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Domains with homogeneous skeletons and invariant algebras
V.M. Gichev Omsk State University, 644077 Omsk, Russia e-mail: gichev@univer.omsk.su
Let N be a complex manifold, G be a real Lie group acting on N by holomorphic automorphisms, and M. be a holomorphically convex domain in N whose skeleton is a single orbit M = G/H. In this article we consider the problem of a description of these domains in the case of compact G (the noncompact case is much more complicated). Because of the lack of a term, we shall call such domain a semihomogeneous domain M. over M. Thus we fix the domain and the boundary — a domain may have several ideal boundaries.
There are many evident examples: the unit disc © and the upper half-plane C+ in C, bounded symmetric domains and Siegel domains in Cn. There are also ’’noncommutative” examples such as Olshanskil semigroups (semigroups of the type Gexp(i'C) where G is a real form of a complex Lie group and C is an invariant cone in the Lie algebra ¿7 of G).
The set of all holomorphic and continuous up to the boundary functions on M. is, for compact M, a closed subalgebra A of the Banach algebra C(M), and the evaluating functional at any point of M defines a maximal ideal of A. Hence there is an embedding of M. to the maximal ideal space A4a of A and the classification problem has a functional-analytic version: describe maximal ideal spaces of invariant algebras on homogeneous spaces of compact Lie groups. These problems are not quite equivalent; the second is, in certain sense, more natural because it often happens that all bounded holomorphic functions on M may be extended to some ideal analytic components at infinity. Furthermore, the functional-analytic approach makes it possible to apply the machinery of Harmonic Analysis and Banach algebras.
The problem has a solution for bi-invariant algebras (i.e., function algebras on groups invariant with respect to left and right shifts). For invariant algebras generated by finite dimensional invariant subspaces the problem is equivalent to the following one: describe polynomially convex hulls for orbits of compact linear groups. This subject is closely connected with actions of linear reductive groups or, in other words, with Invariant Theory.
Classical domains, especially the symmetric ones, are common fields of Complex Analysis, Harmonic Analysis, Representation Theory, Geometry, other mathematical disciplines. In this article only the geometrical part of the subject is considered.
• •
Introduction
Siegel domains and their noncommutative analogues. Siegel domains of the first kind (an equivalent term — tube domains) are domains in Cn of the type Rn + Unt(C) where C is a convex closed pointed generating cone in Rn and Int means ’’interior”. To construct a Siegel domain of the second type, one needs spaces Cm and Cn, a cone Cc ®" with the same properties, and a hermitian form h on Cm with values in Cn such that h(z, z) 6 C for all z £ Cm and h(z,z) ^ 0 for z ^ 0; the domain is defined as the interior of the set {(z, w) £ Cn x Crn : Im z — h(w,w) £ C}. In the both cases skeletons admit simply transitive groups of holomorphic automorphisms: translations in the first case (the skeleton is Mn) and a two-step nilpotent group of affine transformations in the second one (the skeleton is the set Im z — h(w,w) = 0). There is a generalization of Siegel domains of the second kind with a graded nilpotent Lie algebra instead of a two-step one ([21]).
Any bounded homogeneous domain in Cn is biholomorphically equivalent to a Siegel domain ([30]). Domains of the second kind are distinguished by the property to have a nontrivial Cii-structure in skeletons, i.e. nontrivial complex linear subspaces in real tangent spaces (at the point (0, 0) this is the subspace {0} x Cm).
Olshanskii semigroups (or complex Lie semigroups) are noncommutative analogues of Siegel domains of the first kind. Let G be a real form of a complex Lie group Gc , Q be the Lie algebra of G, and C be a convex closed pointed Ad(G)-invariant cone in Q (we shall say ” invariant cone” for a cone with all these properties). Their structure is now well-understood. The set Gexp(iC) is a subsemigroup of Gc and its interior is a semihomogeneous domain with the skeleton G. In a certain sense, Olshanskii semigroups may exists even when the group G has no complexification; for example, this is true for the universal covering group of SL(2,R). These semigroups have the property that representations of discrete holomorphic series extends to them and it was the reason for their consideration in [8] where a program for developing of Harmonic Analysis on them was introduced.
Noncommutative analogues of Siegel domains of the second type are less known. A simple example was recently pointed out by Latypov.
Example 1. Let M. be the intersection of a generic coadjoint orbit in sl(2,C) with the ball of radius r for r greater than the distance from the orbit to the origin (with respect to the SU(2)-invariant hilbertian norm). Then M. is a semihomogeneous domain whose skeleton is a single SU(2)-orbit. The skeleton has a nontrivial C/2-structure (this is easy to check by a comparison of dimensions); M is not holomorphi-cally equivalent to a Siegel domain of the second kind because its automorphisms group coincides with Ad(SU(2)) = SO(3). The algebra Ar of all continuous in clos.M and analytic in M functions coincides with the uniform closure on M. of the algebra of polynomials ([17]).
Invariant algebras. Invariant algebras were studied by many authors. In general, it was a part of attempts to extend the remarkable Function Theory in the Unit Disc ID) to several complex variables. It turns out that the multidimensional theory is quite different from one-dimensional; moreover, there is a significant difference between the Function Theory in the polydisc and in the ball. It was a reason for the consideration of various generalizations and the usage of additional tools such as Harmonic Analysis and Banach algebras. The setting of invariant algebras is a natural field for this machinery. We shall outline only one theme. The Wermer maximality theorem states that the disc-algebra (the algebra of analytic and continuous up to the boundary functions in B) is a maximal subalgebra of C(T) where T is the unit circle. A similar assertion for balls and polydiscs is not true; but it is true in the class of Möbius-invariant function algebras on the skeletons ( [3],[14]; the Möbius group = group of holomorphic automorphisms). In fact, it is possible to give a complete list of these algebras for the ball ([20], see [28]). Here is an example of the usage of the results of this kind: suppose that a continuous in ID) function / has the property to have an analytic continuation from any circle of a fixed hyperbolic radius inside ID), then / is analytic in D. Proof: the set of such functions is a separating Möbius-invariant proper subalgebra of C(ID) closed in the topology of the uniform convergence on compact sets; by [27] (or [13]) it is either algebra of all analytic or algebra of all antianalytic functions, and the second case obviously cannot occur. The results of this type for balls are contained in ([28]). This book also contains a description of U(n)-invariant algebras on spheres in Cn. The maximal ideal spaces of these algebras described in [16] and [12] give examples of semihomogeneous domains of the first kind over spheres which are not equivalent to Siegel domains (see also the remark after the proof of Theorem 1).
Roughly speaking, if the situation is far from abelian then the family of invariant algebras is poor. For example, all bi-invariant algebras on semisimple compact groups are self-adjoint with respect to the complex conjugation; the same is true for SO(n)-invariant algebras on spheres in Mn ([34], [7], [18]). If a bi-invariant algebra has no orthogonal real measures then the group is abelian ([25]). The abelian case, being almost trivial from the point of view of (noncommutative) Harmonic Analysis, is interesting for Function Theory. The set of characters of a compact abelian group G which are contained in an invariant algebra is a semigroup 5 and the maximal ideal space is isomorphic to the semigroup of homomorphisms of S to the multiplicative semigroup closP with pointwise multiplication and convergence. Here is an example of the algebras considered in [4], [19], [6], ch. 7, [15].
Example 2. Let Sa = {(n,ra) 6 1? : n ma > 0} where a > 0 is an irrational number. Then Sa is a semigroup, the maximal ideal space of the corresponding invariant algebra Aa on the dual group T2 is foliated by analytic half-planes whose boundary one-parametrical groups are dense windings of T2. This Aq is a maximal subalgebra of C(T2) and may be realized as an algebra of analytic almost periodic functions on a half-plane (the uniform closure of linear combinations of exponents ei(n+ma)2, n|ma > 0).
Conjectures, a problem, and two theorems
Bi-invariant algebras. The consideration of bi-invariant algebras in the paper [9] was based on the following observation: there is a natural structure of a compact topological semigroup in the maximal ideal space of an invariant algebra. The multiplication may be defined via the convolution of representing measures. The maximal ideal spaces of nontrivial bi-invariant algebras on a compact group G also admit a foliation by analytic leafs isomorphic to Olshanskii semigroups. Only one such subsemigroup has the boundary containing the identity e of G.
The maximal ideal space Ma of a bi-invariant algebra A contains at most countable subsemigroup I consisting of pairwise commuting idempotents such that any idempotent in maximal ideal space is conjugated with some ¿EX. There is the zero ¿0 E X- The algebra A is antisymmetric (a function algebra is called antisymmetric if it contains no real nonconstant functions) if and only if ¿0 is the zero of Ma- Any l 7^ lq corresponds some Olshanskii subsemigroup. We shall call the semigroup corresponding to e the main Olshanskii semigroup.
The author suppose to publish proofs of these results contained in hardly accessible articles [10],[11] in some of forthcoming papers.
A class of invariant algebras. Let H be a closed subgroup of a compact group G and M = G/H. The averaging by right over H (i.e. the operator Anf(g) = f f(gh) dh where dh denotes the Haar measure on H) of any bi-invariant algebra on G gives an invariant algebra on M. Lets denote by 21 the class of all invariant algebras which may be obtained by this way. Their maximal ideal spaces may be received by a kind of holomorphic projection from maximal ideal spaces of bi-invariant algebras. In other words, if A E 21 then there exists a semigroup (the maximal ideal space of a bi-invariant algebra on G) acting on Ma transitively (this will mean that the orbit of any point in M coincides with Ma)- Any {/(n)-invariant algebra on the sphere in Cn belongs to 21 ([12]) but algebras of Example 1 are not in 21. Antisymmetric algebras of class 21 has the property that G has a fixed point in the maximal ideal space (it corresponds to the zero of the maximal ideal space of the bi-invariant algebra).
Let A and B D Abe commutative Banach algebras. The embedding A —► B induces the dual mapping Mb —* M a- We shall say that A and B have the same maximal ideal spaces if this mapping is a bijection.
CONJECTURE 1. For any invariant algebra A on M — G/H which has a G-fixed point in Ma there exists an invariant algebra B E 21 on M which includes A and has the same maximal ideal space.
If G is semisimple then any bi-invariant algebra on G is self-adjont. Thus the conjecture would imply that A has an additional group of symmetry.
Finitely generated invariant algebras and polynomially convex hulls. A Banach algebra A is generated by ai,..., an E A if the algebraically generated by a\,..., an subalgebra is dense in A. If A is commutative then the mapping a : y —* (<p(ai),.. .<p(an)) is a homeomorphic embedding of Ma to Cn. Further, if A is an uniform algebra on compact Q then cx(Ma) — &(Q), where
X = {z E Cn : ]p(z)\ < sup |p(x| for all polynomials p} x£X
is the polynomially convex hull of X CC".
We shall say that A is a finitely generated invariant algebra if it is generated as an uniform algebra by a finite dimensional invariant vector space F C C(M). The evaluating functional at a base point m E M defines an equivariant embedding a : M —► F* and Ma may be identified with ac(M). This means that the problem of the description of maximal ideal spaces for finitely generated invariant algebras is equivalent to the following one.
Problem. Describe polynomially convex hulls of orbits of compact linear groups.
The complexification Gc of G is a reductive algebraic linear group. There is a developed theory for actions of such groups ([31], [32]). Let v £ V = F*, O = Gv, and = Gcv. By the Hilbert-Mumford criterion, if 0 E clos(C9c) then there exists an one-parametrical group 7 in GSvith the same property (i.e. 0 £ clos7u). If 0 ^ clos(C?<E) then there exists a Gc-invariant polynomial which separates from zero. The set clos(C9c) consists of a finite number of orbits and there is the unique closed one among them. The closure of an Gc-orbit in the Zariski topology coincides with the closure in the real one. Hence
6 C clos (Oc).
We may assume that there is no nontrivial G-fixed points in V. Then G has a fixed point in Ma if and only if 0 £ clos(0c). Thus Conjecture 1 means that there exists a semigroup S of contractions of O — M. a such that Sv = O (the action of G may be extended to the action of the maximal ideal space of a bi-invariant algebra; in general, the extension is not linear and the group is greater than G). This would be an analogue of the Hilbert-Mumford criterion for invariant algebras.
THEOREM 1. Let p be a nontrivial irreducible representation of SL(2,C) in a finite dimensional complex linear space V, G = p(SU(2)), v £ V, A be the closure of the algebra of polynomials in C(O). If 0 £ clos(C9<E) and the stabilizer of v in Gc = /?(SL(2,C)) is trivial then A admits an extension B £ 21 with the same maximal ideal space; in other words, Conjecture 1 is true in this setting.
Any invariant algebra may be approximated by finitely generated invariant algebras — it is the closure of an increasing sequence of them. However, their maximal ideal spaces may have new properties. Algebras Aa of Example 2 are not finitely generated ones and such effects as the irrational winding cannot occur for finitely generated invariant algebras. For a general antisymmetric bi-invariant algebra the main Olshanskil semigroup need not be dense in Ma but for finitely generated invariant algebras it is always dense.
Invariant algebras without proper invariant ideals. Algebras of Example 1 has no proper invariant ideals. From the other hand, finitely generated invariant algebras of the class 21 have many proper invariant ideals corresponding to the G-fixed point in .M^the maximal one and its powers).
Conjecture 2. Let G, H, M be as above and A be a separating antisymmetric invariant algebra on M without proper invariant ideals. Then there exist a representation of G in a finite dimensional complex linear space V, a closed orbit O in V, and an equivariant embedding M —*• O such that Ma = O. Furthermore, O contains the unique G-orbit M' such that the algebra A\m> is self-adjoint, there is an involutive antilinear automorphism of A commuting with G such that M' is exactly the set of all fixed points for the corresponding reflection in Ma> and for anU point m' £ M' there exists a point m £ M such that m £ H'm, where H' is the stabilizer of m'.
In Latypov’s example, M' is the nearest to zero SU(2)-orbit in SL(2, C)-orbit in sl(2,C). Let h, e, f be the sl2—triple in the standard realization, Oc and O be orbits of h for SL(2,C) and SU(2) respectively ((9C is the set Tr Z2 = 2, the equation Tt Z*Z = 2 distinguishes O in C?c). The complex linear tangent space T at h to Oc is Ce + C/ and (h + T) fl is the union of two lines h + Ce and h + C/. The line h + Ce intersects the ball Tr Z* Z < 2 -j- r2 by the disc of radius r in C and the same is true for the second line. The homogeneous space M is SU(2)-orbit of h. + re; this is the intersection of Oc and the sphere Tr Z*Z = 2 + r1 (this set is connected). The reflection of the conjecture is Z —» Z*. It transposes the two families of lines above. Further, H' = exp(z'E/i), H = ker Ad = {1, —1}; for m = h one may set m! = h + re or m' = h + rf.
Theorem 2. Let G, v £ V, O, and A be as in Theorem 1, and the stabilizer of v in G® is connected. Then A has no proper invariant ideals if and only if Oc is closed. In this case, either A is isomorphic to an invariant subalgebra of some algebrh of Example 1 or A = C(O).
The following conjecture concerns the reconstruction problem for a general invariant algebra by the two opposite cases considered above.
CONJECTURE 3. Any invariant algebra contains the unique maximal invariant ideal.
This is an analogue of the following fact: the closure of an orbit of a reductive complex linear group contains the unique closed orbit.
Proof of Theorem 1. We use some facts of the theory of uniform algebras ([6], Ch. 3). There is the direct connection between the algebraic constructions and analytic ones; for exampje, the algebra of all bounded on C9C, holomorphic in the relative interior of O fl and continuous on O fl Oc functions is the closure of the normalization of the algebra of polynomials restricted to Oc. The description of maximal ideal spaces if U(n)invariant algebras on balls ([16]) has a great overlap with the classification of algebraic SL(2, (C)-embeddings (the description can be found in [32], Ch. 3, §4) — this fact seems to be not noted else.
If the closure of a three-dimensional SL(2, C)-orbit C7® contains 0 then it consists of three orbits: (9C, the two-dimensional orbit 0% of the highest vector Vh, and {0}. By the Hilbert-Mumford criterion, there exists complex one-dimensional algebraic C-torus C (i-e. homomorphism £ : C* —* Gc where C* = C\{0})
such that lim^o C(z)v = 0- Let £r be the corresponding one-parametrical multiplicative group of right
shifts acting on C?c = Gc. Since (r commutes with G,
lim£r(z)u = 0 for all u £ Oc. (1)
In particular, this means that 0 £ O because this mapping defines the embedding to O of the family
Cr(©\ {0})uU {0}, u £ O, of analytic discs with boundaries in O. Moreover, (1) implies that £r(z)(0 fl 0C) C O fl Oc for all z £ B, z ^ 0. Thus the action of the group G = Gx T (with T = {Cr(-0 : \z\ =1}) on O is well-defined as well as the action of its complexification on Oc. This action extends continuously to closO® by the setting Crlo® = Clojf
Let B be the closure in 0(0) of holomorphic in the relative interior of Ó fl and continuous on closO® functions on clos(9c which coincides with some function in A on 0®. We shall prove that Banach algebra B is the desired G-invariant extension of A and that B £ 21. First off all, note that B is G-invariant. Indeed, the subalgebra B0 of B consisting of functions vanishing on closC?® \ Oc and holomorphic on is ^-invariant because £r continuously extends to closCJc; since ( and (r coincides on clos Oc \ Oc, / o £ — f o £r 6 Bo for all / £ B, hence B is also £r-invariant.
A function / on the maximal ideal space of an uniform algebra A is called A-holomorpkic at the point a: £ -M ,4 if it can be uniformly approximated by functions in A on some neighborhood of x in Ma\ f is ,4-holomorphic on the set X if it is A-holomorphic at any point of this set. By [6], Ch. 3, §9, the following assertion holds: if A is an uniform algebra and f is a continuous function on M a which is A-holomorphic outside the set of its zeroes then the maximal ideal space and the Shilov boundary of the closed subalgebra of C{Ma) generated by A and f coincide with the maximal ideal space and the Shilov boundary of A respectively. The hypothesis that B has an additional maximal ideal or that some function in B attains outside O a value which is greater than the uniform norm on O leads to a contradiction with this result. Indeed, suppose that there exist / £ B, ip 6 Mb, and x £ Ma = O such that <¿>(a) = a(x) for all a £ A but ip(f) ^ f{x). Since B = clos(>l + Bq) we may assume that f = a + b where a £ A and b £ Bq. The uniform algebra B' on O generated by A and b has the same maximal ideal space because b is clearly j4-holomorphic at any point x of (9C fl O (it can be extended analytically to a neighborhood of x in V), and this is the contradiction. The assertion concerning the Shilov boundary is proved by the same arguments with the assumption sup{|/(a:)| : x £ O} > sup{|/(x)| : x £ O} instead of ip(f) ^ f(x).
It remains to prove that B £ 21. Lets consider the standard realization for representations of SL(2,C). The space V can be identified with the space of homogeneous polynomials of the degree n = dim V — 1 of two complex variables z,w and SL(2,C) acts by u(z,w) —*■ u(az + bw,cz + dw), ad — be = 1. The assumption C(0Ü ~* 0 as t —► 0 implies that, in suitable coordinates such that £(<)u(z, u;) = v(tz,t~lw),
v(z, w) = zpwq -|- cizp+lwq~1 ... -f cqzn, p-\-q = n,p>q (2)
Let k,l be positive integers, k > I. The mapping Akt¡ '■ t —*• v(tlz, t~kz + t~lw) defines an analytic disc in Oc for which the following conclusion holds:
lp — kq •<=> lim Ajfc i(t) = Vh and Ip > kq -<=> lim A*/(¿) = 0 (3)
t-fO ’ t-*o
where Vh(z, w) = zn is the highest vector. A direct calculation shows that the same is true for any g from the stabilizer of i>/, and the disc g\k,i{t)v (the stabilizer of zn is the group (z,w) —► (ez,tz + e~l w) where / £ C and en = 1). Let pm be this representation in the space Pm of polynomials of the degree m and ¿r be the r-th power of the natural one-dimensional representation of T. Any irreducible representation of the group G has the form pm<k = pm®^r, with integer r and m £ where Z+ is the set of nonnegative integers, and acts in the space Pm\ if n even then m also must be even. The stationary subgroup a(t) = C(0 °Cr(—0) 1*1 — 1> °f point v acts on monomials by cr(t) : zkwl —*• tk~l~rzkwl. Hence it has a fixed point in Pm if and only if |r| < m and m — r is even and this is the monomial
vm,r(z,w) = zkwl, where k = i(m + r), / = i(m - r). (4)
The spaces H(k, I) of matrix elements of the type (pm,r(g)vm,r, £)> considered as functions on O, defines the decomposition of the quasiregular representation of G in L2(0) (the notation corresponds to the notation of [28], Ch. 12).
By (2) and (3),
lim/(APi9(/)) = 0 for all feH(k,l) •<=>• ^ (5)
In other words, H(k,l) C Bo if and only if Ip < kq. Hence Bo is the closed linear span of H(k,l) where
k + / is even if n is even and Ip < kq. Since for all a = 1,..., q monomials zp+awq~a defines subspaces
of Bo, it follows from (2) that B is generated by Bo and H(sp,sq), s E Z+ (one can exclude these
monomials consequently starting with zn). Let B be the space on G generated by all matrix element of representations pm,r+a, where m, r are as in (4) with Ip < kq, a E Z+ (and is even if n is even), and Ps(p+q),s(p-q)> s £ Then the right averaging of B by a gives B. This set of representations contains all irreducible components of their tensor products, hence the closure of B in C(G) is a bi-invariant algebra, and the proof is finished.
Remark. Since £r is the group of right shifts, the action of G on O may be identified with the action of U(2). It follows from [12] that O D Oc coincides with the orbit of the point v under the action of the semigroup
Sa = {M E GL(2) : \\M\\\\M-1 ||a < 1}
where a = q/p and the norm is the operator norm with respect to the standard scalar product in C2. For any a E [0,1) there exists an invariant algebra on the sphere S3 C C2 whose maximal ideal space is the one-point compactification of an orbit of SQ in some linear space; these algebras are not finitely generated.
Proof of Theorem 2. If 0C is not closed then it contains the unique closed orbit O, and there exist polynomials vanishing on O but nontrivial on O. Conversely, the set of common zeros of polynomials in the proper G-invariant ideal is Gc-invariant and closed; since O C clos Oc, (9® cannot be closed in this case.
If is closed and two-dimensional then either the real dimension of O is 2 or O has a nontrivial complex line in the tangent space to any point of O. In the first case O is the two dimensional sphere or real projective plane and A = C{0) by [18]. In the second one, functions in A satisfies some invariant C.ft-conditions. By [17], A is isomorphic to some subalgebra Example 1 or to a subalgebra of the algebra of analytic functions in the unit ball in C2. The last possibility cannot occur because G has a fixed point in Ma whence A contains a nontrivial invariant ideal.
Let Oc be three-dimensional. Then the stabilizer of a point v E Oc is trivial by the assumption of the theorem, Oc may be identified with Gc and O with its maximal compact subgroup G. Hence there exists an antiholomorphic involution in Oc commuting with G with O as the set of all its fixed points. This means that the restriction to O of the algebra of all analytic in functions is self-adjoint with respect to the complex conjugation. Since Oc is closed and smooth, any analytic in (9® function may be extended analytically to a neighborhood of O in V. It remains to use the following well-known fact from the approximation theory: any analytic in a neighborhood of a polynomially convex compact set K function can be approximated by polynomials, and to finish the proof with the Stone-Weierstrass theorem.
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