Abstract
Let \((X, T^{1,0}X)\) be a compact connected orientable CR manifold of dimension \(2n+1\) with non-degenerate Levi curvature. Assume that X admits a connected compact Lie group G action. Under certain natural assumptions about the group G action, we show that the G-invariant Szegő kernel for (0, q) forms is a complex Fourier integral operator, smoothing away \(\mu ^{-1}(0)\) and there is a precise description of the singularity near \(\mu ^{-1}(0)\), where \(\mu \) denotes the CR moment map. We apply our result to the case when X admits a transversal CR \(S^1\) action and deduce an asymptotic expansion for the mth Fourier component of the G-invariant Szegő kernel for (0, q) forms as \(m\rightarrow +\infty \) and when \(q=0\), we recover Xiaonan Ma and Weiping Zhang’s result about the existence of the G-invariant Bergman kernel for ample line bundles. As an application, we show that if m large enough, quantization commutes with reduction.
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1 Introduction and statement of the main results
Let \((X, T^{1,0}X)\) be a CR manifold of dimension \(2n+1\), \(n\ge 1\). Let \(\Box ^{(q)}_b\) be the Kohn Lalpacian acting on (0, q) forms. The orthogonal projection \(S^{(q)}:L^2_{(0,q)}(X)\rightarrow \mathrm{Ker}\,\Box ^{(q)}_b\) onto \(\mathrm{Ker}\,\Box ^{(q)}_b\) is called the Szegő projection, while its distribution kernel \(S^{(q)}(x,y)\) is called the Szegő kernel. The study of the Szegő projection and kernel is a classical subject in several complex variables and CR geometry. A very important case is when X is a compact strictly pseudoconvex CR manifold. Assume first that X is the boundary of a strictly pseudoconvex domain. Boutet de Monvel-Sjöstrand [2] showed that \(S^{(0)}(x,y)\) is a complex Fourier integral operator.
The Boutet de Monvel-Sjöstrand description of the Szegő kernel had a profound impact in many research areas, especially through [4]: several complex variables, symplectic and contact geometry, geometric quantization, Kähler geometry, semiclassical analysis, quantum chaos, etc. cf. [6, 8, 11, 22, 29, 32], to quote just a few. These ideas also partly motivated the introduction of the recent direct approaches and their various extensions, see [18, 19, 21, 22].
Now, we consider a connected compact Lie group G acting on X. The study of G-invariant Szegő kernel is closely related to Mathematical physics and geometric quantization of CR manifolds. It is a fundamental problem to establish G-invariant Boutet de Monvel-Sjöstrand type theorems for G-invariant Szegő kernels and study the consequence of the G-invariant Szegő kernel. This is the motivation of this work. In this paper, we consider G-invariant Szegő kernel for (0, q) forms and we show that the G-invariant Szegő kernel for (0, q) forms is a complex Fourier integral operator. In particular, \(S^{(q)}(x,y)\) is smoothing outside \(\mu ^{-1}(0)\) and there is a precise description of the singularity near \(\mu ^{-1}(0)\), where \(\mu \) denotes the CR moment map. We apply our result to the case when X admits a transversal CR \(S^1\) action and deduce an asymptotic expansion for the mth Fourier component of the Szegő kernel for (0, q) forms as \(m\rightarrow +\infty \). As an application, we show that, if m large enough, quantization commutes with reduction.
In [20], Ma and Zhang have studied the asymptotic expansion of the invariant Bergman kernel of the \({\text {spin}}^c\) Dirac operator associated with high tensor powers of a positive line bundle on a symplectic manifold admitting a Hamiltonian action of a compact connected Lie group and its relation to the asymptotic expansion of Bergman kernel on symplectic reduced space, also the Toeplitz operator aspect in [20, Section 4.5]. Their approach is inspired by the analytic localization techniques developed by Bismut and Lebeau [3]. About the quantization commutes with reduction problem in symplectic geometry, we refer the readers to [22]. In the second part of [22], Ma described how the G-invariant Bergman kernel concentrates on the Bergman kernel of the reduced space. Note that the “quantization commutes with reduction” in the situations in symplectic case is a very active subject. When the action connected Lie group is compact and the symplectic manifold is also compact, this question was solved finally by Meinrenken [24] and Tian-Zhang [31]. When the action connected Lie group is compact and the symplectic manifold is noncompact, this is a famous conjecture of Vergne and was solved by Ma-Zhang in [23].
It should be mentioned that in [7], Charles relates the Toeplitz operators on a compact complex manifold M with the Toeplitz operators on the “reduced” space for torus action, and in [26], Paoletti studied equivariant Szegő kernels on complex manifolds ( cf. [20, Remark 0.5]).
We now formulate the main results. We refer to Sect. 2 for some notations and terminology used here. Let \((X, T^{1,0}X)\) be a compact connected orientable CR manifold of dimension \(2n+1\), \(n\ge 1\), where \(T^{1,0}X\) denotes the CR structure of X. Fix a global non-vanishing real 1-form \(\omega _0\in C^\infty (X,T^*X)\) such that \(\langle \,\omega _0\,,\,u\,\rangle =0\), for every \(u\in T^{1,0}X\oplus T^{0,1}X\). The Levi form of X at \(x\in X\) is the Hermitian quadratic form on \(T^{1,0}_xX\) given by \(\mathcal {L}_x(U,\overline{V})=-\frac{1}{2i}\langle \,d\omega _0(x)\,,\,U\wedge \overline{V}\,\rangle \), for all \(U, V\in T^{1,0}_xX\). In this work, we assume that
Assumption 1.1
The Levi form 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 \(n_-+n_+=n\).
Let \(HX=\left\{ \mathrm{Re}\,u;\, u\in T^{1,0}X\right\} \) and let \(J:HX\rightarrow HX\) be the complex structure map given by \(J(u+\overline{u})=iu-i\overline{u}\), for every \(u\in T^{1,0}X\). In this work, we assume that X admits a d-dimensional connected compact Lie group G action. We assume throughout that
Assumption 1.2
The G action preserves \(\omega _0\) and J. That is, \(g^*\omega _0=\omega _0\) on X and \(g_*J=Jg_*\) on HX, for every \(g\in G\), where \(g^*\) and \(g_*\) denote the pull-back map and push-forward map of G, respectively.
Let \(\mathfrak {g}\) denote the Lie algebra of G. For any \(\xi \in \mathfrak {g}\), we write \(\xi _X\) to denote the vector field on X induced by \(\xi \). That is, \((\xi _X u)(x)=\frac{\partial }{\partial t}\left( u(\exp (t\xi )\circ x)\right) |_{t=0}\), for any \(u\in C^\infty (X)\).
Definition 1.3
The moment map associated to the form \(\omega _0\) is the map \(\mu :X \rightarrow \mathfrak {g}^*\) such that, for all \(x \in X\) and \(\xi \in \mathfrak {g}\), we have
In this work, we assume that
Assumption 1.4
0 is a regular value of \(\mu \), the action G on \(\mu ^{-1}(0)\) is freely and
where \(\underline{\mathfrak {g}}=\mathrm{Span}\,(\xi _X;\, \xi \in \mathfrak {g})\), \(\underline{\mathfrak {g}}^{\perp _b}=\left\{ v\in HX;\, b(\xi _X,v)=0,\ \ \forall \xi _X\in \underline{\mathfrak {g}}\right\} \), b is the nondegenerate bilinear form on HX given by (2.4).
By Assumption 1.4, \(\mu ^{-1}(0)\) is a d-codimensional submanifold of X. Let \(Y:=\mu ^{-1}(0)\) and let \(HY:=HX\bigcap TY\). Note that if the Levi form is positive at Y, then (1.2) holds. Under the condition (1.2), in Sect. 2.5, we will show that \(\mathrm{dim}\,(HY\bigcap JHY)=2n-2d\) at every point of Y, \(\mu ^{-1}(0)/G=:Y_G\) is a CR manifold with natural CR structure induced by \(T^{1,0}X\) of dimension \(2n-2d+1\) and we can identify \(HY_G\) with \(HY\bigcap JHY\).
Fix a G-invariant smooth Hermitian metric \(\langle \, \cdot \,|\, \cdot \,\rangle \) on \(\mathbb {C}TX\) so that \(T^{1,0}X\) is orthogonal to \(T^{0,1}X\), \(\underline{\mathfrak {g}}\) is orthogonal to \(HY\bigcap JHY\) at every point of Y, \(\langle \, u \,|\, v \, \rangle \) is real if u, v are real tangent vectors, \(\langle \,T\,|\,T\,\rangle =1\) and T is orthogonal to \(T^{1,0}X\oplus T^{0,1}X\), where T is given by (2.2). The Hermitian metric \(\langle \,\cdot \,|\, \cdot \, \rangle \) on \(\mathbb {C}TX\) induces, by duality, a Hermitian metric on \(\mathbb {C}T^*X\) and also on the bundles of (0, q) forms \(T^{*0,q}X, q=0, 1, \cdots , n\). We shall also denote all these induced metrics by \(\langle \, \cdot \,|\, \cdot \,\rangle \). Fix \(g\in G\). Let \(g^*:\Lambda ^r_x(\mathbb {C}T^*X)\rightarrow \Lambda ^r_{g^{-1}\circ x}(\mathbb {C}T^*X)\) be the pull-back map. Since G preserves J, we have \(g^*:T^{*0,q}_xX\rightarrow T^{*0,q}_{g^{-1}\circ x}X\), for all \(x\in X\). Thus, for \(u\in \Omega ^{0,q}(X)\), we have \(g^*u\in \Omega ^{0,q}(X)\). Put \(\Omega ^{0,q}(X)^G:=\left\{ u\in \Omega ^{0,q}(X);\, g^*u=u,\ \ \forall g\in G\right\} \). Since the Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(\mathbb {C}TX\) is G-invariant, the \(L^2\) inner product \((\,\cdot \,|\,\cdot \,)\) on \(\Omega ^{0,q}(X)\) induced by \(\langle \,\cdot \,|\,\cdot \,\rangle \) is G-invariant. Let \(u\in L^2_{(0,q)}(X)\) and \(g\in G\), we can also define \(g^*u\) in the standard way (see the discussion in the beginning of Sect. 3.2). Put \(L^2_{(0,q)}(X)^G:=\left\{ u\in L^2_{(0,q)}(X);\, g^*u=u,\ \ \forall g\in G\right\} \). Let \(\Box ^{(q)}_b : \mathrm{Dom}\,\Box ^{(q)}_b\rightarrow L^2_{(0,q)}(X)\) be the Gaffney extension of Kohn Laplacian (see (3.1)). Put \((\mathrm{Ker}\,\Box ^{(q)}_b)^G:=\mathrm{Ker}\,\Box ^{(q)}_b\bigcap L^2_{(0,q)}(X)^G\). The G-invariant Szegő projection is the orthogonal projection \(S^{(q)}_G:L^2_{(0,q)}(X)\rightarrow (\mathrm{Ker}\,\Box ^{(q)}_b)^G\) with respect to \((\,\cdot \,|\,\cdot \,)\). Let \(S^{(q)}_G(x,y)\in D'(X\times X,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\) be the distribution kernel of \(S^{(q)}_G\). The first main result of this work is the following
Theorem 1.5
With the assumptions and notations above, suppose that \(\Box ^{(q)}_b : \mathrm{Dom}\,\Box ^{(q)}_b\rightarrow L^2_{(0,q)}(X)\) has closed range. If \(q\notin \left\{ n_-, n_+\right\} \), then \(S^{(q)}_G\equiv 0\) on X.
Suppose \(q\in \left\{ n_-, n_+\right\} \). Let D be an open set of X with \(D\bigcap \mu ^{-1}(0)=\emptyset \). Then, \(S^{(q)}_G\equiv 0\) on D.
Let \(p\in \mu ^{-1}(0)\) and let U be an open set of p and let \(x=(x_1,\ldots ,x_{2n+1})\) be local coordinates defined in U. Then, there exist continuous operators \(S^G_-, S^G_+:\Omega ^{0,q}_0(U)\rightarrow \Omega ^{0,q}(U)\) such that
and \(S^G_-(x,y)\), \(S^G_+(x,y)\) satisfy
with
\(a^0_-(x,x)\), \(a^0_+(x,x)\), \(x\in \mu ^{-1}(0)\bigcap U\), are given by (1.8) below, \(\Phi _-(x,y)\in C^\infty (U\times U)\),
there is a constant \(C\ge 1\) such that, for all \((x,y)\in U\times U\),
and \(\Phi _-(x,y)\) satisfies (1.18) below and (1.19) below, and \(\Phi _+(x,y)\in C^\infty (U\times U)\), \(-\overline{\Phi }_+(x,y)\) satisfies (1.4), (1.5), (1.18) below and (1.19) below.
We refer the reader to the discussion before (2.1) and Definition 3.1 for the precise meanings of \(A\equiv B\) and the symbol space \(S^{n-\frac{d}{2}}_{\mathrm{cl}}\,\), respectively.
Let \(\Phi \in C^\infty (U\times U)\). We assume that \(\Phi \) satisfies (1.4), (1.5), (1.18), (1.19). We will show in Theorem 5.2 that the functions \(\Phi \) and \(\Phi _-\) are equivalent on U in the sense of Definition 5.1 if and only if there is a function \(f\in C^\infty (U\times U)\) with \(f(x,x)=1\), for every \(x\in \mu ^{-1}(0)\), such that \(\Phi (x,y)-f(x,y)\Phi _-(x,y)\) vanishes to infinite order at \(\mathrm{diag}\,\Bigr ((\mu ^{-1}(0)\bigcap U)\times (\mu ^{-1}(0)\bigcap U)\Bigr )\). From this observation, we see that the leading term \(a^0_-(x,x)\), \(x\in \mu ^{-1}(0)\), is well-defined. To state the formula for \(a^0_-(x,x)\), we introduce some notations. For a given point \(x_0\in X\), let \(\{W_j\}_{j=1}^{n}\) be an orthonormal frame of \((T^{1,0}X,\langle \,\cdot \,|\,\cdot \,\rangle )\) near \(x_0\), for which the Levi form is diagonal at \(x_0\). Put
We will denote by
Let \(\{T_j\}_{j=1}^{n}\) denote the basis of \(T^{*0,1}X\), dual to \(\{\overline{W}_j\}^{n}_{j=1}\). We assume that \(\mu _j(x_0)<0\) if \(1\le j\le n_-\) and \(\mu _j(x_0)>0\) if \(n_-+1\le j\le n\). Put
and let
be the orthogonal projections onto \(\mathcal {N}(x_0,n_-)\) and \(\mathcal {N}(x_0,n_+)\) with respect to \(\langle \,\cdot \,|\,\cdot \,\rangle \), respectively.
Fix \(x\in \mu ^{-1}(0)\), consider the linear map
Let \(\det R_x=\lambda _1(x)\cdots \lambda _d(x)\), where \(\lambda _j(x)\), \(j=1,2,\ldots ,d\), are the eigenvalues of \(R_x\).
Fix \(x\in \mu ^{-1}(0)\), put \(Y_x=\left\{ g\circ x;\, g\in G\right\} \). \(Y_x\) is a d-dimensional submanifold of X. The G-invariant Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) induces a volume form \(dv_{Y_x}\) on \(Y_x\). Put
Note that the function \(V_{\mathrm{eff}}\,(x)\) was already appeared in Ma-Zhang [23, (0,10)] as exactly the role in the expansion, cf. [23, (0.14)].
Theorem 1.6
With the notations used above, for \(a^0_-(x,y)\) and \(a^0_+(x,y)\) in (1.3), we have
We now assume that X admits an \(S^1\) action: \(S^1\times X\rightarrow X\). We write \(e^{i\theta }\) to denote the \(S^1\) action. Let \(T\in C^\infty (X, TX)\) be the global real vector field induced by the \(S^1\) action given by \((Tu)(x)=\frac{\partial }{\partial \theta }\left( u(e^{i\theta }\circ x)\right) |_{\theta =0}\), \(u\in C^\infty (X)\). We assume that the \(S^1\) action \(e^{i\theta }\) is CR and transversal (see Definition 4.1). We take \(\omega _0\in C^\infty (X,T^*X)\) to be the global real one form determined by \(\langle \,\omega _0\,,\,u\,\rangle =0\), for every \(u\in T^{1,0}X\oplus T^{0,1}X\) and \(\langle \,\omega _0\,,\,T\,\rangle =-1\). In this paper, we assume that
Assumption 1.7
and
Let \(u\in \Omega ^{0,q}(X)\) be arbitrary. Define
For every \(m\in \mathbb {Z}\), let
We denote \(C^\infty _m(X):=\Omega ^{0,0}_m(X)\), \(C^\infty _m(X)^G:=\Omega ^{0,0}_m(X)^G\). From the CR property of the \(S^1\) action and (1.9), it is not difficult to see that \(Tg^*\overline{\partial }_b=g^*T\overline{\partial }_b=\overline{\partial }_bg^*T=\overline{\partial }_bTg^*\) on \(\Omega ^{0,q}(X)\), for all \(g\in G\). Hence,
We now assume that the Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(\mathbb {C}TX\) is \(G\times S^1\) invariant. Then the \(L^2\) inner product \((\,\cdot \,|\,\cdot \,)\) on \(\Omega ^{0,q}(X)\) induced by \(\langle \,\cdot \,|\,\cdot \,\rangle \) is \(G\times S^1\)-invariant. We then have
where \(\overline{\partial }^*_b\) is the \(L^2\) adjoint of \(\overline{\partial }_b\) with respect to \((\,\cdot \,|\,\cdot \,)\).
Let \(L^2_{(0,q), m}(X)^G\) be the completion of \(\Omega _m^{0,q}(X)^G\) with respect to \((\,\cdot \,|\,\cdot \,)\). We write \(L^2_m(X)^G:=L^2_{(0,0),m}(X)^G\). Put \(H^q_{b,m}(X)^G:=(\mathrm{Ker}\,\Box ^{(q)}_b)^G_m:=(\mathrm{Ker}\,\Box ^{(q)}_b)^G\bigcap L^2_{(0,q),m}(X)^G\). It is not difficult to see that, for every \(m\in \mathbb {Z}\), \((\mathrm{Ker}\,\Box ^{(q)}_b)^G_m\subset \Omega ^{0,q}_m(X)^G\) and \(\mathrm{dim}\,(\mathrm{Ker}\,\Box ^{(q)}_b)^G_m<\infty \). The mth G-invariant Szegő projection is the orthogonal projection \(S^{(q)}_{G,m}:L^2_{(0,q)}(X)\rightarrow (\mathrm{Ker}\,\Box ^{(q)}_b)^G_m\) with respect to \((\,\cdot \,|\,\cdot \,)\). Let \(S^{(q)}_{G,m}(x,y)\in C^\infty (X\times X,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\) be the distribution kernel of \(S^{(q)}_{G,m}\). The second main result of this work is the following
Theorem 1.8
With the assumptions and notations used above, if \(q\notin n_-\), then, as \(m\rightarrow +\infty \), \(S^{(q)}_{G,m}=O(m^{-\infty }) \ \text{ on } X\).
Suppose \(q=n_-\). Let D be an open set of X with \(D\bigcap \mu ^{-1}(0)=\emptyset \). Then, as \(m\rightarrow +\infty \), \(S^{(q)}_{G,m}=O(m^{-\infty })\ \ \text{ on } D\).
Let \(p\in \mu ^{-1}(0)\) and let U be an open set of p and let \(x=(x_1,\ldots ,x_{2n+1})\) be local coordinates defined in U. Then, as \(m\rightarrow +\infty \),
and
where \(\tau _{x,n_-}\) is given by (1.7), and \(\Psi (x,y)\in C^\infty (U\times U)\), \(d_x\Psi (x,x)=-d_y\Psi (x,x)=-\omega _0(x)\), for every \(x\in \mu ^{-1}(0)\), \(\Psi (x,y)=0\) if and only if \(x=y\in \mu ^{-1}(0)\) and there is a constant \(C\ge 1\) such that, for all \((x,y)\in U\times U\),
(We refer the reader to Theorem 1.12 for more properties of the phase \(\Psi (x,y)\).)
We refer the reader to the discussion in the beginning of Sect. 2.2 and Definition 2.1 for the precise meanings of \(A=B+O(m^{-\infty })\) and the symbol space \(S^{n-\frac{d}{2}}_{\mathrm{loc}}\,\), respectively.
It is was proved in Theorem 1.12 in [15]) that when X admits a transversal and CR \(S^1\) action and the Levi form is non-degenerate of constant signature on X, then \(\Box ^{(q)}_b\) has \(L^2\) closed range.
Let \(Y_G:=\mu ^{-1}(0)/G\). In Theorem 2.5, we will show that \(Y_G\) is a CR manifold with natural CR structure induced by \(T^{1,0}X\) of dimension \(2n-2d+1\). Let \(\mathcal {L}_{Y_G,x}\) be the Levi form on \(Y_G\) at \(x\in Y_G\) induced naturally from \(\mathcal {L}\). Note that the bilinear form b is non-degenerate on \(\mu ^{-1}(0)\), where b is given by (2.4). Hence, on \((\underline{\mathfrak {g}}, \underline{\mathfrak {g}})\), b has constant signature on \(\mu ^{-1}(0)\). Assume that on \((\underline{\mathfrak {g}}, \underline{\mathfrak {g}})\), b has r negative eigenvalues and \(d-r\) positive eigenvalues on \(\mu ^{-1}(0)\). Hence \(\mathcal {L}_{Y_G}\) has \(q-r\) negative and \(n-d-q+r\) positive eigenvalues at each point of \(Y_G\). Let \(\Box ^{(q-r)}_{b,Y_G}\) be the Kohn Laplacian for \((0,q-r)\) forms on \(Y_G\). Fix \(m\in \mathbb {N}\). Let \(H^{q-r}_{b,m}(Y_G):=\left\{ u\in \Omega ^{0,q-r}(Y_G);\, \Box ^{(q-r)}_{b,Y_G}u=0,\ \ Tu=imu\right\} \). We will apply Theorem 1.8 to establish an isomorphism between \(H^{q}_{b,m}(X)^G\) and \(H^{q-r}_{b,m}(Y_G)\) if m large enough. We introduce some notations.
Since \(\underline{\mathfrak {g}}_x\) is orthogonal to \(H_xY\bigcap JH_xY\) and \(H_xY\bigcap JH_xY\subset \underline{\mathfrak {g}}^{\perp _b}_x\) (see Lemma 2.4 and (2.5) for the meaning of \(\underline{\mathfrak {g}}^{\perp _b}_x\)), for every \(x\in Y\), we can find a G-invariant orthonormal basis \(\left\{ Z_1,\ldots ,Z_n\right\} \) of \(T^{1,0}X\) on Y such that
Let \(\left\{ e_1,\ldots ,e_n\right\} \) denote the orthonormal basis of \(T^{*0,1}X\) on Y, dual to \(\left\{ \overline{Z}_1,\ldots ,\overline{Z}_n\right\} \). Fix \(s=0,1,2,\ldots ,n-d\). For \(x\in Y\), put
and let \(B^{*0,s}X\) be the vector bundle of Y with fiber \(B^{*0,s}_xX\), \(x\in Y\). Let \(C^\infty (Y,B^{*0,s}X)^G\) denote the set of all G-invariant sections of Y with values in \(B^{*0,s}X\). Let
be the natural identification.
Assume that \(\lambda _1<0,\ldots ,\lambda _r<0\), and \(\lambda _{d+1}<0,\ldots ,\lambda _{n_--r+d}<0\). For \(x\in Y\), put
and let
Let \(\iota :Y\rightarrow X\) be the natural inclusion and let \(\iota ^*:\Omega ^{0,q}(X)\rightarrow \Omega ^{0,q}(Y)\) be the pull-back of \(\iota \). Let \(q=n_-\). Let \(S^{(q-r)}_{Y_G,m}:L^2_{(0,q-r)}(Y_G)\rightarrow H^{q-r}_{b,m}(Y_G)\) be the orthogonal projection and let
Let
In Sect. 6.2, we will show that
Theorem 1.9
With the notations and assumptions above, suppose that \(q=n_-\). If m is large, then \(\sigma _m:H^q_{b,m}(X)^G\rightarrow H^{q-r}_{b,m}(Y_G)\) is an isomorphism.
In particular, if m large enough, then \( \mathrm{dim}\,H^q_{b,m}(X)^G=\mathrm{dim}\,H^{q-r}_{b,m}(Y_G). \)
Remark 1.10
Let’s sketch the idea of the proof of Theorem 1.9. W can consider \(\sigma _m\) as a map from \(\Omega ^{0,q}(X)\rightarrow H^{q-r}_{b,m}(Y_G)\):
Let \(\sigma ^*_m:\Omega ^{0,q-r}(Y_G)\rightarrow \Omega ^{0,q}(X)\) be the adjoint of \(\sigma _m\). From Theorem 1.8 and some calculation of complex Fourier integral operators, we will show in Sect. 6.2 that \(F_m=\sigma ^*_m\sigma _m:\Omega ^{0,q}(X)\rightarrow \Omega ^{0,q}(X)\) is the same type of operator as \(S^{(q)}_{G,m}\) and
where \(C_0>0\) is a constant and \(R_m\) is also the same type of operator as \(S^{(q)}_{G,m}\), but the leading symbol of \(R_m\) vanishes at \(\mathrm{diag}\,(Y\times Y)\). By using the fact that the leading symbol of \(R_m\) vanishes at \(\mathrm{diag}\,(Y\times Y)\), we will show in Lemma 6.8 that \( \left\| R_mu\right\| \le \varepsilon _m\left\| u\right\| \), for all \(u\in \Omega ^{0,q}(X)\), for all \(m\in \mathbb {N}\), where \(\varepsilon _m\) is a sequence with \(\lim _{m\rightarrow \infty }\varepsilon _m=0\). In particular, if m is large enough, then the map
is injective. From (1.11) and (1.12), we deduce that, if m is large enough, then \(F_m:H^q_{b,m}(X)^G\rightarrow H^q_{b,m}(X)^G\) is injective. Hence \(\sigma _m:H^q_{b,m}(X)^G\rightarrow H^{q-r}_{b,m}(Y_G)\) is injective.
Similarly, we can repeat the argument above with minor change and deduce that if m is large enough, then the map \(\hat{F}_m=\sigma _m\sigma ^*_m: H^{q-r}_{b,m}(Y_G)\rightarrow H^{q-r}_{b,m}(Y_G)\) is injective. Hence, if m is large enough, then the map \(\sigma ^*_m:H^{q-r}_{b,m}(Y_G)\rightarrow H^{q}_{b,m}(X)^G\) is injective. Thus, \(\mathrm{dim}\,H^q_{b,m}(X)^G=\mathrm{dim}\,H^{q-r}_{b,m}(Y_G)\) and \(\sigma _m\) is an isomorphism if m large enough.
Let’s apply Theorem 1.9 to complex case. Let \((L,h^L)\) be a holomorphic line bundle over a connected compact complex manifold (M, J) with \(\mathrm{dim}\,_{\mathbb {C}}M=n\), where J denotes the complex structure map of M and \(h^L\) is a Hermitian fiber metric of L. Let \(R^L\) be the curvature of L induced by \(h^L\). Assume that \(R^L\) is non-degenerate of constant signature \((n_-,n_+)\) on M. Let K be a connected compact Lie group with Lie algebra \(\mathfrak {k}\). We assume that \(\mathrm{dim}\,_{\mathbb {R}}K=d\) and K acts holomorphically on (M, J), and that the action lifts to a holomorphic action on L. We assume further that \(h^L\) is preserved by the K-action. Then \(R^L\) is a K-invariant form. Let \(\omega =\frac{i}{2\pi }R^L\) and let \(\tilde{\mu }: M \rightarrow \mathfrak {k}^*\) be the moment map induced by \(\omega \). Assume that \(0 \in \mathfrak {k}^*\) is regular and the action of K on \(\tilde{\mu }^{-1}(0)\) is freely. The analogue of the Marsden-Weinstein reduction holds (see [10]). More precisely, the complex structure J on M induces a complex structure \(J_K\) on \(M_0:=\tilde{\mu }^{-1}(0)/K\), for which the line bundle \(L_0:=L/K\) is a holomorphic line bundle over \(M_0\).
For any \(\xi \in \mathfrak {k}\), we write \(\xi _M\) to denote the vector field on M induced by \(\xi \). Let \(\underline{\mathfrak {k}}=\mathrm{Span}\,(\xi _M;\, \xi \in \mathfrak {k})\). On \(\tilde{\mu }^{-1}(0)\), let \(b^L\) be the bilinear form on \(\underline{\mathfrak {k}}\times \underline{\mathfrak {k}}\) given by \(b^L(\,\cdot \,,\,\cdot \,)=\omega (\,\cdot \,,\,J\cdot \,)\). Assume that \(b^L\) has r negative eigenvalues and \(d-r\) positive eigenvalues on \(\tilde{\mu }^{-1}(0)\). Let \(q=n_-\). For \(m\in \mathbb {N}\), let \(H^q(M,L^m)^K\) denote the K-invariant qth Dolbeault cohomology group with values in \(L^m\) and let \(H^{q-r}(M_0,L^m_0)\) denote the \((q-r)\)th Dolbeault cohomology group with values in \(L^m_0\). Theorem 1.9 implies that, if m is large enough, then there is an isomorphism map: \(\widetilde{\sigma }_m:H^q(M,L^m)^K\rightarrow H^{q-r}(M_0,L^m_0)\). In particular, if m is large enough,then
Note that when \(m=1\) and \(q=0\), the equality (1.13) was first proved in [10, §5]. For \(m=1\), the equality (1.13) was established in [30, 33] when L is positive. Zhang [33] combined the methods and results in [31] with Braverman’s idea [5] to construct a suitable quasi-isomorphisim to prove the equality (1.13). The proof of the equality (1.13) in [30] is completely algebraic, while the the proof of the equality (1.13) in [33] is purely analytic where different quasi-homomorphisms between Dolbeault complexes under considerations were constructed to prove the equality (1.13). If m is large enough and \(q=0\), an isomorphism map in (1.13) was also constructed in [20, (0.27), Corollary 4.13].
If m large enough and \(q=0\), an isomorphism map in (1.13) was also constructed in [20, (0.27), Corollary 4.13]. The point of [20, (0.27), Corollary 4.13] is to study the isometric aspect of this map, as an consequence of the asymptotic of G-invariant Bergman kernel of Ma-Zhang [20], they gave another proof that it is an isomorphism for m large, and this approaches of the isomorphism for m large is adopted in this paper. It should be mentioned that in this situation, a version of the full asymptotics of \(S^{(0)}_{G,m}(x,y)\) including (1.10) was established in [20, Theorem 0.1, 0.2].
1.1 The phase functions \(\Phi _-(x,y)\) and \(\Psi (x,y)\)
In this section, we collect some properties of the phase functions \(\Phi _-(x,y)\), \(\Psi (x,y)\) in Theorem 1.5 and Theorem 1.8.
Let \(v=(v_1,\ldots ,v_d)\) be local coordinates of G defined in a neighborhood V of \(e_0\) with \(v(e_0)=(0,\ldots ,0)\). From now on, we will identify the element \(e\in V\) with v(e). Fix \(p\in \mu ^{-1}(0)\). In Theorem 3.7, we will show that there exist local coordinates \(v=(v_1,\ldots ,v_d)\) of G defined in a neighborhood V of \(e_0\) with \(v(e_0)=(0,\ldots ,0)\), local coordinates \(x=(x_1,\ldots ,x_{2n+1})\) of X defined in a neighborhood \(U=U_1\times U_2\) of p with \(0\leftrightarrow p\), where \(U_1\subset \mathbb {R}^d\) is an open set of \(0\in \mathbb {R}^d\), \(U_2\subset \mathbb {R}^{2n+1-d}\) is an open set of \(0\in \mathbb {R}^{2n+1-d} \) and a smooth function \(\gamma =(\gamma _1,\ldots ,\gamma _d)\in C^\infty (U_2,U_1)\) with \(\gamma (0)=0\in \mathbb {R}^d\) such that
where \(a_j(x)\) is a smooth function on \(\mu ^{-1}(0)\bigcap U\), independent of \(x_1,\ldots ,x_{2d}\), \(x_{2n+1}\) and \(a_j(0)=0\), \(j=1,\ldots ,d\),
and
where \(b_{d+1}\in \mathbb {R},\ldots ,b_{2n}\in \mathbb {R}\). Put \(x''=(x_{d+1},\ldots ,x_{2n+1})\), \(\hat{x}''=(x_{d+1}, x_{d+2},\ldots ,x_{2d})\), \(\mathring{x}''=(x_{d+1},\ldots ,x_{2n})\). We have the following (see Theorem 3.11 and Theorem 3.12)
Theorem 1.11
With the notations above, the phase function \(\Phi _-(x,y)\in C^\infty (U\times U)\) is independent of \((x_1,\ldots ,x_d)\) and \((y_1,\ldots ,y_d)\). Hence, \(\Phi _-(x,y)=\Phi _-((0,x''),(0,y'')):=\Phi _-(x'',y'')\). Moreover, there is a constant \(c>0\) such that
Furthermore,
where \(z_j=x_{2j-1}+ix_{2j}\), \(w_j=y_{2j-1}+iy_{2j}\), \(j=d+1,\ldots ,n\), \(\mu _j\), \(j=1,\ldots ,n\), and \(b_{d+1}\in \mathbb {R},\ldots ,b_{2n}\in \mathbb {R}\) are as in (1.17) and f is smooth and satisfies \(f(0,0)=0\), \(f(x, y)=\overline{f}(y, x)\).
We now assume that X admits an \(S^1\) action: \(S^1\times X\rightarrow X\). We will use the same notations as in Theorem 1.8. Recall that we work with Assumption 1.7. Let \(p\in \mu ^{-1}(0)\). We can repeat the proof of Theorem 3.7 with minor change and show that there exist local coordinates \(v=(v_1,\ldots ,v_d)\) of G defined in a neighborhood V of \(e_0\) with \(v(e_0)=(0,\ldots ,0)\), local coordinates \(x=(x_1,\ldots ,x_{2n+1})\) of X defined in a neighborhood \(U=U_1\times (\hat{U}_2\times ]-2\delta ,2\delta [)\) of p with \(0\leftrightarrow p\), where \(U_1\subset \mathbb {R}^d\) is an open set of \(0\in \mathbb {R}^d\), \(\hat{U}_2\subset \mathbb {R}^{2n-d}\) is an open set of \(0\in \mathbb {R}^{2n-d} \), \(\delta >0\), and a smooth function \(\gamma =(\gamma _1,\ldots ,\gamma _d)\in C^\infty (\hat{U}_2\times ]-2\delta ,2\delta [,U_1)\) with \(\gamma (0)=0\in \mathbb {R}^d\) such that \(T=-\frac{\partial }{\partial x_{2n+1}}\) and (1.14), (1.15), (1.16), (1.17) hold. We have the following
Theorem 1.12
With the notations above, the phase function \(\Psi \) satisfies \(\Psi (x,y)=-x_{2n+1}+y_{2n+1}+\hat{\Psi }(\mathring{x}'',\mathring{y}'')\), where \(\hat{\Psi }(\mathring{x}'',\mathring{y}'')\in C^\infty (U\times U)\), \(\mathring{x}''=(x_{d+1},\ldots ,x_{2n})\), \(\mathring{y}''=(y_{d+1},\ldots ,y_{2n})\), and \(\Psi \) satisfies (1.18) and (1.19).
2 Preliminaries
2.1 Standard notations
Let M be a \(C^\infty \) paracompact manifold. We 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 \(\mathbb {C}TM\) and \(\mathbb {C}T^*M\), respectively. Write \(\langle \,\cdot \,,\cdot \,\rangle \) to denote the pointwise duality between TM and \(T^*M\). We extend \(\langle \,\cdot \,,\cdot \,\rangle \) bilinearly to \(\mathbb {C}TM\times \mathbb {C}T^*M\). Let B be a \(C^\infty \) vector bundle over M. The fiber of B at \(x\in M\) will be denoted by \(B_x\). Let E be a vector bundle over a \(C^\infty \) paracompact manifold \(M_1\). We write \(B\boxtimes E^*\) to denote the vector bundle over \(M\times M_1\) with fiber over \((x, y)\in M\times M_1\) consisting of the linear maps from \(E_y\) to \(B_x\). Let \(Y\subset M\) be an open set. From now on, the spaces of distribution sections of B over Y and smooth sections of B over Y will be denoted by \(D'(Y, B)\) and \(C^\infty (Y, B)\), respectively. Let \(E'(Y, B)\) be the subspace of \(D'(Y, B)\) whose elements have compact support in Y.
We recall the Schwartz kernel theorem [12, Theorems 5.2.1, 5.2.6], [19, Thorem B.2.7]. Let B and E be \(C^\infty \) vector bundles over paracompact orientable \(C^\infty \) manifolds M and \(M_1\), respectively, equipped with smooth densities of integration. If \(A: C^\infty _0(M_1,E)\rightarrow D'(M,B)\) is continuous, we write \(K_A(x, y)\) or A(x, y) to denote the distribution kernel of A. The following two statements are equivalent
-
(1)
A is continuous: \(E'(M_1,E)\rightarrow C^\infty (M,B)\),
-
(2)
\(K_A\in C^\infty (M\times M_1,B\boxtimes E^*)\).
If A satisfies (1) or (2), we say that A is smoothing on \(M \times M_1\). Let \(A,\hat{A}: C^\infty _0(M_1,E)\rightarrow D'(M,B)\) be continuous operators. We write
if \(A-\hat{A}\) is a smoothing operator. If \(M=M_1\), we simply write “on M”.
Let \(H(x,y)\in D'(M\times M_1,B\boxtimes E^*)\). We write H to denote the unique continuous operator \(C^\infty _0(M_1,E)\rightarrow D'(M,B)\) with distribution kernel H(x, y). In this work, we identify H with H(x, y).
2.2 Some standard notations in semi-classical analysis
Let \(W_1\) be an open set in \(\mathbb {R}^{N_1}\) and let \(W_2\) be an open set in \(\mathbb {R}^{N_2}\). Let E and F be vector bundles over \(W_1\) and \(W_2\), respectively. An m-dependent continuous operator \(A_m:C^\infty _0(W_2,F)\rightarrow D'(W_1,E)\) is called m-negligible on \(W_1\times W_2\) if, for m large enough, \(A_m\) is smoothing and, for any \(K\Subset W_1\times W_2\), any multi-indices \(\alpha \), \(\beta \) and any \(N\in \mathbb {N}\), there exists \(C_{K,\alpha ,\beta ,N}>0\) such that
In that case we write
If \(A_m, B_m:C^\infty _0(W_2, F)\rightarrow D'(W_1, E)\) are m-dependent continuous operators, we write \(A_m= B_m+O(m^{-\infty })\) on \(W_1\times W_2\) or \(A_m(x,y)=B_m(x,y)+O(m^{-\infty })\) on \(W_1\times W_2\) if \(A_m-B_m=O(m^{-\infty })\) on \(W_1\times W_2\). When \(W=W_1=W_2\), we sometime write “on W”.
Let X and M be smooth manifolds and let E and F be vector bundles over X and M, respectively. Let \(A_m, B_m:C^\infty (M,F)\rightarrow C^\infty (X,E)\) be m-dependent smoothing operators. We write \(A_m=B_m+O(m^{-\infty })\) on \(X\times M\) if on every local coordinate patch D of X and local coordinate patch \(D_1\) of M, \(A_m=B_m+O(m^{-\infty })\) on \(D\times D_1\). When \(X=M\), we sometime write on X.
We recall the definition of the semi-classical symbol spaces
Definition 2.1
Let W be an open set in \(\mathbb {R}^N\). Let
For \(k\in \mathbb {R}\), let
Hence \(a(\cdot ,m)\in S^k_{\mathrm{loc}}(1;W)\) if for every \(\alpha \in \mathbb {N}^N_0\) and \(\chi \in C^\infty _0(W)\), there exists \(C_\alpha >0\) independent of m, such that \(|\partial ^\alpha (\chi a(\cdot ,m))|\le C_\alpha m^{k}\) holds on W.
Consider a sequence \(a_j\in S^{k_j}_{\mathrm{loc}}\,(1)\), \(j\in \mathbb {N}_0\), where \(k_j\searrow -\infty \), and let \(a\in S^{k_0}_{\mathrm{loc}}\,(1)\). We say
if, for every \(\ell \in \mathbb {N}_0\), we have \(a-\sum ^{\ell }_{j=0}a_j\in S^{k_{\ell +1}}_{\mathrm{loc}}\,(1)\). For a given sequence \(a_j\) as above, we can always find such an asymptotic sum a, which is unique up to an element in \(S^{-\infty }_{\mathrm{loc}}\,(1)=S^{-\infty }_{\mathrm{loc}}\,(1;W):=\cap _kS^k_{\mathrm{loc}}\,(1)\).
Similarly, we can define \(S^k_{\mathrm{loc}}\,(1;Y,E)\) in the standard way, where Y is a smooth manifold and E is a vector bundle over Y.
2.3 CR manifolds and bundles
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, that is, \(T^{1,0}X\) is a subbundle of rank n of the complexified tangent bundle \(\mathbb {C}TX\), satisfying \(T^{1,0}X\cap T^{0,1}X=\{0\}\), where \(T^{0,1}X=\overline{T^{1,0}X}\), and \([\mathcal V,\mathcal V]\subset \mathcal V\), where \(\mathcal V=C^\infty (X, T^{1,0}X)\). There is a unique subbundle HX of TX such that \(\mathbb {C}HX=T^{1,0}X \oplus T^{0,1}X\), i.e. HX is the real part of \(T^{1,0}X \oplus T^{0,1}X\). Let \(J:HX\rightarrow HX\) be the complex structure map given by \(J(u+\overline{u})=iu-i\overline{u}\), for every \(u\in T^{1,0}X\). By complex linear extension of J to \(\mathbb {C}TX\), the i-eigenspace of J is \(T^{1,0}X \, = \, \left\{ V \in \mathbb {C}HX \, : \, JV \, = \, \sqrt{-1}V \right\} \). We shall also write (X, HX, J) to denote a compact CR manifold.
We fix a real non-vanishing 1 form \(\omega _0\in C(X,T^*X)\) so that \(\langle \,\omega _0(x)\,,\,u\,\rangle =0\), for every \(u\in H_xX\), for every \(x\in X\). For each \(x \in X\), we define a quadratic form on HX by
We extend \(\mathcal {L}\) to \(\mathbb {C}HX\) by complex linear extension. Then, for \(U, V \in T^{1,0}_xX\),
The Hermitian quadratic form \(\mathcal {L}_x\) on \(T^{1,0}_xX\) is called Levi form at x. We recall that in this paper, we always assume that the Levi form \(\mathcal {L}\) on \(T^{1,0}X\) is non-degenerate of constant signature \((n_-,n_+)\) on X, where \(n_-\) denotes the number of negative eigenvalues of the Levi form and \(n_+\) denotes the number of positive eigenvalues of the Levi form. Let \(T\in C^\infty (X,TX)\) be the non-vanishing vector field determined by
Note that X is a contact manifold with contact form \(\omega _0\), contact plane HX and T is the Reeb vector field.
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, \(\langle \,T\,|\,T\,\rangle =1\) and T 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 \(T^{1,0}X\) and \(T^{0,1}X\), respectively. They can be identified with subbundles of the complexified cotangent bundle \(\mathbb {C}T^*X\). Define the vector bundle of (0, q)-forms by \(T^{*0,q}X := \wedge ^qT^{*0,1}X\). The Hermitian metric \(\langle \, \cdot \, |\, \cdot \,\rangle \) on \(\mathbb {C}TX\) induces, by duality, a Hermitian metric on \(\mathbb {C}T^*X\) and also on the bundles of (0, q) forms \(T^{*0,q}X, q=0, 1, \cdots , n\). We shall also denote all these induced metrics by \(\langle \,\cdot \,|\, \cdot \,\rangle \). Note that we have the pointwise orthogonal decompositions:
For \(x, y\in X\), let d(x, y) denote the distance between x and y induced by the Hermitian metric \(\langle \cdot \mid \cdot \rangle \). Let A be a subset of X. For every \(x\in X\), let \(d(x,A):=\inf \left\{ d(x,y);\, y\in A\right\} \).
Let D be an open set of X. Let \(\Omega ^{0,q}(D)\) denote the space of smooth sections of \(T^{*0,q}X\) over D and let \(\Omega ^{0,q}_0(D)\) be the subspace of \(\Omega ^{0,q}(D)\) whose elements have compact support in D.
2.4 Contact reduction
Let G be a connected compact Lie group with Lie algebra \(\mathfrak {g}\) such that \(\dim _{\mathbb {R}}G = d\). We assume that the Lie group G acts on X preserving \(\omega _0\), i.e. \(g^*\omega _0 =\omega _0\), for any \(g \in G\). For any \(\xi \in \mathfrak {g}\), there is an induced vector field \(\xi _X\) on X given by \((\xi _X u)(x)=\frac{\partial }{\partial t}\left( u( \exp (t\xi )\circ x)\right) |_{t=0}\), for any \(u\in C^\infty (X)\).
Definition 2.2
The contact moment map associated to the form \(\omega _0\) is the map \(\mu :X \rightarrow \mathfrak {g}^*\) such that, for all \(x \in X\) and \(\xi \in \mathfrak {g}\), we have
We now recall the contact reduction from [1, 9]. It was shown in [1, 9] that the contact moment map is G-equivariant, so G acts on \(Y:=\mu ^{-1}(0)\), where G acts on \(\mathfrak {g}^*\) through co-adjoint represent. Since we assume that the action of G on Y is freely, \(Y_G:=\mu ^{-1}(0)/G\) is a smooth manifold. Let \(\pi : Y \rightarrow Y_G\) and \(\iota :Y \hookrightarrow X\) be the natural quotient and inclusion, respectively, then there is a unique induced contact form \(\widetilde{\omega }_0\) on \(Y_G\) such \(\pi ^*\widetilde{\omega }_0 = \iota ^* \omega _0\). We denote by \(HY:= {\text {Ker}} \omega _0 \cap T(\mu ^{-1}(0)) = HX \cap TY\), then the induced contact plane on \(Y_G\) is \(HY_G := \pi _*HY\). In particular, \(\dim HY = 2n-d\) and \(\dim HY_G=2n-2d\).
2.5 CR reduction
In this subsection we study the reduction of CR manifolds with non-degenerate Levi curvature which is a CR analogue of the reduction on complex manifolds considered in [27, §2.1]. For the case of strictly pseudoconvex CR manifolds, the CR reduction was also studied in [17].
Recall that we work with Assumption 1.2. Let b be the nondegenerate bilinear form on HX such that
We denote by \(\underline{\mathfrak {g}} := {\text {Span}} (\xi _X, \xi \in \mathfrak {g})\) the tangent bundle of the orbits in X. Let
Since we assume that \(\underline{\mathfrak {g}}_x \cap \underline{\mathfrak {g}}^{\perp _b}_x = \left\{ 0 \right\} \), for every \(x\in Y\), we immediately get
Lemma 2.3
When restricted to \( \underline{\mathfrak {g}} \times \underline{\mathfrak {g}}\), the bilinear form b is nondegenerate on Y.
For \(x \in Y, V \in H_xX\) and \(\xi \in \mathfrak {g}\), by (2.3) and (2.4), we have
Therefore,
Since \(Y=\mu ^{-1}(0)\), we have
In particular, for \(x \in Y\),
By (2.2), (2.7) and the definition of \(\underline{\mathfrak {g}}\), we have \(\underline{\mathfrak {g}} \subset HX|_Y\). From Lemma 2.3, we can check that \(\underline{\mathfrak {g}}+\underline{\mathfrak {g}}^{\perp _b}=HX|_Y\). Since \(\underline{\mathfrak {g}}_x \cap \underline{\mathfrak {g}}^{\perp _b}_x = \left\{ 0 \right\} \), for every \(x\in Y\), this sum is a direct sum.
Let U be a small open G-invariant neighborhood of Y. Since G acts freely on Y, we can thus also assume that G acts freely on \(\overline{U}\). Since \(\underline{\mathfrak {g}}_x \cap \underline{\mathfrak {g}}^{\perp _b}_x = \left\{ 0 \right\} \), for \(x\in Y\), we have, for \(x \in Y\),
Then, by (2.8), we can choose the horizontal bundles of the fibrations \(U \rightarrow U_G:=U/G\) and \(Y \rightarrow Y_G\) to be
Hence
Lemma 2.4
Proof
The identity (2.10) follows from (2.6) and (2.7). For \(x \in Y, V \in H_xY\) and \(\xi \in \mathfrak {g}\),
Using (2.12), \(\dim H_xU = \dim H_xY + \dim J\underline{\mathfrak {g}}_x\), and the fact that b is nondegenerate on JHY, we obtain (2.11). \(\square \)
By (2.9), and (2.10), we have \(H^HY = JHY \cap HY\). In particular, \(H^HY\) is preserved by J, so we can define the homomorphism \(J_G\) on \(HY_G\) in the following way: For \(V \in HY_G\), we denote by \(V^H\) its lift in \(H^HY\), and we define \(J_G\) on \(Y_G\) by
Hence, we have \(J_G: HY_G \rightarrow HY_G\) such that \(J_G^2 = -{\text {id}}\), where \({\text {id}}\) denotes the identity map \({\text {id}} \, : \, HY_G \rightarrow HY_G\). By complex linear extension of \(J_G\) to \(\mathbb {C}TY_G\), we can define the i-eigenspace of \(J_G\) is given by \(T^{1,0}Y_G \, = \, \left\{ V \in \mathbb {C}HY_G \, : \, J_GV \, = \, \sqrt{-1} V \right\} \).
Theorem 2.5
The subbundle \(T^{1,0}Y_G\) is a CR structure of \(Y_G\).
Proof
Let \(u, v \in C^\infty (Y_G, T^{1,0}Y_G)\), then we can find \(U, V \in C^\infty (Y_G, TY_G)\) such that
By (2.13), we have
Since \(T^{1,0}X\) is a CR structure and it is clearly that \([u^H, v^H] \in \mathbb {C}HY\), we have \([u^H, v^H] \in T^{1,0}X \cap \mathbb {C}HY\). Hence, there is a \(W \in C^\infty (X, HX)\) such that
In particular, \(W, JW \in HY\). Thus, \(W \in HY \cap JHY = H^HY\). Let \(X^H \in H^HY\) be a lift of \(X \in TY_G\) such that \(X^H=W\). Then we have
i.e. we have \([C^\infty (Y_G, T^{1,0}Y_G), C^\infty (Y_G, T^{1,0}Y_G)] \subset C^\infty (Y_G, T^{1,0}Y_G)\). Therefore, \(T^{1,0}Y_G\) is a CR structure of \(Y_G\). \(\square \)
3 G-invariant Szegő kernel asymptotics
In this section, we will establish asymptotic expansion for the G-invariant Szegő kernel. We first review some known results for Szegő kernel.
3.1 Szegő kernel asymptotics
In this subsection, we don’t assume that our CR manifold admits a compact Lie group action but we still assume that the Levi form is non-degenerate of constant signature \((n_-,n_+)\). The Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(\mathbb {C}TX\) induces, by duality, a Hermitian metric on \(\mathbb {C}T^*X\) and also on the bundles of (0, q) forms \(T^{*0,q}X\), \(q=0,1,\ldots ,n\). We shall also denote all these induced metrics by \(\langle \,\cdot \,|\,\cdot \,\rangle \). For \(u\in T^{*0,q}X\), we write \(|u|^2:=\langle \,u\,|\,u\,\rangle \). Let \(D\subset X\) be an open set. Let \(\Omega ^{0,q}(D)\) denote the space of smooth sections of \(T^{*0,q}X\) over D and let \(\Omega ^{0,q}_0(D)\) be the subspace of \(\Omega ^{0,q}(D)\) whose elements have compact support in D.
Let
be the tangential Cauchy-Riemann operator. Let dv(x) be the volume form induced by the Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \). The natural global \(L^2\) inner product \((\,\cdot \,|\,\cdot \,)\) on \(\Omega ^{0,q}(X)\) induced by dv(x) and \(\langle \,\cdot \,|\,\cdot \,\rangle \) is given by
We denote by \(L^2_{(0,q)}(X)\) the completion of \(\Omega ^{0,q}(X)\) with respect to \((\,\cdot \,|\,\cdot \,)\). We write \(L^2(X):=L^2_{(0,0)}(X)\). We extend \((\,\cdot \,|\,\cdot \,)\) to \(L^2_{(0,q)}(X)\) in the standard way. For \(f\in L^2_{(0,q)}(X)\), we denote \(\left\| f\right\| ^2:=(\,f\,|\,f\,)\). We extend \(\overline{\partial }_{b}\) to \(L^2_{(0,r)}(X)\), \(r=0,1,\ldots ,n\), by
where \(\mathrm{Dom}\,\overline{\partial }_{b}:=\{u\in L^2_{(0,r)}(X);\, \overline{\partial }_{b}u\in L^2_{(0,r+1)}(X)\}\) and, for any \(u\in L^2_{(0,r)}(X)\), \(\overline{\partial }_{b} u\) is defined in the sense of distributions. We also write
to denote the Hilbert space adjoint of \(\overline{\partial }_{b}\) in the \(L^2\) space with respect to \((\,\cdot \,|\,\cdot \, )\). Let \(\Box ^{(q)}_{b}\) denote the (Gaffney extension) of the Kohn Laplacian given by
By a result of Gaffney, for every \(q=0,1,\ldots ,n\), \(\Box ^{(q)}_{b}\) is a positive self-adjoint operator (see [19, Proposition 3.1.2]). That is, \(\Box ^{(q)}_{b}\) is self-adjoint and the spectrum of \(\Box ^{(q)}_{b}\) is contained in \(\overline{\mathbb {R}}_+\), \(q=0,1,\ldots ,n\). Let
be the orthogonal projections with respect to the \(L^2\) inner product \((\,\cdot \,|\,\cdot \,)\) and let
denote the distribution kernel of \(S^{(q)}\).
We recall Hörmander symbol space. Let \(D\subset X\) be a local coordinate patch with local coordinates \(x=(x_1,\ldots ,x_{2n+1})\).
Definition 3.1
For \(m\in \mathbb {R}\), \(S^m_{1,0}(D\times D\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\) is the space of all \(a(x,y,t)\in C^\infty (D\times D\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\) 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
Put
Let \(a_j\in S^{m_j}_{1,0}(D\times D\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\), \(j=0,1,2,\ldots \) with \(m_j\rightarrow -\infty \), as \(j\rightarrow \infty \). Then there exists \(a\in S^{m_0}_{1,0}(D\times D\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\) unique modulo \(S^{-\infty }\), such that \(a-\sum ^{k-1}_{j=0}a_j\in S^{m_k}_{1,0}(D\times D\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*\big )\) 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 \(S^{m_0}_{1,0}\big (D\times D\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*\big )\). We write
if \(s(x, y, t)\in S^{m}_{1,0}\big (D\times D\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*\big )\) and
The following was proved in Theorem 4.8 in [15]
Theorem 3.2
Given \(q=0,1,2,\ldots ,n\). Assume that \(q\notin \left\{ n_-,n_+\right\} \). Then, \(S^{(q)}\equiv 0\) on X.
We have the following (see Theorem 1.2 in [13], Theorem 4.7 in [15] and see also [2] for \(q=0\))
Theorem 3.3
We recall that we work with the assumption that the Levi form is non-degenerate of constant signature \((n_-,n_+)\) on X. Let \(q=n_-\) or \(n_+\). Suppose that \(\Box ^{(q)}_b\) has \(L^2\) closed range. Then, \(S^{(q)}(x,y)\in C^\infty (X\times X\setminus {\mathrm{diag}\,(X\times X)},T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\). Let \(D\subset X\) be any local coordinate patch with local coordinates \(x=(x_1,\ldots ,x_{2n+1})\). Then, there exist continuous operators \(S_-, S_+:\Omega ^{0,q}_0(D)\rightarrow D'(D,T^{*0,q}X)\) such that
and \(S_-(x,y)\), \(S_+(x,y)\) satisfy
with
and the phase functions \(\varphi _-\), \(\varphi _+\) satisfy
Remark 3.4
It is well-known that for a strictly pseudoconvec CR manifold of dimension 3, \(\Box ^{(0)}_b\) does not have \(L^2\) closed range in general (see [28]). Kohn [16] proved that if \(q=n_-=n_+\) or \(|n_--n_+|>1\) then \(\Box ^{(q)}_b\) has \(L^2\) closed range.
The following result describes the phase function in local coordinates (see chapter 8 of part I in [13])
Theorem 3.5
For a given point \(p\in X\), let \(\{W_j\}_{j=1}^{n}\) be an orthonormal frame of \(T^{1, 0}X\) in a neighborhood of p such that the Levi form is diagonal at p, i.e. \(\mathcal {L}_{x_{0}}(W_{j},\overline{W}_{s})=\delta _{j,s}\mu _{j}\), \(j,s=1,\ldots ,n\). We take local coordinates \(x=(x_1,\ldots ,x_{2n+1})\), \(z_j=x_j+ix_{d+j}\), \(j=1,\ldots ,d\), \(z_j=x_{2j-1}+ix_{2j}\), \(j=d+1,\ldots ,n\), defined on some neighborhood of p such that \(\omega _0(p)=dx_{2n+1}\), \(x(p)=0\), and, for some \(c_j\in \mathbb {C}\), \(j=1,\ldots ,n\) ,
where \(a_{j,k}(x)\in C^\infty \), \(a_{j,k}(x)=O(|x|)\), for every \(j=1,\ldots ,n\), \(k=1,\ldots ,2n\). Set \(y=(y_1,\ldots ,y_{2n+1})\), \(w_j=y_j+iy_{d+j}\), \(j=1,\ldots ,d\), \(w_j=y_{2j-1}+iy_{2j}\), \(j=d+1,\ldots ,n\). Then, for \(\varphi _-\) in Theorem 3.3, we have
in some neighbourhood of (0, 0) and
where f is smooth and satisfies \(f(0,0)=0\), \(f(x, y)=\overline{f}(y, x)\).
The following formula for the leading term \(s^0_-\) on the diagonal follows from [13, §9]. The formula for the leading term \(s^0_+\) on the diagonal follows similarly.
Theorem 3.6
We assume that the Levi form is non-degenerate of constant signature \((n_-,n_+)\) at each point of X. Suppose that \(\Box ^{(q)}_b\) has \(L^2\) closed range. If \(q=n_{\mp }\), then, for the leading term \(s^0_{\mp }(x,y)\) of the expansion (3.3) of \(s_{\mp }(x,y,t)\), we have
where \(\det \mathcal {L}_{x_0}\) is given by (1.6) and \(\tau _{x_0,n_{\mp }}\) is given by (1.7).
3.2 G-invariant Szegő kernel
Fix \(g\in G\). Let \(g^*:\Lambda ^r_x(\mathbb {C}T^*X)\rightarrow \Lambda ^r_{g^{-1}\circ x}(\mathbb {C}T^*X)\) be the pull-back map. Since G preserves J, we have \(g^*:T^{*0,q}_xX\rightarrow T^{*0,q}_{g^{-1}\circ x}X,\ \forall x\in X\). Thus, for \(u\in \Omega ^{0,q}(X)\), we have \(g^*u\in \Omega ^{0,q}(X)\) and we write \((g^*u)(x) := u(g\circ x)\). Put \(\Omega ^{0,q}(X)^G:=\left\{ u\in \Omega ^{0,q}(X);\, g^*u=u,\ \ \forall g\in G\right\} \). Now, we assume that the Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(\mathbb {C}TX\) is G-invariant and \(\underline{\mathfrak {g}}\) is orthogonal to \(HY\bigcap JHY\) at every point of Y. The Hermitian metric is G-invariant means that, for any G-invariant vector fields U and V, \(\langle \,U\,|\,V\,\rangle \) is G-invariant. Then the \(L^2\) inner product \((\,\cdot \,|\,\cdot \,)\) on \(\Omega ^{0,q}(X)\) induced by \(\langle \,\cdot \,|\,\cdot \,\rangle \) is G-invariant, that is, \((\,u\,|\,v\,)=(\,g^*u\,|\,g^*v\,)\), for all \(u, v\in \Omega ^{0,q}(X)\), \(g\in G\). Let \(u\in L^2_{(0,q)}(X)\) and let \(g\in G\). Take \(u_j\in \Omega ^{0,q}(X)\), \(j=1,2,\ldots \), with \(u_j\rightarrow u\) in \(L^2_{(0,q)}(X)\) as \(j\rightarrow \infty \). Since \((\,\cdot \,|\,\cdot \,)\) is G-invariant, there is a \(v\in L^2_{(0,q)}(X)\) such that \(v=\lim _{j\rightarrow \infty }g^*u_j\). We define \(g^*u:=v\). It is clear that the definition is well-defined. We have \(g^*:L^2_{(0,q)}(X)\rightarrow L^2_{(0,q)}(X)\). Put \(L^2_{(0,q)}(X)^G:=\left\{ u\in L^2_{(0,q)}(X);\, g^*u=u,\ \forall g\in G\right\} \). It is not difficult to see that \(L^2_{(0,q)}(X)^G\) is the completion of \(\Omega ^{0,q}(X)^G\) with respect to \((\,\cdot \,|\,\cdot \,)\). We write \(L^2(X)^G:=L^2_{(0,0)}(X)^G\). Since G preserves J and \((\,\cdot \,|\,\cdot \,)\) is G-invariant, it is straightforward to see that
Put \((\mathrm{Ker}\,\Box ^{(q)}_b)^G:=\mathrm{Ker}\,\Box ^{(q)}_b\bigcap L^2_{(0,q)}(X)^G\). The G-invariant Szegő projection is the orthogonal projection \(S^{(q)}_G:L^2_{(0,q)}(X)\rightarrow (\mathrm{Ker}\,\Box ^{(q)}_b)^G\) with respect to \((\,\cdot \,|\,\cdot \,)\). Let \(S^{(q)}_G(x,y)\in D'(X\times X,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\) be the distribution kernel of \(S^G\). Let \(d\mu \) be a Haar measure on G so that \(|G|_{d\mu }:=\int _Gd\mu =1\).Then,
Note that the integral (3.7) is defined in the sense of distribution.
3.3 G-invariant Szegő kernel asymptotics near \(\mu ^{-1}(0)\)
In this section, we will study G-invariant Szegő kernel near \(\mu ^{-1}(0)\).
Let \(e_0\in G\) be the identity element. Let \(v=(v_1,\ldots ,v_d)\) be the local coordinates of G defined in a neighborhood V of \(e_0\) with \(v(e_0)=(0,\ldots ,0)\). From now on, we will identify the element \(e\in V\) with v(e). We first need
Theorem 3.7
Let \(p\in \mu ^{-1}(0)\). There exist local coordinates \(v=(v_1,\ldots ,v_d)\) of G defined in a neighborhood V of \(e_0\) with \(v(e_0)=(0,\ldots ,0)\), local coordinates \(x=(x_1,\ldots ,x_{2n+1})\) of X defined in a neighborhood \(U=U_1\times U_2\) of p with \(0\leftrightarrow p\), where \(U_1\subset \mathbb {R}^d\) is an open set of \(0\in \mathbb {R}^d\), \(U_2\subset \mathbb {R}^{2n+1-d}\) is an open set of \(0\in \mathbb {R}^{2n+1-d} \) and a smooth function \(\gamma =(\gamma _1,\ldots ,\gamma _d)\in C^\infty (U_2,U_1)\) with \(\gamma (0)=0\in \mathbb {R}^d\) such that
where \(a_j(x)\) is a smooth function on \(\mu ^{-1}(0)\bigcap U\), independent of \(x_1,\ldots ,x_{2d}\), \(x_{2n+1}\) and \(a_j(0)=0\), \(j=1,\ldots ,d\),
and
where \(b_{d+1}\in \mathbb {R},\ldots ,b_{2n}\in \mathbb {R}\).
Proof
From the standard proof of Frobenius Theorem, it is not difficult to see that there exist local coordinates \(v=(v_1,\ldots ,v_d)\) of G defined in a neighborhood V of \(e_0\) with \(v(e_0)=(0,\ldots ,0)\) and local coordinates \(x=(x_1,\ldots ,x_{2n+1})\) of X defined in a neighborhood U of p with \(x(p)=0\) such that
and
Since \(p\in \mu ^{-1}(0)\), we have \(\omega _0(p)(\frac{\partial }{\partial x_j}(p))=0\), \(j=1,2,\ldots ,d\), and hence \( \frac{\partial }{\partial x_j}(p)\in H_pX\), \(j=1,2,\ldots ,d\). Consider the linear map
Since R is self-adjoint, by using linear transformation in \((x_1,\ldots ,x_d)\), we can take \((x_1,\ldots ,x_d)\) such that, for \(j, k = 1, 2, \ldots , d\),
By taking linear transformation in \((v_1,\ldots ,v_d)\), (3.12) still hold.
Let \(\omega _0(\frac{\partial }{\partial x_j})=a_j(x)\in C^\infty (U)\), \(j=1,2,\ldots ,d\). Since \(a_j(x)\) is G-invariant, we have \(\frac{\partial a_j(x)}{\partial x_s}=0\), \(j,s=1,2,\ldots ,d\). By the definition of the moment map, we have
Since p is a regular value of the moment map \(\mu \), the matrix \(\left( \frac{\partial a_j}{\partial x_s}(p)\right) _{1\le j\le d,d+1\le s\le 2n+1}\) is of rank d. We may assume that the matrix \(\left( \frac{\partial a_j}{\partial x_s}(p)\right) _{1\le j\le d,d+1\le s\le 2d}\) is non-singular. Thus, \((x_1,\ldots ,x_d,a_1,\ldots ,a_d,x_{2d+1},\ldots ,x_{2n+1})\) are also local coordinates of X. Hence, we can take \(v=(v_1,\ldots ,v_d)\) and \(x=(x_1,\ldots ,x_{2n+1})\) such that (3.12), (3.13), (3.14) hold and
On \(\mu ^{-1}(0)\bigcap U\), let
Since we only work on \(\mu ^{-1}(0)\), \(b_{j,k}(x)\) is independent of \(x_{d+1},\ldots ,x_{2d}\), for all \(j=1,\ldots ,d\), \(k=1,\ldots ,2n+1\). Moreover, it is easy to see that \(b_{j,k}(x)\) is also independent of \(x_{1},\ldots ,x_{d}\), for all \(j=1,\ldots ,d\), \(k=1,\ldots ,2n+1\). Let \(\widetilde{x}''=(x_{2d+1},\ldots ,x_{2n+1})\). Hence, \(b_{j,k}(x)=b_{j,k}(\widetilde{x}'')\), \(j=1,\ldots ,d\), \(k=1,\ldots ,2n+1\). We claim that the matrix \(\left( b_{j,k}(\widetilde{x}'')\right) _{1\le j\le d,d+1\le k\le 2d}\) is non-singular near p. If not, it is easy to see that there is a non-zero vector \(u\in J\underline{\mathfrak {g}}\bigcap HY\), where \(Y=\mu ^{-1}(0)\). Let \(u=Jv\), \(v\in \underline{\mathfrak {g}}\). Then, \(v\in \underline{\mathfrak {g}}\bigcap JHY=\underline{\mathfrak {g}}\bigcap \underline{\mathfrak {g}}^{\perp _b}\) (see (2.10)). Since \(\underline{\mathfrak {g}} \cap \underline{\mathfrak {g}}^{\perp _b} = \left\{ 0 \right\} \) on \(\mu ^{-1}(0)\), we deduce that \(v=0\) and we get a contradiction. The claim follows. From the claim, we can use linear transformation in \((x_{d+1},\ldots ,x_{2d})\) (the linear transform depends smoothly on \(\widetilde{x}''\)) and we can take \((x_{d+1},\ldots ,x_{2d})\) such that on \(\mu ^{-1}(0)\),
where \(j=1,2,\ldots ,d\). Consider the coordinates change:
Then,
Hence, on \(\mu ^{-1}(0)\bigcap U\), \(J(\frac{\partial }{\partial x_j})\rightarrow \frac{\partial }{\partial u_{d+j}}\), \(j=1,\ldots ,d\). Thus, we can take \(v=(v_1,\ldots ,v_d)\) and \(x=(x_1,\ldots ,x_{2n+1})\) such that (3.8), (3.13), (3.14), (3.15) hold and on \(\mu ^{-1}(0)\bigcap U\),
Let \(Z_j=\frac{1}{2}(\frac{\partial }{\partial x_j}-i\frac{\partial }{\partial x_{d+j}})(p)\in T^{1,0}_pX\), \(j=1,\ldots ,d\). From (3.14), we can check that
Since \(\underline{\mathfrak {g}}_p\) is orthogonal to \(H_pY\bigcap JH_pY\) and \(H_pY\bigcap JH_pY\subset \underline{\mathfrak {g}}^{\perp _b}_p\), we can find an orthonormal frame \(\left\{ Z_1,\ldots ,Z_d,V_1,\ldots ,V_{n-d}\right\} \) for \(T^{1,0}_pX\) such that the Levi form \(\mathcal {L}_p\) is diagonalized with respect to \(Z_1,\ldots ,Z_d,V_1,\ldots ,V_{n-d}\), where \(V_1\in \mathbb {C}H_pY\bigcap J\mathbb {C}H_pY ,\ldots ,V_{n-d}\in \mathbb {C}H_pY\bigcap J\mathbb {C}H_pY\). Write
We claim that \(\alpha _{j,k}=\beta _{j,k}=0\), for all \(k=d+1,\ldots ,2d\), \(j=1,\ldots ,n-d\). Fix \(j=1,\ldots ,n-d\). Since \(\mathrm{Re}\,V_j\in \underline{\mathfrak {g}}^{\perp _b}_p\) and \(\mathrm{span}\,\left\{ \frac{\partial }{\partial x_{d+1}},\ldots ,\frac{\partial }{\partial x_{2d}}\right\} \in \underline{\mathfrak {g}}_p^{\perp _b}\), we conclude that
From (2.10) and (3.16), we deduce that
and hence
From (3.17) and notice that \(J(\mathrm{Re}\,V_j)\in \underline{\mathfrak {g}}^{\perp _b}_p\), we deduce that
Thus, \(\alpha _{j,k}=0\), for all \(k=d+1,\ldots ,2d\), \(j=1,\ldots ,n-d\). Similarly, we can repeat the procedure above and deduce that \(\beta _{j,k}=0\), for all \(k=d+1,\ldots ,2d\), \(j=1,\ldots ,n-d\).
Since \(\mathrm{span}\,\left\{ \mathrm{Re}\,V_j, \mathrm{Im}\,V_j;\, j=1,\ldots ,n-d\right\} \) is transversal to \(\underline{\mathfrak {g}}_p\oplus J\underline{\mathfrak {g}}_p\), we can take linear transformation in \((x_{2d+1},\ldots ,x_{2n+1})\) so that
Consider the coordinates change:
Then,
Thus, we can take \(v=(v_1,\ldots ,v_d)\) and \(x=(x_1,\ldots ,x_{2n+1})\) such that (3.8), (3.9) and (3.10) hold.
Now, we can take linear transformation in \(x_{2n+1}\) so that \(\omega _0(p)=dx_{2n+1}\). Let \(W_j\), \(j=1,\ldots ,n\) be an orthonormal basis of \(T^{1,0}X\) such that \(W_j(p)=Z_j\), \(j=1,\ldots ,n\), where \(Z_j\in T^{1,0}_pX\), \(j=1,\ldots ,n\), are as in (3.10). Let \(\widetilde{x}=(\widetilde{x}_1,\ldots ,\widetilde{x}_{2n+1})\) be the coordinates as in Theorem 3.5. It is easy to see that
We may change \(x_{2n+1}\) be \(x_{2n+1}+h_{2n+1}(0,\ldots ,0,x_{d+1},\ldots ,x_{2n},0)\) and we have
Note that when we change \(x_{2n+1}\) to \(x_{2n+1}+h_{2n+1}(0,\ldots ,0,x_{d+1},\ldots ,x_{2n},0)\), \(\frac{\partial }{\partial x_j}\) will change to \(\frac{\partial }{\partial x_j}+\alpha _j(x)\frac{\partial }{\partial x_{2n+1}}\), \(j=d+1,\ldots ,2n\), where \(\alpha _j(x)\) is a smooth function on \(\mu ^{-1}(0)\bigcap U\), independent of \(x_1,\ldots ,x_{d}\), \(x_{2n+1}\) and \(\alpha _j(0)=0\), \(j=d+1,\ldots ,2n\). Hence, on \(\mu ^{-1}(0)\bigcap U\), we have \(J(\frac{\partial }{\partial x_j})=\frac{\partial }{\partial x_{d+j}}+a_j(x)\frac{\partial }{\partial x_{2n+1}}\), \(j=1,2,\ldots ,d\), where \(a_j(x)\) is a smooth function on \(\mu ^{-1}(0)\bigcap U\), independent of \(x_1,\ldots ,x_{2d}\), \(x_{2n+1}\) and \(a_j(0)=0\), \(j=1,\ldots ,d\).
From (3.4) and (3.18), it is straightforward to see that
where \(\widetilde{b}_j\in \mathbb {R}, \hat{b}_j\in \mathbb {R}\), \(j=1,\ldots ,2n\). Note that \(\omega _0\) is G-invariant. From this observation and (3.20), we deduce that
From (3.21), (3.20) and (3.19), it is straightforward to see that
where \(b_{1}\in \mathbb {R},\ldots ,b_{2n}\in \mathbb {R}\). Since \(\omega _0(p)(\frac{\partial }{\partial x_j})=0\) on \(x_{d+1}=\cdots =x_{2d}=0\), \(j=1,2,\ldots ,d\), we deduce that \(b_1=\cdots =b_d=0\) and we get (3.11). The theorem follows. \(\square \)
We need
Theorem 3.8
Let \(p\in \mu ^{-1}(0)\) and take local coordinates \(x=(x_1,\ldots ,x_{2n+1})\) of X defined in an open set Uof p with \(0\leftrightarrow p\) such that (3.9), (3.10) and (3.11) hold. Let \(\varphi _-(x,y)\in C^\infty (U\times U)\) be as in Theorem 3.3. Then,
where \(z_j=x_j+ix_{d+j}\), \(j=1,\ldots ,d\), \(z_j=x_{2j-1}+ix_{2j}\), \(j=2d+1,\ldots ,2n\), \(\mu _j\), \(j=1,\ldots ,n\), and \(b_{d+1}\in \mathbb {R},\ldots ,b_{2n}\in \mathbb {R}\) are as in (3.11) and f is smooth and satisfies \(f(0,0)=0\), \(f(x, y)=\overline{f}(y, x)\).
Proof
Let \(\widetilde{x}=(\widetilde{x}_1,\ldots ,\widetilde{x}_{2n+1})\) be the coordinates as in Theorem 3.5. It is easy to see that
From (3.4), it is straightforward to see that
where
From (3.25) and (3.11), it is not difficulty to see that (see also (3.20))
From (3.24), (3.26) and (3.6), it is straightforward to check that
where \(\beta _j\in \mathbb {C}\), \(j=1,\ldots ,n\) and f is smooth and satisfies \(f(0,0)=0\), \(f(x, y)=\overline{f}(y, x)\). We now determine \(\beta _j\), \(j=1,\ldots ,n\). We can compute that
Note that \(d_x\varphi _-(x,x)=-\omega _0(x)\). From this observation and (3.11), we deduce that
From (3.28) and (3.29), we deduce that
From (3.30) and (3.27), we get (3.23). \(\square \)
We now work with local coordinates as in Theorem 3.7. From (3.23), we see that near \((p,p)\in U\times U\), we have \(\frac{\partial \varphi _-}{\partial y_{2n+1}}\ne 0\). Using the Malgrange preparation theorem [12, Th. 7.5.7], we have
in some neighborhood of (p, p), where \(\mathring{y}=(y_1,\ldots ,y_{2n})\), \(g, \hat{\varphi }_-\in C^\infty \). Since \(\mathrm{Im}\,\varphi _-\ge 0\), it is not difficult to see that \(\mathrm{Im}\,\hat{\varphi }_-\ge 0\) in some neighborhood of (p, p). We may take U small enough so that (3.31) holds and \(\mathrm{Im}\,\hat{\varphi }_-\ge 0\) on \(U\times U\). From [25, Th. 4.2], we see that since \(\varphi _-(x,y)\) and \(\hat{\varphi }_-(x,y)\) satisfy (3.31), \(\varphi _-(x,y)t\) and \((y_{2n+1}+\hat{\varphi }_-(x,\mathring{y}))t\) are equivalent in the sense of Melin–Sjöstrand. More precisely, for any \(k\in \mathbb {R}\) and any \(b_1(x,y,t)\in S^{k}_{\mathrm{cl}}\,\big (U\times U\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*\big )\), we can find \(b_2(x,y,t)\in S^{k}_{\mathrm{cl}}\,\big (U\times U\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*\big )\) such that
and vise versa. We can replace the phase \(\varphi _-\) by \(y_{2n+1}+\hat{\varphi }_-(x,\mathring{y})\). From now on, we assume that \(\varphi _-(x,y)\) has the form
It is easy to check that \(\varphi _-(x,y)\) satisfies (3.5) and (3.23) with \(f(x,y)=0\).
We now study \(S^{(q)}_G(x,y)\). From Theorem 3.2, we get
Theorem 3.9
Assume that \(q\notin \left\{ n_-,n_+\right\} \).Then, \(S^{(q)}_G\equiv 0\) on X.
Assume that \(q=n_-\) and \(\Box ^{(q)}_b\) has \(L^2\) closed range. Fix \(p\in \mu ^{-1}(0)\) and let \(v=(v_1,\ldots ,v_d)\) and \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates of G and X as in Theorem 3.7. Assume that \(d\mu =m(v)dv=m(v_1,\ldots ,v_d)dv_1\cdots dv_d\) on V, where V is an open neighborhood of \(e_0\in G\) as in Theorem 3.7. From (3.7), we have
where \(\chi \in C^\infty _0(V)\), \(\chi =1\) near \(e_0\). Since G is freely on \(\mu ^{-1}(0)\), if U and V are small, there is a constant \(c>0\) such that
where U is an open set of \(p\in \mu ^{-1}(0)\) as in Theorem 3.7. From now on, we take U and V small enough so that (3.33) holds. In view of Theorem 3.3, we see that \(S^{(q)}(x,y)\) is smoothing away from diagonal. From this observation and (3.33), we conclude that \(\int _G(1-\chi (g))S^{(q)}(x,g\circ y)d\mu (g)\equiv 0\) on U and hence
From Theorem 3.3 and (3.34), we have
Write \(x=(x',x'')=(x',\hat{x}'',\widetilde{x}'')\), \(y=(y',y'')=(y',\hat{y}'',\widetilde{y}'')\), where \(\hat{x}''=(x_{d+1},\ldots ,x_{2d})\), \(\hat{y}''=(y_{d+1},\ldots ,y_{2d})\), \(\widetilde{x}''=(x_{2d+1},\ldots ,x_{2n+1})\), \(\widetilde{y}''=(y_{2d+1},\ldots ,y_{2n+1})\). Since \(S^{(q)}_G(x,y)\) is G-invariant, we have \(S^{(q)}_G(x,y)=S^{(q)}_G((0,x''),(\gamma (y''),y''))\), where \(\gamma \in C^\infty (U_2,U_1)\) is as in Theorem 3.7. From this observation and (3.35), we have
Write \(\mathring{x}''=(x_{d+1},\ldots ,x_{2n})\), \(\mathring{y}''=(y_{d+1},\ldots ,y_{2n})\) From (3.32), (3.36), Theorem 3.7 and Theorem 3.3, we have
From (3.23), it is straightforward to see that
We pause and introduce some notations. Let W be an open set of \(\mathbb {R}^N\), \(N\in \mathbb {N}\). From now on, we write \(W^\mathbb {C}\) to denote an open set in \(\mathbb {C}^N\) with \(W^\mathbb {C}\bigcap \mathbb {R}^N=W\) and for \(f\in C^\infty (W)\), from now on, we write \(\widetilde{f}\in C^\infty (W^\mathbb {C})\) to denote an almost analytic extension of f (see Section 2 in [25]). Let \(h(x'',y'')\in C^\infty (U\times U,\mathbb {C}^d)\) be the solution of the system
and let
It is known that (see page 147 in [25]) \(\mathrm{Im}\,\Phi _-(x'',y'')\ge 0\). Note that
where \(x=(0,(0,\widetilde{x}''))\). We deduce that for \(\hat{x}''=\hat{y}''=0\), \(\widetilde{x}''=\widetilde{y}''\), \(v=-\gamma (y'')\) are real critical points. From this observation, we can calculate that
where \(x=(0,\widetilde{x}'')\) and \(f\in C^\infty \) is a positive function with \(f(p)=1\). By using stationary phase formula of Melin–Sjöstrand [25], we can carry out the v integral in (3.37) and get
where \(a_-(x'',y'',t)\sim \sum ^\infty _{j=0}t^{n-\frac{d}{2}-j}a^0_-(x'',y'')\) in \(S^{n-\frac{d}{2}}_{1,0}(U\times U\times \mathbb {R}_+, T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\),
We now study the property of the phase \(\Phi _-(x'',y'')\). We need the following which is known (see Section 2 in [25])
Theorem 3.10
There exist a constant \(c>0\) and an open set \(\Omega \in \mathbb {R}^d\) such that
for all \(((0,x''),(0,y''))\in U\times U\).
We can now prove
Theorem 3.11
If U is small enough, then there is a constant \(c>0\) such that
Proof
From (3.5), we see that there is a constant \(c_1>0\) such that
where \(\Omega \) is any open set of \(0\in \mathbb {R}^d\). From (3.45) and (3.43), we conclude that there is a constant \(c_2>0\) such that
From (3.23), we see that the matrix
is non-singular. From this observation and notice that \(d_{y'}\hat{\varphi }_-((0,x''),(0,\mathring{x}''))|_{\hat{x}''}=0\), we deduce that if U is small enough then there is a constant \(c_3>0\) such that
From (3.47) and (3.46), the theorem follows. \(\square \)
From now on, we assume that U is small enough so that (3.44) holds.
We now determine the Hessian of \(\Phi _-(x'',y'')\) at (p, p). Let \(\hat{h}(x'',y''):=h(x'',y'')+\gamma (y'')\). From (3.39), we have
From (3.23), we can calculate that
From (3.49) and (3.48), we obtain \(\frac{\partial \hat{h}_1}{\partial x_{d+1}}(p,p)=i\frac{\mu _1}{|\mu _1|}\). We can repeat the procedure above several times and deduce that
From (3.50), (3.23), (3.40) and by some straightforward computation (we omit the details), we get
Theorem 3.12
With the notations used above, we have
where \(z_j=x_{2j-1}+ix_{2j}\), \(j=2d+1,\ldots ,2n\), \(\mu _j\), \(j=1,\ldots ,n\), and \(b_{d+1}\in \mathbb {R},\ldots ,b_{2n}\in \mathbb {R}\) are as in (3.11) and f is smooth and satisfies \(f(0,0)=0\), \(f(x, y)=\overline{f}(y, x)\).
We can change \(\Phi _-(x'',y'')\) be \(\Phi _-(x'',y'')\frac{1}{f(x'')}\), where \(f(x'')\) is as in (3.41). Thus,
where \(x=(0,\widetilde{x}'')\). It is clear that \(\Phi _-(x'',y'')\) still satisfies (3.44) and (3.51).
We now determine the leading term \(a_-^0(p,p)\). In view of (3.42), we only need to calculate m(0). Put \(Y_p=\left\{ g\circ p;\, g\in G\right\} \). \(Y_p\) is a d-dimensional submanifold of X. The G-invariant Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) induces a volume form \(dv_{Y_p}\) on \(Y_p\). Put
For \(f(g)\in C^\infty (G)\), let \(\hat{f}(g\circ p):=f(g)\), \(\forall g\in G\). Then, \(\hat{f}\in C^\infty (Y_p)\). Let \(d\hat{\mu }\) be the measure on G given by \(\int _Gfd\hat{\mu }:=\int _{Y_p}\hat{f}dv_{Y_p}\), for all \(f\in C^\infty (G)\). It is not difficult to see that \(d\hat{\mu }\) is a Haar measure and
Recall that we work with the local coordinates in Theorem 3.7. In view of (3.10), we see that \(\left\{ \frac{1}{\sqrt{2}}\frac{\partial }{\partial x_1},\ldots ,\frac{1}{\sqrt{2}}\frac{\partial }{\partial x_d}\right\} \) is an orthonormal basis for \(\underline{\mathfrak {g}}_p\). Hence \(m(0)=2^{\frac{d}{2}}\frac{1}{V_{\mathrm{eff}}\,(p)}\). From this observation, (3.53) and (3.42), we get
Similarly, we can repeat the procedure above and deduce that
where \(a_+(x'',y'',t)\sim \sum ^\infty _{j=0}t^{n-\frac{d}{2}-j}a^j_+(x'',y'')\) in \(S^{n-\frac{d}{2}}_{1,0}(U\times U\times \mathbb {R}_+, T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\),
and \(\Phi _+(x'',y'')\in C^\infty (U\times U)\), \(\mathrm{Im}\,\Phi _+(x'',y'')\ge 0\), \(-\overline{\Phi }_+(x'',y'')\) satisfies (3.44), (3.51) and (3.52).
Summing up, we get one of the main result of this work
Theorem 3.13
We recall that we work with the assumption that the Levi form is non-degenerate of constant signature \((n_-,n_+)\) on X. Let \(q=n_-\) or \(n_+\). Suppose that \(\Box ^{(q)}_b\) has \(L^2\) closed range. Let \(p\in \mu ^{-1}(0)\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates defined in an open set U of p such that \(x(p)=0\) and (3.8), (3.9), (3.10), (3.11) hold. Write \(x''=(x_{d+1},\ldots ,x_{2n+1})\). Then, there exist continuous operators \(S^G_-, S^G_+:\Omega ^{0,q}_0(U)\rightarrow \Omega ^{0,q}(U)\) such that
and \(S^G_-(x,y)\), \(S^G_+(x,y)\) satisfy
with
\(a^0_-(p,p)\) and \(a^0_+(p,p)\) are given by (3.54) and (3.55) respectively, \(\Phi _-(x'',y'')\in C^\infty (U\times U)\) satisfies (3.52), (3.44) and (3.51), \(\Phi _+(x'',y'')\in C^\infty (U\times U)\), \(-\overline{\Phi }_+(x'',y'')\) satisfies (3.52), (3.44) and (3.51).
3.4 G-invariant Szegő kernel asymptotics away \(\mu ^{-1}(0)\)
The goal of this section is to prove the following
Theorem 3.14
Let D be an open set of X with \(D\bigcap \mu ^{-1}(0)=\emptyset \). Then, \(S^{(q)}_G\equiv 0\ \ \text{ on } D\).
We first need
Lemma 3.15
Let \(p\notin \mu ^{-1}(0)\). Then, there are open sets U of p and V of \(e\in G\) such that for any \(\chi \in C^\infty _0(V)\), we have
Proof
If \(q\notin \left\{ n_-,n_+\right\} \). By Theorem 3.2, we get (3.56). We may assume that \(q=n_-\). Take local coordinates \(v=(v_1,\ldots ,v_d)\) of G defined in a neighborhood V of \(e_0\) with \(v(e_0)=(0,\ldots ,0)\), local coordinates \(x=(x_1,\ldots ,x_{2n+1})\) of X defined in a neighborhood \(U=U_1\times U_2\) of p with \(0\leftrightarrow p\), where \(U_1\subset \mathbb {R}^d\) is an open set of \(0\in \mathbb {R}^d\), \(U_2\subset \mathbb {R}^{2n+1-d}\) is an open set of \(0\in \mathbb {R}^{2n+1-d}\), such that
and
where \(\gamma =(\gamma _1,\ldots ,\gamma _d)\in C^\infty (U_2,U_1)\) with \(\gamma (0)=0\in \mathbb {R}^d\). Note that we don’t use the local coordinates in Theorem 3.7. It should be notice that G needs not act locally freely on near p, (3.33) need not be true. We can not use off-diagonal expansion for the Szegő kernel to get this lemma. We will use some properties of the phase \(\varphi _-\) and integrations by parts to obtain this lemma. From Theorem 3.3, we have
From Theorem 3.3, we have
where \(y''=(y_{d+1},\ldots ,y_{2n+1})\), \(m(v)dv=d\mu |_V\). Since \(p\notin \mu ^{-1}(0)\) and notice that \(d_y\varphi _-(x,x)=\omega _0(x,x)\), we deduce that if V and U are small then \(d_v(\varphi _-(x,(v+\gamma (y''),y'')))\ne 0\), for every \(v\in V\), \((x,y)\in U\times U\). Hence, by using integration by parts with respect to v, we get
Similarly, we have
From (3.57), (3.58) and (3.59), the lemma follows. \(\square \)
Lemma 3.16
Let \(p\notin \mu ^{-1}(0)\) and let \(h\in G\). We can find open sets U of p and V of h such that for every \(\chi \in C^\infty _0(V)\), we have \(\int _GS^{(q)}(x,g\circ y)\chi (g)d\mu (g)\equiv 0\ \ \text{ on } U\).
Proof
Let U and V be open sets as in Lemma 3.15. Let \(\hat{V}=hV\). Then, \(\hat{V}\) is an open set of G. Let \(\hat{\chi }\in C^\infty _0(\hat{V})\). We have
where \(\chi (g):=\hat{\chi }(h\circ g)\in C^\infty _0(V)\). From (3.60) and Lemma 3.15, we deduce that
The lemma follows. \(\square \)
Proof of Theorem 3.14
Fix \(p\in D\). We need to show that \(S^{(q)}_G\) is smoothing near p. Let \(h\in G\). By Lemma 3.16, we can find open sets \(U_h\) of p and \(V_h\) of h such that for every \(\chi \in C^\infty _0(V_h)\), we have
Since G is compact, we can find open sets \(U_{h_j}\) and \(V_{h_j}\), \(j=1,\ldots ,N\), such that \(G=\bigcup ^N_{j=1}V_{h_j}\). Let \(U=D\bigcap \Bigr (\bigcap ^N_{j=1}U_{h_j}\Bigr )\) and let \(\chi _j\in C^\infty _0(V_{h_j})\), \(j=1,\ldots ,N\), with \(\sum ^N_{j=1}\chi _j=1\) on G. From (3.61), we have
The theorem follows. \(\square \)
4 G-invariant Szegő kernel asymptotics on CR manifolds wit \(S^1\) action
Let \((X, T^{1,0}X)\) be a compact CR manifold of dimension \(2n+1\), \(n\ge 1\). We assume that X admits an \(S^1\) action: \(S^1\times X\rightarrow X\). We write \(e^{i\theta }\) to denote the \(S^1\) action. Let \(T\in C^\infty (X, TX)\) be the global real vector field induced by the \(S^1\) action given by \((Tu)(x)=\frac{\partial }{\partial \theta }\left( u(e^{i\theta }\circ x)\right) |_{\theta =0}\), \(u\in C^\infty (X)\). We recall
Definition 4.1
We say that the \(S^1\) action \(e^{i\theta }\) is CR if \([T, C^\infty (X, T^{1,0}X)]\subset C^\infty (X, T^{1,0}X)\) and the \(S^1\) action is transversal if for each \(x\in X\), \(\mathbb {C}T(x)\oplus T_x^{1,0}X\oplus T_x^{0,1}X=\mathbb {C}T_xX\). Moreover, we say that the \(S^1\) action is locally free if \(T\ne 0\) everywhere. It should be mentioned that transversality implies locally free.
We assume now that \((X, T^{1,0}X)\) is a compact connected CR manifold with a transversal CR locally free \(S^1\) action \(e^{i\theta }\) and we let T be the global vector field induced by the \(S^1\) action. Let \(\omega _0\in C^\infty (X,T^*X)\) be the global real one form determined by \(\langle \,\omega _0\,,\,u\,\rangle =0\), for every \(u\in T^{1,0}X\oplus T^{0,1}X\), and \(\langle \,\omega _0\,,\,T\,\rangle =-1\). Note that \(\omega _0\) and T satisfy (2.2). Assume that X admits a compact connected Lie group G action and the Lie group G acts on X preserving \(\omega _0\) and J. We recall that we work with Assumption 1.7.
We now assume that the Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(\mathbb {C}TX\) is \(G\times S^1\) invariant. Then the \(L^2\) inner product \((\,\cdot \,|\,\cdot \,)\) on \(\Omega ^{0,q}(X)\) induced by \(\langle \,\cdot \,|\,\cdot \,\rangle \) is \(G\times S^1\)-invariant. We then have
Let \(L^2_{(0,q),m}(X)^G\) be the completion of \(\Omega ^{0,q}_m(X)^G\) with respect to \((\,\cdot \,|\,\cdot \,)\). We write \(L^2_m(X)^G:=L^2_{(0,0),m}(X)^G\). Put \((\mathrm{Ker}\,\Box ^{(q)}_b)^G_m:=(\mathrm{Ker}\,\Box ^{(q)}_b)^G\bigcap L^2_{(0,q),m}(X)^G\). It is not difficult to see that, for every \(m\in \mathbb {Z}\), \((\mathrm{Ker}\,\Box ^{(q)}_b)^G_m\subset \Omega ^{0,q}_m(X)^G\) and \(\mathrm{dim}\,(\mathrm{Ker}\,\Box ^{(q)}_b)^G_m<\infty \). The mth G-invariant Szegő projection is the orthogonal projection \(S^{(q)}_{G,m}:L^2_{(0,q)}(X)\rightarrow (\mathrm{Ker}\,\Box ^{(q)}_b)^G_m\) with respect to \((\,\cdot \,|\,\cdot \,)\). Let \(S^{(q)}_{G,m}(x,y)\in C^\infty (X\times X,T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\) be the distribution kernel of \(S^{(q)}_{G,m}\). We can check that
The goal of this section is to study the asymptotics of \(S^{(q)}_{G,m}\) as \(m\rightarrow +\infty \).
From Theorem 3.14, (4.1) and by using integration by parts several times, we get
Theorem 4.2
Let \(D\subset X\) be an open set with \(D\bigcap \mu ^{-1}(0)=\emptyset \). Then, \(S^{(q)}_{G,m}=O(m^{-\infty })\ \ \text{ on } D\).
We now study \(S^{(q)}_{G,m}\) near \(\mu ^{-1}(0)\). We can repeat the proof of Theorem 3.7 with minor change and get
Theorem 4.3
Let \(p\in \mu ^{-1}(0)\). There exist local coordinates \(v=(v_1,\ldots ,v_d)\) of G defined in a neighborhood V of \(e_0\) with \(v(e_0)=(0,\ldots ,0)\), local coordinates \(x=(x_1,\ldots ,x_{2n+1})\) of X defined in a neighborhood \(U=U_1\times (\hat{U}_2\times ]-2\delta ,2\delta [)\) of p with \(0\leftrightarrow p\), where \(U_1\subset \mathbb {R}^d\) is an open set of \(0\in \mathbb {R}^d\), \(\hat{U}_2\subset \mathbb {R}^{2n-d}\) is an open set of \(0\in \mathbb {R}^{2n-d} \), \(\delta >0\), and a smooth function \(\gamma =(\gamma _1,\ldots ,\gamma _d)\in C^\infty (\hat{U}_2\times ]-2\delta ,2\delta [,U_1)\) with \(\gamma (0)=0\in \mathbb {R}^d\) such that
where \(a_j(x)\) is a smooth function on \(\mu ^{-1}(0)\bigcap U\), independent of \(x_1,\ldots ,x_{2d}\), \(x_{2n+1}\) and \(a_j(0)=0\), \(j=1,\ldots ,d\),
and
Remark 4.4
Let \(p\in \mu ^{-1}(0)\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates as in Theorem 4.3. We can change \(x_{2n+1}\) be \(x_{2n+1}-\sum ^d_{j=1}a_j(x)x_{d+j}\), where \(a_j(x)\), \(j=1,\ldots ,d\), are as in (4.2). With this new local coordinates \(x=(x_1,\ldots ,x_{2n+1})\), on \(\mu ^{-1}(0)\bigcap U\), we have \(J(\frac{\partial }{\partial x_j})=\frac{\partial }{\partial x_{d+j}},\ \ j=1,2,\ldots ,d\). Moreover, it is clear that \(\Phi _-(x,y)+\sum ^d_{j=1}a_j(x)x_{d+j}-\sum ^{d=1}_{j=1}a_j(y)y_{d+j}\) satisfies (1.19). Note that \(a_j(x)\) is a smooth function on \(\mu ^{-1}(0)\bigcap U\), independent of \(x_1,\ldots ,x_{2d}\), \(x_{2n+1}\) and \(a_j(0)=0\), \(j=1,\ldots ,d\).
We now work with local coordinates as in Theorem 4.3. From (3.51), we see that near \((p,p)\in U\times U\), we have \(\frac{\partial \Phi _-}{\partial y_{2n+1}}\ne 0\). Using the Malgrange preparation theorem [12, Th. 7.5.7], we have
in some neighborhood of (p, p), where \(\mathring{y}''=(y_{d+1},\ldots ,y_{2n})\), \(g, \hat{\Phi }_-\in C^\infty \). Since \(\mathrm{Im}\,\Phi _-\ge 0\), it is not difficult to see that \(\mathrm{Im}\,\hat{\Phi }_-\ge 0\) in some neighborhood of (p, p). We may take U small enough so that (4.3) holds and \(\mathrm{Im}\,\hat{\Phi }_-\ge 0\) on \(U\times U\). From [25, Th. 4.2], we see that since \(\Phi _-(x,y)\) and \(y_{2n+1}+\hat{\Phi }_-(x'',\mathring{y}'')\) satisfy (4.3), \(\Phi _-(x,y)t\) and \((y_{2n+1}+\hat{\Phi }_-(x'',\mathring{y}''))t\) are equivalent in the sense of Melin–Sjöstrand (see the discussion after (3.31), for the meaning of equivalent in the sense of Melin–Sjöstrand). We can replace the phase \(\Phi _-\) by \(y_{2n+1}+\hat{\Phi }_-(x,\mathring{y''})\). From now on, we assume that
It is easy to check that \(\Phi _-(x,y)\) satisfies (3.44) and (3.51) with \(f(x,y)=0\). Similarly, from now on, we assume that
We now study \(S^{(q)}_{G,m}(x,y)\). From Theorem 3.9, we get
Theorem 4.5
Assume that \(q\notin \left\{ n_-,n_+\right\} \). Then, \(S^{(q)}_{G,m}=O(m^{-\infty })\) on X.
Assume that \(q=n_-\). It was proved in Theorem 1.12 in [15] that when X admits a transversal \(S^1\) action, then \(\Box ^{(q)}_b\) has \(L^2\) closed range. Fix \(p\in \mu ^{-1}(0)\) and let \(v=(v_1,\ldots ,v_d)\) and \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates of G and X as in Theorem 4.3 and let U and V be open sets as in Theorem 4.3. We take U small enough so that there is a constant \(c>0\) such that
where \(\delta >0\) is as in Theorem 4.3. We will study \(S^{(q)}_{G,m}(x,y)\) in U. From (4.1), we have
where \(\mathring{x}=(x_1,\ldots ,x_{2n},0)\in U\), \(\mathring{y}=(y_1,\ldots ,y_{2n},0)\in U\), \(\chi \in C^\infty _0(]-2\delta ,2\delta [)\), \(\chi =1\) on \([-\delta , \delta ]\). We first study II. We have
From (4.7), (4.6) and notice that \(S^{(q)}\) is smoothing away from diagonal, we deduce that
We now study I. From Theorem 3.13, (4.1), (4.4) and (4.5), we have
We first study \(I_1\). From \(\frac{\partial }{\partial \theta }\Bigr (i(\theta +\hat{\Phi }_+(\mathring{x}'',\mathring{y}''))t+im\theta \Bigr )\ne 0\), we can integrate by parts with respect to \(\theta \) several times and deduce that
We now study \(I_0\). We have
We can use the complex stationary phase formula of Melin–Sjöstrand [25, Theorem 2.3] to carry the \(dtd\theta \) integration in (4.8) and get (the calculation is similar as in the proof of Theorem 3.17 in [14], we omit the details)
Assume that \(q=n_+\ne n_-\). If \(m\rightarrow -\infty \), then the expansion for \(S^{(q)}_{G,m}(x,y)\) as \(m\rightarrow -\infty \) is similar to \(q=n_-\) case. When \(m\rightarrow +\infty \), we can repeat the method above with minor change and deduce that \(S^{(q)}_{G,m}(x,y)=O(m^{-\infty })\) on X. Summing up, we get Theorem 1.8.
5 Equivalent of the phase function \(\Phi _-(x,y)\)
Let \(p\in \mu ^{-1}(0)\) and let U be a small open set of p. We need
Definition 5.1
With the assumptions and notations used in Theorem 3.13, let \(\Phi _1, \Phi _2\in C^\infty (U\times U)\). We assume that \(\Phi _1\) and \(\Phi _2\) satisfy (3.52), (3.51) and (3.44). We say that \(\Phi _1\) and \(\Phi _2\) are equivalent on U if for any \(b_1(x,y,t)\in S^{n-\frac{d}{2}}_{\mathrm{cl}}\,\big (U\times U\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*\big )\) we can find \(b_2(x,y,t)\in S^{n-\frac{d}{2}}_{\mathrm{cl}}\,\big (U\times U\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*\big )\) such that
and vise versa.
We characterize now the phase \(\Phi _-\).
Theorem 5.2
Let \(\Phi _-(x,y)\in C^\infty (U\times U)\) be as in Theorem 3.13. Let \(\Phi \in C^\infty (U\times U)\). We assume that \(\Phi \) satisfies (3.52), (3.51) and (3.44). The functions \(\Phi \) and \(\Phi _-\) are equivalent on U in the sense of Definition 5.1 if and only if there is a function \(f\in C^\infty (U\times U)\) with \(f(x,x)=1\) such that \(\Phi (x,y)-f(x,y)\Phi _-(x,y)\) vanishes to infinite order at \(\mathrm{diag}\,\Bigr ((\mu ^{-1}(0)\bigcap U)\times (\mu ^{-1}(0)\bigcap U)\Bigr )\).
Proof
The “\(\Leftarrow \)” part follows from global theory of complex Fourier integral operator of Melin–Sjöstrand [25]. We only need to prove the “\(\Rightarrow \)” part. Take local coordinates \(x=(x_1,\ldots ,x_{2n+1})\) defined in some small neighbourhood of p such that \(x(p)=0\) and \(\omega _0(p)=dx_{2n+1}\). Since \(d_y\Phi (x, y)|_{x=y\in \mu ^{-1}(0)}=d_y\Phi _-(x, y)|_{x=y\in \mu ^{-1}(0)}=\omega _0(x)\), we have \(\frac{\partial \Phi }{\partial y_{2n+1}}(p,p)=\frac{\partial \Phi _-}{\partial y_{2n+1}}(p,p)=1\). From this observation and the Malgrange preparation theorem [12, Theorem 7.5.7], we conclude that in some small neighborhood of (p, p), we can find \(f(x,y), f_1(x,y)\in C^\infty \) such that
in some small neighborhood of (p, p), where \(\mathring{y}=(y_1,\ldots ,y_{2n})\). For simplicity, we assume that (5.1) hold on \(U\times U\). It is clear that \(\Phi _-(x,y)\) and \(y_{2n-1}+h(x,\mathring{y})\) are equivalent in the sense of Definition 5.1, \(\Phi (x,y)\) and \(y_{2n+1}+h_1(x,\mathring{y})\) are equivalent in the sense of Definition 5.1, we may assume that \(\Phi _-(x,y)=y_{2n+1}+h(x,\mathring{y})\) and \(\Phi (x,y)=y_{2n+1}+h_1(x,\mathring{y})\). Fix \(x_0\in \mu ^{-1}(0)\bigcap U\). We are going to prove that \(h(x,\mathring{y})-h_1(x,\mathring{y})\) vanishes to infinite order at \((x_0,x_0)\in (\mu ^{-1}(0)\bigcap U)\times (\mu ^{-1}(0)\bigcap U)\). Take
with \(b_0(x,x)\ne 0\) at each \(x\in U\bigcap \mu ^{-1}(0)\). Since \(\Phi \) and \(\Phi _-\) are equivalent on U in the sense of Definition 5.1, we can find \(a(x,y,t)\in S^{n-\frac{d}{2}}_{\mathrm{cl}}\,\big (U\times U\times \mathbb {R}_+,T^{*0,q}X\boxtimes (T^{*0,q}X)^*\big )\) such that
Put \(x_0=(x^1_0,x^2_0,\ldots ,x^{2n+1}_0)\) and \(\mathring{x}_0=(x^1_0,\ldots ,x^{2n}_0)\). Take \(\tau \in C^\infty _0(\mathbb {R}^{2n+1})\), \(\tau _1\in C^\infty _0(\mathbb {R}^{2n})\), \(\chi \in C^\infty _0(\mathbb {R})\) so that \(\tau =1\) near \(x_0\), \(\tau _1=1\) near \(\mathring{x}_0\), \(\chi =1\) near \(x^{2n+1}_0\) and \(\mathrm{Supp}\,\tau \Subset U\), \(\mathrm{Supp}\,\tau _1\times \mathrm{Supp}\,\chi \Subset U'\times \mathrm{Supp}\,\chi \Subset U\), where \(U'\) is an open neighborhood of \(\mathring{x}_0\) in \(\mathbb {R}^{2n}\). For each \(k>0\), we consider the distributions
for \(u\in C^\infty _0(U',T^{*0,q}X)\). By using the stationary phase formula of Melin–Sjöstrand [25], we can show that (cf. the proof of [14, Theorem 3.12]) \(A_k\) and \(B_k\) are smoothing operators and
Since
is smoothing, by using integration by parts with respect to \(y_{2n+1}\), it is easy to see that \(A_k-B_k=O(k^{-\infty })\) (see [14, Section 3]). Thus,
Now, we are ready to prove that \(h(x,\mathring{y})-h_1(x,\mathring{y})\) vanishes to infinite order at \((x_0,\mathring{x}_0)\). We assume that there exist \(\alpha _0\in \mathbb {N}^{2n+1}_0\), \(\beta _0\in \mathbb {N}^{2n}_0\), \(|\alpha _0|+|\beta _0|\ge 1\) such that
and
From (5.2), we have
Since \(h_1(x_0,\mathring{x}_0)=-x^{2n+1}_0\) and \(F_k(x,\mathring{y})=O(k^{-\infty })\), we have
On the other hand, we can check that
since \(g_0(x_0,\mathring{x}_0)\ne 0\). From (5.3), (5.4) and (5.5), we get a contradiction. Thus, \(h(x,\mathring{y})-h_1(x,\mathring{y})\) vanishes to infinite order at \((x_0,\mathring{x}_0)\). Since \(x_0\) is arbitrary, the theorem follows. \(\square \)
6 The proof of Theorem 1.9
6.1 Preparation
Fix \(p\in \mu ^{-1}(0)\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates as in Remark 4.4 defined in an open set U of p. We may assume that \(U=\Omega _1\times \Omega _2\times \Omega _3\times \Omega _4\), where \(\Omega _1\subset \mathbb {R}^d\), \(\Omega _2\subset \mathbb {R}^d\) are open sets of \(0\in \mathbb {R}^d\), \(\Omega _3\subset \mathbb {R}^{2n-2d}\) is an open set of \(0\in \mathbb {R}^{2n-2d}\) and \(\Omega _4\) is an open set of \(0\in \mathbb {R}\). From now on, we identify \(\Omega _2\) with
\(\Omega _3\) with \(\left\{ (0,\ldots ,0,x_{2d+1},\ldots ,x_{2n},0)\in U;\, (x_{d+1},\ldots ,x_{2n})\in \Omega _3\right\} \), \(\Omega _2\times \Omega _3\) with
For \(x=(x_1,\ldots ,x_{2n+1})\), we write \(x''=(x_{d+1},\ldots ,x_{2n+1})\), \(\mathring{x}''=(x_{d+1},\ldots ,x_{2n})\), \(\hat{x}''=(x_{d+1},\ldots ,x_{2d})\),
From now on, we identify \(x''\) with \((0,\ldots ,0,x_{d+1},\ldots ,x_{2n+1})\in U\), \(\mathring{x}''=(x_{d+1},\ldots ,x_{2n})\) with \((0,\ldots ,0,x_{d+1},\ldots ,x_{2n},0)\in U\), \(\hat{x}''\) with \((0,\ldots ,0,x_{d+1},\ldots ,x_{2d},0,\ldots ,0)\in U\), \(\widetilde{x}''\) with \((0,\ldots ,0,x_{2d+1},\ldots ,x_{2n+1})\in U\), \(\widetilde{\mathring{x}}''\) with \((0,\ldots ,0,x_{2d+1},\ldots ,x_{2n},0)\). Since \(G\times S^1\) acts freely on \(\mu ^{-1}(0)\), we take \(\Omega _2\) and \(\Omega _3\) small enough so that if \(x, x_1\in \Omega _2\times \Omega _3\) and \(x\ne x_1\), then
We now assume that \(q=n_-\) and let \(\Psi (x,y)\in C^\infty (U\times U)\) be as in Theorem 1.8. From \(S^{(q)}_{G,m}=(S^{(q)}_{G,m})^*\), we get
where \((S^{(q)}_{G,m})^*:L^2_{(0,q)}(X)\rightarrow L^2_{(0,q)}(X)\) is the adjoint of \(S^{(q)}_{G,m}:L^2_{(0,q)}(X)\rightarrow L^2_{(0,q)}(X)\) with respect to \((\,\cdot \,|\,\cdot \,)\) and \(b^*(x,y,m):T^{*0,q}_xX\rightarrow T^{*0,q}_yX\) is the adjoint of \(b(x,y,m):T^{*0,q}_yX\rightarrow T^{*0,q}_xX\) with respect to \(\langle \,\cdot \,|\,\cdot \,\rangle \). From (6.2), we can repeat the proof of Theorem 5.2 with minor change and deduce that
From \(\overline{\partial }_bS^{(q)}_{G,m}=0\), we can check that
From (6.3), (6.4) and notice that \(\frac{\partial }{\partial x_j}-i\frac{\partial }{\partial x_{d+j}}\in T^{0,1}_xX\), \(j=1,\ldots ,d\), where \(x\in \mu ^{-1}(0)\) (see Remark 4.4), and \(\frac{\partial }{\partial x_j}\Psi (x,y)=\frac{\partial }{\partial y_j}\Psi (x,y)=0\), \(j=1,\ldots ,d\), we conclude that
Let \(G_j(x,y):=\frac{\partial }{\partial y_{d+j}}\Psi (x,y)|_{y_{d+1}=\cdots =y_{2d}=0}\), \(H_j(x,y):=\frac{\partial }{\partial x_{d+j}}\Psi (x,y)|_{x_{d+1}=\cdots =x_{2d}=0}\). Put
Then, for \(j=1, 2, \ldots , d\),
and, for \(j=1,2\),
We also write \(u=(u_1,\ldots ,u_{2n+1})\) to denote the local coordinates of U. Recall that for any smooth function \(f\in C^\infty (U)\), we write \(\widetilde{f}\in C^\infty (U^{\mathbb {C}})\) to denote an almost analytic extension of f (see the discussion after (3.38)). We consider the following two systems
and
where \(\widetilde{\widetilde{u}''}=(0,\ldots ,0,\widetilde{u}_{2d+1},\ldots ,\widetilde{u}_{2n+1})\), \(\widetilde{u''}=(0,\ldots ,0,\widetilde{u}_{d+1},\ldots ,\widetilde{u}_{2n+1})\). From (6.5) and Theorem 1.12, we can take \(\widetilde{\Psi }_1\) and \(\widetilde{\Psi }_2\) so that for every \(j=1,2,\ldots ,d\),
and, for \(j=1, 2\),
where \(\widetilde{\mathring{x}''}=(0,\ldots ,0,\widetilde{x}_{d+1},\ldots ,\widetilde{x}_{2n},0)\), \(\widetilde{\mathring{y}''}=(0,\ldots ,0,\widetilde{y}_{d+1},\ldots ,\widetilde{y}_{2n},0)\).
From Theorem 1.12, (1.19) and \(d_x\Psi (x,x)=-d_y\Psi (x,x)=-\omega _0(x),\ \forall x\in \mu ^{-1}(0)\), it is not difficult to see that
and the matrices
are non-singular. Moreover,
Hence, near (p, p), we can solve (6.7) and (6.8) and the solutions are unique. Let \(\alpha (x,y)=(\alpha _{2d+1}(x,y),\ldots ,\alpha _{2n}(x,y))\in C^\infty (U\times U,\mathbb {C}^{2n-2d})\) and \(\beta (x,y)=(\beta _{d+1}(x,y),\ldots ,\beta _{2n}(x,y))\in C^\infty (U\times U,\mathbb {C}^{2n-d})\) be the solutions of (6.7) and (6.8), respectively. From (6.9), it is easy to see that
From (6.11), we see that the value of \(\widetilde{\Psi }_1(x,\widetilde{\widetilde{u}''})+\widetilde{\Psi }_2(\widetilde{\widetilde{u}''},y)\) at critical point \(\widetilde{\widetilde{u}''}=\alpha (x,y)\) is equal to the value of \(\widetilde{\Psi }_1(x,\widetilde{u''})+\widetilde{\Psi }_2(\widetilde{u''},y)\) at critical point \(\widetilde{u''}=\beta (x,y)\). Put
\(\Psi _3(x,y)\) is a complex phase function. From (6.10), we have
Moreover, we have the following
Theorem 6.1
The function \(\Psi _3(x,y)-\Psi (x,y)\) vanishes to infinite order at \(\mathrm{diag}\,\Bigr ((\mu ^{-1}(0)\bigcap U)\times (\mu ^{-1}(0)\bigcap U)\Bigr )\).
Proof
We consider the kernel \(S^{(q)}_{G,m}\circ S^{(q)}_{G,m}\) on U. Let \(V\Subset U\) be an open set of p. Let \(\chi (\mathring{x}'')\in C^\infty _0(\Omega _2\times \Omega _3)\). From (6.1), we can extend \(\chi (\mathring{x}'')\) to \(W:=\left\{ g\circ e^{i\theta }\circ x;\, (g,e^{i\theta })\in G\times S^1, x\in \Omega _2\times \Omega _3\right\} \) by \(\chi (g\circ e^{i\theta }\circ \mathring{x}''):=\chi (\mathring{x}'')\), for every \((g,e^{i\theta })\in G\times S^1\). Assume that \(\chi =1\) on some neighborhood of V. Let \(\chi _1\in C^\infty _0(U)\) with \(\chi _1=1\) on some neighborhood of V and \(\mathrm{Supp}\,\chi _1\subset \left\{ x\in X;\, \chi (x)=1\right\} \). We have
Let’s first consider \(\chi _1S^{(q)}_{G,m}(1-\chi )\circ S^{(q)}_{G,m}\). We have
If \(u\notin \left\{ x\in X;\, \chi (x)=1\right\} \). Since \(\mathrm{Supp}\,\chi _1\subset \left\{ x\in X;\, \chi (x)=1\right\} \) and \(\chi (x)=\chi (g\circ e^{i\theta }\circ x)\), for every \((g,e^{i\theta })\in G\times S^1\), for every \(x\in X\), we conclude that \(g\circ e^{i\theta }\circ u\notin \mathrm{Supp}\,\chi _1\), for every \((g,e^{i\theta })\in G\times S^1\). From this observation and notice that \(S^{(q)}\) is smoothing away from diagonal, we can integrate by parts with respect to \(\theta \) in (6.14) and deduce that \(\chi _1S^{(q)}_{G,m}\circ (1-\chi )=O(m^{-\infty })\) and hence
From (6.13) and (6.15), we get
We can check that on U,
where \(d\mu (g)d\theta dv(\mathring{u}'')=dv(x)\) on U. We use complex stationary phase formula of Melin–Sjöstrand [25] to carry out the integral (6.17) and get
From (6.16), (6.18) and notice that \((\chi _1S^{(q)}_{G,m}\circ S^{(q)}_{G,m})(x,y)=(\chi _1S^{(q)}_{G,m})(x,y)\), we deduce that
From (6.19), we can repeat the proof of Theorem 5.2 with minor change and deduce that \(\Psi _3(x,y)-\Psi (x,y)\) vanishes to infinite order at \(\mathrm{diag}\,\Bigr ((\mu ^{-1}(0)\bigcap U)\times (\mu ^{-1}(0)\bigcap U)\Bigr )\). \(\square \)
The following two theorems follow from (6.6), (6.12), Theorem 6.1, complex stationary phase formula of Melin–Sjöstrand [25] and some straightforward computation. We omit the details.
Theorem 6.2
With the notations used above, let
where E, F and H are vector bundles over X. Let \(\chi (\mathring{x}'')\in C^\infty _0(\Omega _2\times \Omega _3)\). Then, we have
where \(|\det R_x|\) is in the discussion before Theorem 1.6.
Moreover, if there are \(N_1, N_2\in \mathbb {N}\), such that \(|a_0(x,y)|\le C|(x,y)-(x_0,x_0)|^{N_1}\), \(|b_0(x,y)|\le C|(x,y)-(x_0,x_0)|^{N_2}\), for all \(x_0\in \mu ^{-1}(0)\bigcap U\), where \(C>0\) is a constant, then,
for all \(x_0\in \mu ^{-1}(0)\bigcap U\), where \(\hat{C}>0\) is a constant.
Theorem 6.3
With the notations used above, let
where E, F and H are vector bundles over X. Let \(\chi _1(\widetilde{\mathring{x}}'')\in C^\infty _0(\Omega _3)\). Then, we have
where \(|\det R_x|\) is in the discussion before Theorem 1.6.
Moreover, if there are \(N_1, N_2\in \mathbb {N}\), such that \(|\alpha _0(x,\widetilde{y}'')|\le C|(x,\widetilde{y}'')-(x_0,x_0)|^{N_1}\), \(|\beta _0(x,\widetilde{y}'')|\le C|(x,\widetilde{y}'')-(x_0,x_0)|^{N_2}\), for all \(x_0\in \mu ^{-1}(0)\bigcap U\), where \(C>0\) is a constant, then,
for all \(x_0\in \mu ^{-1}(0)\bigcap U\), where \(\hat{C}>0\) is a constant.
6.2 The proof of Theorem 1.9
Since \(\underline{\mathfrak {g}}_x\) is orthogonal to \(H_xY\bigcap JH_xY\) and \(H_xY\bigcap JH_xY\subset \underline{\mathfrak {g}}^{\perp _b}_x\), for every \(x\in Y\), we can find a G-invariant orthonormal basis \(\left\{ Z_1,\ldots ,Z_n\right\} \) of \(T^{1,0}X\) on Y such that
Let \(\left\{ e_1,\ldots ,e_n\right\} \) denote the orthonormal basis of \(T^{*0,1}X\) on Y, dual to \(\left\{ \overline{Z}_1,\ldots ,\overline{Z}_n\right\} \). Fix \(s=0,1,2,\ldots ,n-d\). For \(x\in Y\), put
and let \(B^{*0,s}X\) be the vector bundle of Y with fiber \(B^{*0,s}_xX\), \(x\in Y\). Let \(C^\infty (Y,B^{*0,s}X)^G\) denote the set of all G-invariant sections of Y with values in \(B^{*0,s}X\). Let
be the natural identification.
Assume that \(\lambda _1<0,\ldots ,\lambda _r<0\), and \(\lambda _{d+1}<0,\ldots ,\lambda _{n_--r+d}<0\). For \(x\in Y\), put
and let
Let \(\iota :Y\rightarrow X\) be the natural inclusion and let \(\iota ^*:\Omega ^{0,q}(X)\rightarrow \Omega ^{0,q}(Y)\) be the pull-back of \(\iota \). Recall that we work with the assumption that \(q=n_-\). Let \(\Box ^{(q-r)}_{b,Y_G}\) be the Kohn Laplacian for \((0,q-r)\) forms on \(Y_G\). Fix \(m\in \mathbb {N}\). Let \(H^{q-r}_{b,m}(Y_G):=\left\{ u\in \Omega ^{0,q-r}(Y_G);\, \Box ^{(q-r)}_{b,Y_G}u=0,\ \ Tu=imu\right\} \). Let \(S^{(q-r)}_{Y_G,m}:L^2_{(0,q-r)}(Y_G)\rightarrow H^{q-r}_{b,m}(Y_G)\) be the orthogonal projection and let \(S^{(q-r)}_{Y_G,m}(x,y)\) be the distribution kernel of \(S^{(q-r)}_{Y_G,m}\). Let
Let
Recall that \(\tau _{x,n_-}\) is given by (1.7). Let \(\sigma ^*_m:\Omega ^{0,q-r}(Y_G)\rightarrow \Omega ^{0,q}(X)\) be the adjoints of \(\sigma _m\). It is easy to see that \(\sigma ^*_mu\in H^q_{b,m}(X)^G:=(\mathrm{Ker}\,\Box ^{(q)}_b)^G_m,\ \forall u\in \Omega ^{0,q-r}(Y_G)\). Let \(\sigma _m(x,y)\) and \(\sigma ^*_m(x,y)\) denote the distribution kernels of \(\sigma _m\) and \(\sigma ^*_m\), respectively.
Let’s pause and recall some results for \(S^{(q-r)}_{Y_G,m}\). We first introduce some notations. Let \(\mathcal {L}_{Y_G,x}\) be the Levi form on \(Y_G\) at \(x\in Y_G\) induced naturally from \(\mathcal {L}\). The Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(T^{1,0}X\) induces a Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(T^{1,0}Y_G\). Let \(\det \,\mathcal {L}_{Y_G,x}=\lambda _1\ldots \lambda _{n-d}\), where \(\lambda _j\), \(j=1,\ldots ,n-d\), are the eigenvalues of \(\mathcal {L}_{Y_G,x}\) with respect to the Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \). For \(x\in Y_G\), let \(\hat{\tau }_x:T^{*0,q-r}_xY_G\rightarrow \hat{\mathcal {N}}(x,n_-)\) be the orthogonal projection.
Let \(\pi :Y\rightarrow Y_G\) be the natural quotient. Let \(S^{(q-r)}_{Y_G}:L^2_{(0,q-r)}(Y_G)\rightarrow \mathrm{Ker}\,\Box ^{(q-r)}_{b,Y_G}\) be the Szegő projection as (3.2). Since \(S^{(q-r)}_{Y_G}\) is smoothing away from diagonal (see Theorem 3.3), it is easy to see that for any \(x, y\in Y\), if \(\pi (e^{i\theta }\circ x)\ne \pi (e^{i\theta }\circ y)\), for every \(\theta \in [0,2\pi [\), then there are open sets U of \(\pi (x)\) in \(Y_G\) and V of \(\pi (y)\) in \(Y_G\) such that for all \(\hat{\chi }\in C^\infty _0(U)\), \(\widetilde{\chi }\in C^\infty _0(V)\), we have
Fix \(p\in Y\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates as in Remark 4.4. We will use the same notations as in the beginning of Sect. 6.1. From now on, we identify \(\widetilde{x}''\) as local coordinates of \(Y_G\) near \(\pi (p)\in Y_G\) and we identify \(W:=\Omega _3\times \Omega _4\) with an open set of \(\pi (p)\) in \(Y_G\). It was proved in Theorem 4.11 in [14] that as \(m\rightarrow +\infty \),
and
where \(p_0\) denotes the principal symbol of \(\Box ^{(q-r)}_{b,Y_G}\), \(z_j=x_{2j-1}+ix_{2j}\), \(j=d+1,\ldots ,n\), and \(\mu _{d+1},\ldots ,\mu _n\) are the eigenvalues of \(\mathcal {L}_{Y_G,p}\).
Note that for any \(\phi _1(\widetilde{x}'',\widetilde{y}'')\in C^\infty (W\times W)\), if \(\phi _1\) satisfies (6.21), then \(\phi _1-\phi \) vanishes to infinite order at \(\widetilde{\mathring{x}}''=\widetilde{\mathring{y}}''\) (see Remark 3.6 in [14]). It is not difficult to see that the phase function \(\Psi (\widetilde{x}'',\widetilde{y}'')\) satisfies (6.21). Hence, we can replace the phase \(\phi (\widetilde{x}'',\widetilde{y}'')\) by \(\Psi (\widetilde{x}'',\widetilde{y}'')\) and we have
We can now prove
Theorem 6.4
With the notations used above, if \(y\notin Y\), then for any open set D of y with \(\overline{D}\bigcap Y=\emptyset \), we have
Let \(x, y\in Y\). If \(\pi (x)\ne \pi (e^{i\theta }\circ y)\), for every \(\theta \in [0,2\pi [\), then there are open sets \(U_G\) of \(\pi (x)\) in \(Y_G\) and V of y in X such that
Let \(p\in \mu ^{-1}(0)\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates as in Remark 4.4. Then,
where U is an open set of p, \(W=\Omega _3\times \Omega _4\), \(\Omega _3\) and \(\Omega _4\) are open sets as in the beginning of Sect. 6.1.
Proof
Note that \(S^{(q)}_{G,m}=O(m^{-\infty })\) away Y. From this observation, we get (6.23). Let \(x, y\in Y\). Assume that \(\pi (x)\ne \pi (e^{i\theta }\circ y)\), for every \(\theta \in [0,2\pi [\). Since
and \(S^{(q)}\) is smoothing away from diagonal, we can integrate by parts with respect to \(\theta \) and deduce that there are open sets \(U_1\) of x in X and \(V_1\) of y in X such that
From (6.20), we see that there are open sets \(\hat{U}_G\) of \(\pi (x)\) in \(Y_G\) and \(\hat{V}_G\) of \(\pi (y)\) in \(Y_G\) such that
From (6.27) and (6.28), we get (6.24).
Fix \(u=(u_1,\ldots ,u_{2n+1})\in Y\bigcap U\). From (6.23) and (6.24), we only need to show that (6.25) and (6.26) hold near u and we may assume that \(u=(0,\ldots ,0,u_{2d+1},\ldots ,u_{2n},0)=\widetilde{\mathring{u}}''\). Let V be a small neighborhood of u. Let \(\chi (\widetilde{\mathring{x}}'')\in C^\infty _0(\Omega _3)\). From (6.1), we can extend \(\chi (\widetilde{\mathring{x}}'')\) to
by \(\chi (g\circ e^{i\theta }\circ \widetilde{\mathring{x}}''):=\chi (\widetilde{\mathring{x}}'')\), for every \((g,e^{i\theta })\in G\times S^1\). Assume that \(\chi =1\) on some neighborhood of V. Let \(V_G=\left\{ \pi (x);\, x\in V\right\} \). Let \(\chi _1\in C^\infty _0(Y_G)\) with \(\chi _1=1\) on some neighborhood of \(V_G\) and \(\mathrm{Supp}\,\chi _1\subset \left\{ \pi (x)\in Y_G;\, x\in Y, \chi (x)=1\right\} \). We have
If \(u\in Y\) but \(u\notin \left\{ x\in X;\, \chi (x)=1\right\} \). Since \(\mathrm{Supp}\,\chi _1\subset \left\{ \pi (x)\in X;\, x\in Y, \chi (x)=1\right\} \) and \(\chi (x)=\chi (g\circ e^{i\theta }\circ x)\), for every \((g,e^{i\theta })\in G\times S^1\), for every \(x\in X\), we conclude that \(\pi (e^{i\theta }\circ u)\notin \mathrm{Supp}\,\chi _1\), for every \(e^{i\theta }\in S^1\). From this observation and (6.20), we get
From (6.29) and (6.30), we get
From (6.22) and Theorem 1.8, we can check that on U,
where \(\hat{b}(\widetilde{\mathring{v}}'',y,m)=\Bigr (\iota _G\circ \hat{p}\circ \tau _{x,n_-}\circ f\circ \iota ^*\circ \chi (\widetilde{\mathring{v}}'')\circ b\Bigr )(\widetilde{\mathring{v}}'',y,m)\). From (6.31) and Theorem 6.3, we see that (6.25) and (6.26) hold near u. The theorem follows. \(\square \)
Let
Let \(F_m(x,y)\) and \(\hat{F}_m(x,y)\) be the distribution kernels of \(F_m\) and \(\hat{F}_m\) respectively. From Theorems 6.2 and 6.3, we can repeat the proof of Theorem 6.4 with minor change and deduce the following two theorems
Theorem 6.5
With the notations used above, if \(y\notin Y\), then for any open set D of y with \(\overline{D}\bigcap Y=\emptyset \), we have \( F_m=O(m^{-\infty })\ \ \text{ on } X\times D\).
Let \(x, y\in Y\). If \(\pi (x)\ne \pi (e^{i\theta }\circ y)\), for every \(\theta \in [0,2\pi [\), then there are open sets \(D_1\) of x in X and \(D_2\) of y in X such that \( F_m=O(m^{-\infty })\ \ \text{ on } D_1\times D_2\).
Let \(p\in \mu ^{-1}(0)\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates as in Remark 4.4. Then,
and
where U is an open set of p.
Theorem 6.6
Let \(x, y\in Y\). If \(\pi (x)\ne \pi (e^{i\theta }\circ y)\), for every \(\theta \in [0,2\pi [\), then there are open sets \(D_G\) of \(\pi (x)\) in \(Y_G\) and \(V_G\) of \(\pi (y)\) in \(Y_G\) such that \( \hat{F}_m=O(m^{-\infty })\ \ \text{ on } D_G\times V_G\).
Let \(p\in \mu ^{-1}(0)\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates as in Remark 4.4. Then,
where \(W=\Omega _3\times \Omega _4\), \(\Omega _3\) and \(\Omega _4\) are open sets as in the beginning of Sect. 6.1.
Let \(R_m:=\frac{1}{C_0}F_m-S^{(q)}_{G,m}:\Omega ^{0,q}(X)\rightarrow H^q_{b,m}(X)^G\), where \(C_0=2^{-3d+3n}\pi ^{\frac{d}{2}}\). Since \(F_m=F_mS^{(q)}_{G,m}\), it is clear that
Our next goal is to show that for m large, \(I+R_m:\Omega ^{0,q}(X)\rightarrow \Omega ^{0,q}(X)\) is injective. From Theorem 6.5 and Theorem 1.8, we see that if \(y\notin Y\), then for any open set D of y with \(\overline{D}\bigcap Y=\emptyset \), we have
Let \(x, y\in Y\). If \(\pi (x)\ne \pi (e^{i\theta }\circ y)\), for every \(\theta \in [0,2\pi [\), then there are open sets \(D_1\) of x in X and \(D_2\) of y in X such that
Let \(p\in \mu ^{-1}(0)\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates as in Remark 4.4. Then,
Moreover, from (6.32) and (1.10), it is easy to see
for all \(x_0\in \mu ^{-1}(0)\bigcap U\), where \(C>0\) is a constant. We need
Lemma 6.7
Let \(p\in \mu ^{-1}(0)\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates as in Remark 4.4 defined in an open set U of p. Let
Assume that \(h(x,y,m)\in C^\infty _0(U\times U, T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\). Then, there is a constant \(\hat{C}>0\) independent of m such that
where \(\delta _m\) is a sequence with \(\lim _{m\rightarrow \infty }\delta _m=0\).
Proof
Fix \(N\in \mathbb {N}\). It is not difficult to see that
where \(H^*_m\) denotes the adjoint of \(H_m\). From Theorem 6.2, we can repeat the proof of Theorem 6.4 with minor change and deduce that
Hence,
where \(\hat{C}>0\) is a constant independent of m. Take N large enough so that \(n-2^{N+1}-\frac{d}{2}<0\). From (6.40) and (6.39), we get (6.38). \(\square \)
We also need
Lemma 6.8
Let \(p\in \mu ^{-1}(0)\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be the local coordinates as in Remark 4.4 defined in an open set U of p. Let
Suppose that
for all \(x_0\in \mu ^{-1}(0)\bigcap U\), where \(C>0\) is a constant. Then,
where \(\varepsilon _m\) is a sequence with \(\lim _{m\rightarrow \infty }\varepsilon _m=0\).
Proof
Fix \(N\in \mathbb {N}\). It is not difficult to see that
where \(B^*_m\) denotes the adjoint of \(B_m\). From Theorem 6.2, we can repeat the proof of Theorem 6.4 with minor change and deduce that
and
for all \(x_0\in \mu ^{-1}(0)\bigcap U\), where \(C>0\) is a constant. Let
where \(h(x,y,m)=\hat{g}(x,y,m)-\hat{g}_0(x,y,m)\). It is clear that \(h(x,y,m)\in S^{n-1-\frac{d}{2}}_{\mathrm{loc}}\,(1; U\times U, T^{*0,q}X\boxtimes (T^{*0,q}X)^*)\). From Lemma 6.7, we see that
where \(\delta _m\) is a sequence with \(\lim _{m\rightarrow \infty }\delta _m=0\).
From (6.43), we see that
where \(C_1>0\) is a constant. From (3.44), we see that
where \(c>0\) is a constant. From (6.45) and (6.46), we conclude that
where \(\hat{C}>0\) is a constant independent of m. From (6.47), we see that if N large enough, then
where \(\hat{\delta }_m\) is a sequence with \(\lim _{m\rightarrow \infty }\hat{\delta }_m=0\).
From (6.42), (6.44) and (6.48), we get (6.41). \(\square \)
From (6.34), (6.35), (6.36), (6.37) and Lemma 6.8, we get
Theorem 6.9
With the notations above, we have \(\left\| R_mu\right\| \le \varepsilon _m\left\| u\right\| ,\ \ \forall u\in \Omega ^{0,q}(X),\ \forall m\in \mathbb {N}\), where \(\varepsilon _m\) is a sequence with \(\lim _{m\rightarrow \infty }\varepsilon _m=0\).
In particular, if m is large enough, then the map \(I+R_m:\Omega ^{0,q}(X)\rightarrow \Omega ^{0,q}(X)\) is injective.
Proof of Theorem 1.9
From (6.33) and Theorem 6.9, we see that if m is large enough, then the map \(F_m=\sigma ^*_m\sigma _m: H^q_{b,m}(X)^G\rightarrow H^q_{b,m}(X)^G\) is injective. Hence, if m is large enough, then the map \(\sigma _m:H^q_{b,m}(X)^G\rightarrow H^{q-r}_{b,m}(Y_G)\) is injective and \(\mathrm{dim}\,H^q_{b,m}(X)^G\le \mathrm{dim}\,H^{q-r}_{b,m}(Y_G)\).
Similarly, we can repeat the proof of Theorem 6.9 with minor change and deduce that, if m is large enough, then the map \(\hat{F}_m=\sigma _m\sigma ^*_m: H^{q-r}_{b,m}(Y_G)\rightarrow H^{q-r}_{b,m}(Y_G)\) is injective. Hence, if m is large enough, then the map \(\sigma ^*_m:H^{q-r}_{b,m}(Y_G)\rightarrow H^{q}_{b,m}(X)^G\) is injective. Thus, \(\mathrm{dim}\,H^q_{b,m}(X)^G=\mathrm{dim}\,H^{q-r}_{b,m}(Y_G)\) and \(\sigma _m\) is an isomorphism if m large enough. \(\square \)
References
Albert, C.: Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact. J. Geom. Phys. 6, 627–649 (1989)
Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegő. Astérisque 34–35, 123–164 (1976)
Bismut, J.-M., Lebeau, G.: Complex immersions and Quillen metrics. Inst. Hautes Études Sci. Publ. Math. 74(1991), ii+298 (1992)
Boutet de Monvel, L., Guillemin, V.: The Spectral Theory of Toeplitz Operators. Annals of Mathematics Studies, Vol. 99. Princeton Univ. Press, Princeton (1981)
Braverman, M.: Cohomology of Mumford Quotient, Quantization of Singular Symplectic Quotients. Programs in Mathematics, 198, pp. 47–59. Birkhäuser, Basel (2001)
Catlin, D.: The Bergman Kernel and a Theorem of Tian. Analysis and Geometry in Several Complex Variables (Katata. 1997), Trends in Mathematics, pp. 1–23. Birkhäuser, Boston (1999)
Charles, L.: Toeplitz operators and Hamiltonian torus actions. J. Funct. Anal. 236(1), 299–350 (2006)
Engliš, M.: Weighted Bergman kernels and quantization. Comm. Math. Phys. 227(2), 211–241,: MR1903645. Zbl 1010, 32002 (2002)
Geiges, H.: Constructions of contact manifolds. Math. Proc. Camb. Philos. Soc. 121(3), 455–464 (1997)
Guillemin, V., Sternberg, S.: Geometric quantization and multiplicities of group representations. Invent. Math. 67(3), 515–538 (1982)
Guillemin, V.: Star products on compact pre-quantizable symplectic manifolds. Lett. Math. Phys. 35(1), 85–89 (1995)
Hörmander, L.: The Analysis of Linear Partial Differential Operators. I, Classics in Mathematics. Springer, Berlin (2003)
Hsiao, C.-Y.: Projections in Several Complex Variables, Mémoirs Society Mathematics. France, Nouv. Sér. Vol. 123, p. 131 (2010)
Hsiao, C.-Y., Marinescu, G.: Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles. Comm. Anal. Geom. 22(1), 1–108 (2014)
Hsiao, C.-Y., Marinescu, G.: On the singularities of the Szegő projections on lower energy forms. J. Differ. Geom. 107(1), 83–155 (2017)
Kohn, J.J.: The range of the tangential Cauchy-Riemann operator. Duke Math. J. 53(2), 307–562 (1986)
Loose, F.: A remark on the reduction of Cauchy-Riemann manifolds. Math. Nachr. 214, 39–51 (2000)
Ma, X., Marinescu, G.: The first coefficients of the asymptotic expansion of the Bergman kernel of the \(spin^c\) Dirac operator. Int. J. Math. 17(6), 737–759 (2006)
Ma, X., Marinescu, G.: Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics, 254. Birkhäuser, Basel (2007)
Ma, X., Zhang, W.: Bergman kernels and symplectic reduction. Astérisque 318, 154 (2008)
Ma, X., Marinescu, G.: Generalized Bergman kernels on symplectic manifolds. Adv. Math. 217(4), 1756–1815 (2008)
Ma, X.: Geometric Quantization on Kähler and Symplectic Manifolds, International Congress of Mathematicians, vol. II, Hyderabad, India, August 19–27, pp. 785–810 (2010)
Ma, X., Zhang, W.: Geometric quantization for proper moment maps: the Vergne conjecture. Acta Math. 212(1), 11–57 (2014)
Meinrenken, E.: Symplectic surgery and the Spinc-Dirac operator. Adv. Math. 134(2), 240–277 (1998)
Melin, A., Sjöstrand, J.: Fourier integral operators with complex-valued phase functions. Springer Lect. Notes Math. 459, 120–223 (1975)
Paoletti, R.: Moment maps and equivariant Szegő kernels. J. Symplectic Geom. 2(1), 133–175 (2003)
Puchol, M.: G-invariant holomorphic Morse inequalities. J. Differ. Geom. (to appear) . arXiv:1506.04526
Rossi, H.: Attaching analytic spaces to an analytic space along a pseudoconcave boundary. In: Proceedings of the Conference on Complex Manifolds (Minneapolis), pp. 242–256. Springer, New York (1965)
Shiffman, B., Zelditch, S.: Distribution of zeros of random and quantum chaotic sections of positive line bundles. Comm. Math. Phys. 200, 661–683 (1999)
Teleman, C.: The quantization conjecture revisited. Ann. Math. (2) 152(1), 1–43 (2000)
Tian, Y., Zhang, W.: An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg. Invent. Math. 132(2), 229–259 (1998)
Zelditch, S.: Szegő kernels and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998)
Zhang, W.: Holomorphic quantization formula in singular reduction. Commun. Contemp. Math. 1(3), 281–293 (1999)
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Communicated by A. Malchiodi.
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The first author was partially supported by Taiwan Ministry of Science and Technology Projects 108-2115-M-001-012-MY5, 109-2923-M-001-010-MY4 and Academia Sinica Career Development Award. This work was initiated when the second author was visiting the Institute of Mathematics at Academia Sinica in the summer of 2016. The second author would like to thank the Institute of Mathematics at Academia Sinica for its hospitality and financial support during his stay. The second author was also supported by Taiwan Ministry of Science of Technology Projects 105-2115-M-008-008-MY2, 107-2115-M-008-007-MY2 and 109-2115-M-008-007-MY2.
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Hsiao, CY., Huang, RT. G-invariant Szegő kernel asymptotics and CR reduction. Calc. Var. 60, 47 (2021). https://doi.org/10.1007/s00526-020-01912-4
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DOI: https://doi.org/10.1007/s00526-020-01912-4