# Asymptotic isometric minimal immersions of conic reductions into spheres for
Hamiltonian actions

Andrea Galasso111Address: Dipartimento di Matematica e Applicazioni,
Università degli Studi di Milano-Bicocca, Via R. Cozzi 55, 20125 Milano,
Italy; ORCID iD: 0000-0002-5792-1674; e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Consider a Hodge manifold and assume that a compact connected Lie group acts
on the manifold in a Hamiltonian and holomorphic manner and that this action
linearises on a given quantizing line bundle. Then, there is an associated
unitary representation on the Hardy space of the circle bundle inside the dual
of the line one. We define conic reductions and we investigate their
equivariant geometry; we introduce maps which are asymptotic isometric minimal
immersions into spheres making use of the corresponding equivariant Szegő
projector.

###### Contents

  1. 1 Introduction
  2. 2 Motivations and preliminaries
    1. 2.1 Conic reductions
    2. 2.2 Equivariant Szegő kernels
  3. 3 Proof of Theorem and Corollary

Keywords: symplectic manifold, group action, coisotropic reduction

Mathematics Subject Classification: 53DXX

##  1 Introduction

Let $(M,\,\omega)$ be a compact connected Hodge manifold of complex dimension
$d$ with complex structure $J$ and quantum line bundle $(A,h)$ whose curvature
form of the unique compatible and holomorphic connection is $-2i\,\omega$; we
shall denote by $g$ the Riemannian structure $\omega(\cdot\,,J\cdot)$. Let
$X\subset A^{*}$ the unit circle bundle, with projection $\pi:X\rightarrow M$.
Then $(X,\,\alpha)$ is a CR manifold and $\alpha$ is the contact form, see
[Z2]. The Hardy space $H(X)$ represents the quantum space and under some
suitable hypothesis classical symmetries on $M$ of a Lie group $G$ of rank
$r_{G}$ give rise to a quantum representation of $G$ on $H(X)$. In the setting
of ladder representations one is led to consider spaces
$H_{k\boldsymbol{\nu}}(X)$ for a fix unitary irreducible representation
$\boldsymbol{\nu}$ as $k\rightarrow+\infty$ and the corresponding projectors
on it. The aim of this paper is to study geometric properties of this
projector.

More precisely, given a compact connected Lie group $G$ and holomorphic and
Hamiltonian action $\mu:G\times M\rightarrow M$, denote the moment map with
$\Phi:M\rightarrow\mathfrak{g}^{*}$ (where $\mathfrak{g}$ is the Lie algebra
of $G$). The Hamiltonian action $\mu$ naturally induces an infinitesimal
contact action of $\mathfrak{g}$ on $X$ (see [Ko]); explicitly, if
$\boldsymbol{\xi}\in\mathfrak{g}$ and $\boldsymbol{\xi}_{M}$ is the
corresponding Hamiltonian vector field on $M$, then its contact lift
$\boldsymbol{\xi}_{X}$ is

$\boldsymbol{\xi}_{X}:=\boldsymbol{\xi}_{M}^{\sharp}-\langle\Phi\circ\pi,\boldsymbol{\xi}\rangle\,\partial_{\theta}\,,$
(1)

where $\boldsymbol{\xi}_{M}^{\sharp}$ denotes the horizontal lift on $X$ of a
vector field $\boldsymbol{\xi}_{M}$ on $M$, and $\partial_{\theta}$ is the
generator of the structure circle action on $X$.

Furthermore, suppose that $\mu$ lifts to an action

$\widetilde{\mu}:G\times X\rightarrow X$

acting via contact and CR automorphism on $X$. Under these assumptions, there
is a naturally induced unitary representation of $G$ on the Hardy space
$H(X)\subset L^{2}(X)$; hence $H(X)$ can be equivariantly decomposed over the
irreducible representations of $G$:

$H(X)=\bigoplus_{\boldsymbol{\nu}\in\widehat{G}}H(X)_{\boldsymbol{\nu}},$

where $\widehat{G}$ is the collection of all irreducible representations of
$G$. If $\Phi(m)\neq 0$ for every $m\in M$, then each isotypical component
$H(X)_{\boldsymbol{\nu}}$ is finite dimensional (see e.g. §2 of [Pa3] where it
is proved for $G$ abelian, the proof still holds for any compact connected Lie
group). Let us fix maximal torus $T$ in $G$, we shall label the irreducible
representations of $G$ with maximal weight
$\boldsymbol{\nu}\in\mathcal{D}^{G}\subset\mathfrak{g}^{*}$ by the one
belonging to the open positive Weyl chamber.

Fix $\boldsymbol{\nu}\in\widehat{G}$, let
$\Pi_{k\boldsymbol{\nu}}:L^{2}(X)\rightarrow H(X)_{k\boldsymbol{\nu}}$ be the
corresponding equivariant Szegő projection; we also denote by
$\Pi_{k\boldsymbol{\nu}}\in\mathcal{C}^{\infty}(X\times X)$ its distributional
kernel, which is given explicitly by the formula:

$\Pi_{k\boldsymbol{\nu}}(x,\,y)=d_{k\boldsymbol{\nu}}\int_{G}\overline{\chi_{k\boldsymbol{\nu}}(g)}\,\Pi(\tilde{\mu}_{g^{-1}}(x),\,y)\,\mathrm{dV}_{G}(g)$
(2)

where $\Pi$ is the Szegő kernel, $\mathrm{dV}_{G}$ is the Haar measure and
$\chi_{k\boldsymbol{\nu}}$ and $d_{k\boldsymbol{\nu}}$ are respectively the
character and the dimension of the representation $k\boldsymbol{\nu}$. In
[Pa5], it is studied the dimensions of isotypes $H(X)_{k\boldsymbol{\nu}}$ and
the behavior of $\Pi_{k\boldsymbol{\nu}}(x,\,x)$ as $k\rightarrow+\infty$.
Now, we need to generalize the results contained in [Pa5] but let us pause to
introduce some notation.

Denote by $\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})$ the cone through the
co-adjoint orbit $\mathcal{O}_{\boldsymbol{\nu}}$ passing through
$\boldsymbol{\nu}$ in $\mathfrak{g}^{*}$. Furthermore, let us denote by
$\sigma_{\boldsymbol{\nu}}$ the Kostant-Kirillov symplectic structure on
$\mathcal{O}_{\boldsymbol{\nu}}$, so that the symplectic volume form and
volume are respectively

$\mathrm{dV}_{\mathcal{O}_{\boldsymbol{\nu}}}:=\frac{\sigma_{\boldsymbol{\nu}}^{n_{G}}}{n_{G}!}\,,\qquad\mathrm{vol}(\mathcal{O}_{\boldsymbol{\nu}}):=\int_{\mathcal{O}_{\boldsymbol{\nu}}}\mathrm{dV}_{\mathcal{O}_{\boldsymbol{\nu}}}$

It turns out that the loci

$M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}:=\Phi^{-1}(\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}}))\quad\text{
and }\quad
X_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}:=\pi^{-1}\left(M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\right)\,$

play an important role in the study of asymptotic expansions. We say that
$\Phi$ is transverse to the cone $\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})$
in $\mathfrak{g}^{*}$ if

$\mathfrak{g}^{*}=\mathrm{d}_{m}\Phi(T_{m}M)+\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})$
(3)

for all $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$. Under transversality
assumption (3), the locus $M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ (if non-
empty) is a connected compact sub-manifold of $M$, of real co-dimension
$r_{G}-1$. Let us recall briefly the geometry of
$M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ as a real sub-manifold in $M$. First,
for a given $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$, there exists a
unique $h_{m}\,T\in G/T$ such that

$\Phi(m)=\varsigma(m)\,\mathrm{Coad}_{h_{m}}({\boldsymbol{\nu}})\,.$

where $\varsigma(m)>0$. For any
$\boldsymbol{0}\neq\boldsymbol{\lambda}\in\mathfrak{g}^{*}$ let
$\boldsymbol{\lambda}^{0}$ be its annihilator subspace in $\mathfrak{g}$, we
have

$\Phi(m)^{0}=\,\mathrm{Ad}_{h_{m}}(\boldsymbol{\nu}^{0})\,.$

Let $\varphi$ be an $\mathrm{Ad}$-invariant Euclidean product on
$\mathfrak{g}$, we shall also use the symbol $\langle\cdot\,,\,\cdot\rangle$,
and denote by $\boldsymbol{\lambda}^{\varphi}\in\mathfrak{g}$ be uniquely
determined by $\varphi(\boldsymbol{\lambda}^{\varphi},\,\cdot)$ and
$\lVert\boldsymbol{\lambda}\rVert$ the corresponding norm. We set

$\mathfrak{t}_{m}^{0}:=\left\\{\boldsymbol{\lambda}\in\mathfrak{g}\,:\,\left[\Phi(m)^{\varphi},\,\boldsymbol{\lambda}\right]=0,\,\langle\Phi(m),\boldsymbol{\lambda}\rangle=0\right\\}\,.$

Thus, we can now introduce the following decomposition of the tangent space
$T_{m}M$ at a given point $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$. For
each $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$, assume that the action of
$G$ at $M$ is free at $m$, we have the following characterization of the
normal bundle (it can be proved by minor adaptations of the arguments used in
Lemma 4.2 and Step 4.3 of [GP1]):

$T_{m}M=N_{m}(M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})\oplus
T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\qquad\text{ and }\qquad
N_{m}(M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})=J_{m}(\mathrm{val}_{m}(\mathfrak{t}_{m}^{0}))$
(4)

where $N_{m}(M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})$ is the normal bundle to
$T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ in $T_{m}M$ and the map
$\mathrm{val}\,:\,\mathfrak{g}\rightarrow T_{m}M$,
$\boldsymbol{\xi}\mapsto\boldsymbol{\xi}_{M}(m)$, is the evaluation which is
injective if the action of $G$ on $M$ is free at $m$. Furthermore, we have a
decomposition

$T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}:=T_{m}^{\mathrm{hor}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\oplus\left(T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\right)^{\perp_{\omega}}\,$

and

$\left(T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\right)^{\perp_{\omega}}:=\mathrm{val}_{m}(\mathfrak{t}_{m}^{0})\,.$
(5)

######  Assumption 1.1.

Let us assume that: $\Phi$ is transverse to the cone
$\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})$, $\boldsymbol{0}\notin\Phi(M)$
and $M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\neq\emptyset$.

Theorem 1.1 below concerns asymptotic expansion for near-diagonal re-scaled
displacements at a fixed diagonal point $(x,x)\in
X^{G}_{\mathcal{O}_{{\nu}}}\times X^{G}_{\mathcal{O}_{{\nu}}}$ and the proof
is very similar to the one of [Pa5, Theorem 1.3]. Suppose $x\in X$, and let
$x+(\mathbf{v},\theta)$ be a Heisenberg local chart for $X$ centered at $x$,
as defined in [SZ]; in particular, if $m:=\pi(x)$ this unitarily identifies
$(T_{m}M,h_{m})$ and $\mathbb{C}^{n}$ with its standard Hermitian structure.
When $\theta=0$, we also write $x+\mathbf{v}$ for $x+(\mathbf{v},0)$.

We recall the last piece of notation:

######  Definition 1.1.

Given an arbitrary orthonormal basis $B_{m}$ of $\mathfrak{t}_{m}^{0}$ for the
restriction of $\varphi$, ${D}^{\varphi}(m)$ is the matrix representing the
restriction of $\mathrm{val}^{*}_{m}g_{m}$ to $\mathfrak{t}_{m}^{0}$ with
respect to $B_{m}$. We can now define a smooth function
$\Psi_{\boldsymbol{\nu}}\,:\,M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\rightarrow\mathbb{R}$,

$\Psi_{\boldsymbol{\nu}}(m):=2^{1+\frac{r_{G}-1}{2}}\pi\cdot\frac{1}{\lVert\Phi(m)\rVert_{\varphi}\sqrt{\det{D}^{\varphi}(m)}}\cdot\frac{\mathrm{vol}(\mathcal{O}_{\boldsymbol{\nu}})^{2}}{\lvert\det(S_{\boldsymbol{\nu}})\rvert}\cdot\frac{\mathrm{vol}(T)}{\mathrm{vol}(G)^{2}}$

where the volumes $\mathrm{vol}(G)$ and $\mathrm{vol}(T)$ are induced by
$\varphi$. Clearly, $\varphi$ restricts to an invariant Riemannian metric on
$T$ whose density and volume is denoted respectively by $\mathrm{dV}_{T}$ and
$\mathrm{vol}(T)$. Eventually, for any $\tau\in\mathfrak{t}$,
$S_{\boldsymbol{\nu}}\,:\,\mathfrak{t}^{\perp_{\varphi}}\rightarrow\mathfrak{t}^{\perp_{\varphi}}$
denotes the restriction of $\mathrm{ad}_{\boldsymbol{\nu}}$.

######  Definition 1.2.

Given $\mathbf{w},\mathbf{v}\in T_{m}M$, we define

$\psi_{2}(\mathbf{w},\mathbf{v}):=-i\,\omega_{m}(\mathbf{w},\,\mathbf{v})-\frac{1}{2}\lVert\mathbf{w}-\mathbf{v}\rVert^{2}\,,$

where $\lVert\cdot\rVert$ is the norm induced by the Riemannian metric $g$ on
$M$.

We can now state the main theorem.

######  Theorem 1.1.

Under Assumption 1.1, let us suppose in addition that $G$ acts freely at $x\in
X_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$, $\pi(x)=m_{x}$. Given $C>0$,
$\epsilon\in(0,\,1/6)$ then uniformly for $\mathbf{w}$ and $\mathbf{v}$ in
$T_{m_{x}}^{\mathrm{hor}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ satisfying
$\lVert\mathbf{v}\rVert\leq C\,k^{\epsilon}$ and $\lVert\mathbf{w}\rVert\leq
C\,k^{\epsilon}$, we have for $k\rightarrow+\infty$ an asymptotic expansion of
the form

$\displaystyle\Pi_{k\boldsymbol{\nu}}\left(x++\frac{\mathbf{w}}{\sqrt{k}},\,x+\frac{\mathbf{v}}{\sqrt{k}}\right)\sim$
$\displaystyle\,\Psi_{\boldsymbol{\nu}}(m_{x})\,\left(\frac{k}{\varsigma(m_{x})\,\pi}\right)^{d+\frac{1-r_{G}}{2}}\cdot
e^{\frac{1}{\varsigma(m_{x})}\,\psi_{2}(\mathbf{w},\mathbf{v})}$
$\displaystyle\cdot\left[1+\sum_{j\geq
1}k^{-j/2}P_{j}(m_{x};\mathbf{v},\mathbf{w})\right]\,,$

here, $P_{j}(m_{x};\cdot,\cdot)$ is a polynomial in $\mathbf{v}$ and
$\mathbf{w}$.

By [GS, Proposition $2.1$], then $M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ is a
co-isotropic submanifold of $M$, we denote by
$\iota\,:\,M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\hookrightarrow M$ the
inclusion. Let $\omega_{0}=\iota^{*}\omega$ denote the pull-back of the
symplectic form to $M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$, the kernel of
$\iota^{*}\omega$ at $m$ is $(T_{m}M_{\varpi})^{\perp_{\omega}}$ and it
defines a distribution called the characteristic distribution of $\omega_{0}$.
It is integrable, we have a smooth fiber mapping

$\pi_{\mathcal{O}_{\boldsymbol{\nu}}}^{M}\,:\,M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\rightarrow\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$

such that the fibers are the leaves of the characteristic distribution. In
view of the decompositions (5) and (4), it is easy to see that the complex
structure $J$ descends to an almost complex structure on
$\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ which, by abuse of
notation, we still denote by $J$. The coisotropic reduction
$\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ is a manifold of complex
dimension $d-r_{G}+1$. The manifold
$\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ inherits a natural almost
Kähler structure $\omega_{1}$ and a Riemannian metric $g_{1}$, since $\omega$
restricts to a symplectic form on
$T_{m_{x}}^{\mathrm{hor}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$; for
simplicity we write $\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ for
the almost Kähler manifold with symplectic form $\omega_{1}$.

Notice that $\varsigma$ descends to a well-defined smooth function on
$\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$. We need to equip the
manifold $\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ with a new
symplectic form:

$\omega_{2}(\cdot\,,\,\cdot):=\frac{1}{\varsigma(m_{x})}\,\omega_{1}(\cdot\,,\,\cdot)\,.$

In view of the decompositions (5) and (4), we introduce a new Riemannian
structure

$g_{2}(\cdot\,,\,\cdot)=\omega_{2}(J\cdot\,,\,\cdot)\,.$

Furthermore, we shall denote the conic reduction with Riemannian structure
$g_{2}$ by $\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$. The key point
here, elucidated in the following corollary, is the following: the equivariant
Szegő kernel $\Pi_{k\boldsymbol{\nu}}$ can be used to study geometric
properties of $\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ rather than
$\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$.

Now, as a corollary of Theorem 1.1 we can investigate geometric properties of
the induced map

$\widetilde{\varphi_{k\boldsymbol{\nu}}}\,:\,X_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\rightarrow
H_{k\boldsymbol{\nu}}(X)^{*}\,,\quad
x\mapsto\frac{1}{k^{d+(1-r_{G})/2}}\,\Pi_{k\boldsymbol{\nu}}(x,\,\cdot)\,,$

as $k$ goes to infinity. By formula (2) and by the fact that the circle action
commutes with $\tilde{\mu}$, the map $\widetilde{\varphi_{k\boldsymbol{\nu}}}$
descends to a map $\varphi_{k\boldsymbol{\nu}}$ on
$\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$. If $k$ is sufficiently
large, we shall prove that $\varphi_{k\boldsymbol{\nu}}$ defines an immersion
of $\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ as a Riemannian sub-
manifold in $\mathbb{S}^{2d_{k}-1}\subset H(X)_{k\boldsymbol{\nu}}^{*}$
endowed with standard round metric $g_{0}$. More precisely we need to give the
following definition.

######  Definition 1.3.

We say that the map $\varphi_{k\boldsymbol{\nu}}$ defines asymptotic isometric
minimal immersion of $\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ into
spheres if there exists $\epsilon>0$ such that the following properties hold:

  * (1)

it is an immersion if $k$ is sufficiently large;

  * (2)

it is an asymptotic isometry into spheres $\mathbb{S}^{2d_{k}-1}\subset
H_{k\boldsymbol{\nu}}(X)^{*}\cong\mathbb{C}^{d_{k}}$:

${\varphi_{k\boldsymbol{\nu}}}^{*}(g_{0})=g_{2}+O(k^{-\epsilon})\,;$

  * (3)

it defines asymptotic minimal immersion into spheres:

$\Delta_{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}{\varphi_{k\boldsymbol{\nu}}}(x)=-(2d-2r_{G}+2)\,{\varphi_{k\boldsymbol{\nu}}}(x)+O(k^{-\epsilon})\,.$

A similar definition was introduced in [Z1], we refer to Section 2 for some
historical remarks and motivations. As a corollary of Theorem 1.1 we can prove
the following.

######  Corollary 1.1.

The map $\varphi_{k\boldsymbol{\nu}}$ defines asymptotic isometric minimal
immersion of $\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ into spheres
in $H(X)_{k\boldsymbol{\nu}}^{*}$.

In [Pa5] scaling asymptotics were investigated by making use of the Kirillov
character formula. In fact, Theorem 1.1 is a simple generalization of the main
Theorem in [Pa5].

Theorem 1.1 is based on micro-local techniques that works in the almost
complex symplectic setting, see [SZ]. Following similar idea as in [SZ], we
study off-locus asymptotic of the projector $\Pi_{k\boldsymbol{\nu}}$ for
arbitrary displacement, as $k$ goes to infinity.

Eventually we recall that, motivated by Kähler geometry, various authors
studied canonical isometric embeddings of Riemannian manifolds into
$\mathbb{S}^{k-1}$ or $\mathbb{R}^{k}$, for $k\gg 1$. In 1994, Bérard, Besson,
and Gallot [BBG] constructed an asymptotically isometric embedding of compact
Riemannian manifolds $M$ into the space of real-valued and square summable
series, see also [Z1], [WZ], [Po]. Those approaches are somehow close in
spirit with the one developed in this paper, in fact the microlocal expression
for Szegő kernel of a CR manifold $X$ is closely related to the heat kernel of
the Kohn Laplacian on it, see the proof in [Hs].

A few words about the organization of the paper. In Section 2 we explain
motivations and preliminaries; in particular in Section 2.1 we investigate the
geometry of $\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$. In Section 3
we prove the main theorem, which is based on computations made in [Pa5] and
eventually we prove the aforementioned corollary.

##  2 Motivations and preliminaries

Given a compact Hodge manifold $(M,\omega)$ the sections of the quantizing
line bundle $L$ can be used to define an embedding
${\phi}_{k}\,:\,M\rightarrow\mathbb{P}(\mathbb{C})^{d_{k}}$ into the complex
projective space $(\mathbb{P}(\mathbb{C})^{d_{k}},\,\omega_{FS})$, where
$\omega_{FS}$ is the Fubini-Study form. It is natural to ask how the pull-back
of $\omega_{FS}$ on $M$ is related to $\omega$. S. Zelditch and D. Catlin
strengthen a previous result of Tian by proving that
$(1/k)\,{\phi_{k}}^{*}(\omega_{FS})$ converges to $\omega$ in the
$C^{\infty}$-topology. B. Shiffman and S. Zelditch generalize this picture in
the symplectic setting in [SZ]. Furthermore in [GH], we obtain an analogue
Kodaira embedding theorem and Tian’s almost-isometry theorem for compact
quantizable pseudo-Kähler manifold making use of the asymptotic expansion of
the Bergman kernels for $L^{\otimes k}$-valued $(0,q)$-forms.

We recall the following theorem in Riemannian geometry.

######  Theorem 2.1 (Proposition $12$ in [L]).

Len $M$ be a Riemannian manifold of dimension $m$ and let
$\phi\,:\,M\rightarrow S^{n}\subset\mathbb{R}^{n+1}$ be an isometric
immersion. Then, $\phi$ is a minimal immersion into $S^{n}$ if and only if

$\Delta\phi=-m\phi\,.$

Now, we claim that ${\phi}_{k}$ defines asymptotic isometric minimal
immersions into spheres. Let us explain why this claim holds motivated by
Theorem 2.1. The map $\phi_{k}$ can be studied by using the Szegő kernel, in
fact its $k$-th Fourier component defines

$\widetilde{\phi}_{k}\,:\,X\rightarrow
H_{k}(X)^{*}\setminus\\{\boldsymbol{0}\\}\,,\quad
x\mapsto\frac{1}{k^{-d}}\Pi_{k}(x,\,\cdot)\,,$

which in turns define a map

$\phi:=\widetilde{\phi}_{k}\circ\pi^{-1}\,:\,M\rightarrow H_{k}(X)^{*}\,.$

By using re-scaling asymptotic expansion in Heisenberg local coordinates as in
[SZ] in a neighborhood of a point $x\in X$, we have an asymptotic expansion

$k^{-d}\,\Pi_{k}\left(x+\frac{\mathbf{v}}{\sqrt{k}},\,x+\frac{\mathbf{w}}{\sqrt{k}}\right)=\frac{1}{\pi^{d}}e^{\mathbf{v}\cdot\mathbf{w}-\frac{1}{2}(\lVert\mathbf{v}\rVert^{2}+\lVert\mathbf{w}\rVert^{2})}\left[1+\sum_{j\geq
1}k^{-j/2}a_{j}(x,\,\mathbf{v},\,\mathbf{w})\right]$

where $a_{j}(x,\,\mathbf{v},\,\mathbf{w})$ are polynomial functions in
$\mathbf{v}$ and $\mathbf{w}$. We can compute

$k^{-d}\,\Delta_{g}\,{\phi}_{k}(x)\,:=\sum_{j=1}^{2{d}}\left(\frac{\partial^{2}\Pi_{k}(x,\,x+\mathbf{v}_{i})}{\partial\mathbf{v}_{i}^{2}}\right)_{\mathbf{v}_{i}=\boldsymbol{0}}=-2d\,\phi_{k}(x)+O(k^{-1/2})\,.$

We are thus interested in studying similar properties for the equivariant map
$\varphi_{k\boldsymbol{\nu}}$ we defined in the introduction.

In a similar way one can use the asymptotics of $G$-invariant Szegő kernel for
horizontal directions contained in [Pa1] and [Pa2] and get a similar result.
More precisely, when $\boldsymbol{0}$ does lie in $\Phi(M)$ then the Marsden-
Weinstein reduction $M_{0}:=\Phi^{-1}(\boldsymbol{0})/G$ is a symplectic
orbifold. Given a point $m\in M^{\prime}:=\Phi^{-1}(\boldsymbol{0})$ let us
denote by $[m]$ the underlying point in $M_{0}$, the tangent space
$T_{[m]}M_{0}$ can be identified with the horizontal tangent space
$T_{m}^{\mathrm{hor}}M^{\prime}$ which is given by the orthogonal direct sum
decomposition:

$T_{m}M^{\prime}:=T_{m}^{\mathrm{hor}}M^{\prime}\oplus\mathfrak{g}_{M}(m)\,,$

where $\mathfrak{g}_{M}(m)$ is the tangent space at the orbit at $m$ (more
precisely when the stabilizer $G_{m}$ is non-trivial the tangent space is
$T_{m}^{\mathrm{hor}}M^{\prime}/G_{m}$). Thus one can get easily that
$G$-invariant asymptotic produce in natural way asymptotic isometric minimal
immersion of the Marsden-Weinstein reduction into the $G$-invariant component
of the Hardy space.

In this paper we shall focus on Szegő kernel asymptotics along rays in weight
space and we aim to obtain analogue of results we have discussed in this
section, we shall now briefly review the literature. In [Pa3] the author
focuses on asymptotic expansions locally reflecting the equivariant
decomposition of $H(X)$ over the irreducible representations of a compact
torus. In [GP1] and [GP2] it was studied respectively the case $G=U(2)$ and
$G=SU(2)$ making use of the Weyl integration formula; in [Pa5] local scaling
asymptotics were generalized to the case of a connected compact Lie group $G$.
Eventually, we refer to [Pa4] where asymptotics of the associated Szegő and
Toeplitz operators were studied in detail for the case of Hamiltonian circle
action lifting to the quantizing line bundle. It is important to notice that,
in general, the spaces $H(X)_{k\boldsymbol{\nu}}$ are not contained in any
space of global sections $H^{0}(M,A^{\otimes k})$.

###  2.1 Conic reductions

In this paper we are considering coisotropic reductions as Example $4$ in
[GS]. In fact consider $G$ a compact connected Lie group and fix a weight
$\boldsymbol{\nu}\in\mathfrak{g}^{*}$ and let $\mathcal{O}_{\boldsymbol{\nu}}$
be the co-adjoint orbit passing through $\boldsymbol{\nu}$. We define

$\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}}):=\\{r\,\boldsymbol{\xi}\in\mathfrak{g}^{*}:\,r\in\mathbb{R}^{+},\,\boldsymbol{\xi}\in\mathcal{O}_{\boldsymbol{\nu}}\\}$

Let $f\in\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})$ and let $h_{f}$ be the
co-normal space to $\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})$ at $f$:

$h_{f}:=\\{\boldsymbol{\xi}\in\mathfrak{g}_{f}\,:\,\langle
f,\,\boldsymbol{\xi}\rangle=0\\}$

where $\mathfrak{g}_{f}$ is the Lie algebra of the stabilizer group $G_{f}$ of
$f$.

Consider $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$, and set

$f=\Phi(m):=\varsigma_{\boldsymbol{\nu}}(m)\,\mathrm{Ad}_{h_{m}}(\,iD_{\boldsymbol{\nu}})\in\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})\,$

By proposition $2.1$ in [GS], $M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ is a
co-isotropic submanifold and the valuation map
$\mathrm{val}_{m}:\mathfrak{g}\rightarrow\mathfrak{g}_{M}$,
$\boldsymbol{\xi}\mapsto\boldsymbol{\xi}_{M}(m)$, maps $h_{f}$ onto the
tangent space to the leaf of the null-foliation through $m$, that is

$(T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})^{\perp_{\omega}}=\mathrm{val}_{m}\left(\ker(\Phi(m))\right)=\mathrm{val}_{m}(h_{f})\,.$
(6)

The sub-bundle
$(T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})^{\perp_{\omega}}$ define a
distribution in the sense of Frobenius which is integrable, in fact, fix
$\mathbf{v},\mathbf{w}\in(T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})^{\perp_{\omega}}$,
then for all $\mathbf{u}\in T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ we
have

$\displaystyle\mathrm{d}\omega_{0}(\mathbf{u},\mathbf{v},\mathbf{w})=$
$\displaystyle\mathbf{u}(\omega_{0}(\mathbf{v},\mathbf{w}))-\mathbf{v}(\omega_{0}(\mathbf{u},\mathbf{w}))+\mathbf{w}(\omega_{0}(\mathbf{u},\mathbf{v}))-\omega_{0}([\mathbf{u},\mathbf{v}],\mathbf{w})+\omega_{0}([\mathbf{u},\mathbf{w}],\mathbf{v})-\omega_{0}([\mathbf{v},\mathbf{w}],\mathbf{u})$
$\displaystyle=-\omega_{0}([\mathbf{v},\mathbf{w}],\mathbf{u})$

and then
$[\mathbf{v},\mathbf{w}]\in(T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})^{\perp_{\omega}}$.
If $(T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})^{\perp_{\omega}}$ is
constant, by Frobenius integrability Theorem
$M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ is foliated by connected submanifold.
As in Introduction we define
$\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ to be the space of the
leaf of the null-foliation and $\pi_{\mathcal{O}_{\boldsymbol{\nu}}}$ taking a
point $m$ in $M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ to the unique leaf $[m]$
in $\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ containing it. If
$(T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})^{\perp_{\omega}}$ has constant
rank then $\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ is a manifold
whose tangent space at a point
$[m]\in\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ can be identified to

$T_{[m]}\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}\cong
T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}/(T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})^{\perp_{\omega}}\cong
T_{m}^{\mathrm{hor}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\,.$

We now show that the symplectic form $\omega_{1}$ defined in the introduction
is well-defined, so that it does not depend on the choice of the point $m$
representing the leaf
$[m]\in\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$. Since the leaves
are connected, it is enough to show that $\omega_{1}$ is invariant under the
flow of vector fields
$\mathbf{v}\in(T_{m}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})^{\perp_{\omega}}$;
in fact we have

${L}_{\mathbf{v}}\omega_{1}=\mathrm{d}\iota_{\mathbf{v}}\omega_{1}+\iota_{\mathbf{v}}\mathrm{d}\omega_{1}=0$

where ${L}$ is the Lie derivative and $\iota$ is the interior product. By
equations (6) and (4), we get that $J$ restricts naturally to a almost complex
structure on $\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$, which is
thus a almost Kähler manifold of complex dimension $d-(r_{G}-1)$.

Furthermore there is a natural action of $G$ on
$\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$. It is easy to see that
this action is Hamiltonian, and that its moment mapping
$\Phi_{\mathcal{O}_{\boldsymbol{\nu}}}$ is related to the original moment
mapping by the identity

$\Phi_{\mathcal{O}_{\boldsymbol{\nu}}}\circ\pi_{\mathcal{O}_{\boldsymbol{\nu}}}=\Phi\circ\iota\,,$

and notice that
$\Phi_{\mathcal{O}_{\boldsymbol{\nu}}}(\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}})\subset\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})$.
Let us denote by $H_{\varsigma_{\boldsymbol{\nu}}}$ the Hamiltonian vector
field associated to $\varsigma_{\boldsymbol{\nu}}$. By Proposition $2.3$ in
[GS], the Hamiltonian flow

$\exp\left(tH_{\varsigma_{\boldsymbol{\nu}}}\right),\qquad-\infty<t<+\infty,$

is periodic of period $2\pi$. We get a Hamiltonian action of $G\times S^{1}$
on $\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$, the moment mapping for
the $S^{1}$-action being the map
$i\,\varsigma_{\boldsymbol{\nu}}:\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}\rightarrow
i\mathbb{R}$; this circle action comes from an action on
$M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$, in the next paragraph we describe
explicitly its lift on $X_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$.

There is also another interesting geometric quotient of
$X_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ which appears naturally in this
picture, it is defined as follows. For each $m\in
M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$, the well-defined circle action (see
also [GS, Proposition 2.3]):

$\mu_{\theta}^{\boldsymbol{\nu}}(m):=\mu_{\exp\left(\theta\Phi(m)^{\varphi}\right)}(m)$
(7)

induces by (1) a free circle action $\tilde{\mu}_{\theta}^{\boldsymbol{\nu}}$
on $X^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}$ traversal to the horizontal bundle
$\pi^{*}(TM^{G}_{\mathcal{O}_{\boldsymbol{\nu}}})$ in
$T_{x}X^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}$. Thus, we have a natural
principal bundle whose projection is denoted by
$\kappa:X^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}\rightarrow
N^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}$ and whose connection $1$-form is
$\beta=\alpha/\lVert\Phi\rVert$. The $2$-form $\mathrm{d}\beta$ is degenerate
and we have a smooth fiber mapping

$\pi_{\mathcal{O}_{\boldsymbol{\nu}}}^{N}\,:\,N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\rightarrow\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$

such that the fibers are the leaves of the characteristic distribution of
$\mathrm{d}\beta$. The quotient
$\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ inherits in a natural way
a symplectic structure $\omega_{2}$ and Riemannian metric $g_{2}$. We
summarize the interplay between the quotients
$\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ and
$\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ in the following diagram:

${X^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$${M^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$${N^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$${\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}}$${\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}}$$\scriptstyle{\pi}$$\scriptstyle{\kappa}$$\scriptstyle{\overline{\pi}}$$\scriptstyle{\overline{\kappa}}$

The manifolds $\overline{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ and
$\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ are diffeomorphic since
$M^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}$ and
$N^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}$ have the same horizontal distribution
in $X^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}$.

In Corollary 1.1 we shall study the Laplacian for a function defined on
$\overline{N^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$. Notice that
$\widetilde{\varphi_{k\boldsymbol{\nu}}}$ can be written locally in a small
shrinking open neighborhood of a point $x$ using Heisenberg local coordinates
for the circle bundle $\pi:X\rightarrow M$ in view of Theorem 1.1. The
function ${\varphi_{k\boldsymbol{\nu}}}$ is defined on
$\overline{N^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$, thus we need a formula
relating the Laplacian on $M$ with the one on
$\overline{N^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$.

The standard circle action $r$ on $X$ descends to, in general non-trivial,
action on $\overline{N^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$, which we denote
by the same symbol. The same is true for $\tilde{\mu}$ which descends to a
circle action on $\overline{M^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$. Given a
function $f\in C^{\infty}({X^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}})$ which is
$r$ and $\tilde{\mu}$-invariant; by abuse of notation (it should be clear from
the context), we denote by $f$ the corresponding function in
$C^{\infty}(\overline{N^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}})^{r}$, denoting
the space of $r$-invariant functions on
$\overline{N^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$ and similarly by $f$ the
corresponding function in
$C^{\infty}(\overline{M^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}})^{\tilde{\mu}}$,
denoting the space of $\tilde{\mu}$-invariant functions on
$\overline{M^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$.

The two circle bundles $\pi$ and $\kappa$ have different CR structures,
because they do not have the same vertical tangent bundle; but notice that
they share the same horizontal distribution, and the same horizontal complex
structure $J$. Thus, in the same way as in [Pa4, Section 2.8], it is easy to
see that the relation between the Laplacian operators $\Delta_{1}$ on
$\overline{M^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$ and $\Delta_{2}$ on
$\overline{N^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$ acting on invariant
functions is given by

$\Delta_{2}(f)=\frac{1}{2}\,g_{2}^{\overline{b}{a}}(x)\partial_{a}{\partial}_{\overline{b}}f(x)=\frac{1}{2}\varsigma(m_{x})\,g_{1}^{\overline{b}{a}}(x)\partial_{a}{\partial}_{\overline{b}}f(x)=\varsigma(m_{x})\,\Delta_{1}(f)(x)\,,$

since

$g_{2,\overline{b}{a}}(x)=\frac{g_{1,\overline{b}{a}}(x)}{\varsigma(m_{x})}\,.$

###  2.2 Equivariant Szegő kernels

Let $\Pi\,:\,L^{2}(X)\rightarrow H(X)$ be the Szegő projector,
$\Pi(\cdot,\cdot)$ its kernel. By [BS], $\Pi$ is a Fourier integral operator
with complex phase:

$\Pi(x,y)=\int_{0}^{+\infty}e^{iu\psi(x,y)}\,s(x,y,u)\,\mathrm{d}u,$ (8)

where the imaginary part of the phase satisfies $\Im(\psi)\geq 0$ and

$s(x,y,u)\sim\sum_{j\geq 0}u^{d-j}\,s_{j}(x,y).$

We shall also making use on the description of the phase $\psi$ in Heisenberg
local coordinates (see §3 of [SZ]).

Now, we recall the property of the equivariant Szegő projector, see [D,
Chapter 15]. For each $\boldsymbol{\nu}\in\hat{G}$, by the formula

$\chi_{\boldsymbol{\nu}}*\chi_{\boldsymbol{\nu}}=d_{\boldsymbol{\nu}}^{-1}\,\chi_{\boldsymbol{\nu}}\,,$

where $*$ is the convolution on $G$, we get that the operator of convolution
by $d_{\boldsymbol{\nu}}\,\chi_{\boldsymbol{\nu}}$ is a non-zero projection in
$L^{2}(G)$ which is Hermitian since $\chi_{\boldsymbol{\nu}}$ is equal to its
complex conjugate. Recall that we have a representation of $G$ on $L^{2}(X)$
given by $(\rho(g)f)(x)=f(\tilde{\mu}_{g^{-1}}(x))$, $g\in G$ and $x\in X$,
thus we get that the orthogonal projection of on the
$\boldsymbol{\nu}$-isotype in $L^{2}(X)$ is

$d_{\boldsymbol{\nu}}\int_{G}\rho(g)\chi_{\boldsymbol{\nu}}(g^{-1})\mathrm{dV}_{G}(g)$

From this, composing with the Szegő projector $\Pi\,:\,L^{2}(X)\rightarrow
H(X)$, it is easy to get

$\Pi_{k\boldsymbol{\nu}}(x,\,y)=d_{k\boldsymbol{\nu}}\int_{G}\overline{\chi_{k\boldsymbol{\nu}}(g)}\,\Pi(\tilde{\mu}_{g^{-1}}(x),\,y)\,\mathrm{dV}_{G}(g)\,.$

##  3 Proof of Theorem and Corollary

###### Proof of Theorem 1.1.

In this section we shall explain how to modify the proof of Theorem 1.3 in
[Pa5] for the case $\mathbf{v},\mathbf{w}\in
T^{\mathrm{hor}}_{m_{x}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$. Since we can
replace $x$ by $\tilde{\mu}_{g}(x)$, we assume
$\Phi(m_{x})=\varsigma(m_{x})\,\boldsymbol{\nu}$ and thus
$\mathfrak{t}_{m}^{0}=\mathfrak{t}\cap\boldsymbol{\nu}^{0}$. By Section 2.2,
we shall study the asymptotics of the oscillatory integral

$\displaystyle\Pi_{k\boldsymbol{\nu}}(x+\mathbf{w},x+\mathbf{v})=$
$\displaystyle
d_{k\boldsymbol{\nu}}\int_{0}^{+\infty}\mathrm{d}u\,\int_{G}\mathrm{dV}_{G}(g)$
(9)
$\displaystyle\left[\overline{\chi_{k\boldsymbol{\nu}}(g)}\,e^{iu\psi(\tilde{\mu}_{g^{-1}}(x+\mathbf{w}),x+\mathbf{v})}\,s(\tilde{\mu}_{g^{-1}}(x+\mathbf{w}),x+\mathbf{v},u)\right]$

as $k$ goes to infinity. First, let us give an analogue of [Pa5, Lemma 7.2].

######  Lemma 3.1.

Given $x\in X_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$, there exists $\delta>0$,
which can be chosen uniformly on $X_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$,
such that for any $\boldsymbol{\xi}\in\mathfrak{g}$ and
$\mathbf{w},\,\mathbf{v}\in
T^{\mathrm{hor}}_{m_{x}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ of
sufficiently small norm, we have

$\mathrm{dist}_{X}(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x+\mathbf{w}),x+\mathbf{v})\geq\delta\,\lVert\boldsymbol{\xi}\rVert^{2}\,.$

###### Proof.

From [Pa3, Section 2], we recall that

$\displaystyle\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x+\mathbf{w})=x+\left(\Theta(\boldsymbol{\xi},\mathbf{w}),\,V(\boldsymbol{\xi},\mathbf{w})\right)$

where

$\Theta(\boldsymbol{\xi},\mathbf{w}):=\langle\Phi(m_{x}),\boldsymbol{\xi}\rangle+\omega_{m}(\boldsymbol{\xi}_{M}(m),\mathbf{w})+O(\lVert(\boldsymbol{\xi},\mathbf{w})\rVert^{3})$

and

$V(\boldsymbol{\xi},\mathbf{w}):=\mathbf{w}-\boldsymbol{\xi}_{M}(m_{x})+O(\lVert(\boldsymbol{\xi},\mathbf{w})\rVert^{2})\,.$

We can use the decomposition

$\mathfrak{g}=\left(\ker(\Phi(m_{x}))\right)^{\perp_{\varphi}}\oplus\ker(\Phi(m_{x}))$

and write $\boldsymbol{\xi}=\boldsymbol{\xi}^{(1)}+\boldsymbol{\xi}^{(2)}$
accordingly. If $\boldsymbol{\xi}^{(1)}\neq 0$ then
$\langle\Phi(m_{x}),\boldsymbol{\xi}\rangle\neq 0$ and we get the statement.
Otherwise, since, for each $m_{x}\in M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$
and $\mathbf{v},\mathbf{w}\in
T^{\mathrm{hor}}_{m_{x}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$ we have

$T_{m_{x}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}=T^{\mathrm{hor}}_{m_{x}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\oplus\mathrm{val}_{m_{x}}(\ker(\Phi(m_{x})))$

we get the statement:

$\lVert\mathbf{w}-\mathbf{v}+\boldsymbol{\xi}^{(2)}\rVert^{2}\geq\lVert\boldsymbol{\xi}^{(2)}_{M}\rVert^{2}\geq
C\lVert\boldsymbol{\xi}^{(2)}\rVert^{2}\,.$

for some $C>0$. ∎

Thus, we may multiply the integrand in (9) by $1+(\varrho-1)$ where $\varrho$
is a cut-off function $\varrho=\varrho(u)$, which is zero outside $(1/D,\,D)$
for some $D\gg 0$. Now, thanks to the previous Lemma 3.1, we can restrict the
integral on $\mathrm{d}u$ to $(1/D,\,D)$ without affecting the asymptotics,
this is done in the standard way by iteratively integrating by parts in
$\mathrm{d}u$.

In a similar way as in [Pa5], we can now apply the results in $\S$3 of [SZ]:

$u\,\psi(\tilde{\mu}_{g^{-1}}(x+\mathbf{w}),x+\mathbf{v})=iu\left[1-e^{i\,\Theta(\boldsymbol{\xi},\mathbf{w})}\right]-iu\,\psi_{2}\left(V(\boldsymbol{\xi},\mathbf{w}),\frac{1}{\sqrt{k}}\mathbf{v}\right)$

where $\Theta(\boldsymbol{\xi},\mathbf{w})$ and
$V(\boldsymbol{\xi},\mathbf{w})$ are as in the proof of the previous Lemma.
Thus, we obtain from (9) an oscillatory integral with the same phase function
as in [Pa5], but slightly different amplitude because we are assuming
$\mathbf{v},\,\mathbf{w}\in
T^{\mathrm{hor}}_{m_{x}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$. Thus, by the
same computations leading to equation (82) and (83) in [Pa5], we obtain that
(9) has an asymptotic expansion in decreasing power of $k$ whose leading term
is obtain explicitly by computing the following integral

$\displaystyle\int_{\mathfrak{t}\cap\boldsymbol{\nu}^{0}}e^{\frac{1}{\varsigma(m_{x})}\left[g_{m_{x}}(\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}-\mathbf{w})+i\,\omega_{m_{x}}(\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}+\mathbf{w})-\frac{1}{2}\lVert\boldsymbol{\xi}_{M}(m_{x})\rVert^{2}_{m_{x}}\right]}\mathrm{d}^{\varphi}\boldsymbol{\xi}\,.$
(10)

Now, notice that $\mathbf{v}-\mathbf{w}$ is orthogonal to every
$\boldsymbol{\xi}_{M}(m_{x})\in\mathrm{val}_{m_{x}}(\mathfrak{t}\cap\boldsymbol{\nu}^{0})$,
see (4). Furthermore, for every
$\boldsymbol{\xi}\in\mathfrak{t}\cap\boldsymbol{\nu}^{0}$ and
$\mathbf{v},\,\mathbf{w}\in
T^{\mathrm{hor}}_{m_{x}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}$

$\omega_{m_{x}}(\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}+\mathbf{w})=\boldsymbol{0}$

since

$(T_{m_{x}}M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G})^{\perp_{\omega}}=\ker(\mathrm{d}\Phi(m_{x}))=\mathrm{val}_{m_{x}}(\mathfrak{t}^{0}_{m})=\mathrm{val}_{m_{x}}(\mathfrak{t}\cap\boldsymbol{\nu}^{0})\,.$

Thus, the integral (10) is of Gaussian type and can be easily computed. ∎

###### Proof of Corollary 1.1.

Let us denote the dimension of $H_{k\boldsymbol{\nu}}(X)$ by
$d_{k\boldsymbol{\nu}}$ and consider an orthonormal basis of
$H_{k\boldsymbol{\nu}}(X)$ given by
${s}_{1}^{k\boldsymbol{\nu}},\,\dots,\,{s}^{k\boldsymbol{\nu}}_{d_{k\boldsymbol{\nu}}}$
with respect to the standard $L^{2}$ product. Now, by definition, the map

$\widetilde{\varphi_{k\boldsymbol{\nu}}}\,:\,X_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}\rightarrow
H_{k\boldsymbol{\nu}}(X)^{*}\,,\quad
x\mapsto\frac{1}{k^{d-(r_{G}-1)/2}}\,\Pi_{k\boldsymbol{\nu}}(x,\,\cdot)\,,$

is explicitly given by

$\widetilde{\varphi_{k\boldsymbol{\nu}}}(x)=\frac{1}{k^{d-(r_{G}-1)/2}}\,\sum_{j=1}^{d_{k\boldsymbol{\nu}}}s_{j}^{k\boldsymbol{\nu}}(x)(s_{j}^{k\boldsymbol{\nu}})^{*}\in
H_{k\boldsymbol{\nu}}(X)^{*}\cong\mathbb{C}^{d_{k\boldsymbol{\nu}}}\,$

and

$\Pi_{k\boldsymbol{\nu}}(x,\,y):=\widetilde{\varphi_{k\boldsymbol{\nu}}}(x)\cdot\overline{\widetilde{\varphi_{k\boldsymbol{\nu}}}(y)}\,.$
(11)

By definition $\mathbb{S}^{2n+1}$ carries the so-called “round metric” endowed
upon it by restriction of the standard Euclidean metric $g_{\mathrm{st}}$, it
is given by

$g_{\mathrm{st}}=g_{\mathrm{st},\,i\overline{j}}\,\mathrm{d}\zeta^{i}\mathrm{d}\overline{\zeta}^{j}\,,\qquad\text{where
}g_{\mathrm{st},\,i\overline{j}}:=\frac{\partial^{2}}{\partial\zeta^{i}\partial\overline{\zeta}^{j}}\ln(1+\delta_{i\overline{j}}\,\zeta^{i}\overline{\zeta}^{j})$

on $\mathbb{C}^{n+1}\setminus\\{0\\}$ with coordinates
$\zeta_{0},\dots,\zeta_{n}$; the corresponding Kähler form is

$\omega_{\mathrm{st}}:=\frac{i}{2}\sum_{i,\,j}g_{\mathrm{st},\,i\overline{j}}\,\mathrm{d}\zeta_{i}\wedge\mathrm{d}\overline{\zeta}_{j}$

We shall follow [SZ] and we consider the 2-form $\Omega$ on
$(\mathbb{C}^{d_{k\boldsymbol{\nu}}}\setminus\\{0\\})\times(\mathbb{C}^{d_{k\boldsymbol{\nu}}}\setminus\\{0\\})$
given by

$\Omega=\frac{i}{2}\mathrm{d}^{(1)}\mathrm{d}^{(2)}\log\zeta\cdot\eta$

where $\zeta$ and $\eta$ are the variables respectively on the first and
second component of

$(\mathbb{C}^{d_{k\boldsymbol{\nu}}}\setminus\\{0\\})\times(\mathbb{C}^{d_{k\boldsymbol{\nu}}}\setminus\\{0\\})$

and analogously the formal differential $\mathrm{d}^{(1)}$ and
$\mathrm{d}^{(2)}$ denoting the exterior differentiation on the first and
second factors, respectively. Furthermore define

$\Phi_{k\boldsymbol{\nu}}=\widetilde{\varphi_{k\boldsymbol{\nu}}}\times\widetilde{\varphi_{k\boldsymbol{\nu}}}\,:\,X\times
X\rightarrow\mathbb{C}^{d_{k\boldsymbol{\nu}}}\times\mathbb{C}^{d_{k\boldsymbol{\nu}}},\quad\Phi_{k\boldsymbol{\nu}}(x,y):=(\widetilde{\varphi_{k\boldsymbol{\nu}}}(x),\widetilde{\varphi_{k\boldsymbol{\nu}}}(y))\,.$

The proof of the following Lemma is left to the reader.

######  Lemma 3.2.

$\Phi_{k\boldsymbol{\nu}}^{*}$ commutes with $\mathrm{d}^{(1)}$ and
$\mathrm{d}^{(2)}$.

Thus we can pullback $\Omega$ via $\Phi_{k\boldsymbol{\nu}}$, by (11) and
Lemma 3.2 we can easily obtain

$\frac{1}{k}\Phi_{k\boldsymbol{\nu}}^{*}\Omega=\frac{i}{2k}\,\Phi_{k\boldsymbol{\nu}}^{*}\mathrm{d}^{(1)}\mathrm{d}^{(2)}\log\zeta\cdot\overline{\eta}=\frac{i}{2k}\,\mathrm{d}^{(1)}\mathrm{d}^{(2)}\Phi_{k\boldsymbol{\nu}}^{*}\log\zeta\cdot\overline{\eta}=\frac{i}{2k}\,\mathrm{d}^{(1)}\mathrm{d}^{(2)}\log\Pi_{k\boldsymbol{\nu}}\,.$

Eventually we compose with the diagonal map $\mathrm{diag}$ and we make use of
Theorem 1.1, using Heisenberg coordinates we get

$\displaystyle\frac{1}{k}\widetilde{\varphi_{k\boldsymbol{\nu}}}^{*}\omega_{\mathrm{st}}$
$\displaystyle=\frac{i}{2k}\,\mathrm{diag}^{*}\mathrm{d}^{(1)}\mathrm{d}^{(2)}\log\Pi_{k\boldsymbol{\nu}}\left(x+\frac{\mathbf{w}}{\sqrt{k}},\,x+\frac{\mathbf{v}}{\sqrt{k}}\right)$
$\displaystyle=\frac{i}{2k}\,\mathrm{diag}^{*}\mathrm{d}^{(1)}\mathrm{d}^{(2)}\left[\frac{1}{\varsigma(m_{x})}\left(\mathbf{w}\cdot\overline{\mathbf{v}}-\frac{1}{2}(\lVert\mathbf{w}\rVert^{2}+\lVert\mathbf{v}^{2}\rVert)\right)\right]+O(1/\sqrt{k})$
$\displaystyle=\frac{1}{\varsigma(m_{x})}\omega_{|x}+O(1/\sqrt{k})$

on $T_{m_{x}}^{\mathrm{hor}}M^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}$ and we get
(2) of Definition 1.3 since in Heisenberg local coordinates $\omega$ is the
standard form on $\mathbb{C}^{d-r_{G}+1}$, see [SZ]. We also get $(1)$ since,
as a consequence of (2) we get that $\mathrm{d}\varphi_{k\boldsymbol{\nu}}$ is
injective for $k$ large, see for example Lemma 5.1.6 in [MM].

Pick $x$ for a given
$m_{x}\in\overline{M^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}$ consider
$\mathbf{v}$, $\mathbf{w}\in
T_{n}\overline{M^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}}\cong
T_{n}^{\mathrm{hor}}M^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}$ as elements in
$T_{x}X^{G}_{\mathcal{O}_{\boldsymbol{\nu}}}$, where we identify
$T_{\kappa(m_{x})}\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ with
$T_{m_{x}}^{\mathrm{hor}}{M_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$. In view of
Theorem 2.1 and Section 2.1 we can study the Laplacian of the Riemannian
immersion ${\varphi_{k\boldsymbol{\nu}}}$ for $k$ large. Recall that the
Laplacian in Heisenberg local coordinates is the standard Euclidean one and
thus, we have in a neighborhood $U$ of the point $n_{x}$,

$\displaystyle\left(\Delta_{2}\varphi_{k\boldsymbol{\nu}}\right)_{|_{U}}$
$\displaystyle=\varsigma(m_{x})\left(\sum_{i,\,j=1}^{d-r_{G}+1}\frac{\partial^{2}\varphi_{k\boldsymbol{\nu}}}{\partial\mathbf{v}_{i}\,\partial\overline{\mathbf{v}_{j}}}\right)_{|_{U}}$
$\displaystyle=-2(d-r_{G}+1)\,\varphi_{k\boldsymbol{\nu}}+O(k^{d-1/2})\,,$

where, for each $\mathbf{v}\in
T_{n_{x}}{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$ is a lift of an element
$[\mathbf{v}]\in T_{[n]}\overline{N_{\mathcal{O}_{\boldsymbol{\nu}}}^{G}}$; we
get the Corollary.

∎

Acknowledgements: The author thanks Prof. R. Paoletti for suggesting him the
study of this problem.

## References

  * [BBG] P. Bérard, G. Besson, S. Gallot, Embedding Riemannian Manifolds by Their Heat kernel, Geom. and Func. Analysis, vol. 4, no. 4, (1994)
  * [BS] L. Boutet de Monvel, J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Astérisque 34-35 (1976), 123–164
  * [D] J. Dixmier, $C^{*}$-algebras, North-Holland (1982), 513 pages
  * [DK] J. J. Duistermaat, J. A. C. Kolk, Lie Groups, Universitext, Springer Berlin, Heidelberg, 344 pages
  * [GH] A. Galasso, C.-Y. Hsiao, Embedding theorems for quantizable pseudo-Kähler manifolds, arXiv
  * [GP1] A. Galasso, R. Paoletti, Equivariant Asymptotics of Szegö kernels under Hamiltonian $U(2)$ actions, Ann. Mat. Pura Appl. (4) 198 (2019), no. 2, 639–683
  * [GP2] A. Galasso, R. Paoletti, Equivariant Asymptotics of Szegö kernels under Hamiltonian $SU(2)$ actions, Asian Journal of Mathematics, Volume 24 (2020), Number 3
  * [GS] V Guillemin, S. Sternberg, Homogeneous quantization and multiplicities of group representations, Journal of Functional Analysis Volume 47, Issue 3, July 1982, pages 344-380
  * [Hs] C.-Y. Hsiao, Projections in several complex variables, Mémoires de la Société Mathématique de France 123 (2010), 1-136
  * [Ko] B. Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, pp. 87–208. Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970
  * [L] H. Blaine Lawson, Lectures on minimal submanifolds (Mathematics lecture series 9), (1980)
  * [MM] X. Ma, G. Marinescu, Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics 254, (2007)
  * [Pa1] R. Paoletti, Scaling limits for equivariant Szegö kernels, Journal of Symplectic Geometry Volume 6, Number 1, 9–32, 2008
  * [Pa2] R. Paoletti, Corrigendum to scaling limits for equivariant Szegö kernels, Journal of Symplectic Geometry Volume 11, Number 2, 317–318, 2013
  * [Pa3] R. Paoletti, Asymptotics of Szegö kernels under Hamiltonian torus actions, Israel Journal of Mathematics 191 (2012), no. 1, 363–403 DOI: 10.1007/s11856-011-0212-4
  * [Pa4] R. Paoletti, Lower-order asymptotics for Szegö and Toeplitz kernels under Hamiltonian circle actions, Recent Advances in Algebraic Geometry 417, 321 (2015)
  * [Pa5] R. Paoletti, Szegö kernel equivariant asymptotics under Hamiltonian Lie group actions, J. Geom. Anal. 32, No. 4, Paper No. 112, 32 p. (2022)
  * [Po] J. W. Portegies, Embeddings of Riemannian Manifolds with Heat Kernels and Eigenfunctions, Communications on Pure and Applied Mathematics, Vol. LXIX, 0478–0518 (2016), Wiley Periodicals, Inc
  * [SZ] B. Shiffman, S. Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds, J. Reine Angew. Math. 544 (2002), 181–222
  * [V] V.S. Varadarajan, An introduction to harmonic analysis on semisimple Lie groups, Corrected reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, vol .16. Cambridge University Press, Cambridge (1999). x+316 pp, ISBN: 0-521-34156-6
  * [WZ] X. Wang, K. Zhu, Isometric embeddings via heat kernel J. Differential Geometry 99 (2015) 497-538
  * [Z1] S. Zelditch, Fine Structure of Zoll Spectra, Journal of Functional Analysis 143 (1996), 415–460
  * [Z2] S. Zelditch, Szegő kernels and a theorem of Tian, Int. Math. Res. Not. 6 (1998), 317–331