ l)/S(U(n) x U(l)) are contained in the following list: C(PnC), C and (in the case n = 1) the algebra of functions which are constant on pairs of orthogonal complex lines.
3 Quaternion spheres, the special cases
Let us consider the action of the group Sp(n) x Sp(l) on S4n_1, Sp(n) acts by the multiplication from the left and Sp(l) acts by the multiplication from the right. We realize Sp(n) as the set of unitary 2n x 2n
matrices such that SlJS = J, J = ^ o* )' ^ements sP(n>^) are matrices X = ^ ^ ^ ^
with B = B*, C = C\ D = — A1. Choose a basis of sp(n, C) in the form
a - — ( ^ ^ 6 • — [ ^ e,J ^ a - — [ ^ ^
17 “ V 0 -<* J ’ 3 ~ V 0 0 ) ’ 3 ~ \ eij+ea 0
The Cartan subalgebra t of sp(n,C) is generated by an, n+ is generated by 6,j and aki, k < I.
Consider the restriction of the representation 7r to sp(n,C), Wi — zn+»:
7r(ai;) = 7r(6jj ) - 7r(en+i,n+i) = -Zj—- + zl— + Wi— - Wj—,
7T(bij) = 7r(ei,n+i) + 7r(e;>+i) = -Wj—+Zi— - Wi— -1- Zj—,
*(ij) = <en+itj) + 7T(en+i|t) = —Zj - + Wi— - Zi— + Wj—.
Since sp(l, C) ® sp(l, C) = so(4, C) and sp(l) © u(l) = u(2) we assume that n > 2. The polynomial
si = w{z\(wiJ2 - W2Zi)r
is annihilated by 7r(n+); denote by P{p,q,r) the corresponding invariant space. P(p,q,r) is the space of the irreducible representation of sp(n, C) with the highest weight (p + q + r, r, 0,..., 0), the lowest vector of this representation is
s2 = z[w\(z2W]_ - ZiWlY.
It was shown in [10] that H(p,q) = ©[’I'q ^p,q^P(p — i, q — i, i). Operators L\ = {z, V^r} — {wy Vr} and
L2 = {w, V2} — {z, Vu,} commute with n (for every n). They map P(p,q,r) on P(p+ 1,9 — l,r) and
P{p— 1, g+1, r) respectively. Operators L\, L2 and L0 = [L\, L2] = + V2} — {w, Vw} — {z, V2}
define the action of Sp(l). The spaces Yli=o‘ ^(z>k — i — 21,1), 0 > I > [k/2] are Sp(n) x Sp(l)-invariant irreducible subspaces of Hk-
Proposition 6. All Sp(n) x Sp(l)-zn varian/ algebras on 54n_1 are contained in the following list: the SO (An)-invariant algebras; the algebra of functions satisfying a condition /(w) = f(wq) where w is a quaternion vector and q is an arbitrary quaternion from Sp(l); in the case n = 2 the algebra of functions satisfying the condition f(w) = f(z) if (z,w) = 0. All these algebras are self-adjoint.
Proof Closures of the spaces -^(0,0, z) and -^*(0,0,2z) in the case n = 2 are the men-
tioned algebras. It is sufficient to show that there are no other ones.
Suppose that P(p, q, r) and P(k, I, m) lie in A, p > q and k < I. Let Si and s2 be the highest and the lowest vectors of representation in P(p,q,r). Since ((ziwi)p~q, sis2) ^ 0, H(2p — 2^,0) C A. Similarly we have //(0,2/ — 2k) C A. It means that A contains H( 1,1) [17], in particular A contains P(0,0,1). Therefore A contains H(p + r,q + r) and H(k + m, I + rn), so A is U(2n)-invariant and self-adjoint.. Moreover, if A is Sp(n) x Sp(l)-invariant then P(2, 0,0), P(0,2,0), P(l, 1, 0) and P(0, 0,1) are contained in A, the space of all even functions lies in A.
Suppose that an algebra A contains P(0,0,/), I > 0, si is the highest vector and s2 is the the lowest vector of the representation in P(0,0,/). P(0,0,2) is contained in A because (S3, (7r(cn)si)s2) ^ 0, where s3 is the lowest vector of representation in P(0,0,2). If n > 3 then P(0,0,1) is contained in the space generated by products of polynomials from P(0,0, 2) and the statement is proved. The exceptional algebra in the case n = 2 also could be described as the algebra of functions which are constant on the fibres of the Hopf fibration S7 —► S4 and even on the base of the fibration (sp(2,C) = so(5,C)).
Corollary. All Sp(n + \)-invariant algebras on the quaternion projective space PnM — Sp(n -f 1)/Sp(n) x Sp(l) are contained in the following list: 5(PnM), C and (in the case n — \) the algebra of functions which are constant on pairs of orthogonal quaternion lines.
Let us consider the action of the group Sp(n) x U(l) on S4"-1. The generating element of u(l) acts on P(p,q,r) by the multiplication on (p — q). Therefore the spaces P(p,q,r) are separated. It means that P(p,q,r) is Sp(n) x U(l)-invariant irreducible subspace of Hp+q+2r, P(p,Q,r) is a subspace of eigenfunctions of operators L\L2 and L2L\ with eigenvalues (p + 1 )q and (q + 1 )p.
Proposition 7. Every antisymmetric Sp(n) x \](I)-invariant algebra on S4n_1 is a subalgebra of some antisymmetric \](2n)-invariant algebra. The only Sp(n) x U(I)-invariant self-adjoint algebra on 54n_1 which is not Sp(n) x Sp(l) and \J(2n)-invariant is the algebra of functions
satisfying conditions f(iv) = f(ocw), |a| = 1 and f(w) = f{wq), where q = ^ ^ ^ j . Every Sp(n) x
U(1 )-invariant algebra A could be represented as A = B © S where B is an antisymmetric invariant algebra and S is a self-adjoint invariant algebra.
Proof. If P(p,p, r) is contained in A then A contains the polynomial si = S2, where si is the highest
vector and s2 is the lowest vector, so A is not antisymmetric. If A is antisymmetric we may assume that A consists of P(p, q,r), such that p > q (the case of the alternative inequality corresponds to the conjugated algebra). Then A is contained in the antisymmetric U(2n)-invariant algebra Up>qH(p,q) U //(0,0) and the first part of the proposition is proved.
Let A be a self-adjoint invariant algebra which is not Sp(n) x Sp(l)- and U(2n)-invariant, as stated in the Proposition 6, A C 0 k, I). Suppose that P(k,k,l) C A and k ^ 0, s\ and s2 are the
highest and the lowest vectors of the representation in P(k,k,l). Since (n(b22S3, (n(c22)si)s2) ^ 0, where S3 is the lowest vector of the representation in P(0, 0, 2), P(0, 0,2) is contained in A. Since (S4, sfs^) ^ 0, where S4 is the highest vector of the representation in P(2Ar, 2k, 0), P(2k,2k,0) is contained in A too. Since (7r(622S6, (^(022)54)55) ^ 0, where S5 is the lowest vector of the representation in P(2k,2k,0) and S6 is the lowest vector of the representation in P(2,2,0), P(2,2,0) is contained in A.
Union of all P(k,k,l) with even k is the set of all polynomials satisfying conditions f(w) = f(ocw),
|o| = 1 and f(w) = f{wq) where q = ^ ^ 0^ ) ’ invarian^ algebra could be described as the set
of all polynomials which are constant on orthogonal complex lines lying on the same quaternion line.
For an invariant algebra A set 5 = A fl A. Then 5 is a self-adjoint invariant algebra. Let B be the orthogonal complement to S in A (in L2), B consists of P(p,q,r) such that either p > q or p < q. If / E B and h £ B then fh E B. B is an antisymmetric invariant algebra and the proposition (and the theorem) is proved.
Corollary. All invariant algebras on P2n+1C = Sp(n+1)/Sp(n) x U(l) are contained in the following list: C(P2n+1 C); C; the algebra of even functions; the algebra of functions which are constant on the
fibres Sp(l)/U(l) of the fibration Sp(n + 1)/Sp(n) x U(l) —> Sp(n + 1)/Sp(n) x Sp(l) and (in the case n=l) its subalgebra of functions which are even on the base of the fibration.
4 Sp(l): a family of invariant algebras
We can identify the group SU(2) = Sp(l) with the set of matrices ^ ^ _ 'j, |a|2 + |fc|2 = 1 or with
the sphere S3 C C2 with the multiplication (a, 6) * (c, d) = (ac — bd, be + ad).
It was shown in [10] that the set of all highest vectors of irreducible representations of sp(l, C) in Hk coincides with the set of polynomials of the form ]T^=0 'YiWl'zk~t, 7 G C.
The vector space generated by polynomials
ai = aw2 — wJ,
a2 = 7r(cn)ai = 2 otzw + ww — zJ,
a3 — ^7r(cn)a2 = az2 + zw
is invariant under the action of SU(2) from the left. Let Aa, a > 0, be the invariant algebra with generating elements ai,a2,a3 (the algebra Aq = Y^H(k,k) is U(2)-invariant and self-adjoint).
Proof of the Theorem 2. There are following relations between the generating elements:
a2 — 4aia3 = 1 (2)
2 |ai |2 + |a212 + 210312 = 1 + 2a2 (3)
An image of the sphere under the mapping T : C2 —> C3 defined by the polynomials a 1, a2 and a3 is the set of points satisfying (2) and (3).
If p is a polynomial from Aa then p is a polynomial on a\, a2 and a3 and satisfies df /\ da\ f\ da2 f\ da3 = 0. This equation is equivalent to
^-{2az + w)^ + w^L + {2aw-j)°-y = o (4)
Every operator commuting with 7r is some polynomial on Lq, L\ and L2. Vector fields iLo, L\ — L2
and i(L\ +L2) generate the space of all invariant real vector fields on the sphere. Vectors dT(i(dz — d~z) + ai(dw — dw)) and dT(—a(dw + dw)) generate a complex tangent line at the point (0, — l,a) = T(1,0). Since (4) is equivalent to
((¿ar(Li + L2) - iLo) + ia(Li - L2))f = 0, (5)
and at the point (1,0) (5) gives
... . d d . .. d d , d 9 n.
^ia^dW + l^d^j + ’ ^ ~ ’
we obtain invariant CR-conditions on S3.
Suppose that a homogeneous harmonic polynomial p ^const is the highest vector of some irreducible component of the quasi-regular representation,
k
p = '£iyiwiJk-i.
1=0
If p satisfies (4) we obtain the relations between 7 70 = 0, (2i - k)ji + 2a(k - 1 + *)7»-i = °-
They implies that 7,- = 0 for all i < k/2 and k is even. Moreover, p is uniquely determined by jk/2-k 12 •
It means that p = 7it/2ai , i. e. p lies in Aa. Since every polynomial on the sphere is the sum of homogeneous harmonic polynomials we prove the first part of the theorem.
For every operator L = toiLo -Mi^i +t2L2 we can choose 7*, z = 1,2,3, such that the polynomial p = loz2 +7itx;1I'1 +72iu2 is annihilated by L. There is a right translation T such that p*T = 7o‘z2 + 7ilwl~zl.
Then L*T = íqÍLq + t\lL\. Since the usual CR-conditions are defined by the equation L\f = 0 suppose that 7i = 1 and to = 1. The right translation by the corresponding diagonal matrix gives a\ for some a > 0. Note that the equation Lof = 0 doesn’t define CR-structure, so the Theorem 2 is proved.
Proof of the Theorem 3. Suppose that there exists / £ Aa such that f = f. Algebra Aa is contained in Up>9>oH(p,q). Therefore / lies in the closure of U£L0//(p, p). If / is not constant function then //(2,2) lies in Aa (see [17]). Since z2w2 £ //(2,2) does not satisfy (4) / is a constant function.
The maximal ideal space of the algebra Aa is the polynomially convex hull of the image of the sphere in C3 [1]. Let z = &(, w = drj, |0| = 1, |C|2 + M2 = 1- Then
ai = o¡62(2 + (rj, a2 = 2a92(r¡+ \rj\2 - |C|2, a3 = q62t]2 - t](.
This mapping extends holomorphically on 6 in the unit disc by a natural way. A calculation shows that the family of mappings fa : ID) —* C3 covers a part of the hyperboloid (2) which is contained in the ellipsoid (3). Thus we have found Ma.
The algebra Aa consists of all analytic in the relative interior and continuous up to the boundary functions since it is generated by the analytic polynomials and every analytic fuction satisfies CR-conditions
(4).
There is a transitive action p of SU(2)C = SL(2,C) on the hyperboloid (2), p(T)M = TMT\ where M = (^ 2flai 2a2 ) ’ emdedding the sphere is equivariant. Choose three subgroups of SU(2):
_ / el<t> 0 \ n _ ( cos t sin t \ n _ ( cos* isint
1 0 e~l<t> / ’ 2 \ — siní cosí J ’ 3 y isint cost
Points ±(0,1,0), ±1/2(1, 0,-1) and ±l/2(¿,0,¿) are the only fixed points with respect to the action of
Gi, G2 and G3 respectively. Hence SU(2) have no fixed points.
Let p. be a linear functional corresponding to the invariant normalized measure on S3, precisely the Haar measure on the group SU(2). Then p(a\) = /z(a2) = p(o.3) = 0 but /¿(1) = 1, so ¡jl is not a multiplicative functional on Aa.
Suppose that Aa — ©]C/t6Z+ an invariant Z+-grading. It means that constant functions lie in
B0. The relation (2) implies that if B is the invariant space generated by ai, a2 and a3 then B2 C Bo-Hence A is contained in B0, and the Theorem 3 is proved.
Remark. A minimal Sp(n)-invariant algebras on the sphere in C2n, n > 2, containing the highest vector aw2 — wi~z\, is not antisymmetric. If n = 2 it contains the algebra of functions which are constant on all pairs of quaternion lines. If n > 3 it contains the algebra of functions which are constant on all quaternion lines.
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