# Szegö kernel equivariant asymptotics under Hamiltonian Lie group actions

Roberto Paoletti111Address: Dipartimento di Matematica e Applicazioni,
Università degli Studi di Milano-Bicocca, Via R. Cozzi 55, 20125 Milano,
Italy; e-mail<EMAIL_ADDRESS>
###### Abstract

Suppose that a compact and connected Lie group $G$ acts on a complex Hodge
manifold $M$ in a holomorphic and Hamiltonian manner, and that the action
linearizes to a positive holomorphic line bundle $A$ on $M$. Then there is an
induced unitary representation on the associated Hardy space and, if the
moment map of the action is nowhere vanishing, the corresponding isotypical
components are all finite dimensional. We study the asymptotic concentration
behavior of the corresponding equivariant Szegö kernels near certain loci
defined by the moment map.

##  1 Introduction

Let $M$ be a connected complex $d$-dimensional projective manifold, and $A$ an
holomorphic ample line bundle on it. There exists an Hermitian metric $h$ such
that the unique covariant $\nabla$ derivative on $A$ that is compatible with
both the complex structure and the metric has curvature
$\Theta=-2\,\imath\,\omega$, where $\omega$ is a Kähler form on $M$. Thus the
triple $(M,J,\omega)$ is a Kähler manifold, with associated Riemannian metric
$\rho^{M}$ and volume form $\mathrm{d}V_{M}:=\omega^{\wedge d}/d!$.

We shall denote by $A^{\vee}$ the dual line bundle of $A$, and by $X\subset
A^{\vee}$ the unit circle bundle; thus $X=\partial D$, where $D\subset
A^{\vee}$ is the unit disc bundle, a strictly pseudoconvex domain. Then
$\nabla$ determines a connection $1$-form $\alpha$ on $X$. If
$\pi:X\rightarrow M$ is the projection, then
$\mathrm{d}V_{X}:=\frac{1}{\textbullet
2\,\pi}\,\alpha\wedge\pi^{*}(\mathrm{d}V_{M})$ is a volume form. Furthermore,
there is on $X$ a natural choice of an $S^{1}$-invariant Riemannian metric
$\rho^{X}$, determined by the conditions that $\pi$ be a Riemannian submersion
with $\ker(\alpha)$ as horizontal tangent bundle, and that the fibers of $\pi$
have unit length. Hence $|\mathrm{d}V_{X}|$ is the Riemannian density of
$\rho^{X}$. We shall denote by $\mathrm{dist}_{X}$ the Riemannian distance
function of $\rho^{X}$.

If $H(X)\subset L^{2}(X)$ is the Hardy space, the orthogonal projector
$\Pi:L^{2}(X)\rightarrow H(X)$ is known as the Szegö projector of $X$, and its
distributional kernel $\Pi\in\mathcal{D}^{\prime}(X\times X)$ as the Szegö
kernel ([BS], [Z]).

Furthermore, let $G$ be a connected compact Lie group, with Lie algebra
$\mathfrak{g}$ and coalgebra $\mathfrak{g}^{\vee}$; we shall denote by $d_{G}$
and $r_{G}$, respectively, the dimension and the rank of $G$. Let us assume
that $\mu:G\times M\rightarrow M$ is a Hamiltonian and holomorphic action on
$(M,J,2\,\omega)$, with moment map $\Phi:M\rightarrow\mathfrak{g}^{\vee}$.

Then to every $\boldsymbol{\xi}\in\mathfrak{g}$ there is associated an
Hamiltonian vector field $\boldsymbol{\xi}_{M}\in\mathfrak{X}_{ham}(M)$, which
canonically lifts to a contact and CR vector field
$\boldsymbol{\xi}_{X}\in\mathfrak{X}_{cont}(X)$, according to the law [Ko]

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

here notation is as follows:

  1. 1.

for any vector field $V$ on $M$, $V^{\sharp}$ denotes its its horizontal lift
to $X$ with respect to $\alpha$ (and similarly for tangent vectors);

  2. 2.

$\partial_{\theta}$ is the generator of the standard circle action on $X$
(fiber rotation).

In other words, the infinitesimal action of $\mathfrak{g}$ on $M$ lifts to an
infinitesimal contact action on $X$.

We shall make the stronger hypothesis that $\mu$ itself lifts to an action
$\tilde{\mu}:G\times X\rightarrow X$, of which the correspondence
$\boldsymbol{\xi}\mapsto\boldsymbol{\xi}_{X}$ is the differential. Then
$\tilde{\mu}$ is a contact and CR action, and naturally determines a unitary
representation of $G$ on $H(X)$. According to the Theorem of Peter and Weyl,
there is an equivariant unitary Hilbert direct sum decomposition of $H(X)$
into isotypical components corresponding to the irreducible representations
(in the following, irreps) of $G$ [St].

For a given choice of a maximal torus $T\leqslant G$ and of a set $R^{+}$ of
positive roots of $\mathfrak{g}$, the irreps of $G$ are determined by the
their maximal weights, which range in the set of dominant weights. This sets
up a bijective correspondence between the family $\hat{G}$ of irreps of $G$
and a subset $\mathcal{D}^{G}$ of the collection $\mathcal{D}$ of all dominant
weights; we have $\mathcal{D}=\mathcal{D}^{G}$ if $G$ is simply connected. We
shall label the irrep with maximal weight
$\boldsymbol{\lambda}\in\mathcal{D}^{G}$ by the regular222i.e., belonging to
the open positive Weyl chamber half-weight
$\boldsymbol{\nu}=\boldsymbol{\lambda}+\boldsymbol{\delta}$, where
$\boldsymbol{\delta}:=2^{-1}\,\sum_{\beta\in R^{+}}\beta$. We shall denote by
$V_{\boldsymbol{\nu}}$ the irrep corresponding to $\boldsymbol{\nu}$, and by
$\chi_{\boldsymbol{\nu}}:G\rightarrow\mathbb{C}$ the corresponding character.
This labeling is consistent with the philosophy of the Kirillov character
formula [Ki], that we shall recall in the course of the paper. Furthermore,
let us set $d_{\boldsymbol{\nu}}:=\dim(V_{\boldsymbol{\nu}})$; if $\varphi$
denotes any Euclidean scalar product on $\mathfrak{g}^{\vee}$ which is
invariant uder the coadjoint action then, by the Weyl dimension formula (see,
e.g., §1 of [Su], §2.5 of [V1]),

$d_{\boldsymbol{\nu}}=\prod_{\boldsymbol{\beta}\in
R^{+}}\frac{\varphi(\boldsymbol{\nu},\boldsymbol{\beta})}{\varphi(\boldsymbol{\delta},\boldsymbol{\beta})}.$
(2)

In particular,
$d_{\boldsymbol{k\,\nu}}=k^{\frac{1}{2}\,(d_{G}-r_{G})}\,d_{\boldsymbol{\nu}}$.
Thus, if we set $\mathcal{E}^{G}:=\mathcal{D}^{G}+\boldsymbol{\delta}$, we
have

$H(X)=\bigoplus_{\boldsymbol{\nu}\in\mathcal{E}^{G}}H(X)^{\tilde{\mu}}_{\boldsymbol{\nu}}$
(3)

where $H(X)^{\tilde{\mu}}_{\boldsymbol{\nu}}$ is the isotypical component
corresponding to $V_{\boldsymbol{\nu}}$. For each
$\boldsymbol{\nu}\in\mathcal{E}^{G}$ we have the associated equivariant Szegö
projector $\Pi_{\boldsymbol{\nu}}^{\tilde{\mu}}:L^{2}(X)\rightarrow
H(X)^{\tilde{\mu}}_{\boldsymbol{\nu}}$.

In general, $H(X)^{\tilde{\mu}}_{\boldsymbol{\nu}}$ may well be infinite-
dimensional, and does not correspond to a space of holomorphic sections of any
tensor power of $A$. Nonetheless, it follows from the theory of [GS2] that if
$\mathbf{0}\not\in\Phi(M)$, then $\dim
H(X)^{\tilde{\mu}}_{\boldsymbol{\nu}}<+\infty$ for every $\boldsymbol{\nu}$
(see §2 [P1]). Thus $\Pi_{\boldsymbol{\nu}}^{\tilde{\mu}}$ is a smoothing
operator, so that its distributional kernel
$\Pi_{\boldsymbol{\nu}}^{\tilde{\mu}}\in\mathcal{C}^{\infty}(X\times X)$.

We are interested in the local asymptotics of
$\Pi_{k\,\boldsymbol{\nu}}^{\tilde{\mu}}$ for a fixed
$\boldsymbol{\nu}\in\mathcal{E}^{G}$ and $k\rightarrow+\infty$ with
$k\,\boldsymbol{\nu}\in\mathcal{E}^{G}$. The latter condition is satisfied for
any $k$ if $\boldsymbol{\delta}\in\mathcal{D}^{G}$, so that
$\mathcal{E}^{G}\subset\mathcal{D}^{G}$, as is the case when $G$ is simply
connected.

This general theme has already been studied in specific cases ([P1], [P2],
[C], [GP1], [GP2]), building on the approach developed in [Z], [BSZ] and [SZ]
to the basic Fourier case where $G=S^{1}$, $\mu$ is trivial and $\Phi=\imath$.
We refer the reader to the introductions of [P1], [P2], [GP1], [GP2] for an
ampler discussion of motivation and general framing. The theme is
geometrically relevant, being related to interesting geometric quotients [P3].

The results in [GP1] and [GP2] are based on the pairing of the Weyl character
and integration formulae with the techniques in [Z] and [SZ]. The new
ingredient here is the Kirillov character formula ([Ki], [R]) which
considerably simplifies some of the arguments, and allows to deal with more
general Lie groups; on the other hand, it forces restrictions on the
stabilizer subgroups.

The following results are governed by the interplay between $\Phi$ and the
cone over the coadjoint orbit through $\boldsymbol{\nu}$,
$\mathcal{O}_{\boldsymbol{\nu}}\subset\mathfrak{g}^{\vee}$. As
$\boldsymbol{\nu}$ is a regular element of $\mathfrak{g}^{\vee}$,
$\mathcal{O}_{\boldsymbol{\nu}}$ is equivariantly diffeomorphic to $G/T$,
hence it has dimension $d_{G}-r_{G}$. Let us set
$\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}}):=\mathbb{R}_{+}\cdot\mathcal{O}_{\boldsymbol{\nu}}$.

We shall need the following hypothesis.

######  Assumption 1.1.

We shall assume that:

  1. 1.

$\mathbf{0}\not\in\Phi(m)$;

  2. 2.

$\Phi$ is transverse to $\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})$
(equivalently, $\Phi$ is transverse to the ray
$\mathbb{R}_{+}\cdot\boldsymbol{\nu}$);

  3. 3.

$M_{\mathcal{O}_{\boldsymbol{\nu}}}:=\Phi^{-1}\big{(}\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})\big{)}\neq\emptyset$.

If Assumption 1.1 holds, $M_{\mathcal{O}_{\boldsymbol{\nu}}}$ is a compact and
connected $G$-invariant submanifold of $M$, of (real) codimension $r_{G}-1$
(see the discussion of [GP1]). Let us set
$X_{\mathcal{O}_{\boldsymbol{\nu}}}:=\pi^{-1}(M_{\mathcal{O}_{\boldsymbol{\nu}}})$.

We shall also make the following assumption on the compact and connected Lie
group $G$.

######  Assumption 1.2.

Let $L(G)\subset\mathfrak{t}^{\vee}$ be the lattice of integral forms on $G$;
then $\boldsymbol{\delta}\in L(G)$.

This condition is satisfied if $G$ is either $U(n)$ for some $n\geq 1$, or a
connected and simply connected compact semisimple Lie group. If $G$ satisfies
this assumption, then it is called acceptable in Harish-Chandra’s terminology
(§2.5 of [V1]). Under Assumption 1.2, $\mathcal{E}^{G}\subset\mathcal{D}^{G}$.

In the following we shall assume throughout that Assumptions 1.1 and 1.2 hold.

######  Theorem 1.1.

Suppose that $\mathcal{O}_{\boldsymbol{\nu}}\cap\mathfrak{t}^{0}=\emptyset$.
Fix $C,\,\epsilon>0$. Then, uniformly for $\mathrm{dist}_{X}\left(G\cdot
x,G\cdot y\right)\geq C\,k^{\epsilon-\frac{1}{2}}$, we have

$\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}(x,y)=O\left(k^{-\infty}\right).$

When $G=U(n)$, the previous hypothesis is satisfied by any
$\boldsymbol{\nu}\in\mathcal{E}^{G}$ with $\sum_{j=1}^{n}\nu_{j}\neq 0$. It is
never satisfied when $G=SU(2)$, but the statement of Theorem 1.1 is
nonetheless true in this case, see [GP2]. More generally, let $\mu^{T}:T\times
M\rightarrow M$ be the restriction of $\mu$, and let
$\Phi^{T}:M\rightarrow\mathfrak{t}^{\vee}$ be the moment map induced by $\Phi$
(that is, the composition of $\Phi$ with the restriction
$\mathfrak{g}^{\vee}\rightarrow\mathfrak{t}^{\vee}$). If
$\mathbf{0}\not\in\Phi^{T}(M)$, then the hypothesis of Theorem 1.1 is
satisfied for any $\boldsymbol{\nu}$ such that
$M_{\mathcal{O}_{\boldsymbol{\nu}}}\neq\emptyset$.

######  Theorem 1.2.

Let us fix $C,\,\epsilon>0$, and assume that $\tilde{\mu}$ is free along
$X_{\mathcal{O}}$. Then, uniformly for

$\max\left\\{\mathrm{dist}_{X}(x,X_{\mathcal{O}_{\boldsymbol{\nu}}}),\mathrm{dist}_{X}(y,X_{\mathcal{O}_{\boldsymbol{\nu}}})\right\\}\geq
C\,k^{\epsilon-\frac{1}{2}},$

we have

$\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}(x,y)=O\left(k^{-\infty}\right).$

If $\tilde{\mu}$ is only generically free along $X_{\mathcal{O}}$, and
$X^{\prime}\subset X$ is the open subset where it is free, the same estimate
holds uniformly on compact subsets of $X^{\prime}$.

We shall focus on the near diagonal asymptotics of
$\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}(x,x)$ for $x$ belonging to a
shrinking tubular neighborhood of $X_{\mathcal{O}_{\boldsymbol{\nu}}}$, of
radius $O\left(k^{\epsilon-1/2}\right)$. Using the normal exponential map, we
may parametrize such a neighborhood by a neighborhood of the zero section in
the normal bundle of $X_{\mathcal{O}_{\boldsymbol{\nu}}}\subset X$, which is
the pull-back of the normal bundle $N(M_{\mathcal{O}_{\boldsymbol{\nu}}}/M)$
of $M_{\mathcal{O}_{\boldsymbol{\nu}}}\subset M$. If
$V_{\mathcal{O}_{\boldsymbol{\nu}}}\subset X$ is a sufficiently small
neighborhood of $X_{\mathcal{O}_{\boldsymbol{\nu}}}$, we shall accordingly
write the general $y\in V_{\mathcal{O}_{\boldsymbol{\nu}}}$ in additive
notation as $y=x+\mathbf{v}$, for unique $x\in
X_{\mathcal{O}_{\boldsymbol{\nu}}}$ and $\mathbf{v}\in
N_{\pi(x)}(M_{\mathcal{O}_{\boldsymbol{\nu}}}/M)$333We may interpret
$x+\mathbf{v}$ in terms of a system of Hesenberg local coordinates on $X$
centered at $x$ [SZ], smoothly varying with $x$, see §4..

In order to state the next Theorem, some further notation is needed.

######  Definition 1.1.

Let $\varphi$ be an $\mathrm{Ad}$-invariant Euclidean product on
$\mathfrak{g}$, with associated norm $\|\cdot\|_{\varphi}$. Let us also denote
by $\varphi$ the induced bi-invariant Riemannian metric on $G$. Clearly,
$\varphi$ restricts to an invariant Riemannian metric $\varphi^{T}$ on $T$. We
shall adopt the following notation:

  1. 1.

$\mathrm{d}^{\varphi}V_{G}$: the Riemannian density on $G$ associated to
$\varphi$;

  2. 2.

$\mathrm{vol}^{\varphi}(G)=\int_{G}\,\mathrm{d}^{\varphi}V_{G}(g)$;

  3. 3.

$\mathrm{d}^{\varphi}V_{T}$: the Riemannian density on $T$

  4. 4.

$\mathrm{vol}^{\varphi}(T):=\int_{T}\,\mathrm{d}^{\varphi}V_{T}(t)$.

  5. 5.

For any $\boldsymbol{\gamma}\in\mathfrak{g}^{\vee}$,

     * •

$\boldsymbol{\gamma}^{\varphi}\in\mathfrak{g}$ is uniquely determined by the
condition
$\boldsymbol{\gamma}=\varphi\big{(}\boldsymbol{\gamma}^{\varphi},\cdot)$;

     * •

$\|\boldsymbol{\gamma}\|_{\varphi}:=\|\boldsymbol{\gamma}^{\varphi}\|_{\varphi}$;

     * •

$\boldsymbol{\gamma}_{\varphi,u}:=\frac{1}{\textbullet\|\boldsymbol{\gamma}^{\varphi}\|_{\varphi}}\,\boldsymbol{\gamma}=\frac{1}{\textbullet\|\boldsymbol{\gamma}\|_{\varphi}}\,\boldsymbol{\gamma}\in\mathfrak{g}^{\vee}$;

     * •

$\boldsymbol{\gamma}^{\varphi}_{u}:=\frac{1}{\textbullet\|\boldsymbol{\gamma}^{\varphi}\|_{\varphi}}\,\boldsymbol{\gamma}^{\varphi}\in\mathfrak{g}$.

  6. 6.

$\mathfrak{t}^{\perp_{\varphi}}\subseteq\mathfrak{g}$: the Euclidean
orthocomplement of $\mathfrak{t}$ with respect to $\varphi$.

  7. 7.

For any $\boldsymbol{\tau}\in\mathfrak{t}$,
$S_{\boldsymbol{\tau}}:\mathfrak{t}^{\perp_{\varphi}}\rightarrow\mathfrak{t}^{\perp_{\varphi}}$
denotes the restriction of $\mathrm{ad}_{\boldsymbol{\tau}}$; when
$\boldsymbol{\tau}$ is regular (as is the case when
$\boldsymbol{\tau}=\boldsymbol{\nu}^{\varphi}$ for
$\boldsymbol{\nu}\in\mathcal{E}^{G}$), $S_{\boldsymbol{\tau}}$ is a linear
automorphism, skew-symmetric with respect to the restriction of $\varphi$.

Let us identify the coalgebra of $T$, $\mathfrak{t}^{\vee}$, with the subspace
of those $\boldsymbol{\lambda}\in\mathfrak{g}^{\vee}$ fixed by $T$ under the
coadjoint action, and let
$\mathfrak{t}^{\vee}_{reg}\subset\mathfrak{t}^{\vee}$ be the open and dense
subset of those elements, called regular, that are fixed precisely by $T$.
Hence $\mathcal{E}^{G}\subset\mathfrak{t}^{\vee}_{reg}$.

By definition, for any $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}$ there exist
$h_{m}\,T\in G/T$ and $\varsigma(m)>0$ such that

$\Phi(m)=\varsigma(m)\,\mathrm{Coad}_{h_{m}}(\boldsymbol{\nu})\in\mathrm{Coad}_{h_{m}}(\mathfrak{t}^{\vee}).$
(4)

Since $\boldsymbol{\nu}$ is regular, $h_{m}\,T$ and $\varsigma(m)$ are
uniquely determined, and the functions $m\in
M_{\mathcal{O}_{\boldsymbol{\nu}}}\mapsto h_{m}\,T\in G/T$ and
$\varsigma:M_{\mathcal{O}_{\boldsymbol{\nu}}}\rightarrow\mathbb{R}$ are
smooth.

For any non-zero $\boldsymbol{\lambda}\in\mathfrak{g}^{\vee}$, let
$\boldsymbol{\lambda}^{0}=\left(\boldsymbol{\lambda}^{\varphi}\right)^{\perp_{\varphi}}\subset\mathfrak{g}$
be its annihilator hyperplane. Then

$\Phi(m)^{0}=\mathrm{Ad}_{h_{m}}\left(\boldsymbol{\nu}^{0}\right).$ (5)

Let us set
$\mathfrak{t}_{\boldsymbol{\nu}}:=\mathfrak{t}\cap\boldsymbol{\nu}^{0}$, so
that we have a $\varphi$-orthogonal direct sum decomposition

$\mathfrak{t}=\mathrm{span}(\boldsymbol{\nu}^{\varphi})\oplus\mathfrak{t}_{\boldsymbol{\nu}}.$

For every $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}$, we shall set
$\mathfrak{t}_{m}:=\mathrm{Ad}_{h_{m}}(\mathfrak{t})$; thus $\mathfrak{t}_{m}$
is the unique Cartan subalgebra of $\mathfrak{g}$ containing $\Phi(m)$, or
equivalently the Lie subalgebra of the (unique) maximal torus
$T_{m}=h_{m}\,T\,h_{m}^{-1}$ stabilizing $\Phi(m)$. Thus,

$\mathfrak{t}_{m}=\left\\{\boldsymbol{\eta}\in\mathfrak{g}:\,\left[\boldsymbol{\eta},\Phi(m)^{\varphi}\right]=0\right\\}.$
(6)

Furthermore, we shall set
$\mathfrak{t}_{m}^{\prime}:=\mathrm{Ad}_{h_{m}}(\mathfrak{t}_{\boldsymbol{\nu}})=\mathfrak{t}_{m}\cap\Phi(m)^{0}$.
More explicitly,

$\mathfrak{t}_{m}^{\prime}=\left\\{\boldsymbol{\eta}\in\mathfrak{g}:\,\left[\boldsymbol{\eta},\Phi(m)^{\varphi}\right]=0,\,\langle\Phi(m),\boldsymbol{\eta}\rangle=0\right\\}.$
(7)

Hence we have the $\varphi$-orthogonal direct sum

$\mathfrak{t}_{m}=\mathrm{span}\big{(}\Phi(m)^{\varphi})\oplus\mathfrak{t}_{m}^{\prime}.$
(8)

Assume $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}$. Then $\mathrm{val}_{m}$ is
injective on $\Phi(m)^{0}$ by Remark 3.1; hence $\mathrm{val}_{m}$ is
injective a fortiori on $\mathfrak{t}_{m}^{\prime}$.

######  Definition 1.2.

In the following, $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}$. We shall adopt
the following notation.

  1. 1.

$\mathrm{val}_{m}^{*}(\rho^{M}_{m})$: the pull-back to $\mathfrak{g}$ of the
Euclidear product $\rho^{M}_{m}=\omega_{m}(\cdot,J_{m}\cdot)$ on $T_{m}M$;
$\rho^{\prime}_{m}$: the restriction of $\mathrm{val}_{m}^{*}(\rho^{M}_{m})$
to $\mathfrak{t}_{m}^{\prime}$. Thus $\rho^{\prime}_{m}$ is non-degenerate
(whence positive definite).

  2. 2.

Given an arbitrary orthonormal basis $\mathcal{R}_{m}$ of
$\mathfrak{t}_{m}^{\prime}$ for the restriction of $\varphi$, let
$D^{\varphi}(m):=M_{\mathcal{R}_{m}}(\rho^{\prime}_{m})$ be the representative
matrix of $\rho^{\prime}_{m}$ w.r.t. $\mathcal{R}_{m}$, and set

$\mathcal{D}^{\varphi}(m):=\sqrt{\det D^{\varphi}(m)};$

then
$\mathcal{D}^{\varphi}:M_{\mathcal{O}_{\boldsymbol{\nu}}}\rightarrow(0,+\infty)$
is well-defined and $\mathcal{C}^{\infty}$;

We can now define a $\mathcal{C}^{\infty}$ function
$\Psi_{\boldsymbol{\nu}}:M_{\mathcal{O}_{\boldsymbol{\nu}}}\rightarrow\mathbb{R}_{+}$
by setting

$\Psi_{\boldsymbol{\nu}}(m):=2^{1+\frac{r_{G}-1}{2}}\,\pi\cdot\frac{1}{\|\Phi(m)\|_{\varphi}\,\mathcal{D}^{\varphi}(m)}\cdot\frac{\mathrm{vol}(\mathcal{O}_{\boldsymbol{\nu}_{\varphi,u}},\sigma_{\boldsymbol{\nu}_{\varphi,u}})^{2}}{\big{|}\det(S_{\boldsymbol{\nu}^{\varphi}_{u}})\big{|}}\cdot\frac{\mathrm{vol}^{\varphi}(T)}{\mathrm{vol}^{\varphi}(G)^{2}}.$

By Theorem 1.3 below, $\Psi_{\boldsymbol{\nu}}$ is actually independent of the
choice of $\varphi$.

We need some further pieces of notation.

Given a real vector subspace $R\subseteq T_{m}M$, we shall denote by
$R^{\perp_{h_{m}}}$ the orthocomplement of $R$ with respect to the Hermitian
structure $h_{m}=\rho^{M}_{m}-\imath\,\omega_{m}$; equivalently,
$R^{\perp_{h_{m}}}$ is the orthocomplement of the complex subspace
$R+J_{m}(R)$ of $T_{m}M$, and is a complex subspace of $(T_{m}M,J_{m})$.
Clearly,

$R^{\perp_{h_{m}}}=R^{\perp_{\rho^{M}_{m}}}\cap R^{\perp_{\omega_{m}}}.$ (9)

where $R^{\perp_{\rho^{M}_{m}}}$ and $R^{\perp_{\omega_{m}}}$ are,
respectively, the Riemannian and symplectic orthocomplements of $R$.

If $m\in M$, let $\mathfrak{g}_{M}(m)\subseteq T_{m}M$ be the vector subspace
given by the evaluations at $m$ of the all the vector fields on $M$ induced by
the elements of $\mathfrak{g}$. We shall see in Lemma 6.1 that for any $m\in
M_{\mathcal{O}_{\boldsymbol{\nu}}}$ the normal space of
$M_{\mathcal{O}_{\boldsymbol{\nu}}}$ at $m$ satisfies

$N_{m}(M_{\mathcal{O}_{\boldsymbol{\nu}}})\subseteq
J_{m}\big{(}\mathfrak{g}_{M}(m)\big{)},\quad\text{hence}\quad
N_{m}(M_{\mathcal{O}_{\boldsymbol{\nu}}})\cap\mathfrak{g}_{M}(m)^{\perp_{h_{m}}}=(0).$
(10)

In this setting, small displacements from a fixed $x\in X$ are conveniently
expressed in Heisenberg local coordinates (HLC) on $X$ centered at $x$ [SZ]. A
choice of HLC at $x$ gives a meaning to the expression $x+\mathbf{v}$, where
$\mathbf{v}\in T_{\pi(x)}M$ has sufficiently small norm. Furthermore, the
curve $\gamma_{x,\mathbf{v}}:\tau\in(-\epsilon,\epsilon)\mapsto
x+\tau\,\mathbf{v}$ is horizontal at $\tau=0$, and in fact
$\gamma_{x,\mathbf{v}}^{\prime}(0)=\mathbf{v}^{\sharp}$. More will be said in
§4.

A further notational ingredient that will go into the statement of Theorem 1.3
is an invariant governing the exponential decay of various asymptotics related
to Szegö kernels [SZ].

######  Definition 1.3.

Let $\|\cdot\|$ and $\omega_{0}$ be the standard norm and symplectic
structures on $\mathbb{R}^{2\,d}$, respectively. Let us define
$\psi_{2}:\mathbb{R}^{2\,d}\times\mathbb{R}^{2\,d}\rightarrow\mathbb{R}$ by
setting

$\psi_{2}(u,v):=-\imath\,\omega_{0}(u,v)-\frac{1}{2\textbullet}\,\|u-v\|^{2}.$

A choice of Heisenberg local coordinates at $x$ entails the choice of a
unitary ismophism $T_{\pi(x)}M\cong\mathbb{C}^{d}$ (with the standard
Hermitian structure), by means of which we shall view $\psi_{2}$ as being
defined on $T_{\pi(x)}M$. For the sake of simplicity, we shall consider
displacements of the form $x+k^{-1/2}\,(\mathbf{v}+\mathbf{w})$, where
$\mathbf{v}$ is normal to $M_{\mathcal{O}_{\boldsymbol{\nu}}}$ and
$\mathbf{w}$ is in $\mathfrak{g}_{M}(m)^{\perp_{h_{m}}}$. Heuristically,
$x+\mathbf{w}$ covers a displacement in a suitable quotient.

######  Theorem 1.3.

Assume that $x\in X_{\mathcal{O}_{\boldsymbol{\nu}}}$, and that $\tilde{\mu}$
is free at $x$. Set $m_{x}=\pi(x)$ and fix $C>0$, $\epsilon\in(0,1/6)$. Then,
uniformly for $\mathbf{v}_{j}\in
N_{m_{x}}(M_{\mathcal{O}_{\boldsymbol{\nu}}})$ and
$\mathbf{w}_{j}\in\mathfrak{g}_{M}(m)^{\perp_{h_{m}}}$ satisfying
$\|\mathbf{v}_{j}\|,\,\|\mathbf{w}_{j}\|\leq C\,k^{\epsilon}$, the following
asymptotic expansion holds as $k\rightarrow+\infty$:

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

where $P_{j}(m_{x};\cdot)$ is a polynomial of degree $\leq 3\,j$ and parity
$j$. If $X^{\prime}_{\mathcal{O}_{\boldsymbol{\nu}}}\subseteq
X_{\mathcal{O}_{\boldsymbol{\nu}}}$ is the open subset on which $\tilde{\mu}$
is free, the estimate holds uniformly on the compact subsets of
$X^{\prime}_{\mathcal{O}_{\boldsymbol{\nu}}}$.

By a Gaussian integral computation in normal Heisenberg coordinates, as in the
proof of Corollary 1.3 of [P1], one can then deduce the following:

######  Corollary 1.1.

Assume that $\tilde{\mu}$ is free along $X_{\mathcal{O}_{\boldsymbol{\nu}}}$.
Then there is an asymptotic expansion

$\dim
H(X)^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}\sim\left(\frac{k}{\pi}\right)^{d+1-r_{G}}\,\left[\delta_{\boldsymbol{\nu},0}+k^{-1}\,\delta_{\boldsymbol{\nu},1}+\ldots\right],$

with

$\delta_{\boldsymbol{\nu},0}:=\frac{1}{2^{\frac{r_{g}-1}{2}}}\,\int_{M_{\mathcal{O}_{\boldsymbol{\nu}}}}\left[\frac{\Psi_{\boldsymbol{\nu}}(m)}{\varsigma(m)^{d+1-r_{G}}}\right]\,\mathrm{d}V_{M_{\mathcal{O}_{\boldsymbol{\nu}}}}(m),$

where $\mathrm{d}V_{M_{\mathcal{O}_{\boldsymbol{\nu}}}}$ is the density on
$M_{\mathcal{O}_{\boldsymbol{\nu}}}$ for the induced Riemannian metric.

In closing this introduction, we mention that there is a wider scope for the
results in this paper. While our focus is on the complex projective setting,
in view of the microlocal theory of almost complex Szegö kernels in [SZ] the
present approach can be naturally extended to the compact symplectic category.

##  2 Examples

We check the statement of Theorem 4.2 against those in [P1] , [GP1], [GP2].

######  Example 2.1.

Suppose $G=T$ is an $r$-dimensional torus. Let us take the standard metric
$\varphi$, so that $\mathrm{vol}^{\varphi}(G)=(2\,\pi)^{r}$. We obtain

$\Psi_{\boldsymbol{\nu}}(m_{x})=\frac{2^{1+\frac{r-1}{2}}\,\pi}{(2\,\pi)^{r}}\cdot\frac{1}{\|\Phi(m)\|_{\varphi}\,\mathcal{D}^{\varphi}(m)}=\frac{1}{\left(\sqrt{2}\,\pi\right)^{r-1}}\,\frac{1}{\|\Phi(m)\|_{\varphi}\,\mathcal{D}^{\varphi}(m)}.$

Thus Theorem 4.2 fits with Theorem 2 of [P1].

######  Example 2.2.

Suppose $G=SU(2)$, so that $d_{G}=3$, $r_{G}=1$. Let
$\varphi:\mathfrak{su}(2)\times\mathfrak{su}(2)\rightarrow\mathbb{R}$ be
defined by

$\varphi(A,B):=\mathrm{trace}\left(A\,\overline{B}^{t}\right)=\mathrm{trace}\left(A\,B\right).$

Let $T\leqslant G$ be the standard torus; $\mathfrak{t}$ is generated by the
diagonal matrix $Z$ with entries $\imath,\,-\imath$, which has norm
$\sqrt{2}$. For $\nu\in\mathbb{Z}$, we shall denote by
$\boldsymbol{\nu}\in\mathfrak{t}^{\vee}$ the weight taking value $\nu$ on $Z$.
We have

$\mathrm{vol}^{\varphi}(G)=2^{3/2}\cdot(2\,\pi^{2}),\quad\mathrm{vol}^{\varphi}(T)=\sqrt{2}\cdot
2\,\pi.$

For any $\nu\in\mathbb{Z}$, let $\boldsymbol{\nu}$ be the weight such that
$\langle\boldsymbol{\nu},Z\rangle=\nu$. Then

$\boldsymbol{\nu}^{\varphi}=\frac{\nu}{2}\,Z,\quad\|\boldsymbol{\nu}\|_{\varphi}=\frac{1}{\sqrt{2}}\,\nu.$

Furthermore,

$\mathrm{vol}(\mathcal{O}_{\boldsymbol{\nu}},\sigma_{\boldsymbol{\nu}})=2\,\pi\,\nu,\quad|\det(S_{\boldsymbol{\nu}^{\varphi}})|=\nu^{2}.$

Finally, let $\lambda(m)>0$ be defined by the condition that
$\Phi(m)^{\varphi}$ be similar to $\lambda(m)\,Z$. Then
$\|\Phi(m)\|_{\varphi}=\left\|\Phi(m)^{\varphi}\right\|_{\varphi}=\sqrt{2}\,\lambda(m)$
and $\varsigma(m)=\big{(}2\,\lambda(m)\big{)}/\nu$. We obtain

$\Psi_{\boldsymbol{\nu}}(m)=2\,\pi\cdot\frac{1}{\sqrt{2}\,\lambda(m)}\cdot
4\,\pi^{2}\cdot\frac{\sqrt{2}\cdot
2\,\pi}{\left(2^{3/2}\cdot(2\,\pi^{2})\right)^{2}}=\frac{1}{2\,\lambda(m)},$

in agreement with [GP2].

######  Example 2.3.

If $G=U(2)$, we have $d_{G}=4$, $r_{G}=2$. Let
$\varphi:\mathfrak{u}(2)\times\mathfrak{u}(2)\rightarrow\mathbb{R}$ be defined
as for $SU(2)$. Let $T\leqslant G$ be the standard maximal torus; then
$\mathfrak{t}$ has orthonormal basis $(R,S)$, where $R$ and $S$ are the
diagonal matrices with diagonal entries $\begin{pmatrix}\imath&0\end{pmatrix}$
and $\begin{pmatrix}0&\imath\end{pmatrix}$, respectively. Let
$\boldsymbol{\nu}=\nu_{1}\,R^{*}+\nu_{2}\,S^{*}$, where $(R^{*},S^{*})$ is the
dual basis. Then $\boldsymbol{\nu}^{\varphi}=\nu_{1}\,R+\nu_{2}\,S$.

We have in this case

$\mathrm{vol}^{\varphi}(\mathcal{O}_{\boldsymbol{\nu}},\sigma_{\boldsymbol{\nu}})=2\,\pi\,(\nu_{1}-\nu_{2}),\quad|\det(S_{\boldsymbol{\nu}^{\varphi}})|=(\nu_{1}-\nu_{2})^{2}.$

Furthermore,

$\mathrm{vol}^{\varphi}(T)=(2\,\pi)^{2},\quad\mathrm{vol}^{\varphi}(G)=8\,\pi^{3}.$

We obtain for $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}$

$\Psi_{\boldsymbol{\nu}}(m)=2^{\frac{3}{2}}\,\pi\cdot\frac{1}{\|\Phi(m)\|_{\varphi}\,\mathcal{D}^{\varphi}(m)}\cdot
4\,\pi^{2}\cdot\frac{4\,\pi^{2}}{64\,\pi^{6}}=\frac{1}{\sqrt{2}\,\pi}\cdot\frac{1}{\|\Phi(m)\|_{\varphi}\,\mathcal{D}^{\varphi}(m)},$
(11)

which tallies with the front factor in the pointwise expansion in Theorem 1.4
of [GP1]. In the latter expansion the numerical factor is written in a
slightly less explicit form, but replacing $V_{3}=2\,\pi^{2}$ it is readily
seen to equal the one in (11).

##  3 Preliminaries

We shall adopt the following notational conventions:

  1. 1.

$R_{j}$ will denote a smooth real, complex or vector valued function defined
in the neighborhood of the origin of some vector space, vanishing to $j$-th
order at the origin, and allowed to vary from line to line;

  2. 2.

if $G$ acts smoothly on a manifold $Z$ and $\boldsymbol{\xi}\in\mathfrak{g}$,
$\boldsymbol{\xi}_{Z}$ will denote the induced vector field on $Z$;

  3. 3.

under the same assumption, if $p\in Z$ we shall denote by
$\mathrm{val}_{p}:\boldsymbol{\xi}\in\mathfrak{g}\rightarrow\boldsymbol{\xi}_{Z}(p)\in
T_{p}Z$ the evaluation map;

  4. 4.

if $m\in M$ and $\mathbf{v}\in T_{m}M$, we shall denote by
$\|\mathbf{v}\|_{m}$ the norm of $\mathbf{v}$ with respect to $\rho^{M}$;

  5. 5.

if $x\in X$ and $\upsilon=a\,\partial_{\theta}|_{x}+\mathbf{v}^{\sharp}\in
T_{x}X$, in computations it will be convenient to set
$\|\upsilon\|_{x}:=\sqrt{a^{2}+\|\mathbf{v}\|_{m}^{2}}$ (this is the norm in
an obvious vertical rescaling of $\rho^{X}$).

######  Remark 3.1.

Arguing as in §2 of [P1] (or §4.1.1 of [GP1]), one verifies that the following
conditions are equivalent:

  1. 1.

Assumption 1.1 holds;

  2. 2.

$\tilde{\mu}$ is locally free along $X_{\mathcal{O}_{\boldsymbol{\nu}}}$;

  3. 3.

for every $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}$,
$\mathrm{val}_{m}:\mathfrak{g}\rightarrow T_{m}M$ is injective on the
annihilator of $\Phi(m)$, that is,

$\ker(\mathrm{val}_{m})\cap\Phi(m)^{0}=(0).$

Let us define

$\displaystyle\mathcal{Z}_{\boldsymbol{\nu}}:=\big{\\{}(x,y)\in
X_{\mathcal{O}}\times X_{\mathcal{O}}:\,y\in G\cdot x\big{\\}}.$ (12)

Then $\mathcal{Z}_{\boldsymbol{\nu}}$ is a $G\times G$-invariant compact and
connected submanifold of $X\times X$.

######  Theorem 3.1.

Uniformly on compact subsets of $(X\times
X)\setminus\mathcal{Z}_{\boldsymbol{\nu}}$, we have

$\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}(x,y)=O\left(k^{-\infty}\right).$

###### Proof.

The argument is a slight modification of the one in §3.1 and §3.2 of [GP1],
based on the theory in [GS1]; hence we shall be somewhat sketchy. The ladder
Szegö projector

$\Pi_{L}:=\bigoplus_{k=1}^{+\infty}\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}:L^{2}(X)\longrightarrow\bigoplus_{k=1}^{+\infty}H(X)^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}$

has a distributional kernel whose wave front satisfies
$\mathrm{WF}(\Pi_{L})\subseteq\mathcal{Z}_{\boldsymbol{\nu}}$.

Let $K\Subset(X\times X)\setminus\mathcal{Z}_{\boldsymbol{\nu}}$. Without
loss, we may assume that $K$ is $G\times G$-invariant. There exists a $G\times
G$-invariant smooth cut-off function $\varrho\geq 0$ on $X\times X$, which is
identically equal to $1$ on a neighborhood of $K$, and vanishes identically on
a neighborhood of $\mathcal{Z}_{\boldsymbol{\nu}}$. Hence
$\varrho\cdot\Pi_{L}\in\mathcal{C}^{\infty}(X\times X)$. Hence, we obtain a
$\mathcal{C}^{\infty}$ function

$F:(g,x,y)\in G\times X\times
X\mapsto(\varrho\cdot\Pi_{L})\left(\tilde{\mu}_{g^{-1}}(x),y\right)\in\mathbb{C}.$

We shall set $F_{x,y}:=F(\cdot,x,y):G\rightarrow\mathbb{C}$.

Let us denote $P_{k\,\boldsymbol{\nu}}:L^{2}(X)\rightarrow
L^{2}(X)_{k\,\boldsymbol{\nu}}$ the projector. Hence,
$\Pi_{k\,\boldsymbol{\nu}}^{\tilde{\mu}}=P_{k\,\boldsymbol{\nu}}\circ\Pi_{L}^{\tilde{\mu}}$.
If $\mathrm{d}^{H}V_{G}(g)$ is the Haar measure on $G$, this means that for
$(x,y)\in X\times X$

$\displaystyle\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}(x,y)$ $\displaystyle=$
$\displaystyle
d_{k\,\boldsymbol{\nu}}\int_{G}\,\overline{\chi_{k\,\boldsymbol{\nu}}(g)}\,\Pi_{L}\left(\tilde{\mu}_{g^{-1}}(x),y\right)\,\mathrm{d}^{H}V_{G}(g).$
(13)

If $(x,y)\in K$, therefore,

$\displaystyle\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}(x,y)$ $\displaystyle=$
$\displaystyle
d_{k\,\boldsymbol{\nu}}\cdot\int_{G}\,\overline{\chi_{k\,\boldsymbol{\nu}}(g)}\,(\varrho\,\Pi_{L})\left(\tilde{\mu}_{g^{-1}}(x),y\right)\,\mathrm{d}^{H}V_{G}(g)$
$\displaystyle=$ $\displaystyle
d_{k\,\boldsymbol{\nu}}\cdot\mathrm{trace}\big{(}\mathcal{F}(F_{x,y})(k\,\boldsymbol{\nu}-\boldsymbol{\delta})\big{)},$

where $\mathcal{F}$ denotes the Fourier transform on $G$ [Su], viewed as a
function on $\mathcal{D}^{G}$. Since $d_{k\,\boldsymbol{\nu}}\leq
C_{\boldsymbol{\nu}}\,k^{d_{G}-r_{G}}$, it suffices to show that
$\mathcal{F}(F_{x,y})(k\,\boldsymbol{\nu}-\boldsymbol{\delta})=O\left(k^{-\infty}\right)$
in Hilbert-Schmidt norm for $k\rightarrow+\infty$. To this end, we apply
arguments in §1 of [Su].

To begin with, for any
$\boldsymbol{\lambda}\in\mathcal{D}^{G}\setminus\\{\mathbf{0}\\}$ we have (see
eq. (1.21) of [Su])

$\left\|\mathcal{F}\left(F_{x,y}\right)(\boldsymbol{\lambda})\right\|_{HS}^{2}\leq\|\boldsymbol{\lambda}\|^{-4\,l}\,\left\|\mathcal{F}\left(\Delta_{G}^{l}F_{x,y}\right)(\boldsymbol{\lambda})\right\|_{HS}^{2};$
(15)

here $\Delta_{G}$ denotes the Laplacian (Casimir) operator on $G$, and
$\left\|\cdot\right\|$ is the Hilbert-Schmidt norm. Hence for $k\gg 0$

$\left\|\mathcal{F}\left(F_{x,y}\right)(k\,\boldsymbol{\nu}-\boldsymbol{\delta})\right\|_{HS}^{2}\leq
2\,k^{-4\,l}\|\boldsymbol{\nu}\|^{-4\,l}\,\left\|\mathcal{F}\left(\Delta_{G}^{l}F_{x,y}\right)(k\,\boldsymbol{\nu}-\boldsymbol{\delta})\right\|_{HS}^{2};$
(16)

On the other hand, by the Parceval identity (eq. (1.16) of [Su]) we also have

$\left\|\mathcal{F}\left(\Delta_{G}^{l}F_{x,y}\right)(\boldsymbol{\lambda})\right\|^{2}_{HS}\leq\frac{1}{d_{\boldsymbol{\lambda}+\boldsymbol{\delta}}}\,\|\Delta_{G}^{l}F_{x,y}\|^{2}_{2}\leq\|\Delta_{G}^{l}F_{x,y}\|^{2}_{2}$
(17)

where $\|\cdot\|_{2}$ denotes the $L^{2}$-norm on $G$
($d_{\boldsymbol{\lambda}+\boldsymbol{\delta}}=d(\boldsymbol{\lambda})$ in the
notation of [Su]).

Furthermore, by compactness for any $l\geq 0$ we can find $C_{l}>0$ such that
$\|\Delta_{G}^{l}F_{x,y}\|_{2}^{2}\leq C_{l}$ for all $(x,y)\in X\times X$.
Hence, by the Parceval identity, for every
$\boldsymbol{\lambda}\in\mathcal{D}^{G}$ we have

$\left\|\mathcal{F}\left(\Delta_{G}^{l}F_{x,y}\right)(\boldsymbol{\lambda})\right\|_{HS}^{2}\leq\frac{1}{d_{\boldsymbol{\lambda}+\boldsymbol{\delta}}}\,C_{l}\leq
C_{l}.$ (18)

Therefore, for $k\gg 0$ we have

$\left\|\mathcal{F}\left(F_{x,y}\right)(k\,\boldsymbol{\nu}-\boldsymbol{\delta})\right\|_{HS}^{2}\leq
k^{-4\,l}\,C_{l}^{\prime}$ (19)

∎

##  4 Proof of Theorem 1.1

In the proof of Theorem 1.1 we shall use the Weyl integration and character
formulae, which we briefly recall below, referring e.g. to [V1] (§2.3-2.5) and
[V2] (§4.13 and 4.14) for a detailed treatment.

Let $W$ denote the Weyl group of $(\mathfrak{g},\mathfrak{t})$; then $W$
naturally acts on $\mathfrak{t}^{\vee}$ preserving the root lattice
$L(R)\subset\mathfrak{t}^{\vee}$.

Let $L(G)\subset\mathfrak{t}^{\vee}$ be the lattice of integral forms of $G$.
Every $\boldsymbol{\gamma}\in L(G)$ defines a character
$E_{\boldsymbol{\gamma}}:T\rightarrow S^{1}$, and we may define

$A_{\boldsymbol{\gamma}}:=\sum_{s\in
W}\epsilon(s)\,E_{s(\boldsymbol{\gamma})}:T\rightarrow\mathbb{C},$

where $\epsilon(s)=\det(s)$ (here $s$ is viewed as a linear map
$\mathfrak{t}\rightarrow\mathfrak{t}$). In particular, since
$\boldsymbol{\delta}\in L(G)$ by Assumption 1.2, we may set
$\Delta:=A_{\boldsymbol{\delta}}$.

Let $\mathrm{d}^{H}V_{G}$, $\mathrm{d}^{H}V_{T}$, $\mathrm{d}^{H}V_{G/T}$ be
the Haar measures on $G$, $T$ and $G/T$, respectively (§2.3 of [V1], §4.13 of
[V2]). Then the following holds (Theorem 4.13.5 of [V2]):

######  Theorem 4.1.

(Weyl Integration Formula) Let us set For every $L^{1}$-function on $G$,

$\int_{G}\,f(g)\,\mathrm{d}^{H}V_{G}(g)=\frac{1}{|W|}\cdot\int_{T}\,f_{T}(t)\,\big{|}\Delta(t)\big{|}^{2}\,\mathrm{d}^{H}V_{T}(t),$

where

$f_{T}(t):=\int_{G/T}\,f\left(g\,t\,g^{-1}\right)\,\mathrm{d}^{H}V_{GT}(gT).$

Similarly, under Assumption 1.2 $A_{\boldsymbol{\nu}}$ is well-defined for any
$\boldsymbol{\nu}\in\mathcal{E}^{G}$. Then we have (Theorem 4.14.4 of [V2]):

######  Theorem 4.2.

(The Weyl Character formula) On the open and dense regular locus
$T^{\prime}\subset T$ (defined by $\Delta\neq 0$), we have

$\left.\chi_{\boldsymbol{\nu}}\right|_{T^{\prime}}=\frac{A_{\boldsymbol{\nu}}}{\textbullet\Delta}.$

Another basic ingredient of the following arguments is the microlocal
representation of $\Pi$ as a Fourier integral operator introduced in [BS], and
its elaboration in the projective (and sympletic) setting in [Z], [BSZ], [SZ].
We refer to the latter papers for a detailed discussion, and simply recall
that the latter description has the form

$\Pi(x,y)=\int_{0}^{+\infty}\,e^{\imath\,u\,\psi(x,y)}\,s(x,y,u)\,\mathrm{d}u,$
(20)

where $\psi$ is a complex phase function (with $\Im(\psi)\geq 0$) and $s$ is a
semiclassical symbol on $X\times X$, admitting an asymptotic expansion of the
form

$s(x,y,u)\sim\sum_{j\geq 0}\,s_{j}(x,y)\,u^{d-j}.$ (21)

We shall invoke the following two properties of $\psi$:

  1. 1.

for any $x\in X$, we have

$\mathrm{d}_{(x,x)}\psi=(\alpha_{x},-\alpha_{x});$ (22)

  2. 2.

there exists a constant $D_{X}>0$ such that ($\Im$ denoting imaginary part)

$\Im\big{(}\psi(x,y)\big{)}\geq
D_{X}\,\mathrm{dist}_{X}(x,y)^{2}\qquad\forall\,x,y\in X.$ (23)

A further useful tool is given by Heisenberg local coordinates on $X$; these
are the local coordinates on $X$, introduced in [SZ], in which the local
scaling asymptotics of Szegö kernels exhibit their universal character. Given
$x\in X$, a system of local Heisenberg coordinates on $X$ centered at $x$ will
be denoted additively, in the form $x+(\theta,\mathbf{v})$; here
$\theta\in(-\pi,\pi)$ and $\mathbf{v}\in\mathbb{C}^{d}$ varies in an open ball
centered at the origin. Fiber rotation is represented by translation in
$\theta$, and the locus $\theta=0$ projects diffeomorphically onto its image
in $M$, hence defines a local section of $X$ which is horizontal at $x$ with
respect to the connection form $\alpha$. Thus a system of Heisenberg local
coordinates for $X$ centered at $x$ entails a choice of local coordinates for
$M$ centered at $m_{x}=\pi(x)$; actually the latter system determines a
unitary isomorphism $\mathbb{C}^{d}\cong T_{m}M$ with respect to
$(\omega_{m},J_{m})$, so when writing $x+(\theta,\mathbf{v})$ it is often
assumed that $\mathbf{v}\in T_{m}M$ (as will be the case below). Further, when
$\theta=0$ we usually write $x+\mathbf{v}$ for $x+(0,\mathbf{v})$ (as in the
statement of Theorem 1.3). Referring to (21), in Heisenberg local coordinates
at $x$ we have

$s_{0}(x,x)=\frac{1}{\pi^{d}}.$ (24)

We refer the reader to [SZ] for a detailed discussion.

###### Proof of Theorem 1.1.

The proof is an adaptation of the argument used when $G=U(2)$ in [GP1], so
we’ll be somewhat sketchy.

To begin with, in view of Theorem 3.1 we may assume without loss that $x$ and
$y$ belong to a small $S^{1}\times G$-invariant neighborhood $V_{\mathcal{O}}$
of $X_{\mathcal{O}}$. Equivalently, $m_{x}:=\pi(x)$ and $m_{y}:=\pi(y)$ belong
to a small $G$-invariant neighborhood of $U_{\mathcal{O}}$ of
$M_{\mathcal{O}}$ in $M$, and . In particular, we may assume that
$\tilde{\mu}$ is free on $V_{\mathcal{O}}$.

Hence $\Phi_{G}(m_{x})$ belongs a small conic neighborhood of
$\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}})$. Furthermore, replacing $(x,y)$
by $\left(\tilde{\mu}_{h}(x),\tilde{\mu}_{h}(y)\right)$ for a suitable $h\in
G$, we may also assume that $\Phi_{G}(m_{y})$ belongs to a small conic
neighborhood of $\mathbb{R}_{+}\cdot\boldsymbol{\nu}$. More precisely, we may
assume that $\Phi_{G}(m_{y})=\lambda_{y}\,\boldsymbol{\nu}+\beta_{y}$, where
$\lambda_{y}>0$, $\beta_{y}\in\boldsymbol{\nu}^{0}$ are smooth (here
$\boldsymbol{\nu}^{\perp}\subset\mathfrak{g}^{\vee}$ is the orthcomplement of
$\boldsymbol{\nu}$ with respect to, say, $\varphi^{H}$), and
$\|\beta_{y}\|\ll\lambda_{y}$.

We have

$\Pi_{k\,\boldsymbol{\nu}}(x,y)=d_{k\,\boldsymbol{\nu}}\,\int_{G}\overline{\chi_{k\,\boldsymbol{\nu}}(g)}\,\Pi\left(\tilde{\mu}_{g^{-1}}(x),y\right)\,\mathrm{d}^{H}V_{G}(g).$
(25)

For a suitably small $\delta_{K}>0$, let us introduce the open cover
$\mathcal{V}=\\{V^{\prime},V^{{}^{\prime\prime}}\\}$ of $G\times X\times X$
given by

$\displaystyle V^{\prime}$ $\displaystyle:=$ $\displaystyle\left\\{(g,x,y)\in
G\times X\times
X\,:\,\mathrm{dist}_{X}\left(\tilde{\mu}_{g^{-1}}(x),y\right)<2\,\delta_{K}\right\\},$
$\displaystyle V^{{}^{\prime\prime}}$ $\displaystyle:=$
$\displaystyle\left\\{(g,x,y)\in G\times X\times
X\,:\,\mathrm{dist}_{X}\left(\tilde{\mu}_{g^{-1}}(x),y\right)>\delta_{K}\right\\}.$

Let $\varrho^{\prime}+\varrho^{\prime\prime}=1$ be a partition of unity on
$G\times X\times X$ subordinate to $\mathcal{V}$. We then have

$\Pi_{k\,\boldsymbol{\nu}}(x,y)=\Pi_{k\,\boldsymbol{\nu}}(x,y)^{\prime}+\Pi_{k\,\boldsymbol{\nu}}(x,y)^{{}^{\prime\prime}},$
(26)

where $\Pi_{k\,\boldsymbol{\nu}}(x,y)^{\prime}$ and
$\Pi_{k\,\boldsymbol{\nu}}(x,y)^{{}^{\prime\prime}}$ are defined as in (25),
except that the integrand has been multiplied by $\varrho^{\prime}$ and
$\varrho^{{}^{\prime\prime}}$, respectively. Therefore,
$\mathcal{G}(g,x^{\prime},y^{\prime}):=\varrho^{{}^{\prime\prime}}\cdot\Pi\left(\tilde{\mu}_{g^{-1}}(x^{\prime}),y^{\prime}\right)$
is a smooth function on $G\times X$. Hence, letting
$\mathcal{G}_{x^{\prime},y^{\prime}}:=\mathcal{G}(\cdot,x^{\prime},y^{\prime})$
arguing as above we have

$\Pi_{k\,\boldsymbol{\nu}}(x^{\prime},y^{\prime})^{{}^{\prime\prime}}=d_{k\,\boldsymbol{\nu}}\,\mathrm{trace}\big{(}\mathcal{F}(\mathcal{G}_{x^{\prime},y^{\prime}})(k\,\boldsymbol{\nu}-\boldsymbol{\delta})\big{)}=O\left(k^{-\infty}\right),$

uniformly in $(x^{\prime},y^{\prime})\in X\times X$. We are thus reduced to
considering the asymptotics of the former summand
$\Pi_{k\,\boldsymbol{\nu}}(x,y)^{\prime}$. On the support of
$\varrho^{\prime}$, $y$ belongs to a small neighborhood of
$\tilde{\mu}_{g^{-1}}(x)$; hence we may represent $\Pi$ as a Fourier integral
operator.

Applying Theorems 4.2 and 4.1, and recalling that
$\Delta\circ\sigma=\epsilon(\sigma)\,\Delta$ for any $\sigma\in W$, we obtain

$\displaystyle\Pi_{k\,\boldsymbol{\nu}}(x,y)$ $\displaystyle\sim$
$\displaystyle
d_{k\,\boldsymbol{\nu}}\,\int_{G}\,\mathrm{d}^{H}V_{G}(g)\,\left[\overline{\chi_{k\,\boldsymbol{\nu}}(g)}\,\varrho^{\prime}(g,x,y)\,\Pi\left(\tilde{\mu}_{g^{-1}}(x),y\right)\right]$
$\displaystyle=$
$\displaystyle\frac{d_{k\,\boldsymbol{\nu}}\textbullet}{|W|\textbullet}\,\int_{T}\,\mathrm{d}^{H}V_{T}(t)\,\int_{G/T}\,\mathrm{d}^{H}V_{G/T}(g\,T)$
$\displaystyle\left[\overline{\chi_{k\,\boldsymbol{\nu}}(t)}\,|\Delta(t)|^{2}\,\varrho^{\prime}\left(g\,t\,g^{-1},x,y\right)\,\Pi\left(\tilde{\mu}_{g\,t^{-1}g^{-1}}(x),y\right)\right]$
$\displaystyle=$
$\displaystyle\frac{d_{k\,\boldsymbol{\nu}}\textbullet}{|W|\textbullet}\,\sum_{\sigma\in
W}\epsilon(\sigma)\,\int_{T}\,\mathrm{d}^{H}V_{T}(t)\,\int_{G/T}\,\mathrm{d}^{H}V_{G/T}(g\,T)$
$\displaystyle\left[\overline{E_{k\,\boldsymbol{\nu}}(t^{\sigma})}\,\Delta(t)\,\varrho^{\prime}\left(g\,t\,g^{-1},x,y\right)\,\Pi\left(\tilde{\mu}_{g\,t^{-1}g^{-1}}(x),y\right)\right]$
$\displaystyle=$ $\displaystyle
d_{k\,\boldsymbol{\nu}}\,\int_{T}\,\mathrm{d}^{H}V_{T}(t)\,\int_{G/T}\,\mathrm{d}^{H}V_{G/T}(g\,T)$
$\displaystyle\left[\overline{E_{k\,\boldsymbol{\nu}}(t)}\,\Delta(t)\,\varrho^{\prime}\left(g\,t\,g^{-1},x,y\right)\,\Pi\left(\tilde{\mu}_{g\,t^{-1}g^{-1}}(x),y\right)\right].$

Choosing a basis of the lattice $L(G)$, we obtain an isomorphism
$B:(S^{1})^{r_{G}}\cong\mathfrak{t}/L(G)\cong T$; we shall write the general
element of $(S^{1})^{r_{G}}$ as
$e^{\imath\,\boldsymbol{\vartheta}}=\begin{pmatrix}e^{\imath\,\vartheta_{1}}&\cdots&e^{\imath\,\vartheta_{r_{G}}}\end{pmatrix}$.
Then with
$\mathrm{d}\boldsymbol{\vartheta}=\mathrm{d}\vartheta_{1}\,\cdots\,\mathrm{d}\vartheta_{r_{G}}$

$B^{*}\big{(}\mathrm{d}V_{T}\big{)}=\frac{1}{(2\,\pi)^{r_{G}}}\,\mathrm{d}\boldsymbol{\vartheta},\quad
E_{k\,\boldsymbol{\nu}}\circ
B\left(e^{\imath\,\boldsymbol{\vartheta}}\right)=e^{\imath\,k\,\langle\boldsymbol{\nu},\boldsymbol{\vartheta}\rangle}$

To simplify notation, we shall simply identify $T$ and $(S^{1})^{r_{G}}$, and
write $e^{\imath\,\boldsymbol{\vartheta}}$ for the corresponding element of
$T$; with the same abuse, we shall think of $\imath\,\boldsymbol{\vartheta}$
as an element of $\mathfrak{t}\leqslant\mathfrak{g}$. Inserting (20) in (4),
we obtain

$\displaystyle\Pi_{k\,\boldsymbol{\nu}}(x,y)$ $\displaystyle\sim$
$\displaystyle\frac{\textbullet
d_{k\,\boldsymbol{\nu}}}{(2\,\pi)^{r_{G}}\textbullet}\,\int_{(-\pi,\pi)^{r_{G}}}\,\mathrm{d}\boldsymbol{\vartheta}\,\int_{G/T}\,\mathrm{d}^{H}V_{G/T}(g\,T)$
$\displaystyle\left[e^{\imath\,\big{[}u\,\psi(\tilde{\mu}_{g\,e^{-\imath\,\boldsymbol{\vartheta}}g^{-1}}(x),y)-k\,\langle\boldsymbol{\nu},\boldsymbol{\vartheta}\rangle\big{]}}\,\mathcal{A}_{k}(gT,\boldsymbol{\vartheta},x,y,u)\right]$
$\displaystyle=$ $\displaystyle\frac{\textbullet
k\,d_{k\,\boldsymbol{\nu}}}{(2\,\pi)^{r_{G}}\textbullet}\,\int_{(-\pi,\pi)^{r_{G}}}\,\mathrm{d}\boldsymbol{\vartheta}\,\int_{G/T}\,\mathrm{d}^{H}V_{G/T}(g\,T)$
$\displaystyle\left[e^{\imath\,k\,\Gamma(g\,T,\boldsymbol{\vartheta},u,x,y)}\,\mathcal{A}_{k}(gT,\boldsymbol{\vartheta},x,y,k\,u)\right]$

where

$\mathcal{A}_{k}(gT,\boldsymbol{\vartheta},x,y,u):=\Delta\left(e^{\imath\,\boldsymbol{\vartheta}}\right)\,\varrho^{\prime}\left(g\,e^{\imath\,\boldsymbol{\vartheta}}\,g^{-1},x,y\right)\,s\left(\tilde{\mu}_{g\,e^{-\imath\,\boldsymbol{\vartheta}}g^{-1}}(x),y,u\right),$
$\Gamma(g\,T,\boldsymbol{\vartheta},u,x,y):=u\,\psi(\tilde{\mu}_{g\,e^{-\imath\,\boldsymbol{\vartheta}}g^{-1}}(x),y)-\langle\boldsymbol{\nu},\boldsymbol{\vartheta}\rangle.$

On the support of $\varrho^{\prime}$ we have
$\tilde{\mu}_{g\,e^{-\imath\,\boldsymbol{\vartheta}}g^{-1}}(x)\sim y$. In view
of (22), in any given coordinate system therefore

$\mathrm{d}_{\big{(}\tilde{\mu}_{g\,e^{-\imath\,\boldsymbol{\vartheta}}g^{-1}}(x),y\big{)}}\psi\sim(\alpha_{y},-\alpha_{y}).$
(29)

On the other hand, by Lemma 2.10 of [P1] in Heisenberg local coordinates
centered at $x$ we have

$\displaystyle\tilde{\mu}_{g\,e^{-\imath\,\boldsymbol{\vartheta}}g^{-1}}(x)=\tilde{\mu}_{e^{-\mathrm{Ad}_{g}(\imath\,\boldsymbol{\vartheta})}}(x)$
$\displaystyle=$ $\displaystyle
x+\Big{(}\big{\langle}\Phi(m_{x}),\mathrm{Ad}_{g}(\imath\,\boldsymbol{\vartheta})\big{\rangle}+R_{3}(\boldsymbol{\vartheta}),-\mathrm{Ad}_{g}(\imath\,\boldsymbol{\vartheta})_{M}(m_{x})+R_{2}(\boldsymbol{\vartheta})\Big{)}$
$\displaystyle=$ $\displaystyle
x+\Big{(}\big{\langle}\Phi\big{(}\mu_{g^{-1}}(m_{x})\big{)},\imath\,\boldsymbol{\vartheta}\big{\rangle}+R_{3}(\boldsymbol{\vartheta}),-\mathrm{Ad}_{g}(\imath\,\boldsymbol{\vartheta})_{M}(m_{x})+R_{2}(\boldsymbol{\vartheta})\Big{)}.$

It follows that on the support of $\varrho^{\prime}$

$\partial_{\boldsymbol{\vartheta}}\Gamma(g\,T,\boldsymbol{\vartheta},u,x,y)\sim
u\,\left.\mathrm{Ad}_{g^{-1}}\Phi(m_{y})\right|_{\boldsymbol{\mathfrak{t}}}-\boldsymbol{\nu}.$
(31)

Since $\mathcal{O}_{\boldsymbol{\nu}}\cap\mathfrak{t}^{0}=\emptyset$, there
exists $r_{0}>0$ such that, with
$\Phi^{T}(m)=\left.\Phi(m)\right|_{\mathfrak{t}}$,

$\|\left.\Phi^{T}(m)\right|_{\mathfrak{t}}\|\geq r_{0},\quad\forall m\in
M_{\mathcal{O}}.$

Hence, if $U_{\mathcal{O}}$ is a sufficiently small open neighborhood of
$M_{\mathcal{O}}$, then

$\left\|\Phi^{T}(m)\right\|\geq\frac{1}{2}\,r_{0},\quad\forall m\in
U_{\mathcal{O}}.$

This applies to
$\mathrm{Ad}_{g^{-1}}\Phi(m_{y})=\Phi\circ\mu_{g^{-1}}(m_{y})$. It then
follows from (31) (arguing as in the proof of Lemma 5.3 of [GP1]) that the
following holds:

######  Lemma 4.1.

Suppose $D\gg 0$, and let
$\rho\in\mathcal{C}^{\infty}_{c}\big{(}(1/(2\,D),2\,D)\big{)}$ be such a bump
function that $\rho\equiv 1$ on $(1/D,D)$. Then only a rapidly decreasing
contribution to the asymptotics of (4) is lost, if the integrand is multiplied
by $\rho(t)$.

Hence we may assume that integration in $u$ is compactly supported. The proof
is the completed by iteratively integrating by parts in $\mathrm{d}u$, as in
the proof of Proposition 5.2 of [GP1].

∎

##  5 Proof of Theorem 1.2

In the proof of Theorem 1.2, we shall rely on the Kirillov character formula
([Ki], [R]), which we briefly recall.

Let $\exp_{G}:\boldsymbol{\xi}\in\mathfrak{g}\mapsto e^{\boldsymbol{\xi}}\in
G$ be the exponential map of $G$, and let
$\mathfrak{g}^{\prime}\subseteq\mathfrak{g}$ and $G^{\prime}\subseteq G$ be
open neighborhoods of the origin $\mathbf{0}\in\mathfrak{g}$ and of the unit
$1_{G}\in G$, respectively, such that $\exp_{G}$ restricts to a diffeomorphism
$\mathfrak{g}^{\prime}\rightarrow G^{\prime}$. We may find an Ad-invariant
Euclidean product $\varphi^{H}$ on $\mathfrak{g}$, such that
$\mathrm{d}^{H}V_{G}(g)$ is the Riemannian density associated to the induced
bi-invariant Riemannian metric on $G$, which with abuse of notation we shall
also denote by $\varphi^{H}$. Let $\mathrm{d}^{H}\boldsymbol{\xi}$ be the
Lebesgue measure on $\mathfrak{g}$ induced by $\varphi^{H}$. Let the
$\mathcal{C}^{\infty}$ function
$\mathcal{P}:\mathfrak{g}^{\prime}\rightarrow(0,+\infty)$ be defined by the
equality

$\exp_{G}^{*}(\mathrm{d}^{H}V_{G})=\mathcal{P}^{2}\,\mathrm{d}^{H}\boldsymbol{\xi}.$

Clearly $\mathcal{P}(0)=1$.

Let us set $n_{G}:=(d_{G}-r_{G})/2$. Then for every
$\boldsymbol{\nu}\in\mathcal{E}^{G}$ and
$\boldsymbol{\xi}\in\mathfrak{g}^{\prime}$ we have

$\dim(\mathcal{O}_{\boldsymbol{\nu}})=\dim(G/T)=d_{G}-r_{G}=2\,n_{G}.$

Furthermore, let us denote by $\sigma_{\boldsymbol{\nu}}$ the Kostant-Kirillov
symplectic structure on $\mathcal{O}_{\boldsymbol{\nu}}$, so that
$\sigma_{\boldsymbol{\nu}}^{n_{G}}/n_{G}!$ is the symplectic volume form on
$\mathcal{O}_{\boldsymbol{\nu}}$. In the following we shall set

$\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}:=\frac{\textbullet\sigma_{\boldsymbol{\nu}}^{n_{G}}}{\textbullet
n_{G}!},\quad\mathrm{vol}\big{(}\mathcal{O}_{\boldsymbol{\nu}}\big{)}:=\int_{\mathcal{O}_{\boldsymbol{\nu}}}\,\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}.$

The Kirillov character formula then says that

$\chi_{\boldsymbol{\nu}}\left(e^{\boldsymbol{\xi}}\right)=\frac{1}{(2\,\pi)^{n_{\boldsymbol{\nu}}}}\,\frac{1}{\textbullet\mathcal{P}(\boldsymbol{\xi})}\,\int_{\mathcal{O}_{\boldsymbol{\nu}}}\,e^{\imath\,\langle\boldsymbol{\lambda},\boldsymbol{\xi}\rangle}\,\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}(\boldsymbol{\lambda})\quad(\boldsymbol{\xi}\in\mathfrak{g}^{\prime}).$
(32)

Given (32) and (2), setting $\boldsymbol{\xi}=\mathbf{0}$ we get

$d_{\boldsymbol{\nu}}=\frac{\mathrm{vol}\big{(}\mathcal{O}_{\boldsymbol{\nu}}\big{)}}{(2\,\pi)^{n_{G}}\textbullet}\,\quad\Rightarrow\quad\mathrm{vol}\big{(}\mathcal{O}_{\boldsymbol{\nu}}\big{)}=(2\,\pi)^{n_{G}}\,\prod_{\boldsymbol{\beta}\in
R^{+}}\frac{\varphi(\boldsymbol{\nu},\boldsymbol{\beta})}{\varphi(\boldsymbol{\delta},\boldsymbol{\beta})}.$
(33)

For every $k\geq 1$, by a rescaling we obtain

$\chi_{k\,\boldsymbol{\nu}}\left(e^{\boldsymbol{\xi}}\right)=\left(\frac{k}{2\,\pi}\right)^{n_{G}}\,\frac{1}{\textbullet\mathcal{P}(\boldsymbol{\xi})}\,\int_{\mathcal{O}_{\boldsymbol{\nu}}}\,e^{\imath\,k\,\langle\boldsymbol{\lambda},\boldsymbol{\xi}\rangle}\,\frac{\textbullet\sigma_{\boldsymbol{\nu}}^{n_{G}}(\boldsymbol{\lambda})}{\textbullet
n_{G}!}\quad(\boldsymbol{\xi}\in\mathfrak{g}^{\prime}).$ (34)

In particular,

$d_{k\,\boldsymbol{\nu}}=\left(\frac{k}{2\,\pi}\right)^{n_{G}}\,\mathrm{vol}\big{(}\mathcal{O}_{\boldsymbol{\nu}}\big{)}.$
(35)

###### Proof of Theorem 1.2.

We may assume without loss that $\epsilon\in(0,1/6)$. Furthermore, it suffices
to prove the Theorem when $x=y$, since by the Cauchy-Schwartz inequality

$\left|\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}(x,y)\right|\leq\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}(x,x)^{\frac{1}{2}}\,\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}(y,y)^{\frac{1}{2}},$

and on the other hand
$\Pi^{\tilde{\mu}}_{k\,\boldsymbol{\nu}}(x,x)^{\frac{1}{2}}$ may be seen to
satisfy an a priori polynomial bound in $k$, by adapting the arguments in
§5.1.2 of [GP1]. By Theorem 3.1, we need only consider the case where $x$
belongs to a small $S^{1}\times G$-invariant neighborhood
$V_{\mathcal{O}_{\boldsymbol{\nu}}}\subseteq X$ of
$X_{\mathcal{O}_{\boldsymbol{\nu}}}$; in particular, we may assume without
loss that $\tilde{\mu}$ is free on $V_{\mathcal{O}_{\boldsymbol{\nu}}}$.
Furthermore, we may replace $x$ by $\tilde{\mu}_{g}(x)$ for any given $g\in
G$, and assume that $\Phi(m_{x})=\lambda_{x}\,\boldsymbol{\nu}+\beta_{x}$,
where $\lambda_{x}>0$, $\beta_{x}\in\boldsymbol{\nu}^{\perp}$, and
$\|\beta_{x}\|\ll\lambda_{x}$.

Let us start from (25) with $x=y$:

$\Pi_{k\,\boldsymbol{\nu}}(x,x)=d_{k\,\boldsymbol{\nu}}\,\int_{G}\overline{\chi_{k\,\boldsymbol{\nu}}(g)}\,\Pi\left(\tilde{\mu}_{g^{-1}}(x),x\right)\,\mathrm{d}^{H}V_{G}(g).$
(36)

Let us set, for some small $\varepsilon_{1}>0$,

$\displaystyle W^{\prime}$ $\displaystyle:=$ $\displaystyle\left\\{(g,x)\in
G\times
X\,:\,\mathrm{dist}_{X}\left(\tilde{\mu}_{g}(x),x\right)<2\,\varepsilon_{1}\right\\}$
$\displaystyle W^{\prime\prime}$ $\displaystyle:=$
$\displaystyle\left\\{(g,x)\in G\times
X\,:\,\mathrm{dist}_{X}\left(\tilde{\mu}_{g}(x),x\right)>\varepsilon_{2}\right\\}.$

Let $\rho^{\prime}+\rho^{\prime\prime}=1$ be a partition of unity on $G\times
X$ subordinate to the open cover $\\{W^{\prime},W^{\prime\prime}\\}$. Then an
argument as in the proof of Theorem 3.1 shows that only a rapidly decreasing
contribution to the asymptotics of (36) is lost, if the integrand is
multiplied by $\rho^{\prime\prime}$. Hence we are reduced to considering the
asymptotics of

$\Pi_{k\,\boldsymbol{\nu}}(x,x)^{\prime}:=d_{k\,\boldsymbol{\nu}}\,\int_{G}\rho^{\prime}(g,x)\,\overline{\chi_{k\,\boldsymbol{\nu}}(g)}\,\Pi\left(\tilde{\mu}_{g^{-1}}(x),x\right)\,\mathrm{d}^{H}V_{G}(g).$
(37)

Since $\tilde{\mu}$ is free on $V_{\mathcal{O}_{\boldsymbol{\nu}}}$, the
partial function $\rho_{x}^{\prime}:=\rho^{\prime}(\cdot,x)$ is supported on a
small open neighborhood of the unit $1_{G}\in G$, which we may assume to be
diffeomorphic to an open neighborhood of $\mathbf{0}\in\mathfrak{g}$ by the
exponential map. Hence on the same neighborhood we may set
$g=e^{\boldsymbol{\xi}}$ and express $\chi_{k\,\boldsymbol{\nu}}(g)$ by the
Kirillov character formula (34). Furthermore, since
$\tilde{\mu}_{g^{-1}}(x)\sim x$ for $\rho^{\prime}_{x}(g)\neq 0$, we may also
replace $\Pi$ by its description as an FIO (20). After the rescaling $u\mapsto
k\,u$, we obtain

$\displaystyle\Pi_{k\,\boldsymbol{\nu}}(x,x)\sim\Pi_{k\,\boldsymbol{\nu}}(x,x)^{\prime}$
$\displaystyle\sim$ $\displaystyle
k\,d_{k\,\boldsymbol{\nu}}\cdot\left(\frac{k}{2\,\pi}\right)^{n_{G}}\,\int_{0}^{+\infty}\,\mathrm{d}u\,\int_{\mathfrak{g}^{\prime}}\,\mathrm{d}^{H}\boldsymbol{\xi}\,\int_{\mathcal{O}_{\boldsymbol{\nu}}}\,\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}(\boldsymbol{\lambda})\left[e^{\imath\,k\,\Gamma_{x}(u,\boldsymbol{\xi},\boldsymbol{\lambda})}\,\mathcal{A}_{x,k}(u,\boldsymbol{\xi},\boldsymbol{\lambda})\right]$

where

$\Gamma_{x}(u,\boldsymbol{\xi},\boldsymbol{\lambda}):=u\,\psi\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x),x\right)-\langle\boldsymbol{\lambda},\boldsymbol{\xi}\rangle$
$\mathcal{A}_{x,k}(u,\boldsymbol{\xi},\boldsymbol{\lambda}):=\rho^{\prime}_{x}\left(e^{\boldsymbol{\xi}}\right)\,\mathcal{P}(\boldsymbol{\xi})\,s\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x),x,k\,u\right).$

Since $\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x)\sim x$ on the support of
$\mathcal{A}_{x,k}$, we have in local coordinates
$\mathrm{d}_{(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x),x)}\psi\sim(\alpha_{x},-\alpha_{x})$.
Hence
$\partial_{\boldsymbol{\xi}}\Gamma_{x}(u,\boldsymbol{\xi},\boldsymbol{\lambda})\sim
u\,\Phi(m_{x})-\boldsymbol{\lambda}$. We then have an analogue of Lemma 4.1,
so that integration in $\mathrm{d}u$ may be assumed to be compactly supported.
We express this by multiplying the amplitude in (5) by a bump function
$\rho=\rho(u)$ compactly supported in $(1/D,D)$ for some $D\gg 0$.

Let $\gamma\in\mathcal{C}^{\infty}(\mathfrak{g})$ be $\geq 0$, supported on a
ball of radius $2$ centered at the origin (say with respect to $\varphi^{H}$)
and $\equiv 1$ on a ball of radius $1$ centered at the origin. Let us define
$\gamma_{k}\in\mathcal{C}_{c}^{\infty}(\mathfrak{g})$ for $k=1,2,\ldots$ by
setting

$\gamma_{k}(\boldsymbol{\xi}):=\gamma\left(k^{1/2-\epsilon}\,\boldsymbol{\xi}\right).$

Let $\Pi_{k\,\boldsymbol{\nu}}(x,x)_{1}$ and
$\Pi_{k\,\boldsymbol{\nu}}(x,x)_{2}$ be given by the second line of (5)
multiplied by, respectively, $\gamma_{k}$ and $1-\gamma_{k}$.

######  Lemma 5.1.

$\Pi_{k\,\boldsymbol{\nu}}(x,x)_{2}=O\left(k^{-\infty}\right)$ as
$k\rightarrow+\infty$.

###### Proof of Lemma 5.1.

On the support of $1-\gamma_{k}$, we have $\|\boldsymbol{\xi}\|^{H}\geq
k^{\epsilon-\frac{1}{\textbullet 2}}$ in $\varphi^{H}$-norm. Hence for a
certain constant $r_{0}>0$ depending only on the choice of an invariant open
neighborhood $V_{\mathcal{O}}\subseteq X$ of $X_{\mathcal{O}}$ we have

$\mathrm{dist}_{X}\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x),x\right)\geq
r_{0}\,k^{\epsilon-\frac{1}{\textbullet 2}}\qquad\big{(}x\in
V_{\mathcal{O}},\,\boldsymbol{\xi}\in\mathrm{supp}(1-\gamma_{D_{1},k})\big{)}.$
(39)

Hence by (23)

$\big{|}\partial_{u}\Gamma_{x}(u,\boldsymbol{\xi},\boldsymbol{\lambda})\big{|}=\big{|}\psi\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x),x\right)\big{|}\geq\big{|}\Im\psi\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x),x\right)\big{|}\geq
D_{X}\,r_{0}^{2}\,k^{2\epsilon-1}.$ (40)

Iteratively integrating by parts in $\mathrm{d}u$ then implies the statement,
since at each step we introduce a factor $O\left(k^{-2\,\epsilon}\right)$ (see
Proposition 5.2 of [GP1]). ∎

Thus we are reduced to considering the asymptotics of
$\Pi_{k\,\boldsymbol{\nu}}(x,x)_{1}$. On the support of $\gamma_{k}$ we have
$\|\boldsymbol{\xi}\|^{H}\leq 2\,k^{\epsilon-\frac{1}{\textbullet 2}}$. Let us
operate the rescaling $\boldsymbol{\xi}\mapsto\boldsymbol{\xi}/\sqrt{k}$, and
rewrite $\Pi_{k\,\boldsymbol{\nu}}(x,x)_{1}$ as

$\displaystyle\Pi_{k\,\boldsymbol{\nu}}(x,x)_{1}$ $\displaystyle=$
$\displaystyle
k^{1-d_{G}/2}\,d_{k\,\boldsymbol{\nu}}\cdot\left(\frac{k}{2\,\pi}\right)^{n_{G}}\,\int_{1/D}^{D}\,\mathrm{d}u\,\int_{\mathfrak{g}^{\prime}}\,\mathrm{d}^{H}\boldsymbol{\xi}\,\int_{\mathcal{O}_{\boldsymbol{\nu}}}\,\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}(\boldsymbol{\lambda})$
(41)
$\displaystyle\left[e^{\imath\,k\,\Gamma_{x}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})}\,\mathcal{A}_{x,k}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})\,\rho(u)\,\gamma\left(k^{-\epsilon}\,\boldsymbol{\xi}\right)\right].$

Now integration in $\boldsymbol{\xi}$ is on an expanding ball of radius
$O\left(k^{\epsilon}\right)$ centered at the origin. In Heisenberg local
coordinates at $x$, by Lemma 2.10 of [P1] we have

$\displaystyle\tilde{\mu}_{e^{-\boldsymbol{\xi}/\sqrt{k}}}(x)$
$\displaystyle=$ $\displaystyle
x+\left(\frac{1}{\sqrt{k}}\,\big{\langle}\Phi(m_{x}),\boldsymbol{\xi})\big{\rangle}+R_{3}\left(\frac{1}{\sqrt{k}}\,\boldsymbol{\xi}\right),-\frac{1}{\sqrt{k}}\,\boldsymbol{\xi}_{M}(m_{x})+R_{2}\left(\frac{1}{\sqrt{k}}\,\boldsymbol{\xi}\right)\right).$

As in the proof of Theorem 1 of [P1], using the expansions in §3 of [SZ] one
gets

$\displaystyle\Gamma_{x}\left(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{k}}\,\langle
u\,\Phi(m_{x})-\boldsymbol{\lambda},\boldsymbol{\xi}\rangle+\frac{\imath\,u}{2\,k}\,\|\boldsymbol{\xi}_{X}(x)\|^{2}+u\,R_{3}\left(\frac{\textbullet\boldsymbol{\xi}}{\sqrt{k}}\right)\,e^{\imath\,\varsigma_{x,k}(\boldsymbol{\xi}/\sqrt{k})},$

where
$\varsigma_{x,k}(\boldsymbol{\xi})=\big{\langle}\Phi(m_{x}),\boldsymbol{\xi})\big{\rangle}+R_{3}(\boldsymbol{\xi})$.
Hence we rewrite (41) as follows

$\displaystyle\Pi_{k\,\boldsymbol{\nu}}(x,x)_{1}$ $\displaystyle=$
$\displaystyle
k^{1-d_{G}/2}\,d_{k\,\boldsymbol{\nu}}\cdot\left(\frac{k}{2\,\pi}\right)^{n_{G}}\,\int_{1/D}^{D}\,\mathrm{d}u\,\int_{\mathfrak{g}^{\prime}}\,\mathrm{d}^{H}\boldsymbol{\xi}\,\int_{\mathcal{O}_{\boldsymbol{\nu}}}\,\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}(\boldsymbol{\lambda})$
(44)
$\displaystyle\left[e^{\imath\,\sqrt{k}\,\Upsilon_{x}(u,\boldsymbol{\xi},\boldsymbol{\lambda})}\,\mathcal{B}_{x,k}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})\,\gamma\left(D_{1}\,k^{-\epsilon}\,\boldsymbol{\xi}\right)\right],$

where now

$\displaystyle\Upsilon_{x}(u,\boldsymbol{\xi},\boldsymbol{\lambda})$
$\displaystyle:=$ $\displaystyle\langle
u\,\Phi(m_{x})-\boldsymbol{\lambda},\boldsymbol{\xi}\rangle$
$\displaystyle\mathcal{B}_{x,k}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})$
$\displaystyle:=$ $\displaystyle
e^{-\frac{u}{2}\,\|\boldsymbol{\xi}_{X}(x)\|^{2}}\cdot\rho(u)\,\mathcal{A}_{x,k}(u,k^{-1/2}\,\boldsymbol{\xi},\boldsymbol{\lambda})\,e^{u\,k\,R_{3}\left(\frac{\textbullet\boldsymbol{\xi}}{\sqrt{k}}\right)\,e^{\imath\,\varsigma_{x,k}(\boldsymbol{\xi}/\sqrt{k})}}$

We have $\boldsymbol{\xi}_{X}(x)\neq 0$ if $x\in
V_{\mathcal{O}_{\boldsymbol{\nu}}}$, $\boldsymbol{\xi}\neq 0$.

Under the present transversality assumption (Assumption 1.1), there exists
$s_{0}>0$ such that

$\big{\|}\Phi(m)-\boldsymbol{\lambda}\big{\|}\geq
s_{0}\cdot\mathrm{dist}_{M}(m,M_{\mathcal{O}}),\qquad\,\forall\,m\in
M,\,\forall\boldsymbol{\lambda}\in\mathcal{C}(\mathcal{O}_{\boldsymbol{\nu}}).$

Therefore, in the situation of the Theorem,

$\|\partial_{\boldsymbol{\xi}}\Upsilon_{x}(u,\boldsymbol{\xi},\boldsymbol{\lambda})\big{\|}=\big{\|}u\,\Phi(m_{x})-\boldsymbol{\lambda}\big{\|}\geq\frac{\textbullet
s_{0}}{\textbullet D}\cdot C\,k^{\epsilon-\frac{1}{\textbullet
2}},\quad\forall\,\boldsymbol{\lambda}\in\mathcal{O}_{\boldsymbol{\nu}},\quad\forall\,u\in\left(\frac{1}{D},D\right).$

The statement of the Theorem then follows by iteratively integrating by parts
in $\mathrm{d}\boldsymbol{\xi}$, since each step introduces a factor
$O\left(k^{-\epsilon}\right)$. ∎

##  6 Proof of Theorem 1.3

Before delving into the proof of Theorem 1.3, let us make the following
remarks.

Suppose $x\in X$, and let $\varpi_{x}:(\theta,\mathbf{v})\mapsto
x+(\theta,\mathbf{v})$ be a system of Heisenberg local coordinates at $x$.
Then $\varpi_{x}$ induces an isomorphism
$T_{x}X\cong\mathbb{R}\times\mathbb{R}^{2n}$, in terms of which we can give a
meaning to the expression $x+\upsilon$, when $\upsilon\in T_{x}X$ is small.
For some $c_{1}>c_{2}>0$ we have

$c_{2}\,\|\upsilon_{1}-\upsilon_{2}\|\leq\mathrm{dist}_{X}(x+\upsilon_{1},x+\upsilon_{2})\leq
c_{1}\,\|\upsilon_{1}-\upsilon_{2}\|$ (45)

if $\upsilon_{j}\sim\mathbf{0}$.

Let $\mathfrak{t}_{m}^{\prime}$ be as in (7); we have the following
characterization of the normal bundle
$N\big{(}M_{\mathcal{O}_{\boldsymbol{\nu}}}\big{)}$ to
$M_{\mathcal{O}_{\boldsymbol{\nu}}}$ in $M$, which can be proved by minor
adaptations of the arguments used in Lemma 4.2 and Step 4.3 of [GP1].

######  Lemma 6.1.

For any $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}$,
$N_{m}\big{(}M_{\mathcal{O}_{\boldsymbol{\nu}}}\big{)}=J_{m}\big{(}\mathfrak{t}_{m}^{\prime}\big{)}$.

Furthermore, we may identify the normal bundle of
$X_{\mathcal{O}_{\boldsymbol{\nu}}}\subseteq X$,
$N\big{(}X_{\mathcal{O}_{\boldsymbol{\nu}}}\big{)}$, with the pull-back of
$N\big{(}M_{\mathcal{O}_{\boldsymbol{\nu}}}\big{)}$; even more explicitly, for
every $x\in X_{\mathcal{O}_{\boldsymbol{\nu}}}$ we have with $m_{x}=\pi(x)$

$N_{x}\big{(}X_{\mathcal{O}_{\boldsymbol{\nu}}}\big{)}=N_{m_{x}}\big{(}X_{\mathcal{O}_{\boldsymbol{\nu}}}\big{)}^{\sharp}.$

Hence there is an orthogonal direct sum

$N_{m_{x}}\big{(}X_{\mathcal{O}_{\boldsymbol{\nu}}}\big{)}^{\sharp}\oplus\mathfrak{g}_{X}(x)\oplus{\mathfrak{g}_{M}(m_{x})^{\perp_{h}}}^{\sharp}\subseteq
T_{x}X.$ (46)

We then have the following consequence, whose proof is omitted.

######  Lemma 6.2.

Suppose $x\in X_{\mathcal{O}_{\boldsymbol{\nu}}}$, and choose a system of
Heisenberg local coordinates at $x$. Then there exists $\delta>0$ such that
for any choice of $\boldsymbol{\xi}\in\mathfrak{g}$, $\mathbf{v}_{j}\in
N_{m_{x}}\big{(}X_{\mathcal{O}_{\boldsymbol{\nu}}}\big{)}$,
$\mathbf{w}_{j}\in{\mathfrak{g}_{M}(m_{x})^{\perp_{h}}}$ of sufficiently small
norm we have

$\mathrm{dist}_{X}\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}\big{(}x+(\mathbf{v}_{1}+\mathbf{w}_{1})\big{)},x+(\mathbf{v}_{2}+\mathbf{w}_{2})\right)\geq\delta\,\|\boldsymbol{\xi}\|^{\varphi}.$

Furthermore, $\delta$ may be chosen uniformly on
$X_{\mathcal{O}_{\boldsymbol{\nu}}}$.

###### Proof of Theorem 1.3.

We may replace $x$ by $\tilde{\mu}_{h}(x)$ for a suitable $h\in G$, and assume
without loss that $\Phi(m_{x})=\varsigma(m_{x})\,\boldsymbol{\nu}$. Hence, we
may assume that

$\mathfrak{t}_{m}=\mathfrak{t},\quad\mathfrak{t}_{m}^{\prime}=\mathfrak{t}_{\boldsymbol{\nu}}$
(47)

(see (6), (7)). Let us set

$x_{j,k}:=\frac{1}{\sqrt{k}}\,(\mathbf{v}_{j}+\mathbf{w}_{j})\qquad(j=1,2;\,k>0).$

and replace (36) by

$\Pi_{k\,\boldsymbol{\nu}}(x_{1,k},x_{2,k})=d_{k\,\boldsymbol{\nu}}\,\int_{G}\overline{\chi_{k\,\boldsymbol{\nu}}(g)}\,\Pi\left(\tilde{\mu}_{g^{-1}}(x_{1,k}),x_{2,k}\right)\,\mathrm{d}^{H}V_{G}(g).$
(48)

Given that $x_{j,k}\rightarrow x$, the argument leading to (37) now implies
$\Pi_{k\,\boldsymbol{\nu}}(x_{1,k},x_{2,k})\sim\Pi_{k\,\boldsymbol{\nu}}(x_{1,k},x_{2,k})^{\prime}$,
where

$\Pi_{k\,\boldsymbol{\nu}}(x_{1,k},x_{2,k})^{\prime}:=d_{k\,\boldsymbol{\nu}}\,\int_{G}\rho^{\prime}(g,x)\,\overline{\chi_{k\,\boldsymbol{\nu}}(g)}\,\Pi\left(\tilde{\mu}_{g^{-1}}(x_{1,k}),x_{2,k}\right)\,\mathrm{d}^{H}V_{G}(g).$
(49)

We then obtain in place of (5)

$\displaystyle\Pi_{k\,\boldsymbol{\nu}}(x_{1,k},x_{2,k})\sim\Pi_{k\,\boldsymbol{\nu}}(x_{1,k},x_{2,k})^{\prime}$
$\displaystyle\sim$ $\displaystyle
k\,d_{k\,\boldsymbol{\nu}}\cdot\left(\frac{k}{2\,\pi}\right)^{n_{G}}\,\,\int_{0}^{+\infty}\,\mathrm{d}u\,\int_{\mathfrak{g}^{\prime}}\,\mathrm{d}^{H}\boldsymbol{\xi}\,\int_{\mathcal{O}_{\boldsymbol{\nu}}}\,\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}(\boldsymbol{\lambda})\left[e^{\imath\,k\,\Gamma_{x,k}(u,\boldsymbol{\xi},\boldsymbol{\lambda})}\,\mathcal{B}_{x,k}(u,\boldsymbol{\xi},\boldsymbol{\lambda})\right]$

where

$\Gamma_{x,k}(u,\boldsymbol{\xi},\boldsymbol{\lambda}):=u\,\psi\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x_{1,k}),x_{2,k}\right)-\langle\boldsymbol{\lambda},\boldsymbol{\xi}\rangle$
(51)
$\mathcal{B}_{x,k}(u,\boldsymbol{\xi},\boldsymbol{\lambda}):=\rho^{\prime}_{x}\left(e^{\boldsymbol{\xi}}\right)\,\mathcal{P}(\boldsymbol{\xi})\cdot
s\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x_{1,k}),x_{2,k},k\,u\right).$ (52)

Since $\tilde{\mu}_{e^{-\boldsymbol{\xi}}}(x_{1,k})\sim x_{2,k}$ on the
support of $\mathcal{B}_{x,k}$, by the same argument used in the proof of
Theorem 1.2 we may multiply the integrand in (6) by the same cut-off function
$\rho=\rho(u)$ without affecting the asymptotics, so as to assume that
integration in $\mathrm{d}u$ is supported in $(1/D,D)$ for some $D\gg 0$.

In view of Lemma 6.2, we have for $k\gg 0$

$\mathrm{dist}_{X}\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}}}\big{(}x_{1,k})\big{)},x_{2,k})\right)\geq\delta\,\|\boldsymbol{\xi}\|^{\varphi}.$

Using this, we obtain an obvious analogue of (39), so that we can reprove
Lemma 5.1 in the present setting. Rescaling in $\boldsymbol{\xi}$, we obtain
in place of (41):

$\displaystyle\Pi_{k\,\boldsymbol{\nu}}(x_{1,k},x_{2,k})$ $\displaystyle\sim$
$\displaystyle
k^{1-d_{G}/2}\,d_{k\,\boldsymbol{\nu}}\cdot\left(\frac{k}{2\,\pi}\right)^{n_{G}}\,\int_{1/D}^{D}\,\mathrm{d}u\,\int_{\mathfrak{g}}\,\mathrm{d}^{H}\boldsymbol{\xi}\,\int_{\mathcal{O}_{\boldsymbol{\nu}}}\,\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}(\boldsymbol{\lambda})$
$\displaystyle\left[e^{\imath\,k\,\Gamma_{x,k}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})}\,\mathcal{B}_{x,k}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})\,\rho(u)\,\gamma\left(k^{-\epsilon}\,\boldsymbol{\xi}\right)\right].$

In view of Corollary 2.2 of [P1], and using that
$\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}(m),\mathbf{w}_{1}\big{)}=0$, in
place of (5) we have

$\displaystyle\tilde{\mu}_{e^{-\boldsymbol{\xi}/\sqrt{k}}}(x_{1,k})=x+\left(\Theta_{k,1},V_{k,1}\right).$
(54)

where

$\displaystyle\Theta_{k,1}$ $\displaystyle=$
$\displaystyle\Theta_{k}(x,\boldsymbol{\xi},\mathbf{v}_{1},\mathbf{w}_{1})$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{k}}\,\big{\langle}\Phi(m_{x}),\boldsymbol{\xi})\big{\rangle}+\frac{1}{k}\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}_{1}\big{)}+R_{3}\left(\frac{1}{\sqrt{k}}\,\boldsymbol{\xi},\frac{1}{\sqrt{k}}\,\mathbf{v}_{1},\frac{1}{\sqrt{k}}\,\mathbf{w}_{1}\right),$
$\displaystyle V_{k,1}$ $\displaystyle=$ $\displaystyle
V_{k}(x,\boldsymbol{\xi},\mathbf{v}_{1},\mathbf{w}_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{k}}\,\big{(}\mathbf{v}_{1}+\mathbf{w}_{1}-\boldsymbol{\xi}_{M}(m_{x})\big{)}+R_{2}\left(\frac{1}{\sqrt{k}}\,\boldsymbol{\xi},\frac{1}{\sqrt{k}}\,\mathbf{v}_{1},\frac{1}{\sqrt{k}}\,\mathbf{w}_{1}\right).$

We then have (see §3 of [SZ])

$\displaystyle
u\,\psi\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}/\sqrt{k}}}(x_{1,k}),x_{2,k}\right)$
$\displaystyle=$
$\displaystyle\imath\,u\,\left[1-e^{\imath\,\Theta_{k}}\right]-\imath\,u\,\psi_{2}\left(V_{k,1},\frac{1}{\sqrt{k}}\,(\mathbf{v}_{2}+\mathbf{w}_{2})\right)+u\,R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{j}}{\sqrt{k}},\frac{\mathbf{w}_{j}}{\sqrt{k}}\right).$

We have

$\displaystyle\imath\,u\,\left[1-e^{\imath\,\Theta_{k,1}}\right]$
$\displaystyle=$ $\displaystyle
u\,\Theta_{k,1}+\frac{\imath\,u}{2}\,\Theta_{k,1}^{2}+u\,R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{1}}{\sqrt{k}},\frac{\mathbf{w}_{1}}{\sqrt{k}}\right)$
$\displaystyle=$
$\displaystyle\frac{u}{\sqrt{k}}\,\big{\langle}\Phi(m_{x}),\boldsymbol{\xi})\big{\rangle}+\frac{u}{k}\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}_{1}\big{)}+\frac{\imath\,u}{2\,k}\,\big{\langle}\Phi(m_{x}),\boldsymbol{\xi}\big{\rangle}^{2}$
$\displaystyle+u\,R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{1}}{\sqrt{k}},\frac{\mathbf{w}_{1}}{\sqrt{k}}\right),$

and

$\displaystyle\psi_{2}\left(V_{k,1},\frac{1}{\sqrt{k}}\,(\mathbf{v}_{2}+\mathbf{w}_{2})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{k}\,\left[-\imath\,\omega_{m_{x}}\big{(}\mathbf{v}_{1}+\mathbf{w}_{1}-\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}_{2}+\mathbf{w}_{2}\big{)}-\frac{1}{2}\,\big{\|}\big{(}\mathbf{v}_{1}-\mathbf{v}_{2})+(\mathbf{w}_{1}-\mathbf{w}_{2})-\boldsymbol{\xi}_{M}(m_{x})\big{\|}_{m_{x}}^{2}\right]$
$\displaystyle+R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{j}}{\sqrt{k}},\frac{\mathbf{w}_{j}}{\sqrt{k}}\right).$

Since $\mathbf{w}_{j}\in\mathfrak{g}_{M}(m)^{\perp_{h_{m}}}$, we have
$\omega_{m}\big{(}\boldsymbol{\xi}_{M}(m_{x}),\mathbf{w}_{2}\big{)}=0$ for any
$\boldsymbol{\xi}\in\mathfrak{g}$.

######  Lemma 6.3.

If $m\in M_{\mathcal{O}_{\boldsymbol{\nu}}}$ and $\mathbf{v}_{j}\in
N_{m}(M_{\mathcal{O}_{\boldsymbol{\nu}}})$, then
$\omega_{m}(\mathbf{v}_{1},\mathbf{v}_{2})=0$.

###### Proof of Lemma 6.3.

By Lemma 6.1, there are
$\boldsymbol{\eta}_{j}\in\mathfrak{t}^{\prime}_{m}\subseteq\mathfrak{t}_{m}$
such that $\mathbf{v}_{j}=J_{m}\big{(}{\boldsymbol{\eta}_{j}}_{M}(m)\big{)}$
(given our previous reduction we may assume $\mathfrak{t}_{m}=\mathfrak{t}$).
Hence
$\omega_{m}(\mathbf{v}_{1},\mathbf{v}_{2})=\omega_{m}\big{(}{\boldsymbol{\eta}_{1}}_{M}(m),{\boldsymbol{\eta}_{2}}_{M}(m)\big{)}$.
On the other hand, $\mu$ restricts to a Hamiltonian action of the maximal
torus $T_{m}$, and therefore the vector fields $\boldsymbol{\eta}_{M}$, with
$\boldsymbol{\eta}\in\mathfrak{t}$, are all in symplectic involution. Hence
$\omega\big{(}{\boldsymbol{\eta}_{1}}_{M},{\boldsymbol{\eta}_{2}}_{M}\big{)}\equiv
0$. Hence $\omega_{m}(\mathbf{v}_{1},\mathbf{v}_{2})=0$. ∎

Given (46) and Lemma 6.3, in view of Definition 1.3 we may rewrite (6) as

$\displaystyle\psi_{2}\left(V_{k,1},\frac{1}{\sqrt{k}}\,(\mathbf{v}_{2}+\mathbf{w}_{2})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{k}\,\left[\psi_{2}(\mathbf{w}_{1},\mathbf{w}_{2})-\frac{1}{2}\,\big{\|}\mathbf{v}_{1}-\mathbf{v}_{2}\|_{m_{x}}^{2}+\imath\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}_{2}\big{)}-\frac{1}{2}\,\|\boldsymbol{\xi}_{M}(m_{x})\big{\|}_{m}^{2}\right]$
$\displaystyle+R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{j}}{\sqrt{k}},\frac{\mathbf{w}_{j}}{\sqrt{k}}\right).$

Hence, (6) may be rewritten

$\displaystyle
u\,\psi\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}/\sqrt{k}}}(x_{1,k}),x_{2,k}\right)$
$\displaystyle=$
$\displaystyle\frac{u}{\sqrt{k}}\,\big{\langle}\Phi(m_{x}),\boldsymbol{\xi})\big{\rangle}+\frac{u}{k}\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}_{1}\big{)}+\frac{\imath\,u}{2\,k}\,\big{\langle}\Phi(m_{x}),\boldsymbol{\xi}\big{\rangle}^{2}$
$\displaystyle+u\,R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{1}}{\sqrt{k}},\frac{\mathbf{w}_{1}}{\sqrt{k}}\right)$
$\displaystyle-\frac{\imath\,u}{k}\,\left[\psi_{2}(\mathbf{w}_{1},\mathbf{w}_{2})-\frac{1}{2}\,\big{\|}\mathbf{v}_{1}-\mathbf{v}_{2}\|_{m_{x}}^{2}+\imath\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}_{2}\big{)}-\frac{1}{2}\,\|\boldsymbol{\xi}_{M}(m_{x})\big{\|}_{m_{x}}^{2}\right]$
$\displaystyle+u\,R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{j}}{\sqrt{k}},\frac{\mathbf{w}_{j}}{\sqrt{k}}\right).$

Whence

$\displaystyle\imath\,k\,u\,\psi\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}/\sqrt{k}}}(x_{1,k}),x_{2,k}\right)$
$\displaystyle=$
$\displaystyle\imath\,\sqrt{k}\,u\,\big{\langle}\Phi(m_{x}),\boldsymbol{\xi}\big{\rangle}+\imath\,u\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}_{1}+\mathbf{v}_{2}\big{)}-\frac{u}{2}\,\big{\langle}\Phi(m_{x}),\boldsymbol{\xi}\big{\rangle}^{2}$
$\displaystyle+u\,\left[\psi_{2}(\mathbf{w}_{1},\mathbf{w}_{2})-\frac{1}{2}\,\big{\|}\mathbf{v}_{1}-\mathbf{v}_{2}\|_{m_{x}}^{2}-\frac{1}{2}\,\|\boldsymbol{\xi}_{M}(m_{x})\big{\|}_{m_{x}}^{2}\right]$
$\displaystyle+\imath\,u\,k\,R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{j}}{\sqrt{k}},\frac{\mathbf{w}_{j}}{\sqrt{k}}\right)$
$\displaystyle=$
$\displaystyle\imath\,\sqrt{k}\,u\,\big{\langle}\Phi(m_{x}),\boldsymbol{\xi}\big{\rangle}+\imath\,u\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}(m_{x}),\mathbf{v}_{1}+\mathbf{v}_{2}\big{)}-\frac{u}{2}\,\|\boldsymbol{\xi}_{X}(x)\big{\|}_{x}^{2}$
$\displaystyle+u\,\left[\psi_{2}(\mathbf{w}_{1},\mathbf{w}_{2})-\frac{1}{2}\,\big{\|}\mathbf{v}_{1}-\mathbf{v}_{2}\|_{m_{x}}^{2}\right]$
$\displaystyle+\imath\,u\,k\,R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{j}}{\sqrt{k}},\frac{\mathbf{w}_{j}}{\sqrt{k}}\right).$

In view of (51)

$\displaystyle\imath\,k\,\Gamma_{x,k}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})$
$\displaystyle=$
$\displaystyle\imath\,k\left[u\,\psi\left(\tilde{\mu}_{e^{-\boldsymbol{\xi}/\sqrt{k}}}(x_{1,k}),x_{2,k}\right)-\left\langle\boldsymbol{\lambda},\frac{1}{\sqrt{k}}\,\boldsymbol{\xi}\,\right\rangle\right]$
$\displaystyle=$
$\displaystyle\imath\,\sqrt{k}\,\big{\langle}u\,\Phi(m_{x})-\boldsymbol{\lambda},\boldsymbol{\xi}\big{\rangle}+\imath\,u\,\omega_{m}\big{(}\boldsymbol{\xi}_{M}(m),\mathbf{v}_{1}+\mathbf{v}_{2}\big{)}-\frac{u}{2}\,\|\boldsymbol{\xi}_{X}(x)\|_{x}^{2}$
$\displaystyle+u\,\left[\psi_{2}(\mathbf{w}_{1},\mathbf{w}_{2})-\frac{1}{2}\,\|\mathbf{v}_{1}-\mathbf{v}_{2}\|_{m}^{2}\right]$
$\displaystyle+\imath\,u\,k\,R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{j}}{\sqrt{k}},\frac{\mathbf{w}_{j}}{\sqrt{k}}\right)$

Let $\mathrm{d}^{\varphi}\boldsymbol{\xi}$ be the Lebesgue measure on
$\mathfrak{g}$ associated to $\varphi$. Then

$\mathrm{d}^{H}\boldsymbol{\xi}=\mathrm{vol}^{\varphi}(G)^{-1}\,\mathrm{d}^{\varphi}\boldsymbol{\xi}.$
(64)

Using this and (35), (6) may be rewritten as an oscillatory integral in
$\sqrt{k}$ with a real phase, in the form

$\displaystyle\Pi_{k\,\boldsymbol{\nu}}(x_{1,k},x_{2,k})$ $\displaystyle\sim$
$\displaystyle\frac{\mathrm{vol}\big{(}\mathcal{O}_{\boldsymbol{\nu}}\big{)}}{\mathrm{vol}^{\varphi}(G)}\cdot
k^{1-d_{G}/2}\,\left(\frac{k}{2\,\pi}\right)^{2\,n_{G}}\,$
$\displaystyle\cdot\int_{1/D}^{D}\,\mathrm{d}u\,\int_{\mathfrak{g}}\,\mathrm{d}^{\varphi}\boldsymbol{\xi}\,\int_{\mathcal{O}_{\boldsymbol{\nu}}}\,\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}(\boldsymbol{\lambda})\left[e^{\imath\,\sqrt{k}\,\Upsilon_{x}(u,\boldsymbol{\xi},\boldsymbol{\lambda})}\,\mathcal{C}_{x,k}(u,\boldsymbol{\xi},\boldsymbol{\lambda};\mathbf{v}_{j},\mathbf{w}_{j})\right],$

with phase $\Upsilon_{x}$ and amplitude $\mathcal{C}_{x,k}$ given by,
respectively,

$\Upsilon_{x}(u,\boldsymbol{\xi},\boldsymbol{\lambda}):=\big{\langle}u\,\Phi(m_{x})-\boldsymbol{\lambda},\boldsymbol{\xi}\big{\rangle},$
(66)
$\displaystyle\mathcal{C}_{x,k}(u,\boldsymbol{\xi},\boldsymbol{\lambda};\mathbf{v}_{j},\mathbf{w}_{j})$
$\displaystyle:=$ $\displaystyle
e^{\imath\,u\,\omega_{m}\big{(}\boldsymbol{\xi}_{M}(m),\mathbf{v}_{1}+\mathbf{v}_{2}\big{)}-\frac{u}{2}\,\|\boldsymbol{\xi}_{X}(x)\|_{x}^{2}+u\,\left[\psi_{2}(\mathbf{w}_{1},\mathbf{w}_{2})-\frac{1}{2}\,\|\mathbf{v}_{1}-\mathbf{v}_{2}\|_{m_{x}}^{2}\right]}$
(67)
$\displaystyle\cdot\mathcal{B}_{x,k}^{\prime}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})$

where

$\displaystyle\mathcal{B}_{x,k}^{\prime}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})$
$\displaystyle=$ $\displaystyle
e^{\imath\,u\,k\,R_{3}\left(\frac{\boldsymbol{\xi}}{\sqrt{k}},\frac{\mathbf{v}_{j}}{\sqrt{k}},\frac{\mathbf{w}_{j}}{\sqrt{k}}\right)}\cdot\mathcal{B}_{x,k}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})\,\rho(u)\,\gamma\left(k^{-\epsilon}\,\boldsymbol{\xi}\right).$

There exists $r_{\boldsymbol{\nu}}>0$, depending only on $\boldsymbol{\nu}$,
such that $\|\boldsymbol{\xi}_{X}(x)\|_{x}\geq
r_{\boldsymbol{\nu}}\,\|\boldsymbol{\xi}\|^{\varphi}$, $\forall\,x\in
X_{\mathcal{O}_{\boldsymbol{\nu}}}$. Furthermore, Taylor expansion at the
origin yields an asymptotic expansion of the form

$\mathcal{B}_{x,k}^{\prime}(u,\boldsymbol{\xi}/\sqrt{k},\boldsymbol{\lambda})\sim\gamma\left(k^{-\epsilon}\,\boldsymbol{\xi}\right)\cdot\sum_{j\geq
0}k^{d-j/2}\,P_{j}(m_{x},u;\boldsymbol{\xi},\mathbf{v}_{j},\mathbf{w}_{j}),$
(69)

where $P_{j}(m_{x},u;\cdot,\cdot,\cdot)$ is a polynomial of degree $\leq
3\,j$, and parity $j$ (recall that
$\|\boldsymbol{\xi}\|^{\varphi},\,\|\mathbf{v}_{j}\|,\,\|\mathbf{w}_{j}\|\leq
C^{\prime}\,k^{\epsilon}$ for some fixed $C^{\prime}>0$, and that
$\epsilon\in(0,1/6)$). In view of (21) and (24),

$P_{j}(m_{x},u;\boldsymbol{\xi},\mathbf{v}_{j},\mathbf{w}_{j})=\left(\frac{u}{\pi}\right)^{d}.$
(70)

The expansion may be integrated term by term.

Recall that we have reduced to the case where
$\Phi(m_{x})\in\mathbb{R}_{+}\,\boldsymbol{\nu}$, hence
$\Phi(m_{x})=\varsigma(m_{x})\,\boldsymbol{\nu}$ (see (4)).

######  Lemma 6.4.

Let
$\mathcal{O}^{\prime}\Subset\mathcal{O}^{\prime\prime}\subset\mathcal{O}_{\boldsymbol{\nu}}$
be suitably small neighborhoods of $\boldsymbol{\nu}$. Let
$\varrho_{\boldsymbol{\nu}}\in\mathcal{C}^{\infty}_{c}(\mathcal{O}^{\prime\prime})$
be $\mathcal{C}^{\infty}$, $\geq 0$, and $\equiv 1$ on $\mathcal{O}^{\prime}$.
Then the asymptotics of (6) are unchanged, it the integrand is multiplied by
$\varrho_{\boldsymbol{\nu}}(\boldsymbol{\lambda})$.

###### Proof of Lemma 6.4.

Since the adjoint action is unitary,
$\mathcal{O}_{\boldsymbol{\nu}}\cap\mathbb{R}_{+}\,\boldsymbol{\nu}=\\{\boldsymbol{\nu}\\}$.
Hence, by (66) there exists $a_{0}>0$ such that

$\|\partial_{\boldsymbol{\xi}}\Upsilon_{x}\|=\|u\,\Phi(m_{x})-\boldsymbol{\lambda}\|^{\varphi}\geq
a_{0},$

for all $u>0$ and
$\boldsymbol{\lambda}\in\mathrm{supp}(1-\varrho_{\boldsymbol{\nu}})$. The
claim follows integrating by parts in $\boldsymbol{\xi}$, which is legitimate
in view of the cut-off and the exponential factor.

∎

In the following, we shall redefine $\mathcal{C}_{x,k}$ implicitly
incorporating the factor $\varrho_{\boldsymbol{\nu}}(\boldsymbol{\lambda})$,
so that integration in
$\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}(\boldsymbol{\lambda})$ is over
$\mathcal{O}^{\prime\prime}\subset\mathcal{O}_{\boldsymbol{\nu}}$.

We have an equivariant diffeomorphism

$\beta:g\,T\in G/T\mapsto
g\cdot\boldsymbol{\nu}:=\mathrm{Coad}_{g}(\boldsymbol{\nu})\in\mathcal{O}_{\boldsymbol{\nu}}.$
(71)

Let $\mathrm{d}^{H}V_{G/T}$ be the Haar measure on $G/T$; then

$\beta^{*}\big{(}\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}}\big{)}=\mathrm{vol}(\mathcal{O}_{\boldsymbol{\nu}})\,\mathrm{d}^{H}V_{G/T}.$
(72)

Furthermore, in view of the factor $\beta^{*}(\varrho_{\boldsymbol{\nu}})$
which is left implicit, integration over
$\mathcal{O}^{\prime\prime}\subset\mathcal{O}_{\boldsymbol{\nu}}$ in (6) gets
replaced by integration over a small neighborhood of $e_{G}\,T\in G/T$,
according to (72). Let us introduce local coordinates on $G/T$ near $e_{G}\,T$
by composing the projection $\pi_{G/T}:G\rightarrow G/T$ with the restriction
of the exponential map of $G$ to the Euclidean orthocomplement
$\mathfrak{t}^{\perp_{\varphi}}\subset\mathfrak{g}$ of $\mathfrak{t}$ w.r.t.
$\varphi$:

$E:\boldsymbol{\gamma}\in\mathfrak{t}^{\perp_{\varphi}}\mapsto
e^{\boldsymbol{\gamma}}\,T\in G/T.$

When restricted to a small open neighborhood of the origin, $E$ is a
diffeomorphism onto its image, hence a local chart for $G/T$ centered at
$e_{G}\,T$; we have an isomorphism $\mathfrak{t}^{\perp_{\varphi}}\cong
T_{e_{G}\,T}(G/T)$. The Lebesgue measure on $\mathfrak{t}^{\perp_{\varphi}}$
associated to the restriction $\varphi^{\prime}$ of $\varphi$ will be denoted
$\mathrm{d}^{\varphi}\boldsymbol{\gamma}$. Being $T$-invariant,
$\varphi^{\prime}$ determines an equivariant Riemannian metric on $G/T$, whose
associated Riemannian density and volume will be denoted
$\mathrm{d}^{\varphi}V_{G/T}$ and $\mathrm{vol}^{\varphi}(G/T)$, respectively.
Then

$E^{*}(\mathrm{d}^{\varphi}V_{G/T})=\mathcal{R}(\boldsymbol{\gamma})\,\mathrm{d}^{\varphi}\boldsymbol{\gamma},\quad\mathcal{R}(\boldsymbol{\gamma})=1+R_{1}(\boldsymbol{\gamma}).$
(73)

Viewing $G$ as a principal $T$-bundle over $G/T$, by fiber integration one
obtains
$\mathrm{vol}^{\varphi}(G/T)=\mathrm{vol}^{\varphi}(G)/\mathrm{vol}^{\varphi}(T)$.
Clearly,

$\mathrm{d}^{\varphi}V_{G/T}=\mathrm{vol}^{\varphi}(G/T)\,\mathrm{d}^{H}V_{G/T}.$
(74)

Hence, if we view $\beta\circ E$, restricted to a small neighborhood of the
origin in $\mathfrak{t}^{\perp_{\varphi}}$ as a local coordinate chart on
$\mathcal{O}_{\boldsymbol{\nu}}$, we obtain by (73) and (74):

$(\beta\circ
E)^{*}(\mathrm{d}V_{\mathcal{O}_{\boldsymbol{\nu}}})=\mathrm{vol}(\mathcal{O}_{\boldsymbol{\nu}})\,E^{*}(\mathrm{d}^{H}V_{G/H})=\mathrm{vol}(\mathcal{O}_{\boldsymbol{\nu}})\,\frac{\mathrm{vol}^{\varphi}(T)}{\mathrm{vol}^{\varphi}(G)}\,\mathcal{R}(\boldsymbol{\gamma})\,\mathrm{d}^{\varphi}\boldsymbol{\gamma}.$
(75)

With these substitutions, recalling (71) we may rewrite (6) in the following
manner:

$\displaystyle\Pi_{k\,\boldsymbol{\nu}}(x_{1,k},x_{2,k})$ $\displaystyle\sim$
$\displaystyle\mathrm{vol}\big{(}\mathcal{O}_{\boldsymbol{\nu}}\big{)}^{2}\cdot\frac{\mathrm{vol}^{\varphi}(T)}{\mathrm{vol}^{\varphi}(G)^{2}}\cdot
k^{1-d_{G}/2}\,\left(\frac{k}{2\,\pi}\right)^{2\,n_{G}}\,$
$\displaystyle\int_{1/D}^{D}\,\mathrm{d}u\,\int_{\mathfrak{g}}\,\mathrm{d}^{\varphi}\boldsymbol{\xi}\,\int_{\mathfrak{t}^{\perp_{\varphi}}}\,\mathrm{d}^{\varphi}\,\boldsymbol{\gamma}\,\left[e^{\imath\,\sqrt{k}\,\Upsilon_{x}(u,\boldsymbol{\xi},e^{\boldsymbol{\lambda}}\cdot\boldsymbol{\nu})}\,\mathcal{C}_{x,k}\left(u,\boldsymbol{\xi},e^{\boldsymbol{\gamma}}\cdot\boldsymbol{\nu};\mathbf{v}_{j},\mathbf{w}_{j}\right)\right].$

Let us set $g\cdot\boldsymbol{\nu}:=\mathrm{Ad}_{g}(\boldsymbol{\xi})$, for
$g\in G$ and $\boldsymbol{\xi}\in\mathfrak{g}$. We have:

$e^{\boldsymbol{\gamma}}\cdot\boldsymbol{\nu}^{\varphi}=\boldsymbol{\nu}^{\varphi}+\left[\boldsymbol{\gamma},\boldsymbol{\nu}^{\varphi}\right]+R_{2}(\boldsymbol{\gamma})=\boldsymbol{\nu}^{\varphi}-\left[\boldsymbol{\nu}^{\varphi},\boldsymbol{\gamma}\right]+R_{2}(\boldsymbol{\gamma}).$

Since $\boldsymbol{\lambda}\mapsto\boldsymbol{\lambda}^{\varphi}$ intertwines
the coadjoint and adjoint actions, (66) may be rewritten as follows:

$\displaystyle\Upsilon_{x}(u,\boldsymbol{\xi},e^{\boldsymbol{\gamma}}\cdot\boldsymbol{\nu})$
$\displaystyle=$
$\displaystyle\big{\langle}u\,\Phi(m_{x})-e^{\boldsymbol{\gamma}}\cdot\boldsymbol{\nu},\boldsymbol{\xi}\big{\rangle}$
(77) $\displaystyle=$
$\displaystyle\varphi\left(u\,\Phi(m_{x})^{\varphi}-e^{\boldsymbol{\gamma}}\cdot\boldsymbol{\nu}^{\varphi},\boldsymbol{\xi}\right)$
$\displaystyle=$
$\displaystyle\varphi\left(u\,\Phi(m_{x})^{\varphi}-\boldsymbol{\nu}^{\varphi}+\left[\boldsymbol{\nu}^{\varphi},\boldsymbol{\gamma}\right]+R_{2}(\boldsymbol{\gamma}),\boldsymbol{\xi}\right)$
$\displaystyle=$
$\displaystyle\varphi\left(\big{(}u\,\varsigma(m_{x})-1\big{)}\,\boldsymbol{\nu}^{\varphi}+\left[\boldsymbol{\nu}^{\varphi},\boldsymbol{\gamma}\right]+R_{2}(\boldsymbol{\gamma}),\boldsymbol{\xi}\right);$

here $R_{2}(\boldsymbol{\gamma})$ is real-valued. In terms of the
$\varphi$-orthogonal direct sum decompositions

$\mathfrak{t}:=\mathrm{span}\left(\boldsymbol{\nu}^{\varphi}\right)\cap\mathfrak{t}_{\boldsymbol{\nu}},\quad\mathfrak{g}=\mathrm{span}\left(\boldsymbol{\nu}^{\varphi}\right)\oplus\mathfrak{t}_{\boldsymbol{\nu}}\oplus\mathfrak{t}^{\perp_{\varphi}},$

we shall write the general element of $\mathfrak{g}$ as

$\boldsymbol{\xi}=s\,\boldsymbol{\nu}^{\varphi}_{u}+\boldsymbol{\xi}^{\prime}+\boldsymbol{\xi}^{\prime\prime},\quad\text{where}\quad
s\in\mathbb{R},\,\boldsymbol{\xi}^{\prime}\in\mathfrak{t}_{\boldsymbol{\nu}},\,\boldsymbol{\xi}^{\prime\prime}\in\mathfrak{t}^{\perp_{\varphi}}.$

Furthermore, we may introduce orthnormal basis of
$\mathfrak{t}_{\boldsymbol{\nu}}$ and $\mathfrak{t}^{\perp_{\varphi}}$ w.r.t.
$\varphi$, so as to unitarily identify
$\mathfrak{t}_{\boldsymbol{\nu}}\cong\mathbb{R}^{r_{G}-1}$ and
$\mathfrak{t}^{\perp_{\varphi}}\cong\mathbb{R}^{2\,n_{\boldsymbol{\nu}}}$. Let
$Z_{\boldsymbol{\nu}^{\varphi}}$ the skew-symmetric and non-degenerate matrix
representing $S_{\boldsymbol{\nu}^{\varphi}}$ w.r.t. the given orthonormal
basis of $\mathfrak{t}^{\perp_{\varphi}}$. Then (77) may be rewritten:

$\displaystyle\Upsilon_{x,\boldsymbol{\xi^{\prime}}}(u,s,\boldsymbol{\xi}^{\prime\prime},\boldsymbol{\gamma}):=\Upsilon_{x}(u,\boldsymbol{\xi},e^{\boldsymbol{\gamma}}\cdot\boldsymbol{\nu})$
$\displaystyle=$
$\displaystyle\varphi\left(\big{(}u\,\varsigma(m_{x})-1\big{)}\,\boldsymbol{\nu}^{\varphi}+\left[\boldsymbol{\nu}^{\varphi},\boldsymbol{\gamma}\right]+R_{2}(\boldsymbol{\gamma}),s\,\boldsymbol{\nu}^{\varphi}_{u}+\boldsymbol{\xi}^{\prime}+\boldsymbol{\xi}^{\prime\prime}\right)$
$\displaystyle=$ $\displaystyle
s\,\big{(}u\,\varsigma(m_{x})-1\big{)}\,\left\|\boldsymbol{\nu}^{\varphi}\right\|_{\varphi}-\varphi\left(\boldsymbol{\gamma},\left[\boldsymbol{\nu}^{\varphi},\boldsymbol{\xi}^{\prime\prime}\right]\right)+\varphi\left(R_{2}(\boldsymbol{\gamma}),s\,\boldsymbol{\nu}^{\varphi}_{u}+\boldsymbol{\xi}^{\prime}+\boldsymbol{\xi}^{\prime\prime}\right)$
$\displaystyle=$ $\displaystyle
s\,\big{(}u\,\varsigma(m_{x})-1\big{)}\,\left\|\boldsymbol{\nu}^{\varphi}\right\|_{\varphi}-\boldsymbol{\gamma}^{t}\,Z_{\boldsymbol{\nu}^{\varphi}}\,\boldsymbol{\xi}^{\prime\prime}+R_{2}(\boldsymbol{\gamma})^{t}\,\left(s\,\boldsymbol{\nu}^{\varphi}_{u}+\boldsymbol{\xi}^{\prime}+\boldsymbol{\xi}^{\prime\prime}\right).$

We write (6) in the form

$\displaystyle\Pi_{k\,\boldsymbol{\nu}}(x_{1,k},x_{2,k})$ $\displaystyle\sim$
$\displaystyle\mathrm{vol}\big{(}\mathcal{O}_{\boldsymbol{\nu}}\big{)}^{2}\cdot\frac{\mathrm{vol}^{\varphi}(T)}{\mathrm{vol}^{\varphi}(G)^{2}}\cdot
k^{1-d_{G}/2}\,\left(\frac{k}{2\,\pi}\right)^{2\,n_{G}}\,$
$\displaystyle\cdot\int_{\mathfrak{t}_{\boldsymbol{\nu}}}\,\mathrm{d}^{\varphi}\boldsymbol{\xi}^{\prime}\,\big{[}\mathcal{I}_{x,k}\left(\boldsymbol{\xi}^{\prime};\mathbf{v}_{j},\mathbf{w}_{j}\right)\big{]},$

where

$\displaystyle\mathcal{I}_{x,k}\left(\boldsymbol{\xi}^{\prime};\mathbf{v}_{j},\mathbf{w}_{j}\right)$
$\displaystyle:=$
$\displaystyle\int_{1/D}^{D}\,\mathrm{d}u\,\int_{-\infty}^{+\infty}\,\mathrm{d}s\,\int_{\mathfrak{t}^{\perp_{\varphi}}}\,\mathrm{d}^{\varphi}\boldsymbol{\xi}^{\prime\prime}\,\int_{\mathfrak{t}^{\perp_{\varphi}}}\,\mathrm{d}^{\varphi}\,\boldsymbol{\gamma}$
(80)
$\displaystyle\left[e^{\imath\,\sqrt{k}\,\Upsilon_{x,\boldsymbol{\xi^{\prime}}}(u,s,\boldsymbol{\xi}^{\prime\prime},\boldsymbol{\gamma})}\,\mathcal{C}_{x,k}\left(u,\boldsymbol{\xi},e^{\boldsymbol{\gamma}}\cdot\boldsymbol{\nu};\mathbf{v}_{j},\mathbf{w}_{j}\right)\right].$

We view $\mathcal{I}_{x,k}\left(\boldsymbol{\xi}^{\prime}\right)$ as an
oscillatory integral depending on the parameter $\boldsymbol{\xi}^{\prime}$,
with real phase $\Upsilon_{x,\boldsymbol{\xi^{\prime}}}$. Using that
$Z_{\boldsymbol{\nu}^{\varphi}}$ is non-degenerate, and that
$\boldsymbol{\gamma}$ is small in norm, one obtains the following.

######  Lemma 6.5.

For any $\boldsymbol{\xi}^{\prime}\in\mathfrak{t}_{\boldsymbol{\nu}}$,
$\Upsilon_{x,\boldsymbol{\xi^{\prime}}}$ has a unique critical point, given by

$P_{0}=(u_{0},s_{0},\boldsymbol{\xi}^{\prime\prime}_{0},\boldsymbol{\gamma}_{0})=\left(\frac{1}{\varsigma(m_{x})},0,\mathbf{0},\mathbf{0}\right).$

Hence $\Upsilon_{x,\boldsymbol{\xi^{\prime}}}(P_{0})=0$. The Hessian matrix at
the critical point is

$H_{P_{0}}(\Upsilon_{x,\boldsymbol{\xi^{\prime}}})=\begin{pmatrix}0&\varsigma(m_{x})\,\left\|\boldsymbol{\nu}^{\varphi}\right\|_{\varphi}&\mathbf{0}^{t}&\mathbf{0}^{t}\\\
\varsigma(m_{x})\,\left\|\boldsymbol{\nu}^{\varphi}\right\|_{\varphi}&0&\mathbf{0}^{t}&\mathbf{0}^{t}\\\
\mathbf{0}&\mathbf{0}&[0]&Z_{\boldsymbol{\nu}^{\varphi}}\\\
\mathbf{0}&\mathbf{0}&-Z_{\boldsymbol{\nu}^{\varphi}}&\left.\partial^{2}_{\boldsymbol{\gamma},\boldsymbol{\gamma}}\Upsilon_{x,\boldsymbol{\xi^{\prime}}}\right|_{P_{0}}\end{pmatrix},$

where $[0]$ denotes the zero matrix of order $(2\,n_{G})\times(2\,n_{G})$.
Hence, its determinant and signature are

$\det\big{(}H_{P_{0}}(\Upsilon_{x,\boldsymbol{\xi^{\prime}}})\big{)}=-\varsigma(m_{x})^{2}\,\left\|\boldsymbol{\nu}^{\varphi}\right\|_{\varphi}^{2}\,\det(Z_{\boldsymbol{\nu}^{\varphi}})^{2},\quad\mathrm{sign}\big{(}H_{P_{0}}(\Upsilon_{x,\boldsymbol{\xi^{\prime}}})\big{)}=0.$

In particular, $P_{0}$ is a non-degenerate critical point.

Integrating by parts in $\mathrm{d}\boldsymbol{\gamma}$ shows that the
asymptotics of (80) are unchanged, if the integrand is multiplied by a cut-off
function in $\boldsymbol{\xi}^{\prime\prime}$, compactly supported and
identically equal to $1$ near the origin.

We can apply the stationary phase Lemma. Recalling (67)-(70), we obtain for
(80) an asymptotic expansion of the form

$\displaystyle\mathcal{I}_{x,k}\left(\boldsymbol{\xi}^{\prime};\mathbf{v}_{j},\mathbf{w}_{j}\right)$
$\displaystyle\sim$
$\displaystyle\gamma\left(k^{-\epsilon}\,\boldsymbol{\xi}^{\prime}\right)\cdot\left(\frac{2\,\pi}{k^{1/2}}\right)^{1+d_{G}-r_{G}}\cdot\frac{e^{\frac{1}{\varsigma(m_{x})}\,\,\left[\psi_{2}(\mathbf{w}_{1},\mathbf{w}_{2})-\frac{1}{2}\,\|\mathbf{v}_{1}-\mathbf{v}_{2}\|_{m_{x}}^{2}\right]}}{\varsigma(m_{x})\,\left\|\boldsymbol{\nu}^{\varphi}\right\|_{\varphi}\,\det(Z_{\boldsymbol{\nu}^{\varphi}})}$
(81)
$\displaystyle\cdot\frac{k^{d}}{\varsigma(m_{x})^{d}\,\pi^{d}}\,e^{\frac{1}{\varsigma(m_{x})}\,\left[\imath\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}^{\prime}(m),\mathbf{v}_{1}+\mathbf{v}_{2}\big{)}-\frac{1}{2}\,\|\boldsymbol{\xi}^{\prime}_{M}(m_{x})\|_{m_{x}}^{2}\right]}$
$\displaystyle\cdot\left[1+\sum_{j\geq
1}k^{-j/2}\,P^{\prime}_{j}(m_{x};\boldsymbol{\xi}^{\prime},\mathbf{v}_{j},\mathbf{w}_{j})\right],$

where again $P_{j}(m_{x};\cdot,\cdot,\cdot)$ is a polynomial of degree $\leq
3\,j$ and parity $j$. We have replaced
$\|\boldsymbol{\xi}^{\prime}_{X}(x)\|_{x}$ by
$\|\boldsymbol{\xi}^{\prime}_{M}(m_{x})\|_{m_{x}}$ in view of the fact that
$\langle\Phi_{M}(m_{x}),\boldsymbol{\xi}^{\prime}\rangle=0$ since
$\boldsymbol{\xi}^{\prime}\in\mathfrak{t}^{\perp_{\varphi}}$, so that
$\boldsymbol{\xi}_{X}^{\prime}(x)=\boldsymbol{\xi}^{\prime}_{M}(m_{x})^{\sharp}$.

The final expansion is obtained by inserting (81) in (6) and integrating term
by term. The front cut-off, in view of the Gaussian type exponential, may be
omitted without affecting the asymptotics. The $j$-th summand in (81), $j\geq
0$, contributes by a factor given by the Gaussian type integral

$k^{-j/2}\,\int_{\mathfrak{t}_{\boldsymbol{\nu}}}\,\mathrm{d}^{\varphi}\boldsymbol{\xi}^{\prime}\,\left[e^{\frac{1}{\varsigma(m_{x})}\,\left[\imath\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}^{\prime}(m_{x}),\mathbf{v}_{1}+\mathbf{v}_{2}\big{)}-\frac{1}{2}\,\big{\|}\boldsymbol{\xi}^{\prime}_{M}(m_{x})\big{\|}_{m_{x}}^{2}\right]}\,P^{\prime}_{j}(m_{x};\boldsymbol{\xi}^{\prime},\mathbf{v}_{j},\mathbf{w}_{j})\right],$
(82)

where we set $P_{0}=1$.

We compute the leading order term. Recall that we have fixed an orthonormal
basis of
$\mathfrak{t}_{m}^{\prime}=\mathfrak{t}_{\boldsymbol{\nu}}\cong\mathbb{R}^{r_{G}-1}$
(47). Let $D^{\varphi}(m_{x})$ be as in Definition 1.2, and let
$P^{\varphi}(m_{x})$ denote its positive definite square root. Furthermore, by
Lemma 6.1 there exist unique
$\boldsymbol{\upsilon}_{j}\in\mathfrak{t}_{\boldsymbol{\nu}}$ such that
$\mathbf{v}_{j}=J_{m_{x}}\big{(}\boldsymbol{{\upsilon}_{j}}_{M}(m_{x})\big{)}$.
Let $\langle\cdot,\cdot\rangle_{st}$ denote the standard scalar product on
$\mathbb{R}^{r_{G}-1}$, then

$\displaystyle\varsigma(m_{x})^{-1}\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}^{\prime}(m),\mathbf{v}_{1}+\mathbf{v}_{2}\big{)}$
$\displaystyle=$
$\displaystyle\varsigma(m_{x})^{-1}\,\omega_{m_{x}}\Big{(}\boldsymbol{\xi}_{M}^{\prime}(m_{x}),J_{m_{x}}\big{(}\boldsymbol{{\upsilon}_{1}}_{M}(m_{x})\big{)}+J_{m_{x}}\big{(}\boldsymbol{{\upsilon}_{2}}_{M}(m_{x})\big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\varsigma(m_{x})^{-1}\,\rho^{M}_{m_{x}}\Big{(}\boldsymbol{\xi}_{M}^{\prime}(m_{x}),{\boldsymbol{\upsilon}_{1}}_{M}(m_{x})+{\boldsymbol{\upsilon}_{2}}_{M}(m_{x})\Big{)}$
$\displaystyle=$
$\displaystyle\varsigma(m_{x})^{-1}\,{\boldsymbol{\xi}^{\prime}}^{T}\,D^{\varphi}(m_{x})\,(\boldsymbol{\upsilon}_{1}+\boldsymbol{\upsilon}_{2})$
$\displaystyle=$
$\displaystyle\left\langle\varsigma(m_{x})^{-1/2}\,P^{\varphi}(m_{x})\,\boldsymbol{\xi}^{\prime},\varsigma(m_{x})^{-1/2}\,P^{\varphi}(m_{x})\,(\boldsymbol{\upsilon}_{1}+\boldsymbol{\upsilon}_{2})\right\rangle_{st}.$

Similarly, if $\|\cdot\|$ is the standard Euclidean norm then

$\displaystyle\varsigma(m_{x})^{-1}\,\left\|\boldsymbol{\xi}^{\prime}_{M}(m_{x})\right\|_{m_{x}}^{2}=\left\|\varsigma(m_{x})^{-1/2}\,P^{\varphi}(m_{x})\,\boldsymbol{\xi}^{\prime}\right\|^{2}.$
(84)

Hence, setting
$\boldsymbol{\eta}=\varsigma(m_{x})^{-1/2}\,P^{\varphi}(m_{x})\,\boldsymbol{\xi}^{\prime}$,
we obtain

$\displaystyle\int_{\mathfrak{t}_{\boldsymbol{\nu}}}\,\mathrm{d}^{\varphi}\boldsymbol{\xi}^{\prime}\,\left[e^{\frac{1}{\varsigma(m_{x})}\,\left[\imath\,\omega_{m_{x}}\big{(}\boldsymbol{\xi}_{M}^{\prime}(m_{x}),\mathbf{v}_{1}+\mathbf{v}_{2}\big{)}-\frac{1}{2}\,\big{\|}\boldsymbol{\xi}^{\prime}_{M}(m_{x})\big{\|}_{m_{x}}^{2}\right]}\,\right]$
(85) $\displaystyle=$
$\displaystyle\frac{\varsigma(m_{x})^{\frac{r_{G}-1}{2}}}{\det\left(P^{\varphi}(m_{x})\right)}\,\int_{\mathbb{R}^{r_{G}-1}}\,\mathrm{d}\boldsymbol{\eta}\,\left[e^{\imath\,\left\langle\boldsymbol{\eta},\varsigma(m_{x})^{-1/2}\,P^{\varphi}(m_{x})\,(\boldsymbol{\upsilon}_{1}+\boldsymbol{\upsilon}_{2})\right\rangle_{st}-\frac{1}{2}\,\|\boldsymbol{\eta}\|^{2}}\,\right]$
$\displaystyle=$
$\displaystyle\frac{(2\,\pi)^{\frac{r_{G}-1}{2}}\,\varsigma(m_{x})^{\frac{r_{G}-1}{2}}}{\mathcal{D}^{\varphi}(m)}\,e^{-\frac{1}{2\,\varsigma(m_{x})}\,\|\mathbf{v}_{1}+\mathbf{v}_{2}\|_{m}^{2}}.$

Plugging (85) into (81) and then in (6) we obtain the leading order term in
the statement of the Theorem. The other terms can be handled similarly.

∎

## References

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