Bergman spaces of natural G-manifolds

Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle G-->M-->X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if there exists a point p in bM such that T_p(G) is contained in the complex tangent space of bM at p, then the Bergman space of M is large. Natural examples include the gauged G-complexifications of Heinzner, Huckleberry, and Kutzschebauch.

Suppose that a Lie group G acts by biholomorphisms on M , and that M is invariant under this action. We assume that the G acts on M freely, properly, and cocompactly. This means that the map G × M → M × M given by (g, p) → (g · p, p) is proper, that the stabilizer of each p ∈ M is trivial, and that the quotient space M /G is compact, where from now on we denote by g · p the action of the group element g ∈ G on the point p ∈ M.
Choosing a G-invariant measure µ, smooth on M , and restricting it toM , we define the Bergman space L 2 O(M, µ) as L 2 (M, µ) ∩ O(M). Compactness of the quotient and invariance of the measure imply that L 2 O(M, µ) does not depend on µ, provided that the measure is smooth on M . Thus we will omit mention of the measure henceforth and write L 2 O(M) for the Bergman space.
In the following result, we use the notation T c p (bM) for the complex tangent space T p (bM) ∩ iT p (bM), and the unimodularity assumption means that the Lie group G admits a biinvariant Haar measure, see [P].
Theorem 1.1. Let G be a unimodular group acting properly, freely and cocompactly by biholomorphisms on a complex manifold M with strongly pseudoconvex boundary bM and suppose that, for some p ∈ bM, we have T p (G · p) ⊂ T c p (bM). Then we have dim G (L 2 O(M)) = ∞. Remark 1.2. A thorough description of the G-dimension can be found in [P], but for our purposes here let us just mention that this result implies that the Bergman space is infinite-dimensional over C as any space of positive G-dimension is infinite-dimensional over C. An example in [GHS] shows that the nontriviality of the Bergman space is not automatic, even in strongly pseudoconvex manifolds admitting a cocompact Lie group action by biholomorphisms.
Since G acts in M by measure-preserving biholomorphisms, its action induces a unitary representation of G in L 2 O(M). By the methods of [DSP], it can be shown that the representation R obtained is nontrivial and in fact we have Corollary 1.3. ker R is compact.
Our method of analyzing the Bergman space is derived from a partial differential equations approach to several complex variables due to Morrey, Spencer, Kohn, and others. This involves the analysis of a self-adjoint boundary value problem for an operator similar to the Hodge Laplacian and is called the∂-Neumann problem.
For any integers p, q with 0 ≤ p, q ≤ n + 1 denote by C ∞ (M, Λ p,q ) the space of all C ∞ forms of type (p, q) on M.
The antiholomorphic exterior derivative∂ =∂| p,q defines a linear map∂ : C ∞ (M, Λ p,q ) → C ∞ (M, Λ p,q+1 ). If it can be established that ∂u = φ has a square-integrable, smooth solution u whenever φ ∈ L 2 belongs to C ∞ (M, Λ 0,1 ) and satisfies the compatibility condition∂φ = 0, then we may construct L 2 holomorphic functions. The first step is to use the pseudoconvexity property of the boundary to construct a function f ∈ L 2 , holomorphic in a neighborhood U x of x inM , that blows up just at x. The function f is usually chosen to be the reciprocal of the Levi polynomial.
Next, we can take a smooth function χ with support in U x that is identically equal 1 close to x. Extending χf by zero on the rest of M, we obtain a function, which we also call χf , defined everywhere and smooth away from x. Furthermore,∂(χf ) = (∂χ)f = 0 near x, sō ∂χf can be extended smoothly to the boundary. If we can now find a smooth solution u to∂u =∂χf , then Φ = χf − u will be holomorphic and must be singular at x since u is smooth up to the boundary. In particular, Φ will be nontrivial.
Let us describe the construction of solutions u ∈ L 2 (M) to∂u = φ with φ ∈ L 2 (M, Λ 0,1 ),∂φ = 0, noting that solutions will only be determined modulo the kernel of∂ (i.e. the square-integrable holomorphic functions). First, we pass to an analysis of a self-adjoint operator as follows. Since the Hilbert space adjoint∂ * of∂ satisfies im∂ * = (ker∂) ⊥ , we will look for u of the form u =∂ * v satisfying In order to eliminate the compatibility condition on φ, let us add a term,∂ * ∂ v, to obtain a new operator and equation in which φ need not be assumed to satisfy∂φ = 0. By von Neumann's theorem, the∂-Neumann Laplacian is self-adjoint with its natural boundary conditions and, when∂φ = 0 is true, Eq. (2) reduces to Eq.
The Laplacian is elliptic but its natural∂-Neumann boundary conditions are not. Still, it turns out that the gain at the boundary depends on the geometry of the boundary, and the best such situation is that which we assume, in which the boundary is strongly pseudoconvex. In this case, the operator gains one degree on the Sobolev scale in neighborhoods of bM and so global estimates including both interior and boundary neighborhoods gain only one degree.
Our method of solution of Eq.
(2) is to establish that satisfies a generalized Fredholm property. Classically, this means that the spaces ker and coker are finite-dimensional, i.e. on a (closed) subspace of finite complex codimension in L 2 (M, Λ p,q ), q > 0, the Laplacian is a Hilbert space isomorphism. In the case of compactM , we can use the Rellich lemma and the operator's Sobolev gain to obtain the finite-dimensionality of those spaces. This completely characterizes the solvability of .
1.2. The G-Fredholm property of the∂-Neumann problem. In the case of a noncompact manifold, an operator A may fail to be Fredholm regardless of its Sobolev gain. That is, the kernel and/or cokernel of A may be infinite-dimensional and/or the image of A may not be closed. Thus, our solvability theory in the G-manifold case will be worked in terms of a generalized Fredholm property valid for Ginvariant subspaces of L 2 . The corresponding dimension is obtained by replacing the ordinary meaning of the complex dimension by the value of von Neumann's G-trace tr G which takes finite values on some closed, G-invariant subspaces of L 2 (M) which are infinite-dimensional over C. For this dimension to be defined, we need that M's symmetry group G be a unimodular group, [P], and that the quotient X = M /G be a compact manifold.
Making appropriate choices of metric on M and in the vector bundles over M and using a Haar measure on G, we obtain Hilbert spaces of sections on which the G-action is unitary. This action allows us to define an trace tr G in the algebra of operators commuting with the action of G.
For the case in which M = G, the G-dimension has a simple definition. The algebra of operators L G ⊂ B(L 2 (G)) commuting with the right action of G is a von Neumann algebra consisting of some left convolutions λ κ against distributions κ on G. On this algebra there is a unique trace tr G agreeing with whenever λ κ ∈ B(L 2 (G)) and κ ∈ L 2 (G), [P, § §5.1, 7.2]. In order to measure invariant projections in B(L 2 (M)) G , one uses the tensor product Tr G = tr G ⊗ tr L 2 (M/G) . Restricting this trace to orthogonal projections P L onto G-invariant subspaces L provides a dimension function dim G , dim G (L) = Tr G (P L ). With this idea of dimension, one generalizes the classical definition of Fredholm operator.
The principal result of [P1] is the following: Theorem 1.5. Let M be a complex manifold with boundary which is strongly pseudoconvex. Let G be a unimodular Lie group acting freely and properly by holomorphic transformations on M so that M/G is compact. Then, for q > 0, the∂-Neumann Laplacian in L 2 (M, Λ p,q ) is G-Fredholm.
In other words, as the Laplacian G-Fredholm, it is an isomorphism in the orthogonal complement of a closed, invariant subspace of finite G-dimension.
Remark 1.6. Examples of manifolds satisfying the hypotheses of the theorem are the gauged G-complexifications of [HHK] for unimodular Lie groups. The unimodularity of G is necessary for the definition of the G-Fredholm property, [P].
1.3. The Levi problem on G-bundles. The G-Fredholm property established earlier provides that the image of the Laplacian nontrivially intersects any closed, invariant subspace of L 2 (M) of large enough dimension. On the other hand, the Paley-Wiener theorem of [AL] provides that if a closed, invariant subspace of L 2 (M) contains an element with compact support, then it is infinite-G-dimensional. Thus, if u denotes the closure of the complex vector space generated by translates of u ∈ C ∞ c , we have im ∩ ∂ χf = {0}, which is the basis of the construction of the Bergman space. To be more precise, we need subspaces in im ∩ ∂ χf ∩ C ∞ (M , Λ 0,1 ) of arbitrarily large G-dimension in order to solve our problem. These are constructed in [P2] as images under∂ of subspaces of L 2 generated by convolutions as follows, where (R ∆ u)(p) = G dt ∆(t)u(pt), R ∆∂ u =∂R ∆ u, and P δ is some invariant projection in L 2 (G). If the projections P δ are chosen appropriately, the convolution kernels ∆ ∈ imP δ will be smooth and so elements u ∈ χf δ will have∂u ∈ C ∞ (M , Λ 0,1 ). If it happens that u / ∈ C ∞ (M ), then Kohn's nontriviality argument can be applied with u replacing χf . This motivates the introduction of the property called amenability in [P2]: Definition 1.7. Let G → M p → X be a principal G-bundle and let ξ :X → M be a piecewise continuous section so that ξ| p(suppχ) is continuous. The action of G on M is called amenable if there exist an x ∈ bM and τ > 0 so that if f is a Levi polynomial at x, then 1) χf −τ ∈ L 2 (M), 2) χf −τ (·, ξ) L 1 (G) < ∞ for all ξ ∈ p(suppχ), and 3) for any nonzero ∆ ∈ C ∞ (G), we have Remark 1.8. Let us point out that our nonstandard use of the term amenable refers to actions rather than groups intrinsically and is unrelated to the existence of an invariant mean on the group as in the property due to von Neumann.
The main result of [P2] and the motivation for much of the present work is Theorem 1.9. Let G and M be as in Thm. 1.5 and assume that the action of G in M is amenable. It follows that the Bergman space Remark 1.10. In [DSP] we apply this theorem to establish the nontriviality of the Bergman spaces of some natural G-manifolds; we describe these in Sect. 5.
1.4. Integrals on the Heisenberg group. Verifying in practice the condition described in Definition 1.7 leads, rather concretely, to the estimation of certain integrals with parameter performed on the group G. In this section we present an example that will serve as a model for the general situation of Theorem 1.1. Note that, since the integrals to estimate have a local character, a global assumptions like cocompactness of the action plays no role and in the following example it is in fact not satisfied.
Let (z 0 , z) be complex coordinates for C n+1 ∼ = C × C n . The Siegel domain D n+1 is defined as There is a convenient coordinatization of bD n+1 in terms of the Heisenberg group in n dimensions H n as follows. That boundary is modeled geometrically as the Lie group whose underlying manifold is R×C n with coordinates (t, ζ 1 , ζ 2 , . . . , ζ n ) = (t, ζ) and whose group law is given by The group H n acts on C n+1 by holomorphic, affine transformations which preserve D n+1 and bD n+1 as follows: if (t, ζ) ∈ H n and z ∈ C n+1 , then define The action of a group element (t, ζ) ∈ H n on the Levi polynomial and, in particular for (z 0 , z) = (0, 0), this expression reduces to Suppose that bM as above coincides with bD n+1 . We will verify that the action of any subgroup G ⊂ {(0, ζ)} ⊂ H n is amenable. The group G consists of points (0, ζ) ∈ H n with ζ belonging to a certain subset S ⊂ C n with the property z, z ′ ∈ S ⇒ z · z ′ ∈ R, for example, S = R n ⊂ C n . Along the path λ(s), we get Taking sufficiently many derivatives and putting s = 0, the resulting integral is manifestly divergent, so from this we conclude that the convolution is not smooth to the boundary.
Note that we obtain a singular convolution from subgroups whose orbits satisfy the tangency condition assumed in Thm. 1.1. Our proof of the theorem depends on the fact that, as it turns out, the general case is a sufficiently small perturbation of this Heisenberg group case to preserve this divergent behavior.

Unitary representations of Lie groups.
Unitary representations of Lie groups in L 2 -spaces of holomorphic functions have been studied intensely, and although the abstract theory of Lie group representations is highly developed, it has been long considered important to provide geometric realizations of these representations.
The Borel-Weil theorem is an important example in which representations are realized as holomorphic functions on a space related to the group. Also, the Mackey program of construction of unitary representations of Lie groups and Harish-Chandra theory are connected to our setting, [Kn].
As our present analytical techniques rely ultimately on the methods of the L 2 -index theorem of Atiyah, it seems worthwhile to mention here that the first example application of that theorem in the original paper [A] was in the construction of L 2 -holomorphic representation spaces for SL(2, R) belonging to the discrete series. Though our method is a long-reaching development of this method, the initial content dates from the index theorem.
To our aesthetic, it seems most attractive to take the natural geometric, complex G-manifolds constructed in [HHK] and investigate their Bergman spaces. Thus these manifolds will provide the starting points of our main class.

Integrals of the Levi polynomial over submanifolds
In this section we discuss the divergence of integrals analogous to (5). The integrals in question will be performed over a submanifold O ⊂ bM, as well as over a 1-parameter family of submanifolds approaching it. We think of O as the orbit through p ∈ bM of some G-action, but for the moment we do not make this assumption. The validity of our approach depends on the fact that the divergence properties of these integrals are invariant under a smooth change of coordinates and of the measure, and as such, they are insensitive to the presence of a group structure on O. We will prove that under a certain order assumption on O, these divergence properties are the same as those of (5). Later, we will see that this assumption is automatically satisfied when O is an orbit.
2.1. Choice of coordinates and Levi polynomial. As we will be interested in only the local picture, we may, without loss of generality, model our situation on a fixed, small neighborhood in C n+1 as follows.
Definition 2.1. Fix p ∈ bM. We will say that a system of local coordinates (z 0 = x 0 + iy 0 , z), where z = (z 1 = x 1 + iy 1 , . . . , z n = Remark 2.2. Obviously, any set of coordinates can be brought to this form by a complex linear transformation. Moreover, if the coordinates are fixed as above, then the complex tangent space If such coordinates are fixed, we select uniquely an equation for bM by solving for Imz 0 : is a smooth, real-valued function. We express the second-order Taylor expansion f 2 of f in the following way, where ℓ is a real-valued, linear function in (Rez 0 , z), while P and L correspond to the parts (involving only z) of type, respectively, (2, 0) and (1, 1) of the second-order expansion. The notation L(z, z) is used to emphasize the fact that L is a real-valued polynomial of degree 2 in z. Since bM is strongly pseudoconvex, L(z, z), the restriction of the Levi form to T c 0 (bM), is positive and Hermitian. We define an adapted Levi polynomial Λ(z 0 , z) by Note that our definition does not coincide with the standard definition of the Levi polynomial associated to the defining function ρ(z 0 , z) = Imz 0 − f (Rez 0 , z) as in [K], for example. Nevertheless, we can show that Λ is still a support function for bM at 0. Lemma 2.3. With Λ(z 0 , z) as defined above, for a small enough neighborhood U of 0 the following hold: If q lies in a sufficiently small neighborhood U of 0, the condition above implies that z q = 0, hence z q 0 = 0 and q = 0, which proves the first point. Let q ∈ M be such that ReΛ(q) = 0. We obtain that . If q belongs to U, the expression on the right-hand side is positive, thus the second point is proved.
2.2. An order 3 vanishing condition. Let O ⊂ bM be a real submanifold of dimension 2n with 0 ∈ O. We will show in Sect. 2.4 how to extend our arguments to manifolds of smaller dimensions. We are now in a position to describe our main assumption.
Definition 2.4. If the restriction of ReΛ to O vanishes to order at least 3 at 0 we will say that the order 3 condition is satisfied.
Remark 2.5. Denote by π the projection of C n+1 onto T 0 (bM), and let Σ = {Λ = 0}. The order 3 condition is then equivalent to the property that π(O) and π(Σ) have order of contact at least two at zero.
We now check that our hypothesis is invariantly defined.
Lemma 2.6. The order 3 condition does not depend on the choice of (adapted) local coordinates about p.
Proof. Let (z ′ 0 , z ′ ) be another set of adapted local coordinates around p ↔ 0. Since its differential at 0 must preserve T 0 (bM), the map giving the change of coordinates between (z 0 , z) and (z ′ 0 , z ′ ) can be expressed up to a real scaling factor as is a holomorphic function and G(z ′ 0 , z ′ ) : C n+1 → C n is a holomorphic mapping whose differential with respect to the variables z ′ has non-vanishing determinant at 0. We denote by F 2 the second order expansion of F : where κ is a complex-linear function and E is a homogeneous polynomial of degree 2 in the variables z ′ . We also write G be a set of equations for the orbit O around 0 in the coordinates (z 0 , z). The order 3 condition gives that h(z) = −2ImP (z) + O(3) and in particular, O must be tangent to T c 0 (bM). In the coordinates (z ′ 0 , z ′ ), the hypersurface bM is locally defined by To this end, we examine the second-order jet of (8). Obviously, we need only to consider f 2 , F 2 and G 1 : . We expand the right-hand side of the previous expression in a polynomial in Rez ′ 0 , Imz ′ 0 and z ′ . Any monomial of this expansion which is of the form QImz ′ 0 with Q = O(1) can be replaced, using (8), by , which is O(3) and thus can be ignored. Performing the computation we obtain , β(z ′ ))+O(3) for a suitable linear form ℓ ′ , which gives in particular P ′ (z ′ ) = iE(z ′ )/2 + P (β(z ′ )).

It follows that the Levi polynomial
). Now, we turn to the second-order jet of the defining equations for O in the new coordinates:

Combining these we obtain
which implies that the order 3 condition holds in the new coordinates (z ′ 0 , z ′ ).
Remark 2.7. The previous lemma also holds for a manifold O ′ of dimension lower than 2n. In fact, π(O ′ ) has order of contact at least 2 with π(Σ) if and only if it is contained in a 2n-dimensional manifold O with the same property.
2.3. Estimation of the integrals. By results of [FS,§18], about a point p ∈ bM there exists a neighborhood U p ⊂ M and local complex coordinates Z = (z 0 , z) ∈ C × C n such that Remark 2.8. Condition (2) means that the hypersurface bM osculates the boundary of the Siegel domain to first order at zero, and to second order along the variables z. This means that the surfaces bM ∩ {z 0 = 0} and bD n+1 ∩ {z 0 = 0} osculate to order 2.
We now perform an analysis on bM analogous to the one in Sect. 1.4 for bD n+1 ∼ = H n . The tangent space at the origin of bM at 0 is given by T 0 (bM) = {Imz 0 = 0}; denote by π : C n+1 → T 0 (bM) the orthogonal projection. We can then express π(O) as where h is a smooth, real-valued function defined in a neighborhood of 0 in (0, z) such that h(0) = 0 and, because of the order 3 condition, h(z) = O(|z| 3 ). Note that, in particular, the tangent space of π(O) at 0 is {z 0 = 0} = T c 0 (bM). Alternatively, we can consider a smooth parametrization Γ : Since bM is given as the zero set of Imz 0 − f with f as in (6), and since O ⊂ bM, it follows that the map C n ∋ ζ → P (ζ) := (h(ζ), f (h(ζ), ζ), ζ) ∈ C n+1 (Rez 0 , Imz 0 , z) gives a smooth parametrization of O.
We are interested in the following integral, keeping the notation R ∆ χΛ −τ only in analogy to (5): where we use the notation ≈ to express that the quotient of the two integrands is a smooth function not vanishing at 0. The last equality follows from the fact that f (Rez 0 , z) = |z| 2 + O(|Rez 0 ||z| + |z| 3 ), and that along the parametrization we have Rez 0 (P (ζ)) = h(ζ) = O(|ζ| 3 ) and z(P (ζ)) = ζ = O(|ζ|). This integral is then a perturbation of the one computed for the Heisenberg group at s = 0, obtained by adding the "high order terms" h(ζ) and O(|ζ| 3 ). Collecting |ζ| 2 in the denominator of (9), we have that h(ζ)/|ζ| 2 → 0 for ζ → 0. This is enough to prove that the integral diverges for large τ , but, as in the case of the Heisenberg group, we actually need to look at the behavior of this integral along a one-parameter family of submanifolds rather than just along O.
Let us now, then, consider the path in C n+1 given by λ(s) = (is, 0), where s ∈ R + , and an arbitrary, smooth 1-parameter family of submanifolds O s such that λ(s) ∈ O s and O 0 = O. We choose a map Γ : R + × C n → C n+1 with the following properties: • Γ(s, 0) = λ(s) • for each fixed s ∈ R + , the map Γ(s, ·) parametrizes O s ; • for each fixed s ∈ R + , the map Γ(s, z) is of the form (A(s, z) + iB(s, z), z). From these properties we deduce the following expression for Γ(s, z) = (A(s, z) + iB(s, z), z): for some smooth maps a, b : R + × C n → C n . We remark that here we employ the notation v · w to denote the Euclidean scalar product of two vectors v, w ∈ C n . In the following, we will replace our Levi polynomial with the more convenient Λ = −iz 0 . With this new definition for Λ we obtain the integral a(s, ζ))) + (f (h(ζ), ζ) + s(1 + ζ · b(s, ζ)))] τ , which we rewrite after collecting (s + |ζ| 2 ) τ as The key observation is the following Lemma 2.9. We have θ(s, ζ) → 0 as (s, ζ) → 0.
Here θ ′ , Q ′ and R ′ are, respectively, the expressions of θ, Q and R in polar coordinates. As a consequence of Lemma 2.9 and of the discussion above, we obtain that I(s, r) → C for r → 0, uniformly in s ∈ R + , where C = Area(S) is a real, positive constant. In particular, the sign of ReI(s, r) is positive and bounded below for (s, r) small enough. It follows that for some C ′ > 0 and s small enough, hence it diverges for s → 0 when τ ≫ 0.

General dimension.
Let us assume that the dimension m of O is lower than 2n. By hypothesis, the tangent space of π(O) at 0 is contained in T c 0 (S) = {z 0 = 0}. Since we are principally interested in the case when O is totally real, we will actually let m ≤ n and suppose that T 0 (π(O)) is spanned by ∂/∂x 1 , . . . , ∂/∂x m . Letting x ′ = (x 1 , . . . , x m ), x ′′ = (x m+1 , . . . , x n ) and y = (y 1 , . . . , y n ), a set of defining functions for π(O) can be written in the following way: where g 1 and g 2 are vector-valued functions that vanish to first order at 0. Moreover, because of the order 3 condition, the function h vanishes up to third order at 0. As before, we will consider a smooth parametrization R m → π(O), of the form where we have split the z-space according to the decomposition z = (x ′ , x ′′ , y). Writing z(ξ) for (ξ, g 1 (ξ), g 2 (ξ)), a parametrization of O is thus given by R m ∋ ξ → P (ξ) = (h(ξ), f (h(ξ), z(ξ)), z(ξ)) ∈ C n+1 (Rez 0 , Imz 0 , z).
Analogously to before, we define a mapping Γ : a(s, ξ)), for some smooth maps a, b : R + × R m → R m and c : R + × R m → C n , parametrizing the manifold O s through the point λ(s) = (is, 0). Again with the choice of the Levi polynomial as Λ = −iz 0 , we then need to evaluate the following integral: (10) We need only observe now that where the first equality is due the order 3 condition and the second follows from the facts that by definition |z(ξ)| 2 = |ξ| 2 +|g 1 (ξ)| 2 +|g 2 (ξ)| 2 and g 1 (ξ), g 2 (ξ) are both O(|ξ| 2 ). This is all that is needed to prove the analogue of Lemma 2.9 for the argument θ(s, ξ) = −(h(ξ) + sξ · a(s, ξ))/(f (h(ξ), z(ξ)) + s + sξ · b(s, ξ)). The divergence of the integral for a sufficiently large τ follows then in the same way as before, namely, where I(s, r) approaches, uniformly in s, a fixed, positive constant for r → 0.

Some properties of the orbit
In this section, we will establish some consequences of the hypothesis T p (G · p) ⊂ T c p (bM). First of all, we derive its implications regarding the dimension of the orbit through p. If S if a hypersurface of C n+1 , we denote by T (S) its tangent bundle, by T c (S) its complex tangent bundle and by CT (S), CT c (S) their respective complexifications.
Lemma 3.1. Let S be a strongly pseudoconvex hypersurface of C n+1 , 0 ∈ S, and let M be a CR submanifold of S, 0 ∈ M, such that T (M) ⊂ T c (S). Then M is totally real and, in particular, dim R M ≤ n.
Proof. Consider the following decompositions of the complexified tangent bundles: where T and R are the following transversal subbundles: (1) T is of dimension 1 and corresponds to the "bad" direction; (2) R can have larger dimension. Assuming, by contradiction, that M is not totally real, it follows that T c (M), and thus T 1,0 (M), are non-trivial. If L is a smooth section of . This is a contradiction since [L, L](0) ∈ CT c 0 (S) as observed above. Corollary 3.2. Let G be a Lie group, acting freely by biholomorphisms on an (n+1)-dimensional complex manifold with strongly pseudoconvex boundary S ∋ 0, and denote by G 0 = 0 · G the orbit of G through 0. If T 0 (G 0 ) ⊂ T c 0 (S) then dim R G ≤ n. Proof. Since G acts by biholomorphisms, T c p (G 0 ) has the same dimension at every p ∈ G 0 , which means that G 0 is a CR submanifold of S. By the same reason the condition T 0 (G 0 ) ⊂ T c 0 (S) implies T (G 0 ) ⊂ T c (S). The previous lemma then yields dim R G 0 ≤ n, thus dim R G ≤ n. Remark 3.4. The results above also hold, with small adaptations to the proof, when we just assume that S is of finite commutator type rather than strongly pseudoconvex.
Next, we will show that, under the same hypothesis, the orbit O = G · 0 must satisfy the order 3 condition. Proof. We start by verifying the claim when dim R O = dim R G = 1. In this case T e G is spanned by a single vector v. The image of v by the differential of the action is a vector field V , defined in a neighborhood of 0 in M, which is tangent to every orbit of G. Since G acts by biholomorphisms, we have V = ReZ for a non singular holomorphic vector field Z. By a choice of complex coordinates (z 0 , z 1 , . . . , z n ) = (z 0 , z), we can assume that Z = ∂/∂z 1 , so that V = ∂/∂x 1 and the orbits of G are parametrized by G ∼ = R ∋ t → (z 0 , t, . . . , z n ) ∈ C n+1 . Note that, by hypothesis, ∂/∂x 1 ∈ T c 0 (bM) hence, up to a linear transformation, we can assume that T c 0 (M) is spanned by ∂/∂x j , ∂/∂y j (1 ≤ j ≤ n), i.e. that (z 0 , z) are adapted coordinates. Choose, as before, a local defining equation for bM of the form {Imz 0 = f (Rez 0 , z)}. Since bM is G-invariant, f does not depend on the variable x 1 . We express the second order expansion f 2 of f according to Eq. (7) and we concentrate on P (z), the homogeneous holomorphic polynomial of degree 2 giving the harmonic part of the expansion. We can write where Q(z ′ ) and ℓ(z ′ ) are, respectively, a homogeneous polynomial of degree 2 and a complex linear function in z ′ = (z 2 , . . . , z n ), and where α ∈ C. We claim that α ∈ R. Indeed, consider the expression of f 2 in the real coordinates (x j , y j ) 0≤j≤n . Since f 2 does not depend on x 1 , the only one of its monomials which includes only the variables (x 1 , y 1 ) is of the form cy 2 1 for some c ∈ R. Since cy 2 1 = −c/4(z 2 1 + z 2 1 − 2z 1 z 1 ), it follows that α = −c/4 ∈ R. In particular, if for t ∈ R we define z(t) = (t, 0, . . . , 0), we have P (z(t)) = αt 2 , which implies ImP (z(t)) ≡ 0. Let γ(t) = (0, t, 0, . . . , 0) = (0, z(t)) ∈ C n+1 parametrize the orbit O of G through 0 -also notice that in these coordinates O coincides with π(O). With Λ(z 0 , z) = z 0 − 2iP (z), we have ReΛ(γ(t)) = 2ImP (z(t)) ≡ 0, thus the order 3 condition is certainly satisfied. By Lemma 2.6 and Remark 2.7, the condition is also verified in the original coordinates.
Let us now turn to the general case. Let g = T e G be the Lie algebra of G, and let U be a neighborhood of 0 in g such that the exponential map is a diffeomorphism U → exp(U) onto a neighborhood of the identity in G. It follows that the map Γ : U → O defined as gives a regular parametrization of a neighborhood of 0 in O. We must verify that the function ReΛ(Γ(v)), defined on U, vanishes to third order at 0, which is equivalent to verifying that, for every fixed w ∈ g with |w| = 1, the function R ∋ t → ReΛ(Γ(tw)) vanishes to third order at 0. This is the same as checking the order 3 condition for the (local) 1-parameter real subgroup G w of G generated by w; since of course the tangent space of the orbit O w of G w is contained in T 0 (O) ⊂ T c 0 (bM), by the discussion above we have that our condition is satisfied for O w .

The tangency condition implies amenability
Let µ be a smooth G-invariant measure on M, and choose p ∈ bM for which the tangency condition is satisfied.
The pull-back of µ| U by a diffeomorphism φ : B k × G → U is then a product measure ν ′ ⊗ ν, where ν ′ is some smooth measure on B k and the Haar measure ν is biinvariant since G is unimodular.
In particular, if Λ is the Levi polynomial at p, χ ∈ C ∞ c ( M ) and 0 < τ < d/2, we have R ∆ χΛ −τ ∈ L 2 (M) since in this case χΛ −τ ∈ L 1 (M), [P2]. We also remark that the set of ∆ which are admissible in the definition of amenability given in there is a G-invariant, smooth subspace of L 2 (G), thus, by translating and rescaling ∆ we can assume ∆(e) = 1. Our aim is to show that R ∆ χΛ −τ does not extend smoothly through p, and in order to do so we will look at its behavior along a certain curve [0, 1] ∋ s → λ(s) ∈ M ending at p. Note that, if χ is suitably chosen, the support of χΛ −τ lies on U.
Also, we can choose the diffeomorphism φ mentioned above in such a way that φ(0, e) = p. Let U be any neighborhood of e in G and let u = χΛ −τ . It follows that and R 2 (s) is a smooth function since u(λ(s) · t) is smooth and bounded for (s, t) ∈ [0, 1] × (G \ U). Thus, we shall concentrate on R 1 (s) for a small enough neighborhood U. Setting v = u • φ and γ(s) = φ −1 (λ(s)), γ(s) = (γ 1 (s), γ 2 (s)) ⊂ B k × U, we can rewrite R 1 (s) as We will show that Re(∂ j R 1 (s)/∂s j ) → ∞ as s → 0 by giving a lower estimate for the real part of the integrand. We point out that this property does not depend on the choice of a smooth measure on B k ×U, or, for our purposes, just of a family of smooth measures µ s on {λ(s)}× U varying smoothly with respect to the parameter s ∈ [0, 1]. After a choice of local adapted coordinates in a neighborhood of p ∈ bM, consider the path λ(s) defined in Sect. 2.3, and the parametrization (s, ξ) → Γ(s, ξ) of the orbits of G through λ(s) given in Sect. 2.4. The push-forward of dξ by Γ induces, in a neighborhood of p, a measure on the orbits of G through λ. By the discussion above, we may define on a small enough neighborhood U, γ 1 ([0, 1]) × U, the pull-back of this measure by the diffeomorphism φ. With this choice, the integral R 1 (s) becomes precisely the one considered in (10). We can now conclude the proof of Thm. 1.1.
Lemma 4.2. Let G act freely on a complex manifold M with boundary, and suppose that the orbit O of G through p ∈ bM satisfies T p (O) ⊂ H p (bM). It follows that the action of G is amenable.
Since ∂I(s, r)/∂s is smooth, I 2 is absolutely convergent for s → 0 by the choice of τ . From Lemma 3.5 we have (in the notation of Sect. 2.4) that h(ξ) = O(|ξ| 3 ), so that the discussion before Eq. (11) applies. Therefore ReI 1 is divergent for s → 0, and we conclude that R 1 (s) is not smooth at s = 0. Hence R ∆ χΛ −τ is not smooth at p along the curve λ(s), which shows that the action is amenable.

5.
Complexification of free G-actions 5.1. HHK tubes. Let X and G be, respectively, a real-analytic manifold of dimension n and a Lie group acting freely and properly on X by real-analytic transformations. In [HHK] it is shown that any such G-action can be extended to a free and proper action by biholomorphisms on a neighborhood of X in its complexification X C ⊃ X.
In the same work, the authors also construct a G-invariant, strongly plurisubharmonic, non-negative function ϕ which vanishes on X, thus by setting for ǫ > 0 sufficiently small, one obtains a strongly pseudoconvex Gmanifold M ǫ on which G acts freely by holomorphic transformations. Note, however, that the construction in [HHK] also applies to the much more general case of a proper, not necessarily free G-action. In the paper, the manifolds M ǫ are called gauged G-complexifications of X. By construction, they are Stein manifolds and so possess a rich collection of holomorphic functions O(M ǫ ) which is invariant under the induced group action. The purpose of this section is to show that the Bergman space L 2 O(M ǫ ) is also non-trivial. In order to achieve this, we prove that the sufficient condition of Thm. 1.1 is satisfied when X does not coincide with the underlying manifold of G. We refer to [DSP] for a treatment of the case G = X.
Proposition 5.1. Let X, G and M ǫ be as in the previous paragraph, with dim R G < dim R X. If ǫ is small enough, there exists a point p ∈ bM ǫ such that the tangency condition of Thm. 1.1 is satisfied at p.
In order to prove the proposition, we fix any point q ∈ X and we choose local complex coordinates z = (z 1 , . . . , z n ), z j = x j + iy j , for a neighborhood U of q in X C such that q ↔ 0 and X ∩ U ↔ {y j = 0, j = 1, . . . , n}. By [DSP,Lemma 3] we may assume that up to a complex linear transformation we have First, however, we will limit ourselves to the case where d = 1 and T 0 (G · 0) is spanned by ∂/∂x 1 . For any q ∈ U \ X, we consider the complex tangent space H(q) = T c q (bM ϕ(q) ) of the level set bM ϕ(q) = {ϕ = ϕ(q)} of the function ϕ through z, and define the set C G = (X ∩ U) ∪ {q ∈ U : T q (G · q) ⊂ H(q)}. In the following lemma, J is the standard complex structure, orthogonality is intended with respect to the standard Euclidean metric in the coordinates z, and ∇ denotes the gradient associated to this metric.
Lemma 5.2. For some small neighborhood U ′ of 0 in C n , the set C G ∩ U ′ is a smooth hypersurface of U ′ .
Proof. Up to a holomorphic change of coordinates, we can assume that the action of G is generated by the vector field ∂/∂x 1 ; notice that we can choose this coordinate change in such a way that its linear part at 0 is the identity, which implies that ϕ still admits the expansion (13). Moreover, in the new coordinates ϕ does not depend on the variable x 1 and ∂/∂x 1 ∈ ∇ϕ(q) ⊥ for all q; hence the set C G coincides with the set {q ∈ U : ∂/∂x 1 ∈ J∇ϕ(q) ⊥ }. Again by (13) where V (z) ∈ ∂/∂x 1 ⊥ and k(z) = O(|z| 2 ). It follows that C G is given by {−2y 1 +k(z) = 0}; by the implicit function theorem, C G is a smooth hypersurface of a small enough neighborhood of 0.
Remark 5.3. From the proof of the previous lemma also follows that the tangent hyperplane of C G at 0 is given by ∂/∂y 1 ⊥ . In fact, the gradient of the defining function −2y 1 +k(z) at 0 is a multiple of ∂/∂y 1 , and the same is true in the original coordinates because the linear part of the coordinate change equals the identity.
Proof of Prop. 5.1. Now, we turn back to the case when the dimension of G is arbitrary, and we define the set C G in the same way as before. Select a collection G 1 , . . . , G d of local 1-parameter subgroups of G with the property that T 0 (G j · 0) is generated by ∂/∂x j for all j = 1, . . . , d. For q in a small neighborhood U ′′ of 0 in C n , then, we have that T q (G·q) is spanned by the union of the T q (G j · q) for 1 ≤ j ≤ d. It follows that By Lemma 5.2, up to shrinking U ′′ , each C G j is a smooth hypersurface; by Remark 5.3, then, we derive that T 0 (C G j ) = ∂/∂y j ⊥ , which implies that the C G j intersect transversally. Since d < n, it follows that C G is a smooth submanifold of U ′′ of real dimension strictly bigger than n.
In particular, C G does not coincide with X ∩ U ′′ , and as a consequence it must intersect bM ǫ for ǫ small enough. Any p ∈ C G ∩ bM ǫ satisfies the claim of the proposition. 5.2. Example. Let X = S 1 θ × R x 1 , G = R t , and let T = C/R be the complex cylinder. The complexification X C of X is given by T × C, in which we consider coordinates (z 0 = θ + iy 0 , z 1 ) where θ ∈ S 1 , y 0 ∈ R, z 1 ∈ C. A tube M ǫ around X can be realized as a domain of X C as follows: M ǫ = {(z 0 , z 1 ) : y 2 0 + y 2 1 < ǫ 2 } ⊂ T × C. Define, now, for any fixed c ∈ R an action φ c of R on X by φ c (t)(θ, x 1 ) = (θ + ct, x 1 + t) ∈ S 1 × R for all t ∈ R. For any fixed c, this action extends by the same formula to an action on X C by biholomorphic transformations which, moreover, preserve each M ǫ . It is also clear that both the action φ c and its extension are free and cocompact on, respectively, X and M ǫ . A computation shows that, indeed, the tangency condition for the action φ c holds along the φ c -invariant submanifold of bM ǫ given by {(z 0 , z 1 ) ∈ T × C : y 2 0 + y 2 1 = ǫ 2 , y 1 = −cy 0 } ⊂ bM ǫ . Thus Thm. 1.1 applies, showing that dim G L 2 O(M ǫ ) = ∞. Moreover, it shows that almost every point of bM ǫ is a (weakly) peaking point. We remark that in this case, because of the presence of cocompact lattices, the methods of [GHS] already apply. However, even under the hypothesis of unimodularity a Lie group does not, in general, admit such a lattice, see [R].