Commutativity of quantization with conic reduction for torus actions on compact CR manifolds

We define conic reductions Xνred\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{\textrm{red}}_{\nu }$$\end{document} for torus actions on the boundary X of a strictly pseudo-convex domain and for a given weight ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} labeling a unitary irreducible representation. There is a natural residual circle action on Xνred\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{\textrm{red}}_{\nu }$$\end{document}. We have two natural decompositions of the corresponding Hardy spaces H(X) and H(Xνred)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(X^{\textrm{red}}_{\nu })$$\end{document}. The first one is given by the ladder of isotypes H(X)kν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(X)_{k\nu }$$\end{document}, k∈Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in {\mathbb {Z}}$$\end{document}; the second one is given by the k-th Fourier components H(Xνred)k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(X^{\textrm{red}}_{\nu })_k$$\end{document} induced by the residual circle action. The aim of this paper is to prove that they are isomorphic for k sufficiently large. The result is given for spaces of (0, q)-forms with L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-coefficient when X is a CR manifold with non-degenerate Levi form.


Introduction
Let X be the boundary of a strictly pseudo-convex domain D in C n+1 .Then (X, T 1,0 X) is a contact manifold of dimension 2n + 1, n ≥ 1, where T 1,0 X is the sub-bundle of T X ⊗ C defining the CR structure.We denote by ω 0 ∈ C ∞ (X, T * X) the contact 1-form whose kernel is the horizontal bundle HX ⊂ T X, we refer to Section 2.1 for definitions.Associated with this data we can define the Hardy space H(X), it is the space of boundary values of holomorphic functions in D which lie in L 2 (X), the Hilbert space of square integrable functions on X. Suppose a contact and CR action of a t-dimensional torus T is given; we denote by µ : X → t * the associated CR moment map.Fix a weight i ν in the lattice i Z t ⊂ t * , if 0 ∈ t * does not lie in the image of the moment map the isotypes H(X) kν = {f ∈ H(X) : (e i θ • f )(x) = e ik ν,θ f (x), θ ∈ R t }, k ∈ Z, are finite dimensional.
Suppose that the ray i R + • ν ∈ t * is transversal to µ, then X ν := µ −1 (i R + • ν) is a sub-manifold of X of codimension t − 1.There is a well-defined locally free action of T t−1 ν := exp T (i ker ν) on X ν , the resulting orbifold X red ν is called conic reduction of X with respect to the weight ν.Let ϕ be an Euclidean product on t, we shall also use the symbol • , • , and denote by λ ϕ ∈ t be uniquely determined by λ = ϕ(λ ϕ , •) and λ the corresponding norm.By abuse of notation we write λ for λ ϕ and we identify t ∼ = iR t with its dual.We set ker ν = ν ⊥ := {λ ∈ t : ν, λ = 0} .
The locus X ν is T-invariant, we will always assume that the action of T on X ν is locally free.After replacing T with its quotient by a finite subgroup, we may and will assume without loss of generality that the action is generically free.In Section 2.1, we show that X red ν is CR manifold with positive definite Levi form of dimension 2n − 2t + 3. Let us define T 1 ν := exp T (i ν), if ν is coprime, we have a Lie group isomorphism between T 1 ν := exp T (i ν) and the circle S 1 .Let us denote by then the character χ ν : T → S 1 , χ ν (e i θ ) := e ik ν, θ , being trivial on T t−1 ν , descends to a character χ ′ ν : T 1 ν → S 1 which is a Lie group isomorphism, see [P3,Lemma 10].Thus, we have a locally free circle action of T 1 ν on X red ν , which induces an action on the Hardy space H(X red ν ).Suppose that the action of T 1 ν on X is transversal to the CR structure.We denote by H(X red ν ) k the corresponding k-th Fourier component and we call the action of T 1 ν on X red ν residual circle action.The aim of this paper is to prove that H(X) kν and H(X red ν ) k are isomorphic for k sufficiently large.We prove the aforementioned result in the more general setting of CR manifolds for spaces of (0, q)-forms when k is large, more precisely we consider (0, q) forms with L 2 coefficients and the corresponding projector S (q) onto the kernel of the Kohn Laplacian H q (X).Now, we make more precise the assumptions on the CR manifold X and on the group action.
Assumption 1.1.Let (X, T 1,0 X) be a compact connected orientable CR manifold of dimension 2n + 1, n ≥ 1, and let ω be the associated contact 1-form.The Levi form L is non-degenerate of constant signature (n − , n + ) on X.That is, the Levi form has exactly n − negative and n + positive eigenvalues at each point of X, where n − + n + = n.

Concerning the group action, we always assume
Assumption 1.2.The action of T preserves the contact form ω 0 and the complex structure J.That is, g * ω 0 = ω 0 on X and g * J = Jg * on the horizontal bundle HX for every g ∈ T where g * and g * denote the pull-back map and push-forward map of T, respectively.
Let X red ν := X ν /T t−1 ν , defined in the same way as before, more precisely we shall assume Assumption 1.3.The moment map µ is transverse to the ray i R + • ν ∈ t * , the action of T on X ν is locally free and for every x ∈ X ν val and it is non-degenerate.
We note that b(U, V ) = 2 L(U, V ) for every U, V ∈ HX.By Assumptions above, we will show that X red ν is a CR manifold with natural CR structure induced by T 1,0 X of dimension 2n − 2(t − 1) + 1.Let L X red ν be the Levi form on X red ν induced naturally from the Levi form L on X.For a given subspace s of t, we denote s X the subspace of infinitesimal vector fields on X.Let us consider Hence, b has constant signature on B × B, suppose b has r negative eigenvalues on B × B where r ≤ n − since L and b have the same number of negative eigenvalues on HX.Fix q = n − , hence by Lemma 2.1, L X red ν has q − r negative eigenvalues at each point of X red ν .We refer to Section 2.1 for definitions, we have: Theorem 1.1.Suppose that ✷ q b has L 2 closed range.Fix a maximal coprime weight ν = 0 in the lattice inside t * and assume that the circle action T 1 ν is a transversal CR action.Fix q = n − , under the assumptions above, X red ν is a compact CR manifold with non-degenerate Levi form having q − r-negative eigenvalues.There is a natural isomorphism of vector spaces σ k : H q (X) kν → H q−r (X red ν ) k for k sufficiently large.
For strictly pseudoconvex domain we have q = n − = r = 0 and thus we have quantization commutes with reduction for spaces of functions for k large.We give a proof in Section 3, which is inspired from [HH] (see [HMM] for the full extension of [HH]) and it is a consequence of the microlocal properties of the projector S (q) kν described in Section 2.2 and calculus of Fourier integral operators of complex type, see [MS].Furthermore, we recall that the way to establish the isometry from kernel expansion for k large comes from [MZ].
The conic reductions defined above appear naturally in geometric quantization.In fact, given a Hamiltonian and holomorphic action with moment map Φ of a compact Lie Group G on a Hodge manifold (M, ω) with quantizing circle bundle π : X → M, one can always define an infinitesimal action of the Lie algebra g on X.If it can be integrated to an action of the whole group G, then one has a representation of G on H(X).In [GS2] it was observed that associated to group actions one can define reductions M Θ red by pulling back a G-invariant and proper sub-manifold Θ of g * via the moment map Φ.When Θ is chosen to be a cone through a co-adjoint orbit C(O ν ) one has associated reduction whose Hardy space of its quantization is where k labels an irreducible representation of the residual circle action described above.Now, it is natural to ask if this spaces are related with the decomposition induced by G on H(X).When G = T Theorem 1.1 states that they are isomorphic to H(X) kν ; the canonical decomposition of the Hardy space H(X) of the quantitation by the built-in circle action does not play any role.The semi-classical parameter k is the one induced by the ladder k ν, k = 0, 1, 2, . . ., labeling unitary irreducible representations.
Further geometrical motivations for this theorem are explained in paper [P3], where it is proved that δ k is an isomorphism for k large enough in the setting when X is the circle bundle of a polarized Hodge manifold whose Grauert tube is D. Thus, Theorem 1.1 generalizes the main theorem in [P3] to compact "quantizable" pseudo-Kähler manifolds.We also refer to [P1] for examples and the explicit expression of the leading term of the asymptotic expansion of dim H(X) kν as k goes to infinity.Along this line of research in [P2] Toeplitz operators were studied for circle action, in [G] the study of asymptotics of compositions of Toeplitz operators with quantomopomorphism is addressed for torus actions.

Preliminaries 2.1 Geometric setting
We recall some notations concerning CR and contact geometry.Let (X, T 1,0 X) be a compact, connected and orientable CR manifold of dimension 2n + 1, n ≥ 1, where T 1,0 X is a CR structure of X.There is a unique sub-bundle HX of T X such that HX ⊗ C = T 1,0 X ⊕ T 0,1 X.Let J : HX → HX be the complex structure map given by J(u + u) = iu − iu, for every u ∈ T 1,0 X.By complex linear extension of J to T X ⊗ C, the i-eigenspace of J is T 1,0 X.We shall also write (X, HX, J) to denote a CR manifold.
Since X is orientable, there always exists a real non-vanishing 1-form ω 0 ∈ C ∞ (X, T * X) so that ω 0 (x) , u = 0, for every u ∈ H x X, for every x ∈ X; ω 0 is called contact form and it naturally defines a volume form on X.For each x ∈ X, we define a quadratic form on HX by Then, we extend L to HX ⊗ C by complex linear extension; for U, V ∈ T 1,0 x X, The Hermitian quadratic form L x on T 1,0 x X is called Levi form at x.In the case when X is the circle bundle of an Hodge manifold (M, ω), the positivity of ω implies that the number of negative eigenvalues of the Levi form is equal to n.The Reeb vector field R ∈ C ∞ (X, T X) is defined to be the non-vanishing vector field determined by For u ∈ CT X, we write |u| 2 := u | u .Denote by T * 1,0 X and T * 0,1 X the dual bundles of T 1,0 X and T 0,1 X, respectively.They can be identified with sub-bundles of the complexified cotangent bundle CT * X.
Assume that X admits an action of t-dimensional torus T. In this work, we assume that the T-action preserves ω 0 and J; that is, t * ω 0 = ω 0 on X and t * J = Jt * on HX.Let t denote the Lie algebra of T , we identify t with its dual t * by means of the scalar product •, • .For any ξ ∈ t, we write ξ X to denote the vector field on X induced by ξ.The moment map associated to the form ω 0 is the map µ : X → t * such that, for all x ∈ X and ξ ∈ t, we have µ(x), ξ = ω 0 (ξ X (x)). (2) Fix a maximal weight ν = 0 in the lattice inside t * .Suppose that iR + • ν is transversal to µ, so we have and X ν is a sub-manifold of X of codimension 2n + 2 − t.We claim that the action of T t−1 ν on X ν is locally free.In fact, by the contrary suppose that there exists ξ ∈ ker ν such that ξ X (x) = 0 on T x X ν , x ∈ X ν .For each v ∈ T x X, we have The action of T restricts to an action of T t−1 whose moment map is given by is the canonical projection onto the t−1-dimensional subspace ν ⊥ in t * .The transversality condition in 1.2 implies that 0 ∈ t t−1 ν is a regular value for the moment µ |T t−1 .Thus, we have and by assumptions 1.2, 1.3 and [HH, Section 2.5] (we shall also refer to [GH2] for definitions concerning CR structures on orbifolds) we have Lemma 2.1.The space of orbits X ν /T t−1 ν is a CR orbifold.Let us denote by π : X ν → X red ν and ι : X ν ֒→ X the natural projection and inclusion, respectively, then there is a unique induced contact form ω red 0 on X ν /T t−1 ν such that π * ω red 0 = ι * ω 0 .

Hardy spaces
We denote by L 2 (0,q) (X), q = 0, 1, . . ., n, the completion of Ω 0,q (X) with respect to ( • | • ).We extend ( • | • ) to L 2 (0,q) (X) in the standard way.We extend ∂ b to L 2 (0,q) (X) by where Dom } and, for any u ∈ L 2 (0,q) (X), ∂ b u is defined in the sense of distributions.We also write to denote the Hilbert space adjoint of ∂ b in the L 2 space with respect to ( • | • ).There is a well-defined orthogonal projection with respect to the L 2 inner product ( • | • ) and let S (q) (x, y) ∈ D ′ (X × X, T * 0,q X ⊠ (T * 0,q X) * ) denote the distribution kernel of S (q) .We will write H q (X) for ker ✷ q b and for functions we simply write H(X) to mean H 0 (X).The distributional kernel S (q) (x, y) was studied in [Hs], before recalling an explicit description describing the oscillatory integral defining the distribution S (q) (x, y) we need to fix some notation.To begin, let us review the concept of the Hörmander symbol space.Let D ⊂ X be a local coordinate patch with local coordinates x = (x 1 , . . ., x 2n+1 ).
Definition 2.1.For every m ∈ R, we denote with the space of all a such that, for all compact K ⋐ D × D and all α, β ∈ N 2n+1 0 , γ ∈ N 0 , there is a constant C α,β,γ > 0 such that For simplicity we denote with S m 1,0 the spaces defined in (7); furthermore, we write Let a j ∈ S m j 1,0 , j = 0, 1, 2, . . .with m j → −∞, as j → ∞.Then there exists a ∈ S m 0 1,0 unique modulo S −∞ , such that for k = 0, 1, 2, . ... If a and a j have the properties above, we write a ∼ ∞ j=0 a j in S m 0 1,0 .It is known that the characteristic set of ✷ q b is given by and Σ + is defined similarly for λ > 0. We recall the following theorem (see [Hs,Theorem 1.2]).
Theorem 2.1.Suppose that the Levi form is non-degenerate and ✷ q b has L 2 closed range.Then, there exist continuous operators S − , S + : L 2 (0,q) (X) → Ker ✷ q b such that ) is the wave front set of S − in the sense of Hörmander.Moreover, consider any small local coordinate patch D ⊂ X with local coordinates x = (x 1 , . . ., x 2n+1 ), then S − (x, y), S + (x, y) satisfy Remark 2.1.Kohn [Kohn] proved that if q = n − = n + or |n − − n + | > 1 then ✷ q b has L 2 closed range.For a description of the phase function in local coordinates see chapter 8 of part I in [Hs].Now we focus on the decomposition induced by the group action on H q (X).Fix t ∈ T and let t * : Λ r x (CT * X) → Λ r t −1 •x (CT * X) be the pull-back map.Since T preserves J, we have t * : T * 0,q x X → T * 0,q g −1 •x X, for all x ∈ X.Thus, for u ∈ Ω 0,q (X), we have t * u ∈ Ω 0,q (X).Put Ω 0,q (X) kν := u ∈ Ω 0,q (X); (e iθ ) * u = e i k ν, θ u, ∀θ ∈ R r .
In fact, σ kν is an isomorphism for k large if we can prove that σ kν : H q (X) kν → H q−r (X red ν ) k and σ * kν : H q−r (X red ν ) k → H q (X) kν are injective for k large.Notice that, by Theorem 3.1, we have on Ũ × ( Ũ3 × Ũ4 ) where we can check β 0 (x ′′ , x′′ ) = α 0 (x ′′ , x′′ ).The injectivity of the map σ kν is a consequence of the following theorem which is an adaptation of proof of the main theorem in [HH].
Theorem 3.2.There exists a Fourier integral operator R k of equivariant Szegő type of degree n − (t − 1)/2 such that where c 0 is a positive constant and 1 + R k : Ω 0,q (X) → Ω 0,q (X) is an injective Fourier integral operator of equivariant Szegő type.
Proof.By Theorem 3.1 and equation ( 12) we can compose σ * kν and σ kν using the complex stationary phase formula of [MS].We can pick e in the definition of σ kν so that the leading term of the symbol of σ * kν σ kν agrees with the one of S (1; Ũ × Ũ , T * (0,q) X ⊠ (T * (0,q) X) * ) and |r 0 (x, y)| ≤ C|(x, y) − (x 0 , x 0 )| for all x 0 ∈ X ν ∩ Ũ, where C > 0 is a constant.The injectivity of the operator 1 + R k follows from direct computations of the leading symbol of R k which vanishes at diag(X ν × X ν ).In fact, as a consequence of this, by Lemma 6.7 and Lemma 6.8 in [HH] we have that R k u ≤ ǫ k u for all u ∈ Ω 0,q (X), for all k ∈ N where ǫ k is a sequence with lim k→+∞ ǫ k = 0; this in turn implies, if k is large enough, that the map 1 + R k is injective.
Analogously, the injectivity of the map σ * kν follows by studying σ kν σ * kν .We don't repeat the proof here since it is an application of the stationary phase formula for Fourier integral operators of complex type, see [MS], and it follows in a similar way as above.