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

Abstract

We define conic reductions $X^{\textrm{red}}_{\nu }$ for torus actions on the boundary X of a strictly pseudo-convex domain and for a given weight $\nu$ labeling a unitary irreducible representation. There is a natural residual circle action on $X^{\textrm{red}}_{\nu }$. We have two natural decompositions of the corresponding Hardy spaces H(X) and $H(X^{\textrm{red}}_{\nu })$. The first one is given by the ladder of isotypes $H(X)_{k\nu }$, $k\in {\mathbb {Z}}$; the second one is given by the k-th Fourier components $H(X^{\textrm{red}}_{\nu })_k$ 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 $L^2$-coefficient when X is a CR manifold with non-degenerate Levi form.

1 Introduction

Let X be the boundary of a strictly pseudo-convex domain D in ${\mathbb {C}}^{n+1}$. Then $(X,\,T^{1,0}X)$ is a contact manifold of dimension $2n+1,\, n\ge 1$, where $T^{1,0}X$ is the sub-bundle of $TX\otimes {\mathbb {C}}$ defining the CR structure. We denote by $\omega _0\in {\mathcal {C}}^{\infty }(X,T^*X)$ the contact 1-form whose kernel is the horizontal bundle $HX\subset TX$; we refer to Sect. 2.1 for definitions. Associated with these 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 ${\mathbb {T}}$ is given; we denote by $\mu : X\rightarrow {\mathfrak {t}}^*$ the associated CR moment map. Fix a weight $i\,\nu$ in the lattice $i\,{\mathbb {Z}}^t\subset {\mathfrak {t}}^*$, if $0\in {\mathfrak {t}}^*$ does not lie in the image of the moment map, the isotypes

$$\begin{aligned}H(X)_{k\nu }=\{f\in H(X):\, (e^{i\, \theta }\cdot f)(x) = e^{ik\,\langle \nu ,\theta \rangle }f(x),\, \theta \in {\mathbb {R}}^t \}, \quad k\in {\mathbb {Z}}, \end{aligned}$$

are finite dimensional.

Suppose that the ray $i\,{\mathbb {R}}_+\cdot \nu \in {\mathfrak {t}}^*$ is transversal to $\mu$, then $X_{\nu }:=\mu ^{-1}(i\,{\mathbb {R}}_+\cdot \nu )$ is a sub-manifold of X of codimension $t-1$. There is a well-defined locally free action of ${\mathbb {T}}^{t-1}_{\nu }:=\exp _{{\mathbb {T}}}(i\,\ker \nu )$ on $X_{\nu }$; the resulting orbifold $X^{\textrm{red}}_{\nu }$ is called conic reduction of X with respect to the weight $\nu$. Let $\varphi$ be an Euclidean product on ${\mathfrak {t}}$, we shall also use the symbol $\langle \cdot ,\,\cdot \rangle$ and denote by ${\lambda }^{\varphi }\in {\mathfrak {t}}$ be uniquely determined by ${\lambda }=\varphi ({\lambda }^{\varphi },\,\cdot )$ and $\Vert {\lambda }\Vert$ the corresponding norm. By abuse of notation we write $\lambda$ for ${\lambda }^{\varphi }$ and we identify ${\mathfrak {t}}\cong i {\mathbb {R}}^t$ with its dual. We set

$$\begin{aligned}\ker \nu = \nu ^{\perp }:=\left\{ {\lambda }\in {\mathfrak {t}}:\,\langle \nu ,\, {\lambda } \rangle =0 \right\} .\end{aligned}$$

The locus $X_{\nu }$ is ${\mathbb {T}}$-invariant; we will always assume that the action of ${\mathbb {T}}$ on $X_{\nu }$ is locally free. After replacing ${\mathbb {T}}$ with its quotient by a finite subgroup, we may and will assume without loss of generality that the action is generically free. In Sect. 2.1, we show that $X^{\textrm{red}}_{\nu }$ is CR manifold with positive definite Levi form of dimension $2n-2t+3$.

Let us define ${\mathbb {T}}^1_{\nu }:=\exp _{{\mathbb {T}}}(i\,\nu )$, if $\nu$ is coprime, we have a Lie group isomorphism

$$\begin{aligned}\kappa _{\nu }:\,S^1\rightarrow {\mathbb {T}}^1_{\nu },\quad e^{i\theta }\mapsto e^{i\theta \nu }\end{aligned}$$

between ${\mathbb {T}}^1_{\nu }:=\exp _{{\mathbb {T}}}(i\,\nu )$ and the circle $S^1$. Let us denote by

$$\begin{aligned} \overline{{\mathbb {T}}^1_{\nu }}:={\mathbb {T}}/ {\mathbb {T}}^{t-1}_{\nu }\cong {\mathbb {T}}^1_{\nu }/ ({\mathbb {T}}^1_{\nu }\cap {\mathbb {T}}^{t-1}_{\nu }), \end{aligned}$$

then the character $\chi _{\nu }\,:\,{\mathbb {T}}\rightarrow S^1,\,\chi _{\nu }(e^{i\,\theta }):=e^{ik\,\langle \nu ,\,\theta \rangle }$, being trivial on ${\mathbb {T}}^{t-1}_{\nu }$, descends to a character $\chi _{\nu }':\,\overline{{\mathbb {T}}^1_{\nu }}\rightarrow S^1$ which is a Lie group isomorphism, see [14, Lemma 10]. Thus, we have a locally free circle action of $\overline{{\mathbb {T}}^1_{\nu }}$ on $X^{\textrm{red}}_{\nu }$, which induces an action on the Hardy space $H(X^{\textrm{red}}_{\nu })$. Suppose that the action of $\overline{{\mathbb {T}}^1_{\nu }}$ on X is transversal to the CR structure. We denote by $H(X^{\textrm{red}}_{\nu })_k$ the corresponding k-th Fourier component, and we call the action of $\overline{{\mathbb {T}}^1_{\nu }}$ on $X^{\textrm{red}}_{\nu }$ residual circle action. The aim of this paper is to prove that $H(X)_{k\nu }$ and $H(X^{\textrm{red}}_{\nu })_k$ 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 $2n+1$, $n\ge 1$, and let $\omega$ 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 ${\mathbb {T}}$ preserves the contact form $\omega _0$ and the complex structure J. That is, $g^*\omega _0=\omega _0$ on X and $g_*J=Jg_*$ on the horizontal bundle HX for every $g\in {\mathbb {T}}$ where $g^*$ and $g_*$ denote the pull-back map and push-forward map of ${\mathbb {T}}$, respectively.

Let $X^{\textrm{red}}_{\nu }:=X_{\nu }/{\mathbb {T}}^{t-1}_{\nu }$, defined in the same way as before, more precisely we shall assume

Assumption 1.3

The moment map $\mu$ is transverse to the ray $i\,{\mathbb {R}}_+\cdot \nu \in {\mathfrak {t}}^*$, the action of ${\mathbb {T}}$ on $X_{\nu }$ is locally free and for every $x\in X_{\nu }$

$$\begin{aligned}\textrm{val}_x(\nu ^{\perp })\cap \textrm{val}_x(\nu ^{\perp })^{\perp _b}=\{0\} \end{aligned}$$

where b is a bilinear form on $H_xX$ such that

$$\begin{aligned} b(\cdot ,\, \cdot ) = \textrm{d}\omega _0(\cdot ,\, J\cdot ) \end{aligned}$$

(1)

and it is non-degenerate.

We note that $b(U,\,V)=2\,L(U,\,V)$ for every $U,\,V\in HX$. By assumptions above, we will show that $X^{\textrm{red}}_{\nu }$ is a CR manifold with natural CR structure induced by $T^{1,0}X$ of dimension $2n - 2(t-1) + 1$. Let $L_{X^{\textrm{red}}_{\nu }}$ be the Levi form on $X^{\textrm{red}}_{\nu }$ induced naturally from the Levi form L on X. For a given subspace ${\mathfrak {s}}$ of ${\mathfrak {t}}$, we denote ${\mathfrak {s}}_X$ the subspace of infinitesimal vector fields on X. Let us consider

$$\begin{aligned}B=\ker {\nu }_X\oplus J\ker {\nu }_X.\end{aligned}$$

Hence, b has constant signature on $B\times B$, suppose b has r negative eigenvalues on $B\times B$ where $r \le n_-$ since L and b have the same number of negative eigenvalues on HX. Fix $q=n_-$; hence, by Lemma 2.1, $L_{X^{\textrm{red}}_{\nu }}$ has $q-r$ negative eigenvalues at each point of ${X^{\textrm{red}}_{\nu }}$. We refer to Sect. 2.1 for definitions; we have:

Theorem 1.1

Suppose that $\Box ^q_b$ has $L^2$ closed range. Fix a maximal coprime weight $\nu \ne 0$ in the lattice inside ${\mathfrak {t}}^*$ and assume that the circle action $\overline{{\mathbb {T}}^1_{\nu }}$ is a transversal CR action. Fix $q=n_-$, under the assumptions above, $X^{\textrm{red}}_{\nu }$ is a compact CR manifold with non-degenerate Levi form having $q-r$-negative eigenvalues. There is a natural isomorphism of vector spaces $\sigma _k\,:\, H^q(X)_{k\nu }\rightarrow H^{q-r}(X^{\textrm{red}}_{\nu })_k$ 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 Sect. 3, which is inspired from [7] (see [8] for the full extension of [7]), and it is a consequence of the microlocal properties of the projector $S^{(q)}_{k\nu }$ described in Sect. 2.2 and calculus of Fourier integral operators of complex type, see [11]. Furthermore, we recall that the way to establish the isometry from kernel expansion for k large comes from [10].

The conic reductions defined above appear naturally in geometric quantization. In fact, given a Hamiltonian and holomorphic action with moment map $\Phi$ of a compact Lie group G on a Hodge manifold $(M,\omega )$ with quantizing circle bundle $\pi :X\rightarrow M$, one can always define an infinitesimal action of the Lie algebra ${\mathfrak {g}}$ 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 [5] it was observed that associated to group actions one can define reductions $M_{\textrm{red}}^{\Theta }$ by pulling back a G-invariant and proper sub-manifold $\Theta$ of ${\mathfrak {g}}^*$ via the moment map $\Phi$. When $\Theta$ is chosen to be a cone through a co-adjoint orbit $C({\mathcal {O}}_{\nu })$, one has associated reduction whose Hardy space of its quantization is

$$\begin{aligned} H(X_{\textrm{red}}^{C({\mathcal {O}}_{\nu })})= \bigoplus _{k\in {\mathbb {Z}}} H(X_{\textrm{red}}^{C({\mathcal {O}}_{\nu })})_k, \end{aligned}$$

where k labels an irreducible representation of the residual circle action described above. Now, it is natural to ask if this spaces are related to the decomposition induced by G on H(X). When $G={\mathbb {T}}$ Theorem 1.1 states that they are isomorphic to $H(X)_{k\nu }$; 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 $k\,\nu$, $k=0,1,2,\dots$, labeling unitary irreducible representations.

Further geometrical motivations for this theorem are explained in paper [14], where it is proved that $\delta _k$ 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 [14] to compact “quantizable” pseudo-Kähler manifolds. We also refer to [12] for examples and the explicit expression of the leading term of the asymptotic expansion of $\dim H(X)_{k\nu }$ as k goes to infinity. Along this line of research in [13] Toeplitz operators were studied for circle action; in [1] the study of asymptotics of compositions of Toeplitz operators with quantomopomorphism is addressed for torus actions.

2 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 $2n+1$, $n\ge 1$, where $T^{1,0}X$ is a CR structure of X. There is a unique sub-bundle HX of TX such that $HX\otimes {\mathbb {C}}=T^{1,0}X \oplus T^{0,1}X$. Let $J:HX\rightarrow HX$ be the complex structure map given by $J(u+\overline{u})=i u-i\overline{u}$, for every $u\in T^{1,0}X$. By complex linear extension of J to $TX\otimes {\mathbb {C}}$, 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 $\omega _0\in {\mathcal {C}}^{\infty }(X,T^*X)$ so that $\langle \,\omega _0(x),\,u\,\rangle =0$, for every $u\in H_xX$, for every $x\in X$; $\omega _0$ is called contact form and it naturally defines a volume form on X. For each $x \in X$, we define a quadratic form on HX by

$$\begin{aligned}{L}_x(U,V) =\frac{1}{2}\textrm{d}\omega _0(JU, V),\qquad \forall \ U, V \in H_xX.\end{aligned}$$

Then, we extend L to $HX\otimes {\mathbb {C}}$ by complex linear extension; for $U, V \in T^{1,0}_xX$,

$$\begin{aligned} {L}_x(U,{\overline{V}}) = \frac{1}{2}\,\textrm{d}\omega _0(JU, {\overline{V}}) = -\frac{1}{2i}\,\textrm{d}\omega _0(U,{\overline{V}}). \end{aligned}$$

The Hermitian quadratic form ${L}_x$ on $T^{1,0}_xX$ is called Levi form at x. In the case when X is the circle bundle of an Hodge manifold $(M,\omega )$, the positivity of $\omega$ implies that the number of negative eigenvalues of the Levi form is equal to n. The Reeb vector field $R\in {\mathcal {C}}^\infty (X,TX)$ is defined to be the non-vanishing vector field determined by

$$\begin{aligned} \omega _0(R)\equiv 1,\quad \textrm{d}\omega _0(R,\cdot )\equiv 0\ \ \hbox { on}\ TX. \end{aligned}$$

Fix a smooth Hermitian metric $\langle \, \cdot \,|\, \cdot \,\rangle$ on ${\mathbb {C}}TX$ so that $T^{1,0}X$ is orthogonal to $T^{0,1}X$, $\langle \, u \,|\, v \,\rangle$ is real if u, v are real tangent vectors, $\langle \,R\,|\,R\,\rangle =1$ and R is orthogonal to $T^{1,0}X\oplus T^{0,1}X$. For $u \in {\mathbb {C}}TX$, we write $|u|^2:= \langle \, u\, |\, u\, \rangle$. 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 ${\mathbb {C}}T^*X$.

Assume that X admits an action of t-dimensional torus ${\mathbb {T}}$. In this work, we assume that the ${\mathbb {T}}$-action preserves $\omega _0$ and J; that is, $t^*\omega _0=\omega _0$ on X and $t_*J=Jt_*$ on HX. Let ${\mathfrak {t}}$ denote the Lie algebra of T, we identify ${\mathfrak {t}}$ with its dual ${\mathfrak {t}}^*$ by means of the scalar product $\langle \cdot ,\cdot \rangle$. For any $\xi \in {\mathfrak {t}}$, we write $\xi _X$ to denote the vector field on X induced by $\xi$. The moment map associated to the form $\omega _0$ is the map $\mu :X \rightarrow {\mathfrak {t}}^*$ such that, for all $x \in X$ and $\xi \in {\mathfrak {t}}$, we have

$$\begin{aligned} \langle \mu (x), \xi \rangle = \omega _0(\xi _X(x)). \end{aligned}$$

(2)

Fix a maximal weight $\nu \ne 0$ in the lattice inside ${\mathfrak {t}}^*$. Suppose that $i{\mathbb {R}}_+\cdot \nu$ is transversal to $\mu$, so we have

$$\begin{aligned} {\mathfrak {t}}^*= i{\mathbb {R}}_+\cdot \nu \oplus \textrm{d}_p\mu (T_pX) \end{aligned}$$

(3)

and $X_{\nu }$ is a sub-manifold of X of codimension $2n+2-t$. We claim that the action of ${\mathbb {T}}^{t-1}_{\nu }$ on $X_{\nu }$ is locally free. In fact, by the contrary suppose that there exists $\xi \in \ker {\nu }$ such that $\xi _X(x)=0$ on $T_xX_\nu$, $x\in X_{\nu }$. For each $v\in T_xX$, we have

$$\begin{aligned}(\textrm{d}_x\mu (v))(\xi )=\textrm{d}_x\omega _0(\xi _X,v)=0 \end{aligned}$$

which contradicts (3).

The action of ${\mathbb {T}}$ restricts to an action of ${\mathbb {T}}^{t-1}$ whose moment map is given by

$$\begin{aligned}\mu _{\vert {\mathbb {T}}^{t-1}}= p_{\nu }\circ \mu ,\quad \text { where } p_{\nu }:\, {\mathfrak {t}}^* \rightarrow {\mathfrak {t}}_{\nu }^* \end{aligned}$$

is the canonical projection onto the $t-1$-dimensional subspace $\nu ^{\perp }$ in ${\mathfrak {t}}^*$. The transversality condition in 1.2 implies that $0\in {\mathfrak {t}}_{\nu }^{t-1}$ is a regular value for the moment $\mu _{\vert {\mathbb {T}}^{t-1}}$. Thus, we have

$$\begin{aligned}X_{\nu }/{\mathbb {T}}^{t-1}_{\nu }=\mu _{\vert {\mathbb {T}}^{t-1}}^{-1}(0)/{\mathbb {T}}^{t-1}_{\nu }\end{aligned}$$

and by assumptions 1.2, 1.3 and [7, Section 2.5] (we shall also refer to [3] for definitions concerning CR structures on orbifolds) we have

Lemma 2.1

The space of orbits $X_{\nu }/{\mathbb {T}}^{t-1}_{\nu }$ is a CR orbifold. Let us denote by $\pi : X_{\nu } \rightarrow X^{\textrm{red}}_{\nu }$ and $\iota : X_{\nu } \hookrightarrow X$ the natural projection and inclusion, respectively, then there is a unique induced contact form $\omega _0^{\textrm{red}}$ on $X_{\nu }/{\mathbb {T}}^{t-1}_{\nu }$ such that

$$\begin{aligned}\pi ^{*}\omega _0^{\textrm{red}}=\iota ^*\omega _0. \end{aligned}$$

In particular, set $HX_{\nu }= TX_{\nu }\cap HX$, we have

$$\begin{aligned}HX= HX_{\nu }\oplus J\,i \nu ^{\perp }_X \quad \text { and }\quad HX_{\nu } = i \nu ^{\perp }_X\oplus \textrm{d}\pi ^{*}HX^{\textrm{red}}_{\nu }. \end{aligned}$$

We will also assume that $\overline{{\mathbb {T}}^1_{\nu }}$-action is transversal CR, that is, the infinitesimal vector field

$$\begin{aligned} (\nu _X u)(x)=\frac{\partial }{\partial t}\left( u(\exp (i t\,\nu )\circ x)\right) |_{t=0},\quad \text {for any }u\in C^\infty (X), \end{aligned}$$

preserves the CR structure $T^{1,0}X$, so that $\nu _X$ and $T^{1,0}X\oplus T^{0,1}X$ generate the complex tangent bundle to X,

$$\begin{aligned} {\mathbb {C}}T_xX={\mathbb {C}}\nu _X(x)\oplus {\mathbb {C}}T_x^{1,0}X\oplus {\mathbb {C}}T_x^{0,1}X \qquad (x\in X). \end{aligned}$$

We define local coordinates that will be useful later. Recall that X admits a CR and transversal $\overline{{\mathbb {T}}^1_{\nu }}$-action which is locally free on $X_{\nu }$, $T\in {\mathcal {C}}^{\infty }(X,\,TX)$ denotes the global real vector field given by this infinitesimal circle action. We will take T to be our Reeb vector field R. In a similar way as in Theorem 3.6 in [7], there exist local coordinates $v=(v_1,\dots ,v_{t-1})$ of ${\mathbb {T}}^{t-1}_{\nu }$ in a small neighborhood $V_0$ of the identity e with $v(e)=(0,\,\dots ,\,0)$, local coordinates $x=(x_1\,\dots ,x_{2n+1})$ defined in a neighborhood $U_1\times U_2$ of $p\in X_{\nu }$, where $U_1\subseteq {\mathbb {R}}^{t-1}$ (resp. $U_{2}\subseteq {\mathbb {R}}^{2n+2-t}$) is an open set of $0\in {\mathbb {R}}^{t-1}$ (resp. $0\in {\mathbb {R}}^{2n+2-t}$) and $p\equiv 0\in {\mathbb {R}}^{2n+1}$, and a smooth function $\gamma =(\gamma _1,\dots ,\,\gamma _{t-1})\in {\mathcal {C}}^{\infty }(U_2,U_1)$ with $\gamma (0)=0$ such that

$$\begin{aligned}&(v_1,\dots ,v_{t-1})\circ (\gamma (x_{t},\dots ,x_{2n+1}),x_{t},\dots ,x_{2n+1}) \\&=(v_1+\gamma _1(x_{t},\dots ,x_{2n+1}),\dots ,\,v_{t-1}+\gamma _d(x_{t},\dots ,x_{2n+1}),\,x_{t},\dots ,x_{2n+1}) \end{aligned}$$

for each $(v_1,\dots ,\,v_{t-1})\in V_0$ and $(x_{t},\dots ,x_{2n+1})\in U_2$. Furthermore, we have

$$\begin{aligned}{\mathfrak {t}}=\textrm{span}\left\{ \partial _{{x}_{j}}\right\} _{j=1,\dots , t-1},\quad \mu ^{-1}(i\,{{\mathbb {R}}}_{\nu }\cdot \nu )\cap U =\{x_{2d-t+1} = \dots =x_{2d}=0 \}, \end{aligned}$$

on $\mu ^{-1}(i\,{{\mathbb {R}}}_{\nu }\cdot \nu )\cap U$ there exist smooth functions $a_j$’s with $a_j(0)=0$ for every $0\le j\le t-1$ and independent on $x_1,\dots ,x_{2(t-1)},\,x_{2n+1}$ such that

$$\begin{aligned}J\left( \partial _{{x}_{j}}\right) =\partial _{{x}_{t-1+j}}+a_j(x)\partial _{{x}_{2n+1}}\qquad j=1,\dots ,t-1, \end{aligned}$$

the Levi form ${L}_p$, the Hermitian metric $\langle \,\cdot \,|\,\cdot \,\rangle$ and the 1-form $\omega _0$ can be written

$$\begin{aligned}{L}_p(Z_j,\,{\overline{Z}}_k)=\mu _j\,\delta _{j,k},\qquad \langle Z_j\vert \,{\overline{Z}}_k \rangle =\delta _{j,k}\qquad (1\le j,k\le n), \end{aligned}$$

and

$$\begin{aligned} \omega _0(x)=&(1+O(|x|))\textrm{d}x_{2n+1}+\sum _{j=1}^{t-1} 4\mu _jx_{t-1+j}\textrm{d}x_j +\sum _{j=t}^n 2\mu _jx_{2j}\textrm{d}x_{2j-1} \\&-\sum _{j=t}^n 2\mu _jx_{2j-1}\textrm{d}x_{2j}+\sum _{j=r}^{2n} b_jx_{2n+1}\textrm{d}x_j+O(|x|^2) \end{aligned}$$

where $b_{r},\dots ,b_{2n}\in {\mathbb {R}}$,

$$\begin{aligned}T_p^{1,0}X=\textrm{span}\{Z_1,\dots ,Z_n\} \end{aligned}$$

and

$$\begin{aligned}Z_j&=\frac{1}{2}\,(\partial _{{x}_{j}}-i\, \partial _{{x}_{t-1+j}})(p)\qquad (j=1,\dots ,t-1)\,,\\ Z_j&=\frac{1}{2}\,(\partial _{{x}_{2j-1}}-i\, \partial _{{x}_{2j}})(p)\qquad (j=t,\dots ,n)\,. \end{aligned}$$

We need to define in local coordinates we just introduced the phase function of the ${\mathbb {T}}^{t-1}_{\nu }$-invariant Szegő kernel $\Phi _-(x,y)\in {\mathcal {C}}^\infty (U\times U)$ which is independent of $(x_1,\ldots ,x_{t-1})$ and $(y_1,\ldots ,y_{t-1})$. Hence, we write $\Phi _-(x,y)=\Phi _-((0,x''),(0,y'')):=\Phi _-(x'',y'')$ and $\mathring{x}'':=(x_{t},\ldots ,x_{2n})$, $\mathring{y}'':=(y_{t},\ldots ,y_{2n})$. Moreover, there is a constant $c>0$ such that

$$\begin{aligned} \mathrm{Im\,}\Phi _-(x'',y'')\ge c\left( |{\mathring{x}''}|^2+|{\mathring{y}''}|^2 +|{\mathring{x}''-\mathring{y}''}|^2\right) ,\ \ \text {for all } ((0,x''),(0,y''))\in U\times U. \end{aligned}$$

(4)

Furthermore,

$$\begin{aligned} \Phi _-&(x'', y'')=-x_{2n+1}+y_{2n+1}+2i\sum ^{t-1}_{j=1}\left| \mu _j\right| y^2_{t-1+j}+2i \sum ^{t-1}_{j=1}\left| \mu _j\right| x^2_{t-1+j} +i\sum ^{n}_{j=t}\left| \mu _j\right| \left| z_j-w_j\right| ^2\nonumber \\&+\sum ^{n}_{j=t}i\mu _j(\overline{z}_jw_j-z_j\overline{w}_j) +\sum ^d_{j=1}(-b_{t-1+j}x_{d+j}x_{2n+1}+b_{t-1+j}y_{t-1+j}y_{2n+1})\nonumber \\&+\sum ^n_{j=t}\frac{1}{2}(b_{2j-1}-ib_{2j})(-z_jx_{2n+1}+w_jy_{2n+1}) +\sum ^n_{j=t}\frac{1}{2}(b_{2j-1}+ib_{2j})(-\overline{z}_jx_{2n+1}+\overline{w}_jy_{2n+1}) \nonumber \\&+(x_{2n+1}-y_{2n+1})f(x, y) +O(\left| (x, y)\right| ^3), \end{aligned}$$

(5)

where $z_j=x_{2j-1}+ix_{2j}$, $w_j=y_{2j-1}+iy_{2j}$, $j=t,\ldots ,n$, $\mu _j$, $j=1,\ldots ,n$, f is smooth and satisfies $f(0,0)=0$, $f(x, y)=\overline{f}(y, x)$.

We now consider $\overline{{\mathbb {T}}^1_{\nu }}$ circle action on X. Let $p\in \mu ^{-1}(i{\mathbb {R}}_+\cdot \nu )$, there exist local coordinates $v=(v_1,\dots ,v_{t-1})$ of ${\mathbb {T}}^{t-1}$ in a small neighborhood $V_0$ of e with $v(e)=(0,\,\dots ,\,0)$, local coordinates $x=(x_1\,\dots ,x_{2n+1})$ defined in a neighborhood $U_1\times U_2$ of p, where $U_1\subseteq {\mathbb {R}}^{t-1}$ (resp. $U_{t-1}\subseteq {\mathbb {R}}^{2n+t}$) is an open set of $0\in {\mathbb {R}}^{t-1}$ (resp. $0\in {\mathbb {R}}^{2n+t}$) and $p\equiv 0\in {\mathbb {R}}^{2n+1}$, and a smooth function $\gamma =(\gamma _1,\dots ,\,\gamma _t)\in {\mathcal {C}}^{\infty }(U_2,U_1)$ with $\gamma (0)=0$ such that $T=-\frac{\partial }{\partial x_{2n+1}}$ and all the properties for the local coordinates defined before hold. The phase function $\Psi$ satisfies $\Psi (x,y)=-x_{2n+1}+y_{2n+1}+{\hat{\Psi }}(\mathring{x}'',\mathring{y}'')$, where ${\hat{\Psi }}(\mathring{x}'',\mathring{y}'')\in {\mathcal {C}}^\infty (U\times U)$ and $\Psi$ satisfies (5).

2.2 Hardy spaces

We denote by $L^2_{(0,q)}(X)$, $q=0,1,\ldots ,n$, the completion of $\Omega ^{0,q}(X)$ with respect to $(\,\cdot \,|\,\cdot \,)$. We extend $(\,\cdot \,|\,\cdot \,)$ to $L^2_{(0,q)}(X)$ in the standard way. We extend ${\overline{\partial }}_{b}$ to $L^2_{(0,q)}(X)$ by

$$\begin{aligned} {\overline{\partial }}_{b}:\mathrm{Dom\,}{\overline{\partial }}_{b}\subset L^2_{(0,q)}(X)\rightarrow L^2_{(0,q+1)}(X), \end{aligned}$$

where $\mathrm{Dom\,}{\overline{\partial }}_{b}:=\{u\in L^2_{(0,q)}(X);\, {\overline{\partial }}_{b}u\in L^2_{(0,q+1)}(X)\}$ and, for any $u\in L^2_{(0,q)}(X)$, ${\overline{\partial }}_{b} u$ is defined in the sense of distributions. We also write

$$\begin{aligned} \overline{\partial }^{*}_{b}:\mathrm{Dom\,}\overline{\partial }^{*}_{b}\subset L^2_{(0,q+1)}(X)\rightarrow L^2_{(0,q)}(X) \end{aligned}$$

to denote the Hilbert space adjoint of ${\overline{\partial }}_{b}$ in the $L^2$ space with respect to $(\,\cdot \,|\,\cdot \, )$. There is a well-defined orthogonal projection

$$\begin{aligned} S^{(q)}:L^2_{(0,q)}(X)\rightarrow \mathrm{Ker\,}\Box ^q_b \end{aligned}$$

(6)

with respect to the $L^2$ inner product $(\,\cdot \,|\,\cdot \,)$ and let

$$\begin{aligned} S^{(q)}(x,y)\in {\mathcal {D}}'(X\times X,T^{*0,q}X\boxtimes (T^{*0,q}X)^*) \end{aligned}$$

denote the distribution kernel of $S^{(q)}$. We will write $H^q(X)$ for $\ker \Box ^q_b$, and for functions, we simply write H(X) to mean $H^0(X)$. The distributional kernel $S^{(q)}(x,y)$ was studied in [6], 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 $D\subset X$ be a local coordinate patch with local coordinates $x=(x_1,\ldots ,x_{2n+1})$.

Definition 2.1

For every $m\in {\mathbb {R}}$, we denote with

$$\begin{aligned} S^m_{1,0}(D\times D\times {\mathbb {R}}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\subseteq {\mathcal {C}}^\infty (D\times D\times {\mathbb {R}}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\nonumber \\ \end{aligned}$$

(7)

the space of all a such that, for all compact $K\Subset D\times D$ and all $\alpha , \beta \in {\mathbb {N}}^{2n+1}_0$, $\gamma \in {\mathbb {N}}_0$, there is a constant $C_{\alpha ,\beta ,\gamma }>0$ such that

$$\begin{aligned}\left| \partial ^\alpha _x\partial ^\beta _y\partial ^\gamma _t a(x,y,t)\right| \le C_{\alpha ,\beta ,\gamma }(1+\left| t\right| )^{m-\gamma },\ \ \hbox { for every}\ (x,y,t)\in K\times {\mathbb {R}}_+, t\ge 1.\end{aligned}$$

For simplicity we denote with $S^m_{1,0}$ the spaces defined in (7); furthermore, we write

$$\begin{aligned} S^{-\infty }(D\times D\times {\mathbb {R}}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*):=\bigcap _{m\in {\mathbb {R}}}S^m_{1,0}(D\times D\times {\mathbb {R}}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*). \end{aligned}$$

Let $a_j\in S^{m_j}_{1,0}$, $j=0,1,2,\ldots$ with $m_j\rightarrow -\infty$, as $j\rightarrow \infty$. Then there exists $a\in S^{m_0}_{1,0}$ unique modulo $S^{-\infty }$, such that

$$\begin{aligned} a-\sum ^{k-1}_{j=0}a_j\in S^{m_k}_{1,0}(D\times D\times {\mathbb {R}}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*\big ) \end{aligned}$$

for $k=0,1,2,\ldots$. If a and $a_j$ have the properties above, we write $a\sim \sum ^{\infty }_{j=0}a_j$ in $S^{m_0}_{1,0}$.

It is known that the characteristic set of $\Box ^q_b$ is given by

$$\begin{aligned} \Sigma =\Sigma ^-\cup \Sigma ^+,\quad \Sigma ^-=\left\{ (x,\lambda \omega _0(x))\in T^*X;\,\lambda <0\right\} , \end{aligned}$$

and $\Sigma ^+$ is defined similarly for $\lambda >0$. We recall the following theorem (see [6, Theorem 1.2]).

Theorem 2.1

Suppose that the Levi form is non-degenerate and $\Box ^q_b$ has $L^2$ closed range. Then, there exist continuous operators $S_-, S_+: L^2_{(0,q)}(X)\rightarrow \mathrm{Ker\,}\Box ^q_b$ such that

$$\begin{aligned} S^{(q)}=S_-+S_+,\quad S_+\equiv 0\ \ \hbox { if}\ q\ne n_+\end{aligned}$$

and

$$\begin{aligned}\mathrm{WF'\,}(S_-)=\mathrm{diag\,}(\Sigma ^-\times \Sigma ^-),\quad \mathrm{WF'\,}(S_+)=\mathrm{diag\,}(\Sigma ^+\times \Sigma ^+)\ \ \hbox { if}\ q=n_-=n_+, \end{aligned}$$

where $\mathrm{WF'\,}(S_-)=\left\{ (x,\xi ,y,\eta )\in T^*X\times T^*X;\,(x,\xi ,y,-\eta )\in \mathrm{WF\,}(S_-)\right\}$, $\mathrm{WF\,}(S_-)$ is the wave front set of $S_-$ in the sense of Hörmander.

Moreover, consider any small local coordinate patch $D\subset X$ with local coordinates $x=(x_1,\ldots ,x_{2n+1})$, then $S_-(x,y)$, $S_+(x,y)$ satisfy

$$\begin{aligned} S_{\mp }(x, y)\equiv \int ^{\infty }_{0}e^{i\varphi _{\mp }(x, y)t}s_{\mp }(x, y, t)dt\ \ \hbox { on}\ D, \end{aligned}$$

with

$$\begin{aligned}s_{\mp }(x,y,t)\sim \sum ^\infty _{j=0}s^j_{\mp }(x, y)t^{n-j}\text { in }S^{n}_{1, 0}(D\times D\times {\mathbb {R}}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*),\end{aligned}$$

$s_+(x,y,t)=0$ if $q\ne n_+$ and $s^0_-(x,x)\ne 0$ for all $x\in D$. The phase functions $\varphi _-$, $\varphi _+$ satisfy

$$\begin{aligned}\varphi _+, \varphi _-\in {\mathcal {C}}^\infty (D\times D),\ \ \mathrm{Im\,}\varphi _{\mp }(x, y)\ge 0,\,\varphi _-(x, x)=0,\ \ \varphi _-(x, y)\ne 0\ \ \text{ if }\ \ x\ne y,\end{aligned}$$

and

$$\begin{aligned} d_x\varphi _-(x, y)\big |_{x=y}=-\omega _0(x), \ \ d_y\varphi _-(x, y)\big |_{x=y}=\omega _0(x), \,-\overline{\varphi }_+(x, y)=\varphi _-(x,y). \end{aligned}$$

Remark 2.1

Kohn [9] proved that if $q=n_-=n_+$ or $\left| n_--n_+\right| >1$ then $\Box ^q_b$ has $L^2$ closed range. For a description of the phase function in local coordinates see chapter 8 of part I in [6].

Now we focus on the decomposition induced by the group action on $H^q(X)$. Fix $t\in {\mathbb {T}}$ and let $t^*:\Lambda ^r_x({\mathbb {C}}T^*X)\rightarrow \Lambda ^r_{t^{-1}\circ x}({\mathbb {C}}T^*X)$ be the pull-back map. Since ${\mathbb {T}}$ preserves J, we have $t^*:T^{*0,q}_xX\rightarrow T^{*0,q}_{g^{-1}\circ x}X$, for all $x\in X.$ Thus, for $u\in \Omega ^{0,q}(X)$, we have $t^*u\in \Omega ^{0,q}(X)$. Put

$$\begin{aligned} \Omega ^{0,q}(X)_{k\nu }:=\left\{ u\in \Omega ^{0,q}(X);\, (e^{i\theta })^*u=e^{i\,k\langle \nu ,\,\theta \rangle }u,\ \ \forall \theta \in {\mathbb {R}}^r\right\} . \end{aligned}$$

Since the Hermitian metric $\langle \,\cdot \,|\,\cdot \,\rangle$ on ${\mathbb {C}}TX$ is ${\mathbb {T}}$-invariant, the $L^2$ inner product $(\,\cdot \,|\,\cdot \,)$ on $\Omega ^{0,q}(X)$ induced by $\langle \,\cdot \,|\,\cdot \,\rangle$ is ${\mathbb {T}}$-invariant. Let $u\in L^2_{(0,q)}(X)$ and $t\in {\mathbb {T}}$, we can also define $t^*u$ in the standard way. We introduce the following notation

$$\begin{aligned}L^2_{(0,q)}(X)_{k\nu }:=\left\{ u\in L^2_{(0,q)}(X);\, (e^{i\theta })^*u=e^{i\,k\langle \nu ,\,\theta \rangle }u,\ \ \forall \theta \in {\mathbb {R}}^r\right\} ,\end{aligned}$$

and put

$$\begin{aligned}(\mathrm{Ker\,}\Box ^q_b)_{k\nu }:=\mathrm{Ker\,}\Box ^q_b\cap L^2_{(0,q)}(X)_{k\nu }.\end{aligned}$$

The equivariant Szegő projection is the orthogonal projection

$$\begin{aligned}S^{(q)}_{k\nu }:L^2_{(0,q)}(X)\rightarrow (\mathrm{Ker\,}\Box ^q_b)_{k\nu }\end{aligned}$$

with respect to $(\,\cdot \,|\,\cdot \,)$. Let $S^{(q)}_{k\nu }(x,y)\in {\mathcal {D}}'(X\times X,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)$ be the distribution kernel of $S^{(q)}_{k\nu }$. The asymptotic expansion for the distributional kernel of the projector $S^{(0)}_{k\nu }$ was studied in [12] when X is the quantizing circle bundle of a given Hodge manifold in Heisenberg local coordinates. Using similar ideas as in [7] one can generalize the results for (0, q)-forms in the following theorem, we give a sketch of the proof, which is similar to the proof of [7, Theorem 1.8], here the group G in [7] is ${\mathbb {T}}^{t-1}_{\nu }$, and the circle action in [7] is given by the action of $\overline{{\mathbb {T}}^1_{\nu }}$.

Since the action is locally free we need to recall some notations from [2]. Since the action is locally free $\mu ^{-1}(0)/\overline{{\mathbb {T}}^1_{\nu }}$ is an orbifold, let us denote with $\pi \,:\,\mu ^{-1}(0)\rightarrow \mu ^{-1}(0)/\overline{{\mathbb {T}}^1_{\nu }}$ the projection. Furthermore the action of $\overline{{\mathbb {T}}^1_{\nu }}$ commutes with the one of ${\mathbb {T}}^{t-1}_{\nu }$; then, we have a smooth locally free action of ${\mathbb {T}}^{t-1}_{\nu }$ on $\mu ^{-1}(0)/\overline{{\mathbb {T}}^1_{\nu }}$. Given $y\in X$ and g in the stabilizer $({\mathbb {T}}^{t-1}_{\nu })_{\pi (y)}$, there exist $|(\overline{{\mathbb {T}}^1_{\nu }})_y^1 |$ elements $e^{ i \,\theta _{g,j}}\in \overline{{\mathbb {T}}^1_{\nu }}$ ($j=1,\dots ,|(\overline{{\mathbb {T}}^1_{\nu }})_y^1 |$) such that

$$\begin{aligned}g\circ y= e^{- i \,\theta _{g,j}}\circ y. \end{aligned}$$

Thus, all the elements $(e^{ i \,\theta },\,g)\in \overline{{\mathbb {T}}^1_{\nu }}\times {\mathbb {T}}^{t-1}_{\nu } ={\mathbb {T}}$ satisfying

$$\begin{aligned} e^{ i \,\theta }\cdot g\circ y=y. \end{aligned}$$

are of the form $(e^{ i \theta _{g,j}}, g)$ for each $g\in ({\mathbb {T}}^{t-1}_{\nu })_{\pi (y)}$.

Theorem 2.2

Suppose that $\Box ^q_b$ has $L^2$ closed range. Then, there exist continuous operators $S^-_{k\nu }, S^+_{k\nu }: L^2_{(0,q)}(X)\rightarrow (\mathrm{Ker\,}\Box ^q_b)_{k\nu }$ such that

$$\begin{aligned} S^{(q)}_{k\nu }=S^-_{k\nu }+S^+_{k\nu },\quad S_+\equiv 0\ \ \hbox { if}\ q\ne n_+\end{aligned}$$

Suppose for simplicity that $q\ne n_+$. If $q\ne n_-$, then $S^{(q)}_{k\nu }\equiv O(k^{-\infty })$ on X.

Suppose $q= n_-$ and let D be an open set in X such that the intersection $\mu ^{-1}(i\,{\mathbb {R}}_+\cdot \nu )\cap D= \emptyset$. Then $S^{(q)}_{k\nu }\equiv O(k^{-\infty })$ on D.

Let $p\in \mu ^{-1}(i\,{\mathbb {R}}_+\cdot \nu )$ and let U a local neighborhood of p with local coordinates $(x_1,\dots \,x_{2n+1})$. Then, if $q= n_-$, for every fix $y\in U$, we consider $S^{(q)}_{k\nu }(x,y)$ as a k-dependent smooth function in x, then

$$\begin{aligned}S^{(q)}_{k\nu }(x,\,y)= \sum _{h\in G_{\pi (y)}} \sum _{j=1}^{|S^1_x|} e^{ik\, \theta _{h,j}} e^{i k\Vert \nu \Vert \,\Psi (x,\,y)}\,b(x,\,y,\,k\Vert \nu \Vert )+O(k^{-\infty }). \end{aligned}$$

for every $x\in U_y$, where $U_y$ is a small open neighborhood of y. The phase function $\Psi (x,\,y)$ is defined in local coordinates in the end of Sect. 2.1; the symbol satisfies

$$\begin{aligned}b(x,\,y,\,k\Vert \nu \Vert )\in S^{n+(1-t)/2}_{\textrm{loc}}(1,\,U\times U,\,T^{*\,(0,q)}X\boxtimes (T^{*\,(0,q)})^*)\, \end{aligned}$$

and the leading term of $b(x,x,\Vert \nu \Vert )$ is nonzero.

Proof

Suppose $q=n_-$, on small local neighborhood D of a point $p\in X_\nu$ we have

$$\begin{aligned} S_{k\nu }^{(q)}(x,\,y)=\frac{1}{(2\,\pi )^t}\int _{-\pi }^{\pi }\dots \int _{-\pi }^{\pi } e^{i k\,\langle \nu ,\,\theta \rangle }\,S^{(q)}( x,\,e^{i \,\theta }\cdot y) \,\textrm{d}\theta \, \end{aligned}$$

where $\theta =(\theta _1,\dots ,\theta _t)$. It is easy to prove that the oscillatory integral has a rapidly decreasing asymptotic as $k\rightarrow +\infty$ far away from a local neighborhood of those elements $(e^{ i \,\theta },\,g)\in \overline{{\mathbb {T}}^1_{\nu }}\times {\mathbb {T}}^{t-1}_{\nu } ={\mathbb {T}}$ such that

$$\begin{aligned} e^{ i \,\theta }\cdot g\circ y=y. \end{aligned}$$

We shall consider the case of a local neighborhood of the identity in ${\mathbb {T}}$; we set $\theta _t$ the variable for circle action of $\overline{{\mathbb {T}}^1_{\nu }}$ and $(\theta _1,\dots ,\theta _{t-1})$ the variables for ${\mathbb {T}}^{t-1}_{\nu }$. We can then use local coordinates defined in 2.1; we have

$$\begin{aligned} S_{k\nu }^{(q)}(x,\,y)=\frac{1}{2\,\pi }\int _{-\pi }^{\pi } e^{i k\, \Vert \nu \Vert \theta _t -i\,kx_{2n+1}+i\,ky_{2n+1}}\,S_{{\mathbb {T}}^{t-1}}^{(q)}(\mathring{x},\,e^{i \,\theta _t}\cdot \mathring{y}) \,\textrm{d}\theta _t \end{aligned}$$

(8)

where

$$\begin{aligned} S_{{\mathbb {T}}^{t-1}}^{(q)}({x},\,{y})=\frac{1}{(2\,\pi )^{t-1}}\int _{-\pi }^{\pi }\dots \int _{-\pi }^{\pi } \,S^{(q)}( x,\,e^{i \,\textrm{diag}(\theta _1,\dots ,\theta _{t-1})}\cdot y) \,\textrm{d}\theta _1\,\dots \textrm{d}\theta _{t-1}, \end{aligned}$$

(9)

and $\mathring{x}=(x_1,\dots ,x_n,0)$ and $\mathring{y}=(y_1,\dots ,y_n,0)$. The action of ${\mathbb {T}}$ restricts to an action of ${\mathbb {T}}^{t-1}$ whose moment map is given by

$$\begin{aligned}\mu _{\vert {\mathbb {T}}^{t-1}}= p_{\nu }\circ \mu ,\quad \text { where } p_{\nu }:\, {\mathfrak {t}}^* \rightarrow {\mathfrak {t}}_{\nu }^* \end{aligned}$$

is the canonical projection onto the $t-1$-dimensional subspace $\nu ^{\perp }$ in ${\mathfrak {t}}^*$. Since we are assuming $X_{\nu } \ne \emptyset$, we have that $0\in {\mathfrak {t}}_{t-1}^*$ does lie in the image of the moment map $\mu _{\vert {\mathbb {T}}^{t-1}}$. The proof follows in a similar way as in Theorem 1.8 in [7] where here we have the action of $G={\mathbb {T}}^{t-1}$ with moment map $\mu _{{\mathbb {T}}^{t-1}}$ and the $k\nu$ Fourier components are the ones induced by the transversal $\overline{{\mathbb {T}}^1_{\nu }}$-action; notice that [7, Assumption 1.7] is satisfied. We shall also refer to [2, Theorem 5.4] for the locally free action case. $\square$

For ease of notation we introduce a definition to say that an operator behaves microlocally as equivariant Szegő projector we just studied. For simplicity we assume $q=n_-$.

Definition 2.2

(Equivariant Szegő-type operator) Suppose that $q=n_-$ and consider $H: \Omega ^{0,q}(X)\rightarrow \Omega ^{0,q}(X)$ be a continuous operator with distribution kernel

$$\begin{aligned} H(x,y)\in {\mathcal {D}}'(X\times X,T^{*0,q}X\boxtimes (T^{*0,q}X)^*). \end{aligned}$$

We say that H is a complex Fourier integral operator of equivariant Szegő type of order $n+(1-t)/2 \in {\mathbb {Z}}$ if H is smoothing away $\mu ^{-1}(i\,{\mathbb {R}}_+\cdot \nu )$ and for given $p\in \mu ^{-1}(i\,{\mathbb {R}}_+\cdot \nu )$ let D a local neighborhood of p with local coordinates $(x_1,\dots \,x_{2n+1})$. Then, the distributional kernel of H satisfies

$$\begin{aligned}H_{k}(x,y)= e^{i k\,\Psi (x,\,y)}\,a(x,\,y,\,k)+O(k^{-\infty })\quad \text { on }D\end{aligned}$$

where $a\in S^{n+(1-t)/2}_{1,0}(D\times D\times {\mathbb {R}}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)$, and $\Psi$ is as in the end of Sect. 2.1.

For $k=0$, $H^{(q)}(X)_0$ is the space of ${\mathbb {T}}$-fixed vectors, by Theorem 1.5 in [7] we have $\dim H_0^{(q)}(X)< +\infty$ since when $0\notin \mu (X)$ the projector $S^{(q)}_{{\mathbb {T}}}$ is smoothing. Now, given $k_1, k_2\in {\mathbb {Z}}$ with $k_1, k_2\ne 0$, consider $f_1\in H^{(q)}(X)_{k_1\nu }$ and $f_2\in H^{(q)}(X)_{k_2\nu }$, we have $f_1\cdot f_2 \in H^{(q)}(X)_{(k_1+k_2)\nu }$; we can study $\dim H(X)_{k\nu }$ for k large; we have

$$\begin{aligned}\dim H^{(q)}(X)_{k\nu } = \int _X S_{k\nu }^{(q)}(x,\,x)\,\textrm{dV}_X(x). \end{aligned}$$

These dimensions can be studied as $k\rightarrow +\infty$ by using the microlocal properties of the Szegő kernel and Stationary Phase Lemma in a similar way as in Corollary 1.3 in [12] we have $\dim H^{(q)}(X)_{k\nu }= O(k^{d+1-t})$. So, we have the following generalization of Theorem A.3 in [4], where it is proved for spaces of functions $H(X)_{k\nu }$.

Lemma 2.2

If $0\notin \mu (X)$, then $H^q(X)_{k\nu }$ are finite dimensional.

3 Proof of Theorem 1.1

In this section we shall explain how to prove asymptotic commutativity for quantization and reduction for spaces of (0, q)-forms. We recall that $X^{\textrm{red}}_{\nu }$ is a CR orbifold whose Levi form is non-degenerate, with $n_--r$ negative eigenvalues; let us denote with $S^{(q)}_{\textrm{red}}$ the corresponding Szegő kernel for (0, q)-forms whose k-th Fourier components are the one induced by the $\overline{{\mathbb {T}}^1_{\nu }}$-action on $X^{\textrm{red}}_{\nu }$. We shall recall briefly its microlocal expression by [3, Theorem 1.2]. Now, let us denote by $e^{i\theta }\cdot$ the transversal and CR locally free $\overline{{\mathbb {T}}^1_{\nu }}$-action and we take the Reeb vector field R to be the vector field on X induced by it.

Let $q=n_--r$, and consider an open set $U\subset X$, $p\in U$, and an orbifold chart $({\widetilde{U}},G_U)\rightarrow U$; we denote by ${\widetilde{x}}$ the coordinates on ${\widetilde{U}}$. For every $\ell \in {\mathbb {N}}$, put

$$\begin{aligned}X_\ell :=\{x\in X;\, e^{i\theta }x\ne x, \theta \in [0,{2\pi }/{\ell }[, e^{i\,{2\pi }/{\ell }}x=x\}.\end{aligned}$$

With the assumptions and notations used above, assume that $p\in X_\ell$, for some $\ell \in {\mathbb {N}}$. We have as $k\rightarrow +\infty$,

$$\begin{aligned} S^{(q)}_{\textrm{red},k}(x,y)=\sum ^{\ell -1}_{j=0}\sum _{g\in G_U} e^{\frac{2\pi kj}{\ell }}\, e^{ik\, \Psi ({\widetilde{x}}, e^{i\frac{2\pi j}{\ell }}\cdot g\cdot {\widetilde{y}})}b({\widetilde{x}},\,e^{i\frac{2\pi j}{\ell }}\cdot g\cdot {\widetilde{y}},k)+O(k^{-\infty }) \end{aligned}$$

(10)

where the phase function

$$\begin{aligned} \begin{aligned}&\Psi \in \,{C}^\infty ({{\widetilde{U}}}\times {{\widetilde{U}}})\,,\qquad \Psi ({{\widetilde{x}}},{{\widetilde{x}}})=0,\ \ \hbox { for all}\ {{\widetilde{x}}}\in {{\widetilde{U}}},\\&\inf _{e^{i\theta }\in S^1}\left\{ \textrm{dist}^2({{\widetilde{x}}},e^{i\theta }\,{{\widetilde{y}}})\right\} /C\le \mathrm{Im\,}\Psi ({{\widetilde{x}}},{{\widetilde{y}}})\le C\inf _{e^{i\theta }\in S^1}\left\{ \textrm{dist}^2({{\widetilde{x}}},e^{i\theta }\,{{\widetilde{y}}})\right\} \, \end{aligned} \end{aligned}$$

for each $({{\widetilde{x}}},{{\widetilde{y}}})\in {{\widetilde{U}}}\times {{\widetilde{U}}}$, $C>1$ is a constant, and the symbol satisfies

$$\begin{aligned} b({{\widetilde{x}}},{{\widetilde{y}}},k)\sim \sum ^{+\infty }_{j=0}b_j({{\widetilde{x}}},{{\widetilde{y}}})\,k^{n-j} \quad \text {in} \quad S^{n-(t-1)}(1;{{\widetilde{U}}}\times {{\widetilde{U}}},T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\, \end{aligned}$$

and $b_0({{\widetilde{x}}},{{\widetilde{x}}})$ is nonzero.

Let us recall briefly why we have a local chart $({\widetilde{U}},\,G_U)\rightarrow U$, for ease of notation let us put $G={\mathbb {T}}^{t-1}_{\nu }$. We recall that, for every $x\in X_{\nu }$, by the slice theorem a neighborhood of any orbit ${\mathbb {T}}^{t-1}_{\nu }\cdot x=x_0$ is equivariantly diffeomorphic to a neighborhood of the zero section of the associated principal bundle

$$\begin{aligned} G\times _{G_x} N_x, \end{aligned}$$

where $N_x$ is the normal space to $G\cdot x$ in $X_{\nu }$ and $G_x$ is the stabilizer of x for the action of G, which is finite. Therefore, for some $\epsilon >0$ and for an open ball $B_{2{e}}(\epsilon )\subseteq N_x$, one has a homeomorphism $B_{2{e}}(\epsilon )/G_x \cong {\overline{U}}$ onto some neighborhood of $x_0$ in $X^{\textrm{red}}_{\nu }$.

Since $\nu ^{\perp }_X:=\textrm{val}(\nu ^{\perp })$ is orthogonal to

$$\begin{aligned} H_xX_{\nu }\cap J_x H_xX_{\nu }\quad \text { and } \quad H_xX_{\nu }\cap J_x H_xX_{\nu }\subset (\nu ^{\perp }_X)^{\perp _b} \end{aligned}$$

for every $x\in X_{\nu }$, we can find a ${\mathbb {T}}$-invariant orthonormal basis $\{Z_1,\,\dots ,\, Z_n\}$ of $T^{*\,0,1}X$ on $X_{\nu }$ such that for each $j,\,k=1,\dots n$,

$$\begin{aligned}L_x(Z_j(x),\,{\overline{Z}}_j(x))= \delta _{jk}\,\lambda _j(x), \end{aligned}$$

where

$$\begin{aligned}{} & {} Z_j(x)\in (\nu ^{\perp }_{X,x}+i J \nu ^{\perp }_{X,x})\hbox { for each}\ j=1,\,\dots ,\, t-1, \\{} & {} Z_j(x)\in H_xX_{\nu }\cap J_x(H_xX_{\nu }) \hbox { for each}\ j=r,\,\dots ,\, n. \end{aligned}$$

Let $\{Z_1^*,\,\dots ,\, Z_n^*\}$ denote the orthonormal basis of $T^{*\,0,1}X$ on $X_{\nu }$, dual to $Z_1,\,\dots ,\, Z_n$. Fix $s = 0,\, 1,\, 2,\dots ,\, n-r+1$. For $x \in X_{\nu }$, put

$$\begin{aligned}B_x^{*\,0,s}X=\left\{ \sum _{r\le j_1<\dots <j_s\le n} a_{j_1,\dots ,j_s}Z_{j_1}^*,\,\dots , Z_{j_s}^*;\,a_{j_1,\dots ,j_s}\in {\mathbb {C}},\, \right\} \end{aligned}$$

and let $B^{*\,0,s}X$ be the vector bundle of $X_{\nu }$ with fiber $B_x^{*\,0,s}$, $x\in X_{\nu }$. Let $C^{\infty }(X_{\nu }, B^{*\,0,s}X)^{{\mathbb {T}}}$ denote the set of all ${\mathbb {T}}$-invariant sections of $X_{\nu }$ with values in $B_x^{*\,0,s}X$; let

$$\begin{aligned}\iota _{\textrm{red}} :\,C^{\infty }(X_{\nu }, B^{*\,0,s}X)^{{\mathbb {T}}} \rightarrow \Omega ^{0,s}(X^{\textrm{red}}_{\nu }) \end{aligned}$$

be the natural identification. Before defining the map between $H^q(X)_{k\nu }$ and $H^{q-r}(X^{\textrm{red}}_{\nu })_{k}$ we need one more piece of notation. We can assume that $\lambda _1< 0, \cdots , \lambda _{r} < 0$ and also $\lambda _{t}< 0,\dots ,\, \lambda _{n_--r+t-1} < 0$. For $x\in X_{\nu }$, set

$$\begin{aligned}\hat{{\mathcal {N}}}(x,n_-)=\{c\,Z_t^*\wedge \,\cdots \,\wedge Z_{n_--r+t-1}^*,c\in {\mathbb {C}} \} \end{aligned}$$

and let ${\hat{p}}\,:\,{\mathcal {N}}(x,n_-)\rightarrow \hat{{\mathcal {N}}}(x,n_-)$ to be

$$\begin{aligned}{\hat{p}}(c\, Z_1^*\wedge \,\cdots \,\wedge Z_r^*\wedge Z_t^*\wedge \,\cdots \,\wedge Z_{n_--r+t-1}^* ):= c\,Z_t^*\wedge \,\cdots \,\wedge Z_{n_--r+t-1}^* \end{aligned}$$

where c is a complex number. Put $\iota _{\nu } \,:\, X_{\nu } \rightarrow X$ be the natural inclusion and let $\iota _{\nu }^* \,:\, \Omega ^{0,q}(X)\rightarrow \Omega ^{0,q}(X_{\nu })$ be the pull-back of $\iota _{\nu }$.

Let $q = n_-$; now, inspired by [7], we define the map

$$\begin{aligned}\sigma _{k\nu }:\,H^{q}(X)_{k\nu }\rightarrow H^{q-r}(X^{\textrm{red}}_{\nu })_k \end{aligned}$$

given by

$$\begin{aligned} \sigma _{k\nu }(u):=(k\nu )^{(t-1)/4}\,S^{(q-r)}_{\textrm{red},k} \circ \iota _{\textrm{red}}\circ {\hat{p}} \circ \tau _{x,n_-}\circ e \circ \iota ^*_{\nu } \circ S^{(q)}_{k\nu }(u) , \end{aligned}$$

(11)

here, e(x) is a ${\mathbb {T}}$-invariant smooth function on $X_{\nu }$ that can be found explicitly. Notice that operators $S^{(q-r)}_{\textrm{red},k}$ and $S^{(q)}_{k\nu }$ appearing in (11) are known explicitly. $S^{(q-r)}_{\textrm{red},k}$ is the k-th Fourier component of the standard Szegő kernel for the CR manifold $X^{\textrm{red}}_{\nu }$; on the other hand, $S^{(q)}_{k\nu }$ is described in Sect. 2.2.

Before stating the next theorem we shall specialize the local coordinates defined in Sect. 2.1. Let us consider $p\in X_{\nu }$ and let $x=(x_1,\,\dots ,\,x_{2n+1})$ be the local coordinates in an open neighborhood U of p defined in Sect. 2.1. We may assume that $U=U_1\times U_2 \times U_3\times U_4$, where $U_1\subset {\mathbb {R}}^{t-1}$, $U_2\subset {\mathbb {R}}^{t-1}$ are open sets of $0\in {\mathbb {R}}^{t-1}$, $U_3\subset {\mathbb {R}}^{2n-2(t-1)}$ is an open set of $0\in {\mathbb {R}}^{2n-2(t-1)}$ and $U_4$ is an open set of $0\in {\mathbb {R}}$. From now on, we can identify $U_2$ with

$$\begin{aligned} \{(0,\,\dots ,\,0,\,x_{t},\dots ,\,x_{2(t-1)},\,0,\,\dots ,\,0)\in U:\,(x_t,\,\dots ,\,x_{2(t-1)})\in U_2\}\end{aligned}$$

and $U_3$ with

$$\begin{aligned}\{(0,\,\dots ,\,0,\,x_{2t-1},\dots ,\,x_{2n},\,0)\in U:\,(x_t,\,\dots ,\,x_{2n})\in U_3\}.\end{aligned}$$

For a given orbifold chart $({\widetilde{U}},\,G_U)\rightarrow U$ we write ${\tilde{U}}_i$ for the corresponding open sets in ${\tilde{U}}$. Eventually we recall that ${\widetilde{x}}'':=(x_{2t-1},\,\dots ,\,x_{2n+1})$.

Theorem 3.1

Under the assumptions above, if $x\notin X_{\nu }$, then for every sufficiently small open set D of x with ${\overline{D}}\cap X_{\nu }=\emptyset$, we have $\sigma _{k\nu }=O(k^{-\infty })$ on $X^{\textrm{red}}_{\nu }\times D$.

Let $\pi :\,X_{\nu } \rightarrow X^{\textrm{red}}_{\nu }$ the projection. If $x,y\in X_{\nu }$ and $\pi (x)\ne \pi (e^{i\theta }\cdot y)$ for every $e^{i\theta }\in \overline{{\mathbb {T}}^1_{\nu }}$, then there exist open set U of $\pi (x)$ in $X^{\textrm{red}}_{\nu }$ and V of y in X such that $\sigma _{k\nu }=O(k^{-\infty })$ on $U\times V$.

Eventually, let $p\in X_{\nu }\cap X_{\ell }$, using the local coordinates defined above, we have

$$\begin{aligned}\sigma _{k\nu }({\tilde{x}}'',y'')= \sum ^{\ell -1}_{j=0}\sum _{g\in G_U} e^{\frac{2\pi kj}{\ell }} e^{ik\, \Psi ({\widetilde{x}}, \,e^{\frac{2\pi j}{\ell }}\cdot g\cdot {\widetilde{y}})} \,\alpha ({\widetilde{x}},\, e^{\frac{2\pi j}{\ell }}\cdot g\cdot {\widetilde{y}},k)+O(k^{-\infty })\quad \text {on }({\tilde{U}}_3\times {\tilde{U}}_4)\times {\tilde{U}}\end{aligned}$$

where

$$\begin{aligned}\alpha ({\tilde{x}}'',y'',k)\in S^{n-\frac{3}{4}(t-1)}_{\textrm{loc}}(1;\,({\tilde{U}}_3\times {\tilde{U}}_4)\times {\tilde{U}},\,T^{*,(0,q-r)}X^{\textrm{red}}_{\nu }\boxtimes (T^{*,(0,q)}X)^*) \end{aligned}$$

and the leading term $\alpha _0$ in the expansion of $\alpha$ can be computed explicitly along the diagonal.

Proof

Since, by Theorem 2.2, $S^{(q)}_{k\nu }$ has rapidly decreasing asymptotics as k goes to infinity away $X_{\nu }$, we obtain that $\sigma _k=O(k^{-\infty })$ on $X^{\textrm{red}}_{\nu }\times D$.

Now, let us prove the second statement. If $x,y\in X_{\nu }$ and $\pi (x)\ne \pi (e^{i\theta }\cdot y)$ for every $e^{i\theta }\in \overline{{\mathbb {T}}^1_{\nu }}$, since $S^{(q-r)}_{\textrm{red},k}$ is smoothing away from diagonal, then there exist open set U of $\pi (x)$ in $X^{\textrm{red}}_{\nu }$ and V of y in X such that $\chi \,S^{(q-r)}_{\textrm{red},k}\eta = O(k^{-\infty })$ on $X^{\textrm{red}}_{\nu }$, where ${\chi }\in C^{\infty }_0(U)$ and $\eta \in C^{\infty }_0(V)$. Furthermore by (8), we see that, for sufficiently small neighborhoods of x and y, we can integrate by parts in $\textrm{d}\theta _t$ and see that $S^{(q)}_{k\nu }$ has rapidly decreasing asymptotic. The second statements follow.

Eventually, let $p\in X_{\nu }$ and consider a small open neighborhood U with coordinates defined as above. Let $\chi \in C^{\infty }_0({\tilde{U}}_3)$ be a $G_U$ invariant bump function and assume $\chi =1$ on some neighborhood of p, it extends naturally to a function on ${\mathbb {T}}\cdot {\tilde{U}}_3$. Let us put $U^{\sharp } = \{\pi (x):\,x\in U\}$. Let $\eta \in C^{\infty }_0(X^{\textrm{red}}_{\nu })$ such that $\eta =1$ on some neighborhood of $U^{\sharp }$ and

$$\begin{aligned} \textrm{supp}(\eta )\subset \{\pi (x)\in X^{\textrm{red}}_{\nu } :\, x\in X_{\nu },\,\chi (x)=1 \}. \end{aligned}$$

Thus, we get

$$\begin{aligned} \eta \,\sigma _{k\nu }\sim (k\nu )^{(t-1)/4}\,\eta \,S^{(q-r)}_{\textrm{red},k}\circ \iota _{\textrm{red}}\circ {\hat{p}} \circ \tau _{x,n_-}\circ e \circ \iota ^*_{\nu } \circ \chi \,S^{(q)}_{k\nu } . \end{aligned}$$

Now, we can compose the operators and we can use the complex stationary phase formula of [11]; we get the theorem. $\square$

In fact, $\sigma _{k\nu }$ is an isomorphism for k large if we can prove that

$$\begin{aligned}\sigma _{k\nu }:\, H^{q}(X)_{k\nu }\rightarrow H^{q-r}(X^{\textrm{red}}_{\nu })_k\quad \text { and }\quad \sigma _{k\nu }^*:\, H^{q-r}(X^{\textrm{red}}_{\nu })_k\rightarrow H^{q}(X)_{k\nu } \end{aligned}$$

are injective for k large. Notice that, by Theorem 3.1, we have

$$\begin{aligned} \sigma _{k\nu }^*({x}'',{\tilde{y}}'')= \sum ^{\ell -1}_{j=0}\sum _{g\in G_U} e^{\frac{2\pi kj}{\ell }} e^{-i k\,\overline{\Psi ({x}'',e^{\frac{2\pi j}{\ell }}\cdot g\cdot {\tilde{y}}'')}} \,\beta ({x}'',\,e^{\frac{2\pi j}{\ell }}\cdot g\cdot {\tilde{y}}'',k)+O(k^{-\infty }) \end{aligned}$$

(12)

on ${\tilde{U}}\times ({\tilde{U}}_3\times {\tilde{U}}_4)$ where we can check $\beta _0({\tilde{x}}'',{\tilde{x}}'')=\alpha _0({\tilde{x}}'',{\tilde{x}}'')$.

The injectivity of the map $\sigma _{k\nu }$ is a consequence of the following theorem which is an adaptation of proof of the main theorem in [7].

Theorem 3.2

There exists a Fourier integral operator $R_k$ of equivariant Szegő type of degree $n-(t-1)/2$ such that

$$\begin{aligned} \sigma _{k\nu }^*\sigma _{k\nu } \equiv c_0 \,(1+R_{k})\,S^{(q)}_{k\nu } \end{aligned}$$

(13)

where $c_0$ is a positive constant and $1+R_{k}:\,\Omega ^{0,q}(X)\rightarrow \Omega ^{0,q}(X)$ is an injective Fourier integral operator of equivariant Szegő type.

Proof

By Theorem 3.1 and equation (12) we can compose $\sigma _{k\nu }^*$ and $\sigma _{k\nu }$ using the complex stationary phase formula of [11]. We can pick e in the definition of $\sigma _{k\nu }$ so that the leading term of the symbol of $\sigma _{k\nu }^*\sigma _{k\nu }$ agrees with the one of $S^{(q)}_{k\nu }$ along the diagonal. Thus we can find an operator

$$\begin{aligned}R_k(x,\,y)= \sum ^{\ell -1}_{j=0}\sum _{g\in G_U} e^{\frac{2\pi kj}{\ell }} e^{ik\,\Psi (x'',e^{\frac{2\pi j}{\ell }}\cdot g\cdot y'')}\,r(x'',\,e^{\frac{2\pi j}{\ell }}\cdot g\cdot y'',k)+O(k^{-\infty })\quad \text {on }{\tilde{U}}\times {\tilde{U}} \end{aligned}$$

such that

$$\begin{aligned}r\in S^{n-(t-1)/2}(1;\,{\tilde{U}}\times {\tilde{U}},\,T^{*(0,q)}X\boxtimes (T^{*(0,q)}X)^*) \end{aligned}$$

and

$$\begin{aligned}|r_0(x,\,y) |\le C|(x,y)-(x_0,x_0) |\end{aligned}$$

for all $x_0\in X_{\nu }\cap {\tilde{U}}$, 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 $\textrm{diag}(X_{\nu } \times X_{\nu } )$. In fact, as a consequence of this, by Lemma 6.7 and Lemma 6.8 in [7] we have that $\Vert R_k u \Vert \le \epsilon _k \Vert u\Vert$ for all $u \in \Omega ^{0,q}(X)$, for all $k\in {\mathbb {N}}$ where $\epsilon _k$ is a sequence with $\lim _{k\rightarrow +\infty } \epsilon _k = 0$; this in turn implies, if k is large enough, that the map $1+R_{k}$ is injective. $\square$

Analogously, the injectivity of the map $\sigma _{k\nu }^*$ follows by studying $\sigma _{k\nu }\sigma _{k\nu }^*$. We do not repeat the proof here since it is an application of the stationary phase formula for Fourier integral operators of complex type, see [11], and it follows in a similar way as above.