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V. M. Gichev Sobolev Institute of Mathematics, Omsk Branch Omsk, Russia gichev@iitam.omsk.net.ru

Invariant function algebras on homogeneous spaces
Invariant function algebras on a homogeneous space contain many information (particularly, of the complex geometric nature) about it. Let G be a Lie group, which acts transitively on a manifold M . Suppose that G is compact. The space C (M ) of all continuous complex functions on M endowed with the sup-norm is a commutative Banach algebra. The group G acts on C (M ) by translations. We say that A is an invariant function algebra on M if it is a closed G-invariant subalgebra of C (M ) that contains the constant functions. The maximal ideal space MA = Hom(A, C) is a geometric ob ject, which keeps the most essential information on A. For example, if MA = M , then A = C (M ) (M is naturally embedded to MA as its Shilov boundary). If G GL(V ), where V is a finite dimensional complex linear space, and M = Gv , v V , is an orbit, then MA may be identified with the polynomial hull of M . For any compact set Q V , its polynomial hull Q is defined as Q = {z V : |p(z )| sup |p( )| for all
Q

p P (V )}.

If Q = Q, then Q is called polynomially convex. The hull of a "generic" Q may be very irregular. The problem of determination of the hulls is certainly insoluble in general. For the orbits Gv , this problem seems to be difficult but soluble. The answer is known if G is the isotropy group of a bounded symmetric domain (Kaup and Zaitzev, 2003; Kaup, 2004). Then the problem can be reduced to the case of the group Sn Tn acting naturally in Cn , where Sn is the group of all permutations of the coordinates. The hull of the group G in L(V ) is a semigroup; it is determined by the hull of its maximal torus T : G = GT G. The polynomially convex orbits are exactly the real forms of closed orbits of GC (Gichev and Latypov, 2001). A homogeneous space admits an invariant function algebra which is not self-conjugated with respect to the complex conjugation if and only if the isotropy representation has no trivial component (Latypov, 1999; this generalizes results of Gangolli, de Leew, and Wolf of 60s). Any invariant function algebra A contains the unique maximal invariant ideal J , which is necessarily closed. Adding the constant functions to J , we get an algebra B with the invariant maximal ideal of codimension one (equivalently, G has a fixed point in MB ); factorizing by J , we get an algebra without proper invariant ideals (the norm in A/J is not the sup-norm in general). Thus, one has to consider these opposite cases before the general one. Any invariant function algebra without proper invariant ideals can be realized as the closure of P (V ) in C (M ), where M = Gv and GC v is closed. The algebra is the closure of the set of all smooth CR-functions on M ; thus, it is completely determined by the CR-structure and the inner complex geometry of GC v . This also uniquely defines an equivariant embedding of the homogeneous CR-manifold M and its hull M to the complex manifold GC v . The complexified flag manifolds (the closed adjoint orbits in the complex semisimple Lie algebras) is the simplest case. The hulls can be described if the flag manifold is a compact hermitian symmetric space (using the results on the bounded symmetric domains). For the full flags in Cn the answer is not known yet, even for n = 3. We say that A is finitely generated if it is generated as a Banach algebra by its finite dimensional invariant subspace. Then M , A, and MA = Gv can be realized as above in some finite dimensional space V . The invariant function algebras which are not finitely generated have many new properties. For example, if A is finitely generated and G has a fixed point in MA , then we may assume without loss of generality that 0 Gv . Due to the Hilbert­Mumford criterion, there exists g, such that limt+ eit v = 0, where g is the Lie algebra of G. In general, this is not true but some analog holds with a chain of one parameter semigroups instead of the single semigroup eit . 1


The case of noncompact G is much more complicated but some of the results and constructions of the compact case also hold for it. For example, the maximal ideal spaces of bi-invariant algebras on Lie groups, under some analytic restrictions, has a natural semigroup structure.

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