Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions

The asymptotic results for Berezin-Toeplitz operators yield a strict quantization for the algebra of smooth functions on a given Hodge manifold. It seems natural to generalize this picture for quantizable pseudo-Kähler manifolds in the presence of a group action. Thus, in this setting, we introduce...

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2025
1. Verfasser:	Galasso, Andrea
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citation_txt	Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions. Andrea Galasso. SIGMA 21 (2025), 048, 25 pages
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description	The asymptotic results for Berezin-Toeplitz operators yield a strict quantization for the algebra of smooth functions on a given Hodge manifold. It seems natural to generalize this picture for quantizable pseudo-Kähler manifolds in the presence of a group action. Thus, in this setting, we introduce a Berezin transform which has a complete asymptotic expansion on the preimage of the zero set of the moment map. It leads in a natural way to proving that certain quantization maps are strict.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 048, 25 pages Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions Andrea GALASSO Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via R. Cozzi 55, 20125 Milano, Italy E-mail: andrea.galasso@unimib.it Received January 29, 2025, in final form June 17, 2025; Published online June 25, 2025 https://doi.org/10.3842/SIGMA.2025.048 Abstract. The asymptotic results for Berezin–Toeplitz operators yield a strict quantization for the algebra of smooth functions on a given Hodge manifold. It seems natural to generalize this picture for quantizable pseudo-Kähler manifolds in presence of a group action. Thus, in this setting we introduce a Berezin transform which has a complete asymptotic expansion on the preimage of the zero set of the moment map. It leads in a natural way to prove that certain quantization maps are strict. Key words: CR manifolds; Toeplitz operators; star products; group actions 2020 Mathematics Subject Classification: 32V20; 32A25; 53D50 1 Introduction This paper is a continuation of [7]. Here, we further investigate the properties of the star product induced by Toeplitz operators, with a focus on establishing conditions under which certain quantization maps are strict. The present paper aims to extend known results for Kähler manifolds to the setting of pseudo-Kähler manifolds equipped with a group action. We emphasize that the purpose of this introduction is to provide a broad overview of the results. Consider a compact connected complex manifold (M,J) of real dimension 2n, where J is an integrable complex structure. Let ω be a real, non-degenerate closed 2-form on M , and assume that it satisfies the compatibility condition ω(JX, JY ) = ω(X,Y ), ∀X,Y ∈ TM . In this case, the triple (M,ω, J) is called a pseudo-Kähler manifold. Unlike in the standard Kähler case, we do not assume that ω is positive definite. Instead, ω has constant signature (n−, n+), meaning that at each point m ∈ M , the symmetric bilinear form gm(X,Y ) := ωm(X, JY ) has signature (n−, n+). Let (L, hL) be a Hermitian line bundle over M , endowed with a Hermitian connection ∇L whose curvature RL satisfies RL = −2πiω. Definition 1.1. For each positive integer k, the quantization space Qk(M) of (M,ω) with respect to the polarization J is defined as the space of harmonic (0, n−)-forms with values in the k-th tensor power L⊗k of L, Qk(M) := ker□(n−) k , where □(n−) k is the Kodaira Laplacian acting on L⊗k-valued (0, n−)-forms. In the special case when n− = 0 (i.e., when g is positive definite), this construction recovers the classical geometric quantization: Qk(M) coincides with the space H0 ( M,L⊗k ) of holomor- phic sections. Hence, this framework generalizes the Kähler case to pseudo-Kähler manifolds mailto:andrea.galasso@unimib.it https://doi.org/10.3842/SIGMA.2025.048 2 A. Galasso with indefinite metrics. There is a natural inner product on Qk(M), we refer to Section 2 for the definition and for more details. Let {·, ·} denote the Poisson bracket on C∞(M) induced by the symplectic form ω. We now recall the definition of a star product. Definition 1.2. A star product for the algebra C∞(M) is given by the formal power series g ⋆ h = +∞∑ j=0 Cj(g, h)ν −j such that ⋆ is an associative C[[ν]]-linear product, that is, (g ⋆ h) ⋆ l = g ⋆ (h ⋆ l), for all g, h, l ∈ C∞(M) and C0(g, h) = g·h, C1(g, h)−C1(h, g) = i{g, h}, for all g, h ∈ C∞(M). The star product is said to be of Wick type if and only if the function appearing as first argument is differentiated in holomorphic directions while the one appearing as second argument is differentiated in anti- holomorphic directions. In addition to the defining conditions, a star product may enjoy further properties, such as (1) (Unit) Let 1 be the constant function equals to 1 everywhere, then 1 ⋆ f = f ⋆ 1 = f for every smooth function on M . (2) (Parity) For every smooth functions f and g on M , then f ⋆ g = f ⋆ g. (3) (Trace) Denote the trace on Qk(M) by Trk. It induces a C[[ν]]-linear map Tr: C∞(M)[[ν]] → ν−nC[[ν]] such that Tr(f ⋆ g) = Tr(g ⋆ f). Let G be a compact connected Lie group having dimension dimRG = nG. Assume that M admits a Hamiltonian and holomorphic action of G with moment map Φ. Suppose that 0, in the dual of the Lie algebra, is a regular value for Φ. Assume that the pseudo-metric g restricted to the orbit through any m ∈ Φ−1(0) is non-degenerate and the action of G on Φ−1(0) is free. Then the symplectic reduction MG := Φ−1(0)/G is a quantizable pseudo-Kähler manifold, see, for example, [31, Appendix B]. More precisely, let ι : Φ−1(0) ↪→ M denote the inclusion map. Then the following holds: (1) the topological quotient MG := Φ−1(0)/G is a smooth manifold of dimension 2n − 2nG, and the quotient map π : Φ−1(0) → MG is a principal G-bundle; (2) there exist a unique pseudo-Riemannian metric gG and complex structure JG on MG such that π∗gG = ι∗g, π∗JG = ι∗J , and the 2-form ωG := gG(·, JG·) defines a symplectic form on the quotient. Furthermore, we assume that the symmetric bilinear form gG has signature ( nG −, n G + ) . Suppose that the action of G on M lifts on (L, hL). Then there exists an unitary represen- tation of G on Qk(M). Let Qk(M)G be the space of G-fixed vectors in Qk(M). In Section 3, for every smooth function f on MG, we define a G-invariant Toeplitz operators TG k [f ] acting on Qk(M)G. Our next result show that G-invariant Toeplitz operators induces a star product ⋆ which defines a strict deformation quantization in the sense of [18, Chapter 2]. The following theorem is a consequence of Theorem 2.3 which is formulated further down. Theorem 1.3. There exists a unique formal star-product of Wick type f ⋆ g := +∞∑ j=0 νjCj(f, g), Cj(f, g) lies in C∞(MG), Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 3 such that for every smooth functions f and g on MG, for every N ∈ N we have with suitable constants KN (f, g), and for all k ∈ N∥∥∥∥∥∥TG k [f ]TG k [g]− N−1∑ j=0 k−jTG k [Cj(f, g)] ∥∥∥∥∥∥ = KN (f, g)k−N . (1.1) Furthermore, the star product ⋆ satisfies the properties (1), (2), and (3) of Definition 1.2. To a certain extent the strategy to show Theorem 1.3 is similar to the strategy done in [32], details are given in Section 7. Now, if we put G = {e}, where e is the identity element, we recover previous results in [32], and we generalize it to pseudo-Kähler manifolds. Furthermore, we refer to the surveys [15, 33], and [38] for detailed discussions on the induced star products. In order to better contextualize the aim of this work and to motivate our main results, we recall some relevant developments in the literature. In [1], the authors studied Berezin– Toeplitz quantization on Hodge manifolds, their proofs are based on the theory of generalized Toeplitz structures developed by L. Boutet de Monvel, V. Guillemin, and J. Sjöstrand in the framework of microlocal analysis, see [2]. In [33], additional properties of the Berezin–Toeplitz star product were established, and it was shown that the structure of Berezin–Toeplitz operators naturally leads to the strictness of certain quantization maps. The idea of considering pseudo- Kähler manifolds is suggested by results contained in [13, 23], in particular see [22, Theorem 1.7] and [23, Section 8.2], where the authors worked out the asymptotic expansion for the Bergman kernel for (0, q)-forms under the assumption that the curvature is non-degenerate. In the geometric quantization of pseudo-Kähler manifolds, two main strategies are available: one based on the theory of Berezin–Toeplitz quantization with vector bundles, as developed by X. Ma and G. Marinescu [23], and another one employing Toeplitz operators on the associated circle bundle. In this work, we adopt the second approach. This choice is primarily motivated by the fact that it aligns with the framework we have developed in a previous article [7]. The main approach of X. Ma and G. Marinescu is the local index theory, especially the analytic localization techniques developed by J.-M. Bismut and G. Lebeau. Our approach is based of Fourier integral operators of complex type as developed by A. Melin and J. Sjöstrand. It is worth noting that, in principle, both methods could be employed in the context of this article. Our method can be applied to study Berezin–Toeplitz star product for Reeb invariant smooth functions on non-degenerate CR manifolds with transversal and CR R-action. We now introduce the setting for our quantization commutes with reduction result in the pseudo-Kähler case. Suppose that the action of G on M lifts to an action on (L, hL). Then (L, hL) descends to a Hermitian line bundle ( LG, h G L ) over MG with curvature RLG = −2πiωG. Recall that the standard Kähler case is recovered when n− = 0, and consequently nG − = 0. This allows us to define the quantization of the reduced space as follows. Definition 1.4. The quantization space Qk(MG) is defined as the space of harmonic ( 0, nG − ) - forms with values in L⊗k G Qk(MG) := ker2 (nG −) k , where □ (nG −) k is the Kodaira Laplacian acting on L⊗k G -valued ( 0, nG − ) -forms. There is a canonical map Qk(M)G → Qk(MG), and our main result is the following (the proof is given in Section 5.2). Theorem 1.5 (quantization commutes with reduction). Under the above assumptions, for every k ∈ N, the canonical map Qk(M)G −→ Qk(MG) is an isomorphism of finite-dimensional Hilbert spaces. 4 A. Galasso The principle that “quantization commutes with reduction” was originally formulated by V. Guillemin and S. Sternberg in the Kähler setting [9]. The present result extends this principle to the pseudo-Kähler setting, of which the Kähler case is a particular instance. In the context of general symplectic manifolds, proofs of this principle were developed by E. Meinrenken [26], M. Vergne [36, 37], and E. Meinrenken and R. Sjamaar [27]. Independently, Y. Tian and W. Zhang [35] gave a different analytic approach. For a survey on the topic, we refer to [21], where the connection with the CR setting introduced in [12] is also highlighted. Finally, let us also mention that in many of the aforementioned approaches, the quantization space is defined as the index of a twisted Spinc-Dirac operator. In contrast, our proof of The- orem 1.5 follows the geometric strategy introduced in [9], adapted to the setting of indefinite curvature. We refer to Section 5.1 for the detailed construction and background. 2 Background and statement of the results 2.1 Examples There are many examples of pseudo-Kähler manifolds that serve as natural candidates for quan- tization. First, we refer to the thesis [31], where the author investigates the symplectic and pseudo-Riemannian geometry of the PSL(3,R) Hitchin component associated with a closed ori- entable surface. When the genus is at least 2, the author defines a mapping class group-invariant pseudo-Kähler metric on the Hitchin component. A second example is given by the Grauert tube of an analytic pseudo-Riemannian manifold; see [39] for an introduction in the classical Riemannian setting. Finally, we present a natural example arising in the context of the Heisenberg group. Before describing this example, we recall some material from [6]. A CR manifold ( X,T 1,0X ) , with contact form ω0, is said to be nondegenerate if its Levi form is nondegenerate at every point, for precise definitions, we refer the reader to Section 2.3. In this case, replacing ω0 by −ω0 if necessary, we may assume that the (constant) dimension k of a maximal positive-definite subspace for the Levi form is at least n/2. We then say that X is k-strongly pseudoconvex. When k = n, we simply say that X is strongly pseudoconvex. If X is k-strongly pseudoconvex with k < n, then by [6, Lemma 13.1], there exist smooth subbundles E+ and E− of T 1,0X such that T 1,0X = E+ ⊕E−, the Levi form is positive definite on E+, negative definite on E−, and E+ ⊥ E− with respect to the Levi form. Example 2.1. Let Hn = Cn × R denote the Heisenberg group. Occasionally, as in [5], it is useful to modify the group law on Hpol n = R2n+1 via (p, q, t) · (p′, q′, t′) = (p+ p′, q + q′, t+ t′ + pq′) where z = p+ iq. Moreover, following [6, pp. 436–437], we may define a left-invariant CR structure on Hpol n which is k-strongly pseudoconvex, with n/2 ≤ k ≤ n, by taking T 1,0Hpol n to be the bundle spanned by Z1, . . . , Zk, Zk+1, . . . , Zn, where Zj = ∂ ∂zj + izj ∂ ∂t . By compactifying this example as in [5, p. 68], one obtains a natural compact pseudo- Kähler manifold. Let Γ be the subgroup of Hpol n consisting of points with integer coordinates Γ = { (p, q, t) ∈ Hpol n \| p, q ∈ Zn and t ∈ Z } . Then Γ is a discrete subgroup of Hpol n , and the right coset space M = Γ\Hpol n is a compact mani- fold. It is easy to verify that the half-open unit cubeQ2n+1 inHpol n is a fundamental domain for Γ, meaning that each right coset of Γ contains precisely one point of Q2n+1. Hence, topologically, M can be viewed as the closed unit cube Q2n+1 with certain boundary faces identified—a kind of “twisted torus”. Moreover, by the above discussion, M inherits a pseudo-Kähler structure. Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 5 2.2 Toeplitz operators in the pseudo-Kähler setting Let (M,ω, J) be a pseudo-Kähler manifold and dimRM = 2n. Let (L, hL) be a Hermitian line bundle on M having Hermitian connection whose curvature is RL = −2πiω. Fix a Hermitian metric ⟨·\|·⟩ on the holomorphic tangent bundle T 1,0M . The Hermitian metric ⟨·\|·⟩ on T 1,0M induces a Hermitian metric on TM ⊗ C and also on⊕ q∈N∪{0}, 0≤q≤n T ∗0,qM, where T ∗0,qM is the bundle of (0, q) forms. Consider the vector bundle T ∗0,qM ⊗ L⊗k whose space of smooth sections is denoted by Ω0,q ( M,L⊗k ) . The Hermitian metrics ⟨·\|·⟩ and hL induce a new Hermitian metric ⟨·\|·⟩ hL⊗k on T ∗0,qM⊗L⊗k, the corresponding norm is denoted by \| · \| hL⊗k . We can define the L2-inner product as follows (s1, s2)k = ∫ M ⟨s1\|s2⟩hL⊗kdVM , s1, s2 ∈ Ω0,q ( M,L⊗k ) , and we denote by ∥·∥k the corresponding norm, where dVM is the volume form on M induced by ⟨·\|·⟩. Let L2 0,q ( M,L⊗k ) be the completion of Ω0,q ( M,L⊗k ) with respect to (·, ·)k. Let □(q) k : Ω0,q ( M,L⊗k ) → Ω0,q ( M,L⊗k ) be the Kodaira Laplacian. We denote by the same symbol the L2 extension of □(q) k □(q) k : Dom□(q) k ⊂ L2 0,q ( M,L⊗k ) → L2 0,q ( M,L⊗k ) , where Dom□(q) k = { u ∈ L2 0,q ( M,L⊗k ) \| □(q) k u ∈ L2 0,q ( M,L⊗k )} . The projection P (q) k : L2 0,q ( M,L⊗k ) → Hq ( M,L⊗k ) := Ker□(q) k onto the kernel of □(q) k is called Bergman projector or Bergman projection. Since □(q) k is elliptic, its distributional kernel satisfies P (q) k (·, ·) ∈ C∞(M ×M,Lk ⊗ (( T ∗0,qM ) ⊠ ( T ∗0,qM )∗)⊗ (Lk )∗) . Now, we shall follow [14] and define Toeplitz operators in this framework. Fix a smooth function f on M . Define the Berezin–Toeplitz operator T (q) k [f ] := P (q) k ◦ f ◦ P (q) k : L2 0,q ( M,L⊗k ) → Hq ( M,L⊗k ) := Ker□(q) k ; we always fix q = n− and we write Tk[f ] = T (n−) k [f ]. Note that ifM is compact complex manifold endowed with a positive line bundle L, we put q = n− = 0 and we recover the standard Kähler setting. Thus, the quantizing Hilbert space is Ker□(q) k , for Riemann–Roch-type results we refer to [14] and [8, Section 1.2]. 2.3 CR formulation The circle bundle X inside the dual of the line bundle L is a compact, connected and orientable CR manifold of dimension 2n+1. We now provide a more precise formulation of these concepts. Let ( X,T 1,0X ) be a compact, connected and orientable CR manifold of dimension 2n+1, n ≥ 1, where T 1,0X is a subbundle of the complexified tangent bundle TX ⊗ C. There is a unique sub-bundle HX of TX such that HX ⊗ C = T 1,0X ⊕ T 0,1X which is called the horizontal 6 A. Galasso bundle. Let J : HX → HX be the complex structure map given by J(u + u) = iu − iu, for every u ∈ T 1,0X. By complex linear extension of J to TX ⊗ C, the i-eigenspace of J is T 1,0X. Since X is orientable, there always exists a real non-vanishing 1-form ω0 ∈ C∞(X,T ∗X) so that ⟨ω0(x), u⟩ = 0, for every u ∈ HxX, for every x ∈ X. The one form ω0 is called contact form and it naturally defines a volume form dVX on X. For each x ∈ X, we define a quadratic form on HX with respect to ω0 by Lx(U, V ) = 1 2dω0(JU, V ), ∀U, V ∈ HxX. Then, we extend L to HX ⊗ C by complex linear extension; for U, V ∈ T 1,0 x X, Lx ( U, V ) = 1 2 dω0 ( JU, V ) = − 1 2i dω0 ( U, V ) . The Hermitian quadratic form Lx on T 1,0 x X is called Levi form at x (with respect to ω0). Hereafter we shall always assume that the Levi form is non-degenerate with n− negative and n+ positive eigenvalues, n+ + n− = n. The Reeb vector field R ∈ C∞(X,TX) is defined to be the non-vanishing vector field determined by ω0(R) ≡ −1, dω0(R, ·) ≡ 0 on TX. There is a Reeb vector field such that the flow of it induces a transversal CR R-action on X. Suppose that all orbits of the flow of R are compact. Then X admits a transversal CR circle action, which we denote by eiθ·, where θ ∈ R. Note that this is the case of the unit circle bundle X ⊂ L∨, with projection π : X → M and connection contact form ω0 such that dω0 = 2π∗(ω). Then (X,ω) is a CR manifold and ω0 is the contact form. Now, we define the quantizing space and we identify it later with the kernel of the Kodaira Laplacian. For all x ∈ X, let T ∗0,qX be the q-th exterior power of ( T (0,1)X )∗ , whose space of sections is denoted by Ω0,q(X) and its elements are called (0, q)-forms. The Hermitian metric ⟨·\|·⟩ on TX ⊗ C induces in a natural way a L2 inner product (·\|·) on Ω0,q(X). Denote as ∥·∥ the corresponding norm of (·\|·). Let L2 (0,q)(X) be the completion of Ω0,q(X) with respect to (·\|·). Let 2q b be the Gaffney extension of the Kohn Laplacian. The Szegő projection is the orthogonal projection S(q) : L2 (0,q)(X) → Ker2q b with respect to (·\|·). Let S(q)(x, y) ∈ D′(X ×X,T ∗0,qX ⊠ ( T ∗0,qX )∗) be the distributional kernel of S(q), D′ denotes the space of distribution sections, see Section 3.2 for notation. In this paper, we will assume that 2 q b has L2 closed range. This hypothesis is always satisfied for the circle bundle case we are considering. Fix q = n− and let f ∈ C∞(X,C), then the Toeplitz operator is given by TX [f ] := S(n−) ◦ f · ◦S(n−) : L2 (0,n−)(X) → Ker2 n− b . The transversal circle action can be used to define the corresponding Fourier components which we denote by TX k [f ], see [7] where we use pseudodifferential operators of order zero (furthermore we refer to [34] for composition with certain classical pseudodifferential operators of order one). Let u ∈ Ω0,q(X) be arbitrary. Define Ru := ∂ ∂θ (( eiθ )∗ u ) \|θ=0 ∈ Ω0,q(X). For every k ∈ Z, let Ω0,q k (X) := { u ∈ Ω0,q(X) \| Ru = iku } , q = 0, 1, 2, . . . , n. Thus, we denote C∞ k (X) := Ω0,0 k (X). Then, the L2 inner product (·\|·) on Ω0,q(X) induced by ⟨·\|·⟩ is S1-invariant. Let L2 (0,q),k(X) be the completion of Ω0,q k (X) with respect to (·\|·). The k-th Szegő projection is the orthogonal projection S (q) k : L2 (0,q)(X) → ( Ker2q b ) k with respect to (·\|·). For Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 7 ease of notation we put H(X) := Ker2 n− b and Hk(X) := ( Ker2 n− b ) k . Eventually, for every S1 invariant function f ∈ C∞(X), the k-th Toeplitz operator is given by TX k [f ] := S (n−) k ◦ f · ◦S(n−) k : L2 (0,n−)(X) → Hk(X). Now, recall that X is the circle bundle inside the dual of the line bundle (L, hL). We have defined two important operators: TX k [f ] ( constructed with the k-th Fourier component of the Szegő projector S (n−) k ) and Tk[f ] ( constructed with the Bergman projector P (n−) k ) . They can be identified, in fact we claim S (n−) k (xm, x′m′) = (xm)k ( 1 2π P (n−) k (m,m′)(x′m′)k ) , where π(xm) = m, π(x′m′) = m′. Proof of the claim. We adopt the following conventions and notations: every element x∨ in (L∨)∨ = L defines a map x∗ : L∨ → C which is the dual of an element x ∈ L∨. We also write fm for the restriction of f ∈ L2 (0,n−),k(X) along the fibre Sm of the standard circle action onX. First, notice that any function f ∈ L2 (0,n−),k(X) is uniquely defined at m along the fibre Sm ⊂ L∨ by f ( e−iθ · xm ) = eikθf(xm); and thus it defines a linear map fm : (L∨ m)⊗k → T ∗0,qX, which is the same as an element σ(m) ∈ T ∗0,q m M ⊗ L⊗k m . We have the following isomorphism: Sk : L2 (0,n−),k(X) → L2 0,n− ( M,L⊗k ) , f 7→ σf where the inverse of Sk is given by σ 7→ fσ, with fσ(xm) = (xm)k(σ(m)). Finally, the operator given by taking the k-th Fourier component along the fibers S of π : X → M is given explicitly by Fk : L2 (0,n−)(X) → L2 (0,n−),k(X), f 7→ 1 2π ∫ S f(x)(x∗)kdx. Now, the claim follows by the formula S (n−) k = S−1 k ◦ P (n−) k ◦Sk ◦ Fk. ■ 2.4 Group action framework A Hamiltonian and holomorphic action of a compact Lie group G on M induces an infinitesimal action of its Lie algebra on X via the Kostant formula, see [17]. Under suitable topological conditions—such as when G is semisimple or the manifold M is simply connected—this in- finitesimal action can be integrated to an action of G on the circle bundle X inside the dual line bundle L∗. This action of G on X preserves both ω0 and J . We are thus led to introduce the main definitions in the CR setting, but before doing so we give more details on the Kostant formula and the quantization of group actions on pseudo-Kähler manifolds. Let M be a pseudo-Kähler manifold with complex structure J and Hodge form ω. Re- call that we denote by X the circle bundle lying in the dual line bundle L∗, with circle ac- tion eiθ· : S1 ×X → X, projection π : X → M and ∂θ the generator of the structure circle action on it. Hence X is naturally a contact and Cauchy–Riemann manifold; ω0 is the contact form. Suppose given, in addition, an action µ : G × M → M of a compact Lie group G with Lie algebra g, which is holomorphic with respect to the complex structure J and Hamiltonian with moment map Φ: M → g∨, 2ω(ξM , ·) = dΦξ(·). By [17], the action µ naturally induces an infinitesimal contact action of g on the circle bundle X. Explicitly, if ξ ∈ g and ξM is the corresponding Hamiltonian vector field on M , then its contact lift ξX is as follows. Let v♯ denote the horizontal lift on X of a vector field v on M , we have ξX := ξ♯M − ⟨Φ ◦ π, ξ⟩R. (2.1) 8 A. Galasso Furthermore, suppose that the infinitesimal action (2.1) can be integrated to an action of G on X. By hypothesis, it preserves the Cauchy–Riemann structure and the contact form α. There is a naturally induced unitary representation of G on the Hardy space H(X) ⊂ L2 (0,n−)(X) given by g : f 7→ f ◦ g−1·. Thus, adopting the convection of the previous subsection, we find that X admits an action of a compact connected Lie group G of dimension d, and the G action preserves ω0 and the CR structure. Furthermore, for any ξ ∈ g, the vector field ξX appearing in equation (2.1) is (ξXu)(x) = ∂ ∂t [ u ( etξx )] \|t=0 , for any u ∈ C∞(X). Fix a smooth Hermitian metric ⟨·\|·⟩ invariant under the action of G and S1 on TX ⊗ C so that T 1,0X is orthogonal to T 0,1X, ⟨u\|v⟩ is real if u, v are real tangent vectors, and ⟨R\|R⟩ = 1. Now, we define the quantizing G-invariant space. Fix g ∈ G and x ∈ X, for any r ∈ N let g∗ : Λr x(T ∗X ⊗ C) → Λr g−1◦x(T ∗X ⊗ C) be the pullback map. Since G preserves J , we have g∗ : T ∗0,q x X → T ∗0,q g−1◦xX for all x ∈ X. Thus, for u ∈ Ω0,q(X), we have g∗u ∈ Ω0,q(X). Put Ω0,q(X)G := { u ∈ Ω0,q(X) \| g∗u = u, ∀g ∈ G } . Since the Hermitian metric ⟨·\|·⟩ on TX⊗C is G-invariant, the L2 inner product (·\|·) on Ω0,q(X) induced by ⟨·\|·⟩ is G-invariant. Denote as ∥·∥ the corresponding norm of (·\|·). Let u ∈ L2 (0,q)(X) and g ∈ G, we define g∗u in the standard way. Put L2 (0,q)(X)G := { u ∈ L2 (0,q)(X) \| g∗u = u, ∀g ∈ G } . Put ( Ker2q b )G := Ker2q b ∩ L2 (0,q)(X)G. The G-invariant Szegő projection is the orthogonal pro- jection S (q) G : L2 (0,q)(X) → ( Ker2q b )G with respect to (·\|·). Let S(q) G (x, y) ∈ D′(X ×X,T ∗0,qX ⊠ ( T ∗0,qX )∗) be the distributional ker- nel of S (q) G , D′ denotes the space of distribution sections, see Section 3.2 for notation. Let f ∈ C∞(X,C) be invariant under the circle action and the action of G, then the G- invariant Toeplitz operator is given by TG[f ] := S (n−) G ◦ f · ◦S(n−) G : L2 (0,n−)(X) → ( Ker2 n− b )G . The transversal circle action can be used to define the corresponding Fourier components. First, we note that eiθ · (gx) = g ( eiθ · x ) , for every x ∈ X, θ ∈ [0, 2π], g ∈ G. For every k ∈ Z, let Ω0,q k (X)G = { u ∈ Ω0,q(X)G \| Ru = iku } , q = 0, 1, 2, . . . , n. Thus, we denote C∞ k (X)G := Ω0,0 k (X)G. Then, the L2 inner product (·\|·) on Ω0,q(X) induced by ⟨·\|·⟩ is G and S1-invariant. Let L2 (0,q),k(X)G be the completion of Ω0,q k (X)G with respect to (·\|·). The k-th G-invariant Szegő projection is the orthogonal projection S (q) G,k : L2 (0,q)(X) → ( Ker2q b )G k with respect to (·\|·). For ease of notation, we put HG(X) := ( Ker2 n− b )G and HG k (X) :=( Ker2 n− b )G k . Eventually, for every G and S1 invariant function f ∈ C∞(X), the k-th G-invariant Toeplitz operator is given by TG k [f ] := S (n−) G,k ◦ f · ◦S(n−) G,k : L2 (0,n−)(X) → HG k (X). Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 9 2.5 A key result on Toeplitz operators The goal of this subsection is to establish a key result concerning Toeplitz operators. This result will serve as the main analytical tool in the proof of the principal theorem of the paper. Definition 2.2. The contact moment map associated to the form ω0 is the map µ : X → g∗ such that, for all x ∈ X and ξ ∈ g, we have ⟨µ(x), ξ⟩ = ω0(ξX(x)). Note that µ is Φ ◦ π : X → g∗. Let b be the nondegenerate bilinear form on HX such that b(·, ·) = dω0(·, J ·). Clearly b is the pullback via π of the pseudo-Riemannian metric on M . Let us denote µ−1(0) by Y . Assume that 0 is a regular value of µ, assume that the action of G on µ−1(0) is free and g x ∩ g⊥b x = {0} at every point x ∈ Y, (2.2) where g = Span(ξX ; ξ ∈ g) and g⊥b = { v ∈ HX \| b(ξX , v) = 0, ∀ξX ∈ g } . Note that (2.2) is one of the main assumptions in [11]. By [11, Section 2.5], we observe that when restricted to g× g, the bilinear form b is nondegenerate on Y . Note that in Theorem 1.5 we assume that the pseudo-metric g restricted to the orbit through any m ∈ Φ−1(0) is non-degenerate. This assumption implies equation (2.2). Then, µ−1(0) is a d-codimensional submanifold of X. Let Y := µ−1(0) and let HY := HX ∩ TY . Now, recall that R is the Reeb vector field of the contact form ω0 and let L denote the Lie derivative. A vector field V : X → TX such that LV ω0 = gω0 for some smooth function g : X → R is called a contact vector field. By [25, Lemma 3.5.14], for every function g : X → R there exists a unique contact vector field Xg which satisfies ω0(Xg) = g and dω0(Xg, V ) = (dg(R))α(V )− dg(V ) for each vector field V on X. Thus, there is a one-to-one correspondence between contact vector fields and smooth functions on (X,ω0). The analog of the Poisson bracket in contact geometry is {g, h} := ω0([Xg, Xh]) = dg(Xh)− dh(Xg) + dω0(Xg, Xh) for f and g smooth function on X. Now, we are interested in the subspace of H-invariant smooth functions on X. Thus, for any S1-invariant smooth function g we have dg(R) = R(g) = 0 and dω0(Xg, V ) = −dg(V ). Eventually, we note that for two H-invariant smooth functions g and h on X, we obtain {g, h} = −dω0(Xg, Xh) + dω0(Xh, Xg) + dω0(Xg, Xh) = dω0(Xh, Xg). The CR reduction is the CR manifold XG := µ−1(0)/G [11]. Let πX : µ−1(0) → XG be the projection. Then, XG is a circle bundle over the symplectic reduction MG := Φ−1(0)/G. In Section 5.1, we explain how to generalize some of the results in [9] to the setting of pseudo- Riemannian manifolds. In particular, there is an identification MG ∼= Ms/G C, where GC is the complexification of the group G and Ms is the saturation by GC of Φ−1(0) in M . Now, consider a smooth function f̃ on X invariant under the action of G and S1, it gives rise to a smooth function f̃\|Y on Y = µ−1(0). Then, f̃\|Y defines a smooth function f on MG. Thus, we define the sup-norm of a G-invariant function on Φ−1(0) to be∥∥f̃∥∥∞ := ∥f∥∞ := sup {∣∣f̃(x)∣∣ \| x ∈ µ−1(0) } , which is the uniform norm of f on the quotient MG. Vice versa, let f be a smooth S1 in- variant function on XG, its lift defines a smooth G and S1 invariant function on Y . By [20, 10 A. Galasso Lemma 5.34 (b)], the extension Lemma for functions on submanifolds, since Y is a smooth prop- erly embedded submanifold of X, then there exists a smooth function f̃ on X whose restriction to Y coincides with f . As the last piece of notation, let f̃ be a G-invariant function on X, we denote the operator norm of its corresponding level k G-invariant Toeplitz operator ∥∥TG k [ f̃ ]∥∥ = sup { ∥TG k [ f̃ ] s∥ ∥s∥ \| s ∈ HG k (X), s ̸= 0 } . (2.3) In the following theorem, we state an analogue of the main result in [1]. Parts (c) and (b) of the following theorem are a consequence of the results contained in [7], see also [23, 24], where the Bergman kernels and the corresponding Toeplitz operators on symplectic manifolds, for the mixed curvature case, are studied. In Section 6, we prove part (a) by adapting the strategy of [33] to our setting. Theorem 2.3. Under the assumptions and notations above. Let f̃ , g̃ be G and S1 invariant functions on X, then (a) there exists a positive constant C, such that ∥f∥∞ − C k ≤ ∥∥TG k [ f̃ ]∥∥ ≤ ∥f∥∞ and in particular limk ∥∥TG k [ f̃ ]∥∥ = ∥f∥∞; (b) the semi-classical Dirac condition holds ∥∥ki[TG k [ f̃ ] , TG k [ g̃ ]] − TG k [{ f̃ , g̃ }]∥∥ = O ( 1 k ) ; (c) the semi-classical von Neumann’s condition holds ∥∥TG k [ f̃ ] TG k [ g̃ ] − TG k [ f̃ · g̃ ]∥∥ = O ( 1 k ) . A main consequence of Theorem 2.3 is Theorem 1.3 stated in the introduction. This is shown in Section 7. Finally, note that the Berezin–Toeplitz quantization is also a positive quantization in the following sense, see [19]. Let f ∈ C∞(XG) S1 be such that f(x) ≥ 0 for every x ∈ XG. Then TG k [f ] is a non-negative operator, in the sense that ( TG k [f ]s, s ) ≥ 0 for every s ∈ HG k (X). Equivalently, by the spectral theorem, a self-adjoint linear operator A is positive if and only if Spec(A) ⊂ [0,+∞). On the other hand, the Weyl quantization of a non-negative function need not be a non-negative operator. 3 Preliminaries 3.1 Standard notations The following notations are adopted through this article: N is the set of natural numbers, N0 = N ∪ {0}, R is the set of real numbers, and we denote by R+ = {x ∈ R \| x > 0} and R+ = {x ∈ R \| x ≥ 0}. Furthermore, we write α = (α1, . . . , αn) ∈ Nn 0 if αj ∈ N0, j = 1, . . . , n. Let M be a smooth paracompact manifold and let TM and T ∗M denote the tangent bundle of M and the cotangent bundle of M , respectively. The complexified tangent bundle of M and the complexified cotangent bundle of M will be denoted by TM ⊗C and T ∗M ⊗C, respectively. Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 11 Let ⟨·, ·⟩ denote the pointwise duality between TM and T ∗M and we extend ⟨·, ·⟩ bi-linearly to TM ⊗ C× T ∗M ⊗ C. Let B be a smooth vector bundle over M whose fiber at x ∈ M will be denoted by Bx. If E is another vector bundle over a smooth paracompact manifold N , we write B⊠E∗ to denote the vector bundle over M ×N with fiber over (x, y) ∈ M ×N consisting of the linear maps from Ey to Bx. Let Y ⊂ M be an open set, the spaces of distribution sections of B over Y and smooth sections of B over Y will be denoted by D′(Y,B) and C∞(Y,B), respectively. Let E′(Y,B) be the subspace of D′(Y,B) whose elements have compact support in Y . Now, we recall the Schwartz kernel theorem. Let B and E be smooth vector bundles over paracompact orientable smooth manifolds M and N , respectively, equipped with smooth den- sities of integration. If A : C∞ 0 (N,E) → D′(M,B) is continuous, we write A(x, y) to denote the distribution kernel of A. The following two statements are equivalent (1) A is continuous: E′(N,E) → C∞(M,B), (2) the distributional kernel A(x, y) lies in C∞(M ×N,B ⊠ E∗). If A is continuous, we say that A is smoothing on M ×N . Let A, Â : C∞ 0 (N,E) → D′(M,B) be continuous operators, then we write A ≡ Â (on M ×N) if A − Â is a smoothing operator. If M = N , we simply write “on M”. We say that A is properly supported if the restrictions of the two projections (x, y) 7→ x, (x, y) 7→ y to supp(A(x, y)) are proper. Eventually, let H(x, y) ∈ D′(M × N,B ⊠ E∗) then we write H to denote the unique con- tinuous operator C∞ 0 (N,E) → D′(M,B) with distribution kernel H(x, y) and we identify H with H(x, y). 3.2 Operators and symbols Let ( X,T 1,0X ) be a compact, connected and orientable CR manifold of dimension 2n+1, n ≥ 1. Let D be an open set of X. Recall that Ω0,q(D) denote the space of smooth sections of T ∗0,qX over D and let Ω0,q c (D) be the subspace of Ω0,q(D) whose elements have compact support in D. Let ∂b : Ω 0,q(X) → Ω0,q+1(X) be the tangential Cauchy–Riemann operator. Let dv(x) be the volume form on X induced by the Hermitian metric ⟨·\|·⟩. The natural global L2 inner product (·\|·) on Ω0,q(X) induced by dv(x) and ⟨·\|·⟩ is given by (u\|v) := ∫ X ⟨u(x)\|v(x)⟩dv(x), u, v ∈ Ω0,q(X). Recall that L2 (0,q)(X) denote the completion of Ω0,q(X) with respect to (·\|·). We extend (·\|·) to L2 (0,q)(X) in the standard way. For f ∈ L2 (0,q)(X), we denote ∥f∥2 := (f \|f). We extend ∂b to L2 (0,r)(X), r = 0, 1, . . . , n, by ∂b : Dom ∂b ⊂ L2 (0,r)(X) → L2 (0,r+1)(X), where Dom ∂b := { u ∈ L2 (0,r)(X) \| ∂bu ∈ L2 (0,r+1)(X) } and, for any u ∈ L2 (0,r)(X), ∂bu is defined in the sense of distributions. We also write ∂ ∗ b : Dom ∂ ∗ b ⊂ L2 (0,r+1)(X) → L2 (0,r)(X) 12 A. Galasso to denote the Hilbert space adjoint of ∂b in the L2 space with respect to (·\|·). Let 2 q b denote the Gaffney extension of the Kohn Laplacian whose domain Dom2 q b is given by{ s ∈ L2 (0,q)(X) \| s ∈ Dom ∂b ∩Dom ∂ ∗ b , ∂bs ∈ Dom ∂ ∗ b , ∂ ∗ bs ∈ Dom ∂b } and 2 q bs = ∂b∂ ∗ bs+ ∂ ∗ b∂bs for s ∈ Dom2 q b . Let S(q) : L2 (0,q)(X) → Ker2q b be the orthogonal projection with respect to the L2 inner product (·\|·) and let S(q)(x, y) ∈ D′(X ×X,T ∗0,qX ⊠ ( T ∗0,qX )∗) denote the distribution kernel of S(q). Now, we recall Hörmander symbol space. Let D ⊂ X be a local coordinate patch with local coordinates x = (x1, . . . , x2n+1). Definition 3.1. For m ∈ R, Sm 1,0 ( D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) is the space of all a(x, y, t) ∈ C∞(D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) such that, for all compactK ⋐ D×D and all α, β ∈ N2n+1 0 , γ ∈ N0, there is a constant Cα,β,γ > 0 such that∣∣∂α x ∂ β y ∂ γ t a(x, y, t) ∣∣ ≤ Cα,β,γ(1 + \|t\|)m−γ for every (x, y, t) ∈ K × R+, t ≥ 1. Put S−∞(D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) to be⋂ m∈R Sm 1,0 ( D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) . Now, let aj ∈ S mj 1,0 ( D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) , j = 0, 1, 2, . . . with mj → −∞, as j → ∞. Then there exists a ∈ Sm0 1,0 ( D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) unique modulo S−∞ such that a− k−1∑ j=0 aj ∈ Smk 1,0 ( D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) for k = 0, 1, 2, . . . . If a and aj have the properties above, we write a ∼ ∑∞ j=0 aj in Sm0 1,0 ( D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) . Furthermore, we write s(x, y, t) ∈ Sm cl ( D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) if s(x, y, t) ∈ Sm 1,0 ( D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 13 and s(x, y, t) ∼ ∞∑ j=0 sj(x, y)tm−j in Sm 1,0 ( D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) , where sj(x, y) ∈ C∞(D ×D,T ∗0,qX ⊠ ( T ∗0,qX )∗) , j ∈ N0. Sometimes, we simply write Sm 1,0 to denote Sm 1,0 ( D ×D × R+, T ∗0,qX ⊠ ( T ∗0,qX )∗) , where m ∈ R ∪ {−∞}. Let E be a smooth vector bundle over X. Let m ∈ R, 0 ≤ ρ, δ ≤ 1. Let Sm ρ,δ(X,E) denote the Hörmander symbol space on X with values in E of order m type (ρ, δ) and let Sm cl (X,E) denote the space of classical symbols on X with values in E of order m. For a ∈ Sm cl (X,E), we write Op(a) to denote the pseudodifferential operator on X of order m from sections of E to sections of E with full symbol a. Let D ⊂ X be an open set. Let Lm cl ( D,T ∗0,qX ⊠ ( T ∗0,qX )∗) denote the space of classical pseudodifferential operators on D of order m from sections of T ∗0,qX to sections of T ∗0,qX respectively. Let P ∈ Lm cl ( D,T ∗0,qX ⊠ ( T ∗0,qX )∗) and we write σP and σ0 P to denote the full symbol of P and the principal symbol of P , respectively. 3.3 Toeplitz operators In this subsection, we briefly recall some results from [7]. Here, the Reeb vector field R is transversal to the space g at every point p ∈ µ−1(0), eiθ · g ◦ x = g ◦ eiθ · x, for every x ∈ X, θ ∈ [0, 2π], g ∈ G, and the action of G and S1 is free near µ−1(0). For every smooth function g ∈ C∞(X)G invariant under the action of S1, the k-th G-invariant Toeplitz operator is given by TG k [g] := SG k ◦ g · ◦SG k : L2 (0,n−)(X) → L2 (0,n−),k(X)G. The main interest for us is the case when g = f̃ and f is in C∞(XG) S1 . For the following theorem see in [11, Theorem 1.8] and [7]. Theorem 3.2. Let f ∈ C∞(X)G and suppose q = n−. Let D be an open set in X such that the intersection µ−1(0) ∩D = ∅. Then, TG k [f ] ≡ O(k−∞) on D. Let p ∈ µ−1(0) and let U a local neighborhood of p. Then, if q = n−, for every fixed y ∈ U , we consider TG k [f ](x, y) as a k-dependent smooth function in x. There exist local coordinates (x1, . . . , x2n+1) on U such that TG k [f ](x, y) = eikΨ(x,y)b(x, y, k) +O(k−∞) for every x, y ∈ U . For an explicit expression of the phase function Ψ we refer to [7], the symbol satisfies b(x, y,m) ∈ S n−d/2 loc ( 1, U × U, T ∗0,qX ⊠ ( T ∗0,qX )∗) and the leading term of b(x, y,m) along the diagonal at x ∈ µ−1(0) is given by b0(x, x) := 2d−1 1 Veff(x) π−n−1+ d 2 \|detRx\|− 1 2 \|detLx\| τx,n− , where Veff(x) := ∫ Ox dVOx, Ox is the orbit through x, and Rx is the linear map Rx : g x → g x , u → Rxu, ⟨Rxu\|v⟩ = ⟨dω0(x), Ju ∧ v⟩. 14 A. Galasso Now we recall the definition of the operator Ĥ, which appears in the description of the Cj ’s, see [7]. It is convenient to introduce a class of operator which looks microlocally like the Szegő kernel. Here we recall the definition from [7] for q = n− and action free case. Definition 3.3. Let H : Ω0,n−(X) → Ω0,n−(X) be a continuous operator with distribution kernel H(x, y) ∈ D′(X ×X,T ∗0,n−X ⊠ ( T ∗0,n−X )∗) . We say that H is a complex Fourier integral operator of G-invariant Szegő type of order k ∈ Z if H is smoothing away µ−1(0) and H has the following microlocal expression. Let p ∈ µ−1(0), and let D a local neighborhood of p with local coordinates (x1, . . . x2n+1). Then, the distribu- tional kernel of H satisfies H(x, y) ≡ ∫ ∞ 0 eitΦ−(x,y)a−(x, y, t)dt on D where a− ∈ S k− d 2 cl ( D ×D × R+, T ∗0,n−X ⊠ ( T ∗0,qX )∗) , a+ = 0 if q ̸= n+, where Φ− is the phase of the G-invariant Szegő kernel. Furthermore, we write σ0 H,−(x, y) to denote the leading term of the expansion of a−. Let Ψk(X)G denote the space of all complex Fourier integral operators of Szegő type of order k. Here we adapt the discussion before [7, Theorem 4.9] to G-action case. Let R̂ := 1 2 SG(−iR+ (−iR)∗)SG : Ω0,n−(X) → Ω0,n−(X), where (−iR)∗ is the adjoint of −iR with respect to (·\|·). Then, we note that R̂ ∈ Ψn+1(X)G with σ0 R̂,−(x, x) ̸= 0, for every x ∈ X. Let Ĥ ∈ Ψn−1(X)G with SGĤ = Ĥ = ĤSG, R̂Ĥ ≡ ĤR̂ ≡ SG. (3.1) Note that Ĥ is uniquely determined by (3.1), up to some smoothing operators. The operator Ĥ can always be found by setting Ĥ = SGH̃SG for a given pseudodifferential operator H̃. In fact, the first equation in (3.1) is always satisfied. Furthermore, the second equation can be solved by an easy application of the stationary phase formula. Now, we recall the G-invariant version of [7, Theorem 4.9]. Theorem 3.4. Let q = n−. Let f, g ∈ C∞(X), then we have TG[f ] ◦ TG[g]− N∑ j=0 ĤjTG [ Ĉj(f, g) ] ∈ ΨG n−N−1(X) for every N ∈ N0, where Ĉj(f, g) ∈ C∞(X), Ĉj is a universal bidifferential operator of order ≤ 2j, j = 0, 1, . . . , and Ĉ0(f, g) = f · g, Ĉ1(f, g)− Ĉ1(g, f) = i{f, g}. Eventually, we recall the following theorem from [7]. Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 15 Theorem 3.5. With the same assumptions as above, let f, g ∈ C∞(X)G. Let q = n−. Then, as k ≫ 1,∥∥∥∥TG f,k ◦ TG g,k − TG g,k ◦ TG f,k − 1 k TG i{f,g},k ∥∥∥∥ = O ( k−2 ) , and ∥∥∥∥∥∥TG f,k ◦ TG g,k − N∑ j=0 k−jTG Cj(f,g),k ∥∥∥∥∥∥ = O ( k−N−1 ) in L2 operator norm, for every N ∈ N, where Cj(f, g) ∈ C∞(X)G, Cj is a universal bidifferential operator of order ≤ 2j, j = 0, 1, . . . , and C0(f, g) = f · g, C1(f, g)− C1(g, f) = i{f, g}. Moreover, the star product f ∗ g = +∞∑ j=0 Cj(f, g)ν −j , f, g ∈ C∞(X)G0 , is associative. 4 Berezin transform 4.1 Vector valued reproducing kernel Here, we recall some definitions concerning reproducing kernel Hilbert spaces for vector valued functions, we refer to [3, 28] (see also [4, Section 2]). Definition 4.1. Let X be a manifold and E be a vector bundle on X. A E-valued reproducing kernel Hilbert space on X is a Hilbert space H such that (1) the elements of H are sections of the bundle E → X, (2) for all x ∈ X there exists a positive constant Cx such that ∥s(x)∥Ex ≤ Cx∥s∥H for each s ∈ H. In this article, we shall consider the case where X is CR manifold satisfying the assumption of the Introduction and H is the Hilbert space HG(X). Now, let q = n−, x ∈ X, and s ∈ HG k (X) then s(x) is in the vector space T ∗(0,q) x X which is endowed with the Hermitian metric ⟨·\|·⟩. The Hilbert space HG k (X) is a vector valued reproducing kernel Hilbert space on X with inner product (·\|·). The evaluation map evx : HG k (X) → T ∗(0,q)X, evx(s) = s(x) is a bounded operator and the reproducing kernel associated with HG k (X) is SG. The ker- nel SG reproduces the value of a section s ∈ HG k (X) at point x ∈ X. Indeed, for all x ∈ X and v ∈ T ∗(0,q) x X ev∗xv = SG k (·, x)v so that ⟨s(x)\|v⟩ = ( s, SG k (·, x)v ) . Thus, for each x ∈ X and v ∈ T ∗(0,q) x X the (0, q)-forms sx,v = SG k (·, x)v is called the coherent state (of level k) associ- ated to x and v. Definition 4.2. The covariant Berezin symbol σ (k) k (A) of an equivariant operator of Szegő type A ∈ ΨG k (X) is σ(k) v (A) : X → C, x 7→ σv(A)(x) := (sx,v\|Asx,v) (sx,v\|sx,v) for a given fixed v ∈ T ∗(0,q) x X. 16 A. Galasso Remark 4.3. The definition of coherent states goes back to Berezin. A coordinate independent version and extensions to line bundles were given by Rawnsley [30], we mainly refer to [33] where the coherent vectors are parameterized by the elements of L∗ \ 0, where L is a positive line bundle of a given Hodge manifold. In the following we shall generalize the coherent states in our setting. Let f be a G invariant function on µ−1(0), for any extension f̃ on X, we assign to it its Toeplitz operator TG k [ f̃ ] and then assign to it the covariant symbol, as in Definition 4.2 σ(k) v ( TG k [ f̃ ]) : X → C which is again an element of C∞(X)G and thus it defines a circle invariant function on the CR reduction XG. 4.2 Berezin transforms for Toeplitz operators Recall that π : µ−1(0) → XG is the projection. Given a smooth function f ∈ C∞(XG) invariant under the circle action, we define the Berezin transform of level k. Definition 4.4. The map C∞(XG) S1 → C∞(XG) S1 , f 7→ IGk [f ] := σ (k) v ( TG k [f ] ) is called the Berezin transform of level k. In particular, we are interested in studying IGk [f ] on a neighborhood U of p in µ−1(0). Theorem 4.5. Let p ∈ µ−1(0) and let U a local neighborhood of p. Then, if q = n−, there exist local coordinates (x1, . . . , x2n+1) on U such that the Berezin transform IGk [f ] evaluated at the point x ∈ U has a complete asymptotic expansion in decreasing integer powers of k IGk [f ](x) ∼ +∞∑ j=0 Ij [f ](x) 1 kj , as k goes to infinity, where Ij : C ∞(XG) S1 → C∞(XG) S1 are maps such that I0[f ] = f . Proof. Let f ∈ C∞(XG) S1 and let us first study the numerator of σ (k) v ( TG k [ f̃ ]) which is ( SG k (·, x)v ∣∣TG k [ f̃ ] S (q) k (·, x)v ) = ∫ X v†S (q) k (x, y)f̃(y)S (q) k (y, x)vdVX(y). The denominator is obtain by taking f ≡ 1. Thus, by Proposition 3.2, we have a complete asymptotic expansion in decreasing integer power of k, and we obtain( S (q) k (·, x)v ∣∣T (q) k [ f̃ ] S (q) k (·, x)v ) ∼ kdv†σ0 TG[f̃ ] (x, x)v +O ( kd−1 ) where σ0 TG[f̃ ] (x, x) = 1 2 π−n−1 \|detLx\| f̃(x)τx,n− , for every x ∈ µ−1(0). Thus, taking the quotient we get IGk [ f̃ ] (x) = f̃(x) + O ( kd−1 ) for each x ∈ µ−1(0). ■ Theorem 4.5 is consequence of the asymptotic expansion of Toeplitz operators for (0, q)-forms and the stationary phase formula. Here, we shall mainly refer to [7] where we study Toeplitz operators on CR manifolds, see also [10, 23] for the asymptotic behavior of the Bergman kernel for (0, q)-forms, and references therein. Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 17 5 Quantization commutes with reduction 5.1 Quotients and complexification of groups Let G be a compact connected Lie group and let g be its Lie algebra. Here, we recall some results proved in [9, Section 4] and we explain how to extend it to pseudo Kähler manifolds. There exists a unique connected complex Lie group, GC, with the following two properties: � its Lie algebra is g⊕ √ −1g, � G is a maximal compact subgroup of GC. Now, we shall prove some properties of G-actions on pseudo-Kähler manifolds. A polarization of X is an integrable Lagrangian subbundle, F , of TM ⊗C. The next lemma, an analogue of [9, Lemma 4.3], will be used to reconcile this definition with the standard one. Lemma 5.1. Let V be a (real) symplectic vector space with symplectic form, Ω. Let E be a Lagrangian subspace of V ⊗C Then there exists a unique linear mapping K : V → V such that (i) K2 = −I, (ii) E = { v + √ −1Kv, v ∈ V } , (iii) Ω(Kv,Kw) = Ω(v, w), (iv) The quadratic form B(v, w) = Ω(v,Kw) is symmetric and non-degenerate. Let (M,ω) be a symplectic manifold and F a polarization. By the lemma we get for each m ∈ M a mapping K = Jm : TmM → TmM with the properties (i), (ii), and (iii) and a quadratic form, B = gm, on M . Furthermore, J defines an almost-complex structure on M and g a pseudo-Riemannian structure. The integrability of F implies that the almost-complex structure is complex. Therefore, the quadruple (M,J, g, ω) is a pseudo-Kähler manifold in the usual sense. Let (M,ω) be a compact pseudo-Kähler manifold and G a compact connected Lie group which acts on M , preserving F . We shall prove Theorem 5.2 below, which is an analogue of [9, Theorem 4.4] and in fact the first part of the proof is identical. However, note that [9, Theorem 4.4] relies on the fact that the group of analytic diffeomorphisms ofM which preserve F is a finite-dimensional Lie group, see [16]. Here, we prove that the former result still holds true in the setting of pseudo-Riemannian manifolds (in Theorem 5.2, we assume that GC is simply connected, for the general case we refer to [9, Theorem 4.4]). Let Iso(X, g) be the group of isometries from (M, g) onto itself, that is f ∈ Iso(X, g) if and only if f is a smooth diffeomorphism and f∗g = g. Let ( M,T 1,0M ) be a connected complex manifold and denote by Iso(M, g) the group of isometries onM with respect to some Riemannian metric g. Let AutJ(M) be the group of complex automorphisms on M , that is f ∈ AutJ(M) if and only if f : M → M is a smooth diffeomorphism satisfying df ( T 1,0M ) ⊆ T 1,0M . Theorem 5.2. Let GC be simply-connected. The action of G can be canonically extended to an action of GC, preserving the pseudo-Kähler structure of M . Proof. Let τ : gC → (real) vector fields on M be the mapping, ξ(1) + √ −1ξ(2) 7→ ξ (1) M + Jξ (2) M where ξ(1), ξ(2) ∈ g. By [9, equations (4.2) and (4.3)], τ is a morphism of Lie algebras. Moreover, by (4.2), if η ∈ gC, τ(η) is a vector field preserving F . By Lemma 5.3 below, Iso(M, g)∩AutJ(M) is a (finite-dimensional) Lie group; therefore, if GC is simply-connected, τ can be extended uniquely to a morphism of Lie groups. ■ Lemma 5.3. The group Iso(M, g) ∩AutJ(M) is a Lie group. 18 A. Galasso Proof. By [16, 29], we have that Iso(M, g) is a finite-dimensional Lie group, since (M, g) is a pseudo-Riemannian manifold. Recall that a subgroup of a Lie group is itself a Lie group if it is a closed subgroup. Now, consider the subgroup Iso(M, g, J) ⊂ Iso(M, g), consisting of those diffeomorphisms that preserve both the metric g and the complex structure J . The preservation of g and J by a diffeomorphism are smooth conditions. Thus, we consider the set Iso(M, g, J) as the solution set of these equations within the group Iso(M, g). If a sequence of diffeomorphisms in Iso(M, g, J) converges to a diffeomorphism in the topology of Iso(M, g) (which is the compact-open topology), the limit will also preserve g and J . Thus, Iso(M, g, J) is a closed subgroup of the Lie group Iso(M, g), it is itself a Lie group. ■ Recall now that (M,ω) is the complex pseudo-Kähler manifold obtained by quotienting out by the circle action on X, M := X/S1. The CR action of G on X descends to a complex and Hamiltonian action on M with moment map Φ: M → g∗. Thus, we form the reduced space MG := Φ−1(0)/G. Let Ms be the saturation of Φ−1(0) with respect to GC, Ms := { g ◦ x \| x ∈ Φ−1(0), g ∈ GC}, the points of Ms are called stable points. In the same way as in [9, Theorem 4.5], one can prove the following theorem. Here, we shall retrace the proof again since we use the assumption that the pseudo-metric g restricted to the orbit through any m ∈ Φ−1(0) is non-degenerate. Theorem 5.4. Assume that the pseudo-metric g restricted to the orbit through any m ∈ Φ−1(0) is non-degenerate and the action of G on Φ−1(0) is free. The set Ms is an open subset of M and GC acts freely on it and MG can be represented as the quotient space MG := Ms/G C. Proof. Let V be a real symplectic vector space and F a Lagrangian subspace of the complexi- fication V ⊗ C. Let J and B as in Lemma 5.1. Since the origin in g is a regular value of Φ, then Φ−1(0) is a G-invariant co-isotropic sub- manifold of M . Moreover, the action of G on Φ−1(0) is locally free and the orbits of G are the leaves of the null-foliation. Recall that b is the nondegenerate bilinear form on HX such that b(·, ·) = dω0(·, J ·). Let us denote µ−1(0) by Y . There is a natural projection πY : Y → Φ−1(0). Recall that we assume that 0 is a regular value of µ, and that the action of G on µ−1(0) is free. Furthermore, by hypothesis the bilinear form b is nondegenerate on Y . Then, πY (gx) = Tπ(x)Φ −1(0) and{ ξM (π(x)) ∈ Tπ(x)M \| ξ ∈ √ −1g } is a complementary space to Tπ(x)Φ −1(0) in Tπ(x)M . This shows that Ms contains an open neighborhood, U , of Φ−1(0). Since Ms = ⋃ g∈GC gU, then Ms is itself open. Thus, the stabilizer algebra of x in gC is zero; so the action of GC on Ms is locally free. To show that GC acts freely on Ms one can proceed as in [9, the second part of Theorem 4.5, p. 527]. ■ 5.2 Proof of Theorem 1.5 The action of G on the line bundle, L, can be canonically extended to an action of GC on L. The proof of this fact is identical with the proof of [9, Theorem 5.1], and we will omit it. Here, Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 19 we just recall the infinitesimal action of an element η = √ −1ξ ∈ √ −1g on a harmonic L-valued (0, n−)-form s −∇ηM s− 2π⟨Φ(·), ξ⟩s. (5.1) Let s be a harmonic L-valued (0, n−)-form, and let (s, s)(m) be the norm of s. By defini- tion (s, s) is a non-negative real-valued function. By assumption, ( , ) is invariant with respect to parallel transport; so for all η = √ −1ξ ∈ √ −1g, ηM (s, s) = (∇ηM s, s)+(s,∇ηM s). Suppose now that s is GC-invariant. Then, by (5.1), ∇ηM s = −2π⟨Φ(·), ξ⟩s so ηM ⟨s, s⟩ = −4π⟨Φ(·), ξ⟩⟨s, s⟩. Now let Q(M) be the space of harmonic L-valued (0, n−)-forms and let Q(Ms) be the space of harmonic L-valued (0, n−)-forms over Ms. Let Q(M)G band Q(Ms) G. be the set of G-fixed vectors in these two spaces. The following theorem follows as in [9, Theorem 5.2]. Theorem 5.5. The canonical mapping Q(Ms) G → Q(MG) is bijective. It is clear that the restriction mapping r : Q(M)G → Q(Ms) G is injective; so, by Theorem 5.5, to prove “quantization commutes with reduction”, it is enough to prove that r is surjective. Now, we shall need the following theorem. Theorem 5.6. The set M \Ms is contained in a complex sub-manifold of M of complex codi- mension greater than one. This theorem is an analogue of [9, Theorem 5.7] for pseudo-Riemannian manifolds. Note that here the existence theorem [9, Theorem 5.6] is not true in general for the pseudo-Riemannian case: we have to replace the spaces of smooth sections of Lk with space of Lk-valued (0, n−)- forms, see Theorem 5.9. Then, Theorem 5.6 follows by combining Theorems 5.7 and 5.9 below. Theorem 5.7. Let s be a G-invariant Lk-valued (0, n−)-forms on M . Then every m ∈ M fulfilling s(m) ̸= 0 lies in Ms. Proof. See the proofs of [9, Theorems 5.3 and 5.4], they hold true in our setting up to replacing the space of G-invariant holomorphic sections of Lk with the space of G-invariant Lk-valued (0, n−)-forms. ■ Before stating Theorem 5.9, we recall the following result from [11]. Although it can be deduced from the theory of Toeplitz operators discussed in the previous sections, here we present Theorem 5.8 in terms of the Bergman kernel, rather than the Szegő kernel. As we explained in Section 2, we identify the G-invariant Bergman projector P (q),G k of Lk with the k-th Fourier component of the G-invariant Szegő kernel S (q) G,k. Let us premise a further piece of notation. We will identify the curvature form RL with the Hermitian matrix ṘL ∈ C∞(M,End ( T 1,0M )) , 〈 RL(z), U ∧ V 〉 = 〈 ṘL(z)U \|V 〉 , for every U and V in T 1,0 z M , z ∈ M . We denote by W the subbundle of rank q = n− of T 1,0M generated by the eigenvectors corresponding to negative eigenvalues of ṘL. Then, detW ∗ := ΛqW ∗ is a rank one subbundle, where W ∗ is the dual bundle of the complex conjugate bundle of W and ΛqW ∗ is the vector space of all finite sums v1 ∧ · · · ∧ vq, v1, . . . , vq ∈ W ∗ . We denote by PdetW ∗ the orthogonal projection from T ∗0,qM onto detW ∗ . Let s and s1 be local holomorphic trivializing sections of L defined on open sets D ⊂ M and D1 ⊂ M , respectively, \|s\|2hL = e−2ϕ, \|s1\|2hL = e−2ϕ1 , ϕ ∈ C∞(D,R), ϕ1 ∈ C∞(D1,R) which can be assumed to be G-invariant. The localization of P (q),G k with respect to s, s1 is given by P (q),G k,s,s1 : Ω0,q c (D) → Ω0,q(D1), u → s−k 1 e−kϕ1 ( P (q),G k ( skekϕu )) . 20 A. Galasso Let P (q),G k,s,s1 (x, y) ∈ C∞(D1 ×D,T ∗0,qM ⊠ ( T ∗0,qM )∗) be the distribution kernel of P (q),G k,s,s1 . When D = D1, s = s1, we write P (q),G k,s := P (q),G k,s,s , P (q),G k,s (x, y) := P (q),G k,s,s (x, y). Theorem 5.8. With the notations and assumptions used above and recall that we let q = n−. Let s be a local holomorphic trivializing section of L defined on an open set D ⊂ M , \|s\|2hL = e−2ϕ. Let D be an open set of X with D∩Φ−1(0) = ∅. Then, as k → +∞, P (q),G k,s = O(k−∞) on D. Let m ∈ Φ−1(0) and let U be an open set of m. Then, as k → +∞, in local coordinates defined in U (as introduced in [11]), P (q),G k,s (x, y) = eikΨ(x,y)b(x, y, k) +O(k−∞) on D ×D, where b ∈ S n−d/2 cl ( 1;D ×D,T ∗0,qM ⊠ ( T ∗0,qM )∗) , b(x, y, k) ∼ ∑ j=0 bj(x, y)k n−j in S n−d/2 cl ( 1;D ×D,T ∗0,qM ⊠ ( T ∗0,qM )∗) , and bj ∈ C∞(D ×D,T ∗0,qM ⊠ ( T ∗0,qM )∗) , b0 = b0(m)PdetW ∗ with b0(m) > 0 and j = 0, 1, . . . , and we refer to [11, Theorem 1.8] for the properties of the phase function Ψ ∈ C∞(D ×D). Theorem 5.9. If the set Φ−1(0) is non-empty and zero is a regular value of Φ, then for some k, there exists a global nonzero holomorphic G-invariant Lk-valued (0, n−)-forms. Proof. Let S0, . . . , Sdk be an orthonormal basis of the vector space Hq ( M,L⊗k )G . In the standard situation, when the line bundle is positive, we consider q = 0. Let m ∈ Φ−1(0); since all the components transform by the same scalar under a change of frame and since P (0),G k (m,m) = dk∑ j=0 \|Sj(m)\|2 hL⊗k = O ( kn−d/2 ) , it is easy to see that the sections Sj do not share common zeros. In the pseudo-Riemannian setting, we assume q = n−. Let m ∈ Φ−1(0), in view of Theo- rem 5.8, we have P (q) k (m,m) = b0(m)PdetW ∗kn−d/2 +O ( kn−d/2−1 ) , (5.2) with b0(m) > 0. By (5.2), it follows that the sections Sj do not share common zeros. Since M is compact, there exists k0 such that P (n−),G k (m,m) for all k ≥ k0 and m ∈ M . Thus, by equation (5.2), there do not exist m ∈ M such that for each s in Qk(M)G, s(m) = 0 for all k > 0. In particular, since Φ−1(0) is non-empty, then for some k, there exists a global non-zero holomorphic G-invariant Lk-valued (0, n−)-forms. ■ “Quantization commutes with reduction” follows as in [9] by combining [9, Theorem 5.7] (the analogous result here is Theorem 5.6) and the following analogue of [9, Theorem 5.8]. Theorem 5.10. Let s be a vector in Qk(M)G. Then ⟨s, s⟩ is bounded and takes its maximum value on Φ−1(0). Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 21 Proof. Let m be a point of Ms. Then m = gm0 with m0 ∈ Φ−1(0) and g ∈ GC. By the Cartan decomposition of GC, we have GC = PG, where G is a maximal compact subgroup of GC and the Lie algebra of P is √ −1g. Thus we write g = eηk with η = √ −1ξ ∈ √ −1g and k ∈ G. Replacing m0 with km0, we can assume that m = eηm0. Consider the curve γ : (−∞,+∞) → M , t 7→ γ(t) = etηm0. Hence, by the formula ηM ⟨s, s⟩ = −4π⟨Φ(·), ξ⟩⟨s, s⟩ we have, along γ(t), d dt⟨s, s⟩ = −4π⟨Φ(·), ξ⟩⟨s, s⟩. As in [9, Lemma 4.7], we note that ηM is the gradient vector field associated with the function Φξ := ⟨Φ(·), ξ⟩. Thus Φξ is strictly increasing along γ(t), so it is positive for t > 0 and negative for t < 0. Therefore, ⟨s, s⟩ has a unique maximum mum at t = 0. ■ Finally, we note that Theorem 5.10 implies the surjectivity of r. Indeed, if m is a point of M , then we can find a neighborhood U of m in M and a non-zero holomorphic G-invariant Lk-valued (0, n−)-forms, s0 : U → Lk ⊗ T ∗0,n−M . Then s = fs0 on U ∩Ms, f being a bounded holomorphic function. Since (M \Ms)∩U is of complex codimension greater than one in U , the singularity of f at m is removable. Thus s extends to a holomorphic section of Lk ⊗ T ∗0,n−M over all of M . 6 Proof of Theorem 2.3 Let f̃ ∈ C∞(X) be a G and S1 smooth invariant function on X. First, we prove that∥∥IGk [f ] ∥∥ ∞ ≤ ∥∥TG k [ f̃ ]∥∥ ≤ ∥f∥∞. (6.1) Recall that given two G invariant (0, n−) forms s and t in H(X)Gk , we denote their inner product by (s\|t). Now, using the Cauchy–Schwartz inequality, we obtain that for each ele- ment x ∈ µ−1(0) ∥∥IGk [f ](x) ∥∥2 ∞ = ∣∣(SG k (·, x)v ∣∣TG k [f ]SG k (·, x)v )∣∣2( SG k (·, x)v ∣∣SG k (·, x)v )2 ≤ ( TG k [f ]SG k (·, x)v ∣∣TG k [f ]SG k (·, x)v )( SG k (·, x)v ∣∣SG k (·, x)v ) ≤∥TG k [f ]∥2, where the last inequality follows by the definition of the operator norm, see (2.3). Taking the sup both sides over µ−1(0), it shows the first inequality in (6.1). For the second inequality, we note that for s ∈ HG k (X), ∥TG k [f ]s∥2 ∥s∥2 = ( TG k [f ]s ∣∣TG k [f ]s )∫ X⟨s, s⟩dVX = ∥f∥2∞ ( SG k [f ]s ∣∣SG k [f ]s )∫ X⟨s, s⟩dVX ≤ ∥f∥2∞. Hence, it proves the second inequality (6.1). Now, we are ready to prove the theorem. Since X is compact, then µ−1(0) is closed and hence compact. Choose xe ∈ µ−1(0) a point with ∣∣f̃(xe)∣∣ = ∥f∥∞. Since the Berezin transform has as a leading term the identity, we have ∣∣(IGk [f ] ) (xe)− f̃(xe) ∣∣ ≤ C k for a suitable C > 0. Thus, ∥f∥∞ − C k = ∣∣f̃(xe)∣∣− C k ≤ ∣∣(IGk [f ] ) (xe) ∣∣ ≤ ∥∥IGk [f ] ∥∥ ∞. Putting (6.1) in this last inequality, we get part (a) of Theorem 2.3. Parts (b) and (c) of Theorem 2.3 are consequence of Theorem 3.5. 22 A. Galasso 7 Proof of Theorem 1.3 It is an easy consequence of [7] that the star product defined by Toeplitz operators is of Wick type. Let f and g be two G and S1 invariant smooth function on XG. For ease of notation, we write TG k [f ] for TG k [ f̃ ] . It is clear by Theorem 2.3 and [7] that ⋆ is a well-defined associative star product. In fact, by part (b) of Theorem 2.3, we get∥∥TG k [{f, g} − i(C1(f, g)− C1(g, f))] ∥∥ = O ( 1 k ) , and taking the limit for k → +∞ and using part (a) we get ∥{f, g}− i(C1(f, g)−C1(g, f))∥∞ = 0 and thus we get part (b). Similarly, by Theorem 2.3 (c) and (a), we get that C0(g, h) = g · h. In a similar way the associativity property is proved. Now, we prove the uniqueness. Let Cj and C̃j be two systems of bi-differential operators inducing star products and both fulfilling the asymptotic condition (1.1). We show that Cj = C̃j , for all j ∈ N0, hence the induced star products coincide. Note C0 = C̃0 by hypothesis. By induction, assume that Cj = C̃j for each j ≤ N − 2. By the results in [7], we have TG k [f ] ◦ TG k [g] ∼ N−1∑ j=0 k−jTG k [Cj(f, g)] +O ( k−N ) and TG k [f ] ◦ TG k [g] ∼ N−1∑ j=0 k−jTG k [ C̃j(f, g) ] +O ( k−N ) , where ∼ stand for “as the same asymptotic as”. For every f, g ∈ C∞(XG) S1 , by the inductive hypothesis and subtracting these two expression, we obtain∥∥∥∥ 1 kN−1 TG k [ CN−1(f, g)− C̃N−1(f, g) ]∥∥∥∥ ≤ K kN . Eventually, taking the limit on both sides as k goes to infinity and applying Theorem 2.3 (a), we obtain CN−1(f, g) = C̃N−1(f, g). Now we recall that AN = R̂lTG[f ]TG[g]− N−1∑ j=0 R̂N−jTG[Cj(f, g)] which is an operator of Szegő type of order zero by Theorem 3.4 and it is invariant under the circle action. Thus, its symbol is a G and S1 invariant function on XG which, by restriction on µ−1(0), defines a S1-invariant function on XG. By definition, it is the next element CN (f, g) in the star-product. The unit of the algebra C∞(XG) S1 , the constant function 1, will also be the unit in the star-product. In fact, it is equivalent to Cj(1, f) = Cj(f, 1) = 0 (7.1) for j ≥ 1 and for every f ∈ C∞(XG) S1 . First, by Theorem 3.2 note that C0(1, g) = C0(1, g) = g for every k. Now, if we put g = 1, A1 = R̂TG[f ]TG[g]− R̂TG[f · g] = 0, and hence the symbol of A1 vanishes. By induction, the claim (7.1) follows. Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions 23 Let us now prove that the Berezin–Toeplitz star product fulfills parity. Considering the formal parameter to be real, ν = ν, this is equivalent to Ck(f, g) = Ck(g, f) (7.2) for k ≥ 0. Note that, as an easy consequence of the definition of Berezin–Toeplitz star product, we have TG k [f ]† = TG k [f ]. The star product g ⋆ f is given by the asymptotic expansion of the composition TG k [g] · TG k [ f ] = TG k [g]† · TG k [f ]† = ( TG k [f ] · TG k [g] )† ∼ +∞∑ j=0 k−jTG k [Cj(f, g)] † ∼ +∞∑ j=0 k−jTG k [ Cj(f, g) ] . Thus, the claim (7.2) follows. By [7], see also [8], the trace Trk on H(X)Gk admits a complete asymptotic expansion in decreasing integer power of k Trk ( TG k [f ] ) ∼ kd ( +∞∑ k=0 k−jτj(f) ) , with τj(f) ∈ C. It induces a C[[ν]]-linear map Tr: C∞(XG) S1 [[ν]] → ν−nC[[ν]] such that Tr(f) := +∞∑ j=0 νj−nτj(f), where for f ∈ C∞(M) the τj(f) are given by the asymptotic expansion above and for arbitrary elements by C[[ν]]-linear extension, ν = k−1. Now, we shall prove Tr(f ⋆ g) = Tr(g ⋆ f). (7.3) By C[[ν]]-linearity, we prove (7.3) for f and g in C∞(XG) S1 . Note that f ⋆ g − g ⋆ f is given by the asymptotic expansion of TG k [f ]TG k [g] − TG k [g]TG k [f ]. Hence the trace of f ⋆ g − g ⋆ f is given by the expansion of Trk ( TG k [f ]TG k [g] − TG k [g]TG k [f ] ) . 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Pure Math., Vol. 84, American Mathematical Society, Providence, RI, 2012, 299– 339, arXiv:1107.0463. https://iris.sissa.it/handle/20.500.11767/134510 https://doi.org/10.1007/978-94-015-1276-3_22 http://arxiv.org/abs/math.QA/9910137 https://doi.org/10.1155/2010/927280 https://doi.org/10.1155/2010/927280 http://arxiv.org/abs/1003.2523 https://doi.org/10.1007/s002220050223 https://doi.org/10.1215/S0012-7094-96-08206-X https://doi.org/10.1215/S0012-7094-96-08207-1 https://doi.org/10.4171/emss/31 http://arxiv.org/abs/1901.11327 https://doi.org/10.1090/pspum/084/1363 http://arxiv.org/abs/1107.0463 1 Introduction 2 Background and statement of the results 2.1 Examples 2.2 Toeplitz operators in the pseudo-Kähler setting 2.3 CR formulation 2.4 Group action framework 2.5 A key result on Toeplitz operators 3 Preliminaries 3.1 Standard notations 3.2 Operators and symbols 3.3 Toeplitz operators 4 Berezin transform 4.1 Vector valued reproducing kernel 4.2 Berezin transforms for Toeplitz operators 5 Quantization commutes with reduction 5.1 Quotients and complexification of groups 5.2 Proof of Theorem 1.5 6 Proof of Theorem 2.3 7 Proof of Theorem 1.3 References
id	nasplib_isofts_kiev_ua-123456789-213528
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-15T20:34:39Z
publishDate	2025
publisher	Інститут математики НАН України
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spelling	Galasso, Andrea 2026-02-18T11:25:48Z 2025 Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions. Andrea Galasso. SIGMA 21 (2025), 048, 25 pages 1815-0659 2020 Mathematics Subject Classification: 32V20; 32A25; 53D50 arXiv:2410.03322 https://nasplib.isofts.kiev.ua/handle/123456789/213528 https://doi.org/10.3842/SIGMA.2025.048 The asymptotic results for Berezin-Toeplitz operators yield a strict quantization for the algebra of smooth functions on a given Hodge manifold. It seems natural to generalize this picture for quantizable pseudo-Kähler manifolds in the presence of a group action. Thus, in this setting, we introduce a Berezin transform which has a complete asymptotic expansion on the preimage of the zero set of the moment map. It leads in a natural way to proving that certain quantization maps are strict. We are indebted to the referees for various interesting comments and for suggesting several improvements. The author thanks Chin-Yu Hsiao and Herrmann Hendrik for their valuable conversations. The author is a member of GNSAGA, part of the Istituto Nazionale di Alta Matematica, and expresses gratitude to the group for its support. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions Article published earlier
spellingShingle	Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions Galasso, Andrea
title	Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions
title_full	Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions
title_fullStr	Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions
title_full_unstemmed	Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions
title_short	Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions
title_sort	strict quantization for compact pseudo-kähler manifolds and group actions
url	https://nasplib.isofts.kiev.ua/handle/123456789/213528
work_keys_str_mv	AT galassoandrea strictquantizationforcompactpseudokahlermanifoldsandgroupactions

Strict Quantization for Compact Pseudo-Kähler Manifolds and Group Actions

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