Quasi-Polynomials and the Singular $[Q,R]=0$ Theorem

Quasi-Polynomials and the Singular $\bm{[Q,R]=0}$
Theorem

Yiannis LOIZIDES

Y. Loizides

Pennsylvania State University, USA<EMAIL_ADDRESS>
Received July 16, 2019, in final form November 13, 2019; Published online
November 18, 2019

In this short note we revisit the ‘shift-desingularization’ version of the
$[Q,R]=0$ theorem for possibly singular symplectic quotients. We take as
starting point an elegant proof due to Szenes–Vergne of the quasi-polynomial
behavior of the multiplicity as a function of the tensor power of the
prequantum line bundle. We use the Berline–Vergne index formula and the
stationary phase expansion to compute the quasi-polynomial, adapting an early
approach of Meinrenken.

symplectic geometry; Hamiltonian $G$-spaces; symplectic reduction; geometric
quantization; quasi-polynomials; stationary phase

53D20; 53D50

##  1 Introduction

Let $(M,\omega)$ be a compact connected symplectic manifold equipped with an
action of a compact connected Lie group $G$ by symplectomorphisms. Suppose
that the action of $G$ is Hamiltonian, meaning that there is a $G$-equivariant
map, the moment map,

$\mu_{\mathfrak{g}}\colon\ M\rightarrow\mathfrak{g}^{\ast},$

where $\mathfrak{g}^{\ast}$ is the dual of the Lie algebra
$\mathfrak{g}=\textnormal{Lie}(G)$, satisfying the moment map condition

$\displaystyle\iota(X_{M})\omega=-{\rm
d}\langle\mu_{\mathfrak{g}},X\rangle,\qquad X\in\mathfrak{g}.$ (1.1)

Let $\big{(}L,\nabla^{L}\big{)}$ be a $G$-equivariant prequantum line bundle
with connection on $M$, i.e., $L$ is a $G$-equivariant Hermitian line bundle
with compatible connection $\nabla^{L}$,
$\big{(}\nabla^{L}\big{)}^{2}=-2\pi{\mathrm{i}}\omega$ and the derivative of
the $G$-action on $L$ satisfies Kostant’s condition

$\mathcal{L}^{L}_{X}-\nabla^{L}_{X_{M}}=2\pi{\mathrm{i}}\langle\mu_{\mathfrak{g}},X\rangle.$

Choose a compatible almost complex structure $J$ on $M$, i.e.,
$\omega(Jw,Jv)=\omega(w,v)$ and $\omega(w,Jv)\allowbreak=:g(w,v)$ is a
Riemannian metric. Let $D_{L}$ denote the Dolbeault–Dirac operator twisted by
$\big{(}L,\nabla^{L}\big{)}$, an elliptic differential operator acting on
sections of the spinor bundle $\wedge T^{\ast}_{0,1}M\otimes L$. The kernel of
$D_{L}$ carries an action of $G$, and the $G$-equivariant index is defined to
be the difference
$\textnormal{index}_{G}(D_{L}):=\textnormal{ker}\big{(}D_{L}^{\textnormal{even}}\big{)}-\textnormal{ker}\big{(}D_{L}^{\textnormal{odd}}\big{)}$
of the kernel of $D_{L}$ on even/odd degree forms, regarded as an element of
the representation ring $R(G)$.

The quantization-commutes-with-reduction theorem ($[Q,R]=0$ theorem) describes
the multiplicity of the trivial representation in
$\textnormal{index}_{G}(D_{L})$ in terms of the symplectic quotient
$M^{\textnormal{red}}:=\mu_{\mathfrak{g}}^{-1}(0)/G$. When $0$ is a regular
value of $\mu_{\mathfrak{g}}$, $M^{\textnormal{red}}$ is an orbifold and the
theorem states that $\textnormal{index}_{G}(D)^{G}$ equals the index of the
twisted Dolbeault–Dirac operator $D^{\textnormal{red}}_{L^{\textnormal{red}}}$
on $M^{\textnormal{red}}$. The theorem was first conjectured by
Guillemin–Sternberg [3], and the general case ($M$, $G$ both compact, $0$ a
regular value) was first proved by Meinrenken [8]. Different proofs of the
$[Q,R]=0$ theorem were given by Tian–Zhang [15] and Paradan [11]. The theorem
has since been extended in various directions.

There are versions of the $[Q,R]=0$ theorem when $0$ is not necessarily a
regular value, due to Meinrenken–Sjamaar [10]; below we will give a precise
statement of one of these results, involving a partial _shift
desingularization_ , i.e., $\textnormal{index}_{G}(D_{L})^{G}$ is related to
the index on the symplectic quotient at a nearby weakly regular value. At the
same time, we introduce some notation that will be of use later on.

Fix a maximal torus $T$ with Lie algebra $\mathfrak{t}$. Let
$\Lambda\subset\mathfrak{t}^{\ast}$ be the (real) weight lattice. Given
$\lambda\in\Lambda$, the corresponding character $T\rightarrow U(1)$ is
written $t\mapsto t^{\lambda}={\rm
e}^{2\pi{\mathrm{i}}\langle\lambda,X\rangle}$ where $t={\rm e}^{X}$,
$X\in\mathfrak{t}$. Let $\mathcal{R}\subset\Lambda$ be the set of roots. We
also fix a closed positive Weyl chamber $\mathfrak{t}_{+}$, which determines a
set of positive (resp. negative) roots $\mathcal{R}_{\pm}$. For each
relatively open face $\sigma\subset\mathfrak{t}_{+}^{\ast}$, the stabilizer
$G_{\xi}$ of points $\xi\in\sigma$ under the coadjoint action, does not depend
on $\xi$, and will be denoted $G_{\sigma}$. If
$\sigma_{1}\subset\overline{\sigma}_{2}$ then $G_{\sigma_{1}}\supset
G_{\sigma_{2}}$. Note also that $G_{\sigma}$ is connected and contains the
maximal torus $T$. The Lie algebra $\mathfrak{g}_{\sigma}$ decomposes into its
semi-simple and central parts
$\mathfrak{g}_{\sigma}=[\mathfrak{g}_{\sigma},\mathfrak{g}_{\sigma}]\oplus\mathfrak{z}_{\sigma}$.
The subspace $\mathfrak{z}_{\sigma}^{\ast}\subset\mathfrak{t}^{\ast}$ is
defined to be the annihilator of
$[\mathfrak{g}_{\sigma},\mathfrak{g}_{\sigma}]$, or equivalently the fixed
point set of the coadjoint $G_{\sigma}$ action. The face $\sigma$ is an open
subset of $\mathfrak{z}_{\sigma}^{\ast}$.

Let $\Delta=\mu_{\mathfrak{g}}(M)\cap\mathfrak{t}_{+}^{\ast}$ be the moment
polytope. A well-known theorem in symplectic geometry states that there is a
unique face $\sigma\subset\mathfrak{t}^{\ast}_{+}$ of minimal dimension such
that $\Delta\subset\overline{\sigma}$ (briefly, this is a consequence of
(1.1), which implies that ${\rm d}\mu_{\mathfrak{g}}$ has constant rank on the
top dimensional $G$-orbit type stratum, and the complement of the latter has
codimension at least $2$); $\sigma$ is called the _principal face_ or
_principal wall_. The corresponding symplectic cross-section, called the
_principal cross-section_ , $Y=\mu_{\mathfrak{g}}^{-1}(\sigma)$ is a
Hamiltonian $G_{\sigma}$-space. Moreover the semi-simple part
$[G_{\sigma},G_{\sigma}]$ of $G_{\sigma}$ acts trivially on $Y$. For further
details, see for example [5] and references therein.

Let $I\subset\mathfrak{z}_{\sigma}^{\ast}$ be the smallest affine subspace
containing $\Delta$. Let $\mathfrak{t}_{I}\subset\mathfrak{t}$ be the
annihilator of the subspace parallel to $I$, and let
$T_{I}=\exp(\mathfrak{t}_{I})\subset T$ be the corresponding subtorus. By
equation (1.1), $\mathfrak{t}_{I}$ is the generic infinitesimal stabilizer of
$Y$. In particular $T_{I}$ acts trivially, hence the quotient torus $T/T_{I}$
acts on $Y$. The moment map $\mu_{\mathfrak{g}}$ may have no non-trivial
regular values. But the restriction

$\mu_{\mathfrak{g}}|_{Y}\colon\ Y\rightarrow I$

viewed as a map with codomain $I$, always has non-trivial regular values, and
we will refer to these as _weakly-regular values_. If $\xi$ is a weakly-
regular value, then the reduced space
$M_{\xi}=\mu_{\mathfrak{g}}^{-1}(\xi)/G_{\sigma}$ is an orbifold. Let
$L_{\xi}=L|_{\mu_{\mathfrak{g}}^{-1}(\xi)}/G_{\sigma}$ be the corresponding
(orbifold) line bundle over $M_{\xi}$.

######  Theorem 1.1 ([10], see also [11, 13]).

Let $(M,\omega,\mu_{\mathfrak{g}})$ be a compact connected Hamiltonian
$G$-space with moment polytope $\Delta$. If $0\notin\Delta$ then
$\textnormal{index}_{G}(D_{L})^{G}=0$. Otherwise for every weakly-regular
value $\xi\in\Delta$ sufficiently close to $0$,
$\textnormal{index}_{G}(D_{L})^{G}$ equals the index of the Dolbeault–Dirac
operator $D^{\textnormal{red}}_{L_{\xi}}$ on the reduced space $M_{\xi}$.

We will now describe the main result of this article and its relation to
Theorem 1.1. Consider tensor powers $L^{k}$, $k\in\mathbb{Z}_{>0}$ of the
prequantum line bundle. For a dominant weight $\lambda$, let
$\chi_{\lambda}\in R(G)$ denote the character of the irreducible
representation of $G$ with highest weight $\lambda$. We define the
_multiplicity function_ $m_{G}(k,\lambda)$ by the expression

$\displaystyle\textnormal{index}_{G}(D_{L^{k}})=\sum_{\lambda\in\Lambda\cap\mathfrak{t}_{+}^{\ast}}m_{G}(k,\lambda)\chi_{\lambda}.$
(1.2)

An important theme in the work of Szenes–Vergne [14] and also in our approach,
is that the function $m_{G}(k,\lambda)$ has more coherent behavior than its
restriction to any fixed value of $k$.

The statement of the result requires some further background on orbifolds, for
which we refer the reader to, for example, [2, Appendix A], [8, Section 2]. A
small warning is that we will not require the action of isotropy groups in
orbifold charts to be effective (this is in agreement with the references [2,
8] mentioned above). One advantage of permitting this, is that for a locally
free action of a compact Lie group $K$ on a manifold $P$, the corresponding
orbifold $P/K$ has orbifold charts given automatically by the slice theorem,
with the isotropy groups being simply the isotropy groups for the action of
$K$ on $P$.

In fact all the orbifolds that we will encounter arise naturally as such
quotients $P/K$, and one could avoid mentioning orbifolds altogether by
working instead with suitable $K$-basic structures on $P$. An example is the
description of characteristic forms for orbifold vector bundles, which can be
defined in terms of orbifold charts for $P/K$, or alternatively in terms of
$K$-basic differential forms on $P$. In brief, the latter approach goes as
follows. One can take the complex $(\Omega_{\textnormal{bas}}(P),{\rm d})$ of
$K$-basic differential forms on $P$ as a working definition of the de Rham
complex of $P/K$ (if $K$ acts freely then $P/K$ is a manifold and pullback of
forms from $P/K$ to $P$ is an isomorphism of complexes $(\Omega(P/K),{\rm
d})\simeq(\Omega_{\textnormal{bas}}(P),{\rm d})$). A $K$-equivariant vector
bundle $E\rightarrow P$ determines an orbifold vector bundle $E/K$ over $P/K$.
Let $\theta$ be a connection on $P$ with curvature $F_{\theta}$. The choice of
connection determines a _Cartan map_ (cf. [9]) from closed $K$-equivariant
forms $\alpha(X)$ on $P$ to closed $K$-basic forms:
$\alpha(X)\mapsto\textnormal{Car}_{\theta}(\alpha):=\Pi_{\textnormal{hor}}\alpha(F_{\theta})$,
where $\Pi_{\textnormal{hor}}$ is the projection onto the horizontal part
relative to the connection. The Cartan map induces an isomorphism from the
$K$-equivariant cohomology of $P$ to the cohomology of the complex of basic
differential forms on $P$. If $\alpha(X)$ is a $K$-equivariant characteristic
form (constructed via the $K$-equivariant analogue of the usual Chern–Weil
construction cf. [1, 9]), then one may take
$\textnormal{Car}_{\theta}(\alpha)\in\Omega_{\textnormal{bas}}(P)$ as the
definition of the corresponding characteristic form for $E/K$.

Let $\xi\in\Delta$ be a weakly-regular value. By the moment map equation
(1.1), the action of $K=T/T_{I}$ on the level set

$P=\mu_{\mathfrak{g}}^{-1}(\xi)$

is locally free. The set $S_{P}$ of elements $g\in T/T_{I}$ such that
$P^{g}\neq\varnothing$ is finite. For each $g\in S_{P}$, we obtain an orbifold

$\Sigma_{g}=P^{g}/(T/T_{I}),\qquad\Sigma=\bigsqcup_{g\in S_{P}}\Sigma_{g}.$

Note that $\Sigma_{1}=P/(T/T_{I})=M_{\xi}$ identifies with the reduced space
itself, and more generally $\Sigma_{g}$ identifies with a symplectic quotient
of $Y^{g}$. For each $g\in S_{P}$ there is an immersion
$\Sigma_{g}\hookrightarrow\Sigma$ induced by $P^{g}\hookrightarrow P$. Let
$\nu_{\Sigma_{g},\Sigma}$ denote the (orbifold) normal bundle (the quotient
$\nu_{P^{g},P}/(T/T_{I})$), which inherits a complex structure from the almost
complex structures on $Y$, $Y^{g}$. Define the characteristic form

$\mathcal{D}_{\mathbb{C}}^{g}(\nu_{\Sigma_{g},\Sigma})=\textnormal{det}_{\mathbb{C}}\big{(}1-g_{\nu}^{-1}{\rm
e}^{-\frac{{\mathrm{i}}}{2\pi}F_{\nu}}\big{)},$

where $g_{\nu}$ denotes the action of $g$ on the normal bundle (defined in
terms of an orbifold chart, or in terms of $\nu_{P^{g},P}$), and $F_{\nu}$
denotes the curvature. Taking the quotient of $L|_{P^{g}}$ we obtain
(orbifold) line bundles

$L_{\Sigma_{g}}=(L|_{P^{g}})/(T/T_{I}),\qquad L_{\Sigma}=\bigsqcup_{g\in
S_{P}}L_{\Sigma_{g}}.$

There is a locally constant function

$g_{L}\colon\ \Sigma_{g}\rightarrow U(1)$

giving the phase of the action of $g$ on $L_{\Sigma_{g}}$ (or equivalently on
$L|_{P^{g}}$). Let $d\colon\Sigma\rightarrow\mathbb{Z}$ be the locally
constant function giving the size of a generic isotropy group for $\Sigma$ (or
equivalently the number of elements in the generic stabilizer for the
$T/T_{I}$ action on $\sqcup P^{g}$).

Let $\theta$ be a connection for the locally free $K=T/T_{I}$-action on
$\sqcup_{g\in S_{P}}P^{g}$. The curvature $F_{\theta}$ is horizontal and
$\mathfrak{t}/\mathfrak{t}_{I}$-valued, hence for any
$\lambda\in(\mathfrak{t}/\mathfrak{t}_{I})^{\ast}=I$, the form
$\langle\lambda,F_{\theta}\rangle$ is $K$-basic, hence descends to $\Sigma$.
With the preparations above, we can state the main result of this note.

######  Theorem 1.2.

If $0\notin\Delta$ then $m(k,0)=0$ for all $k\geq 1$. If $0\in\Delta$ then
there is a closed polytope $\mathfrak{p}\subset\Delta$ of the same dimension
as $\Delta$ and containing the origin such that the following is true. Let
$C_{\mathfrak{p}}$ denote the cone

$C_{\mathfrak{p}}=\\{(t,t\tau)\,|\,t\in(0,\infty),\tau\in\mathfrak{p}\\}\subset\mathbb{R}\times\mathfrak{t}^{\ast}.$

Fix a weakly regular value $\xi\in\Delta$ sufficiently close to $0$ as in
Theorem 1.1. Let $P=\mu_{\mathfrak{g}}^{-1}(\xi)$ and define $\Sigma$,
$L_{\Sigma}$, etc. as above. Then for all
$(k,\lambda)\in(\mathbb{Z}_{>0}\times\Lambda)\cap C_{\mathfrak{p}}$,

$\displaystyle m_{G}(k,\lambda)=\sum_{g\in
S_{P}}g^{-\lambda}\int_{\Sigma_{g}}\frac{1}{d}\frac{g_{L}^{k}\textnormal{Ch}(L_{\Sigma})^{k}\textnormal{Td}(\Sigma)}{\mathcal{D}_{\mathbb{C}}^{g}(\nu_{\Sigma_{g},\Sigma})}{\rm
e}^{\langle\lambda,F_{\theta}\rangle}.$ (1.3)

Of course this result is also originally due to Meinrenken–Sjamaar [10].
Theorem 1.1 follows immediately from Theorem 1.2 by applying Kawasaki’s index
theorem for orbifolds to
$\textnormal{index}\big{(}D^{\textnormal{red}}_{L_{\xi}}\big{)}$ and comparing
with the evaluation of (1.3) at $(k,\lambda)=(1,0)$.

Let us give a brief summary of our approach to deriving Theorem 1.2. Recall
that a function $f$ on a lattice $\Gamma$ in a real vector space $V$ is said
to be _quasi-polynomial_ if there is a sublattice $\Gamma^{\prime}$ with
$\Gamma/\Gamma^{\prime}$ finite and $f$ restricts to a polynomial function on
each coset of $\Gamma^{\prime}$. More generally, one says $f$ is quasi-
polynomial on a subset $\Gamma_{0}\subset\Gamma$ if
$f\upharpoonright\Gamma_{0}=q\upharpoonright\Gamma_{0}$ for some quasi-
polynomial $q$. A fundamental fact, originally derived from Theorem 1.1 by
Meinrenken–Sjamaar [10], is that $m_{G}$ is quasi-polynomial on the subset
$C_{\mathfrak{p}}\cap(\mathbb{Z}_{>0}\times\Lambda)$. Our first goal, in
Section 2, is to give an independent proof of this fact, taking as a starting
point a formula for $m_{G}$ due to Szenes–Vergne [14] (inspired by work of
Paradan [11]), which they obtained by a combinatorial rearrangement of the
fixed-point formula for the index.

Then in Section 3 we adapt an idea of Meinrenken [7] to compute the quasi-
polynomial $m_{G}\upharpoonright C_{\mathfrak{p}}$ using the Berline–Vergne
index formula and the principle of stationary phase. The output of the
stationary phase formula is an asymptotic expansion for $m_{G}(k,k\xi)$ in
powers of $k$ (allowing coefficients that are periodic in $k$). As one knows
in advance that $m_{G}(k,k\xi)$ is quasi-polynomial in $k$, one concludes that
the expansion is exact, yielding Theorem 1.2.

The article of Meinrenken–Sjamaar [10] contains, besides Theorem 1.1, a wealth
of detailed information about singular reduction and $[Q,R]=0$. Our goal in
this short note is much more modest. We also do not make a great claim of
originality, and in particular the debt to [14] and [7] will be apparent. Part
of our motivation stems from the hope that the article of Szenes–Vergne [14],
in combination with this note, will provide a more elementary treatment of the
$[Q,R]=0$ theorem than was previously available.

##  2 Quasi-polynomials and the multiplicity function

The goal of this section is Theorem 2.2 on the quasi-polynomial behavior of
the multiplicity function, which we prove using results of Szenes–Vergne [14]
reviewed below.

The quotient $\mathfrak{g}/\mathfrak{t}$ can be identified with the unique
$\textnormal{Ad}(T)$-invariant complement to $\mathfrak{t}$ in $\mathfrak{g}$.
Let $\mathfrak{t}\subset\mathfrak{h}\subset\mathfrak{g}$ be a $T$-invariant
subspace. We may similarly identify $\mathfrak{h}/\mathfrak{t}$ and
$\mathfrak{g}/\mathfrak{h}$ with subspaces of $\mathfrak{g}$. The choice of
positive roots $\mathcal{R}_{+}$ determines a complex structure on
$\mathfrak{g}/\mathfrak{t}$, whose $+{\mathrm{i}}$-eigenspace is identified
with the direct sum of the positive root spaces:

$(\mathfrak{g}/\mathfrak{t})^{1,0}\simeq\bigoplus_{\alpha\in\mathcal{R}_{+}}\mathfrak{g}_{\alpha}.$

We obtain similar complex structures on $\mathfrak{g}/\mathfrak{h}$,
$\mathfrak{h}/\mathfrak{t}$, whose $+{\mathrm{i}}$-eigenspaces are direct sums
of positive roots spaces. We will write
$\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(a)$ (resp.
$\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{h}}(a)$,
$\textnormal{det}_{\mathbb{C}}^{\mathfrak{h}/\mathfrak{t}}(a)$) for the
determinant of a complex linear endomorphism $a$ of
$\mathfrak{g}/\mathfrak{t}$ (resp. $\mathfrak{g}/\mathfrak{h}$,
$\mathfrak{h}/\mathfrak{t}$). An example is the endomorphism
$\textnormal{Ad}_{t}$, $t\in T$ (resp. $\textnormal{ad}_{X}$,
$X\in\mathfrak{t}$); in this case we will simply write
$\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(t)$ instead of
$\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(\textnormal{Ad}_{t})$
(resp. $\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(X)$ instead
of
$\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(\textnormal{ad}_{X})$),
the action of $T$ (resp. $\mathfrak{t}$) on $\mathfrak{g}/\mathfrak{t}$ being
understood. Then for example if $t={\rm e}^{X}\in T$,

$\displaystyle\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}\big{)}=\prod_{\alpha\in\mathcal{R}_{+}}\big{(}1-t^{-\alpha}\big{)}=\prod_{\alpha\in\mathcal{R}_{+}}\big{(}1-{\rm
e}^{-2\pi{\mathrm{i}}\langle\alpha,X\rangle}\big{)},$
$\displaystyle\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(-X)=\prod_{\alpha\in\mathcal{R}_{+}}-2\pi{\mathrm{i}}\langle\alpha,X\rangle.$

For $\lambda\in\Lambda\cap\mathfrak{t}_{+}^{\ast}$, the Weyl character formula
says that for $t\in T$,

$\displaystyle\chi_{\lambda}(t)\cdot\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}\big{)}=\sum_{w\in
W}(-1)^{l(w)}t^{w(\lambda+\rho)-\rho},$ (2.1)

where $W$ is the Weyl group, $l(w)$ is the length of the element $w\in W$, and
$\rho$ is the half sum of the positive roots. The right-hand-side is an
element of $R(T)$ with multiplicity function $m_{\lambda}$ obtained by Fourier
transform. Note that

  * •

$m_{\lambda}$ is anti-symmetric under the $\rho$-_shifted action of the Weyl
group_ :

$\displaystyle m_{\lambda}(w(\mu+\rho)-\rho)=(-1)^{l(w)}m_{\lambda}(\mu).$

  * •

The support of $m_{\lambda}|_{\Lambda\cap\mathfrak{t}_{+}^{\ast}}$ is
$\\{\lambda\\}$, where it takes the value $1$.

Conversely these two properties determine $m_{\lambda}$: it is the unique
$W$-anti-symmetric function on $\Lambda$ extending the multiplicity function
of $\chi_{\lambda}$. Applying these observations to the multiplicity function
$m_{G}$ defined in (1.2), we make the following definition.

######  Definition 2.1.

Let $m(k,-)\colon\Lambda\rightarrow\mathbb{Z}$ be the unique $\rho$-shifted
$W$-anti-symmetric function such that $m(k,\lambda)=m_{G}(k,\lambda)$ for all
$\lambda\in\Lambda\cap\mathfrak{t}_{+}^{\ast}$. The corresponding character
$Q(k,-)\colon T\rightarrow\mathbb{C}$ is defined as the inverse Fourier
transform:

$Q(k,t)=\sum_{\lambda\in\Lambda}m(k,\lambda)t^{\lambda}.$

Using the Weyl character formula (2.1) and the definition of $m_{G}$, it is
easy to verify that

$\displaystyle
Q(k,t)=\sum_{\lambda\in\Lambda}m(k,\lambda)t^{\lambda}=\textnormal{index}_{T}(D_{L^{k}})(t)\cdot\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}\big{)}.$

We define

$\mu=\textnormal{pr}_{\mathfrak{t}^{\ast}}\circ\mu_{\mathfrak{g}}$

to be the composition of the moment map $\mu_{\mathfrak{g}}$ with the
projection to $\mathfrak{t}^{\ast}$. Then $\mu$ is a moment map for the action
of $T$ on $M$. Suppose $t\in T$ is sufficiently generic, so that
$M^{t}=M^{T}$. The Atiyah–Bott–Segal formula for the index yields

$\displaystyle Q(k,t)=\sum_{F\subset M^{T}}t^{k\mu_{F}}\int_{F}\frac{{\rm
e}^{k\omega}\textnormal{Td}(F)}{\mathcal{D}_{\mathbb{C}}^{t}(\nu_{F})}\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}\big{)},$
(2.2)

where the sum is over connected components $F$ of $M^{T}$, and $\mu_{F}$
denotes the constant value of the moment map $\mu$ on $F$. The multiplicity
$m$ is obtained by Fourier transform of (2.2).

Key to the approach in [14] is a different expression for $m(k,\lambda)$ that
we briefly describe here. The formula depends on the choice of an invariant
inner product on $\mathfrak{g}$, as well as a generic point $\gamma$ contained
in $\mathfrak{t}^{\ast}_{+}$ and sufficiently close to $0$ (see [14, Section
4.1] for the meaning of ‘generic’ here). Using the inner product we identify
$\mathfrak{t}\simeq\mathfrak{t}^{\ast}$. We need some additional notation:

  * •

Let $\textnormal{Comp}_{T}(M)$ denote the set of connected components of
$M^{H}$, as $H$ ranges over all (connected) sub-tori of $T$.

  * •

For $C\in\textnormal{Comp}_{T}(M)$, let $\mathfrak{t}_{C}\subset\mathfrak{t}$
be its generic infinitesimal stabilizer. Let $A_{C}$ be the smallest affine
subspace containing the image $\mu(C)$. In particular $A_{M}$ is the smallest
affine subspace containing $\mu(M)$. Note that $A_{C}$ is a translate of the
annihilator of $\mathfrak{t}_{C}$.

  * •

Let $\gamma_{C}\in A_{C}$ be the orthogonal projection of $\gamma$ onto
$A_{C}$, and let $\tau_{C}=\gamma_{C}-\gamma$.

The Szenes–Vergne–Paradan formula [14, equation (39)] (see also [14,
Proposition 41, Theorem 48]) is a sum of contributions:

$\displaystyle m=\sum_{C}m_{C},$ (2.3)

where $C$ ranges over components $C\in\textnormal{Comp}_{T}(M)$ such that
$\gamma_{C}\in\mu_{\mathfrak{g}}(C)$. Szenes–Vergne derive this formula
directly from (2.2) using an interesting combinatorial rearrangement, the main
ingredient of which is a decomposition formula for Kostant-type partition
functions. The formula is inspired by, and closely related to, the work of
Paradan [11]. The fact that only a subset of the components in
$\textnormal{Comp}_{T}(M)$ contribute is non-trivial and quite important for
$[Q,R]=0$. The proof given by Szenes–Vergne involves studying the asymptotic
behavior of the $m_{C}$’s using the Berline–Vergne formula and the principle
of stationary phase. It goes back to results of Paradan [11], who proved a
closely related result using transversally elliptic symbols and K-theoretic
methods. Note that Szenes–Vergne assume for simplicity that $M^{T}$ consists
of isolated fixed points, but it is not difficult to handle the general case
with the same methods; see for example [6, Section 7] for some indications of
how this can be done.

For the proof of Theorem 2.2 we do not need the precise definition of the
terms $m_{C}$ in (2.3), but we will need the following two crucial properties:

  1. 1.

The function $m_{C}$ restricts to a quasi-polynomial on each
$\Lambda$-translate of the set $(\mathbb{Z}\times\Lambda)\cap\bm{A}_{C}$,
where

$\bm{A}_{C}=\\{(t,t\tau)\,|\,t\in\mathbb{R}_{>0},\,\tau\in
A_{C}\\}\subset\mathbb{R}\times\mathfrak{t}^{\ast}.$

  2. 2.

Let $\textnormal{wt}(\nu_{C})$ denote the list of complex weights (for the
compatible almost complex structure $J$) for the $\mathfrak{t}_{C}$ action on
the normal bundle $\nu_{C}$. If $\lambda\in\Lambda$ is in the support of
$m_{C}(k,-)$ then $\lambda$ satisfies the inequality

$\displaystyle\langle\tau_{C},\lambda\rangle\geq
k\langle\tau_{C},\gamma_{C}\rangle+\langle\tau_{C},\sigma_{C}\rangle,\qquad\sigma_{C}:=\sum_{\begin{subarray}{c}\delta\in\textnormal{wt}(\nu_{C})\\\
\langle\tau_{C},\delta\rangle>0\end{subarray}}\delta-\sum_{\begin{subarray}{c}\alpha\in\mathcal{R}_{+}\\\
\langle\tau_{C},\alpha\rangle>0\end{subarray}}\alpha.$ (2.4)

See the proof of [14, Theorem 49]. Note that, except for the special case
$\tau_{C}=0$, (2.4) defines a half-space in $\mathfrak{t}^{\ast}$.

We will refer to these two properties as ‘property (a)’, ‘property (b)’ in the
proof of the next result. Theorem 2.2 is a strengthening of [14, Theorem 49]
(which says that the function $k\mapsto m(k,0)$ is quasi-polynomial), and our
arguments are based on their elegant approach.

######  Theorem 2.2 ([10], see also [11, 12, 13]).

If $0\notin\Delta$ then $m(k,0)=0$ for all $k\geq 1$. If $0\in\Delta$ then
there is a closed polytope $\mathfrak{p}\subset\Delta$ of the same dimension
as $\Delta$ and containing the origin such that $m(k,\lambda)$ is quasi-
polynomial on the set of integral points $(\mathbb{Z}\times\Lambda)\cap
C_{\mathfrak{p}}$ contained in the cone

$C_{\mathfrak{p}}=\\{(t,t\tau)\,|\,t\in(0,\infty),\,\tau\in\mathfrak{p}\\}\subset\mathbb{R}\times\mathfrak{t}^{\ast}.$

###### Proof.

The strategy is based on choosing a suitable
$\gamma\in\mathfrak{t}^{\ast}_{+}$ and then analyzing the supports of the
contributions $m_{C}$ to $m$ in the corresponding Szenes–Vergne–Paradan
formula (2.3) using property (b). The contribution $m_{C}$ appears in (2.3)
only if
$\gamma_{C}\in\mu_{\mathfrak{g}}(M)\cap\mathfrak{t}^{\ast}=W\cdot\Delta\subset
W\cdot I$ (recall by definition $I$ is the smallest affine subspace containing
$\Delta$). Because $\gamma$ is chosen generically, the only
$C\in\textnormal{Comp}_{T}(M)$ which may contribute to (2.3) are those such
that the affine subspace $A_{C}$ is entirely contained in $I$ or one of its
Weyl reflections, and throughout the proof we assume this is the case.

Suppose $0\in\Delta$. We argue that by a suitable choice of $\gamma$, one can
arrange that _for all but one_ of the contributions, (i)
$\langle\tau_{C},\gamma_{C}\rangle\geq 0$ with equality if and only if $0\in
A_{C}$, (ii)
$\langle\tau_{C},\gamma_{C}\rangle>\langle\tau_{C},\gamma_{I}\rangle$, where
$\gamma_{I}$ is the orthogonal projection of $\gamma$ onto $I$, and (iii)
$\langle\tau_{C},\sigma_{C}\rangle>0$. The one special contribution is denoted
$m_{C_{I}}$ below and corresponds to the subspace $A_{C_{I}}=I$. By property
(b), (i) and (iii) imply that for $C\neq C_{I}$, the support of $m_{C}(k,-)$
lies outside $kH_{C}$ where $H_{C}$ is the half-space

$H_{C}=\\{\xi\,|\,\langle\tau_{C},\xi\rangle\leq\langle\tau_{C},\gamma_{C}\rangle\\}.$

Let $\mathfrak{p}$ be the intersection of $I$ with all of the half-spaces
$H_{C}$ for $C\neq C_{I}$. By (ii), the relative interior of $\mathfrak{p}$,
viewed as a polytope in $I$, contains the point $\gamma_{I}$, hence in
particular is non-empty. By construction $m\upharpoonright
C_{\mathfrak{p}}=m_{C_{I}}\upharpoonright C_{\mathfrak{p}}$. Then property (a)
implies that $m_{C_{I}}$ is quasi-polynomial on $C_{\mathfrak{p}}$, hence the
result.

We claim that one can ensure (i) holds for all $C$ by choosing
$\gamma\in\mathfrak{t}^{\ast}_{+}$ sufficiently close to $0$. Indeed let
$A_{C}^{0}$ be the subspace parallel to $A_{C}$, and let $a_{C}\in A_{C}$ be
the nearest point in $A_{C}$ to $0$. Then $\gamma_{C}-a_{C}\in A_{C}^{0}$
while $\tau_{C}$, $a_{C}$ are both orthogonal to $A_{C}^{0}$, hence
$\langle\tau_{C},\gamma_{C}-a_{C}\rangle=0=\langle
a_{C},\gamma_{C}-a_{C}\rangle$. These imply
$\langle\tau_{C},\gamma_{C}\rangle=\|a_{C}\|^{2}-\langle a_{C},\gamma\rangle$.
If $0\in A_{C}$ then $a_{C}=0$ and this vanishes. Otherwise we can ensure
$\langle\tau_{C},\gamma_{C}\rangle>0$ by choosing $\|\gamma\|<\|a_{C}\|$.
Since only finitely many $C$ occur, we can choose $\gamma$ such that this
holds for all $C$ with $0\notin A_{C}$. We now turn to verifying (ii), (iii),
and also handle the case $0\notin\Delta$ along the way.

Suppose $\gamma_{C}\in\mu_{\mathfrak{g}}(C)$, so that $m_{C}$ indeed appears
in (2.3). If $\alpha\in\mathcal{R}_{+}$ and $\langle\tau_{C},\alpha\rangle>0$,
then since $\gamma\in\mathfrak{t}^{\ast}_{+}$ it follows that
$\langle\gamma_{C},\alpha\rangle>0$. It is a consequence of the cross-section
theorem (cf. [5]) that $\alpha|_{\mathfrak{t}_{C}}$ appears in the list of
weights $\textnormal{wt}(\nu_{C})$. Hence

$\displaystyle\sigma_{C}=\sum_{\begin{subarray}{c}\delta\in\textnormal{wt}(\nu_{C})-\mathcal{R}_{+}^{\tau_{C}}\\\
\langle\tau_{C},\delta\rangle>0\end{subarray}}\delta,$ (2.5)

where $\mathcal{R}_{+}^{\tau_{C}}$ denotes the set of positive roots $\alpha$
such that $\langle\tau_{C},\alpha\rangle>0$, and
$\textnormal{wt}(\nu_{C})-\mathcal{R}_{+}^{\tau_{C}}$ denotes the list of
weights on $\nu_{C}$ with one copy of $\alpha|_{\mathfrak{t}_{C}}$ removed for
each $\alpha\in\mathcal{R}_{+}$ satisfying $\langle\tau_{C},\alpha\rangle>0$.
Hence

$\displaystyle\langle\tau_{C},\sigma_{C}\rangle\geq 0$ (2.6)

and the inequality is strict if at least one weight $\delta$ contributes in
(2.5).

If $0\notin\Delta$ then, choosing $\gamma$ sufficiently close to $0$, we can
ensure that for each $C$ such that $0\in A_{C}$ we have
$\gamma_{C}\notin\mu_{\mathfrak{g}}(M)$ (a fortiori
$\gamma_{C}\notin\mu_{\mathfrak{g}}(C)$), hence $m_{C}$ does not appear in
(2.3) at all. On the other hand, by (i), (2.6) and property (b), if $0\notin
A_{C}$ then $m_{C}(k,0)=0$ for all $k\geq 1$. We conclude that if
$0\notin\Delta$ then $m(k,0)=0$ for all $k\geq 1$.

We turn to the case $0\in\Delta\subset I$. In this case we may choose $\gamma$
such that it is simultaneously close to $0$ and arbitrarily close to
$\gamma_{I}$, the orthogonal projection of $\gamma$ onto $I$. Since
$\tau_{C}=\gamma_{C}-\gamma$,
$\langle\tau_{C},\gamma\rangle\leq\langle\tau_{C},\gamma_{C}\rangle$ with
equality if and only if $\gamma_{C}=\gamma$. By taking $\gamma$ sufficiently
close to $I$, one can ensure that
$\langle\tau_{C},\gamma_{I}\rangle\leq\langle\tau_{C},\gamma_{C}\rangle$ with
equality if and only if $\gamma_{C}=\gamma_{I}$.

We first consider contributions from components $C\in\textnormal{Comp}_{T}(M)$
such that $\gamma_{C}\notin\mathfrak{t}_{+}^{\ast}$. In this case there exists
a negative root $\alpha\in\mathcal{R}_{-}$ such that
$\langle\gamma_{C},\alpha\rangle>0$. It follows from the cross-section theorem
that $\alpha|_{\mathfrak{t}_{C}}\in\textnormal{wt}(\nu_{C})$. Since
$\gamma\in\mathfrak{t}_{+}^{\ast}$, $\langle\gamma,\alpha\rangle\leq 0$ and so

$\langle\tau_{C},\alpha\rangle=\langle\gamma_{C},\alpha\rangle-\langle\gamma,\alpha\rangle>0.$

As $\alpha\notin\mathcal{R}_{+}$, we see that $\delta=\alpha$ indeed
contributes in (2.5), hence $\langle\tau_{C},\sigma_{C}\rangle>0$. Moreover
since $\gamma_{C}\notin\mathfrak{t}_{+}^{\ast}$, $\gamma_{C}\neq\gamma_{I}$,
hence $\langle\tau_{C},\gamma_{I}\rangle<\langle\tau_{C},\gamma_{C}\rangle$.
This establishes (ii), (iii) for this case.

We are left to consider contributions from $C\in\textnormal{Comp}_{T}(M)$ such
that $\gamma_{C}\in\Delta=\mu_{\mathfrak{g}}(M)\cap\mathfrak{t}_{+}^{\ast}$.
Let $\Delta_{\textnormal{reg}}\subset\Delta$ be the relatively open dense
subset of weakly regular values. The connected components of
$\Delta_{\textnormal{reg}}$ are relatively open polytopes inside the subspace
$I$. Choose a connected component
$\mathfrak{a}\subset\Delta_{\textnormal{reg}}$ containing $0$ in its closure.
We may choose $\gamma\in\mathfrak{t}^{\ast}_{+}$ such that the orthogonal
projection $\gamma_{I}$ onto $I$ lies in $\mathfrak{a}$. The fibre
$\mu_{\mathfrak{g}}^{-1}(\gamma_{I})$ is connected and contained in
$M^{T_{I}}$, hence there is a unique connected component $C_{I}\subset
M^{T_{I}}$ containing $\mu_{\mathfrak{g}}^{-1}(\gamma_{I})$. Then
$A_{C_{I}}=I$ and by property (a), $m_{C_{I}}$ is quasi-polynomial on the set
of integral points in $\bm{A}_{C}=\\{(t,t\tau)\,|\,t>0,\,\tau\in I\\}\supset
C_{\mathfrak{p}}$.

The final situation to consider consists of the contributions from
$C\in\textnormal{Comp}_{T}(M)$ such that
$\gamma_{C}\in\Delta\setminus\Delta_{\textnormal{reg}}$. In particular
$\gamma_{C}\neq\gamma_{I}$ hence

$\displaystyle\langle\tau_{C},\gamma_{I}\rangle<\langle\tau_{C},\gamma_{C}\rangle$
(2.7)

establishing (ii) for this case. Let $\sigma$ be the face of
$\mathfrak{t}^{\ast}_{+}$ containing $\gamma_{C}$. The subset

$U=G_{\sigma}\cdot\bigcup_{\overline{\tau}\supset\sigma}\tau,$

where the union is taken over relatively open faces of
$\mathfrak{t}^{\ast}_{+}$ whose closure contains $\sigma$, is a slice for the
coadjoint $G_{\sigma}$-action. Let $Y=\mu_{\mathfrak{g}}^{-1}(U)$ be the
corresponding symplectic cross-section, cf. [5, Remark 3.7, Theorem 3.8].
Consider the function $f=\langle\tau_{C},\mu\rangle|_{Y}\colon
Y\rightarrow\mathbb{R}$, for which $C\cap Y\subset
Y^{\tau_{C}}=\textnormal{Crit}(f)$ is a critical submanifold. Note that
$f|_{C\cap Y}=\langle\tau_{C},\gamma_{C}\rangle$. A result from symplectic
geometry says that in a suitable tubular neighborhood of $C\cap Y$, the
function $f$ takes the form

$\displaystyle
f(z_{1},\dots,z_{n})=\langle\tau_{C},\gamma_{C}\rangle-\pi\sum_{j}|z_{j}|^{2}\langle\tau_{C},\delta_{j}\rangle,$
(2.8)

where $\delta_{j}\in\textnormal{wt}(\nu_{C\cap Y,Y})$,
$\langle\tau_{C},\delta_{j}\rangle\neq 0$, $z_{j}$ is a vector in the
subbundle of $\nu_{C\cap Y,Y}$ where $\mathfrak{t}_{C}$ acts with weight
$\delta_{j}$, and $|z_{j}|$ denotes its norm with respect to a suitable
Hermitian structure.

Let $S$ be the line segment with endpoints $\gamma_{I}$ and $\gamma_{C}$. By
convexity $S\subset\Delta$. The inverse image
$\mu_{\mathfrak{g}}^{-1}(S)\subset Y$ is connected since $\mu_{\mathfrak{g}}$
has connected fibres. By (2.7), along the line segment $S$, $f$ varies between
its absolute minimum $\langle\tau_{C},\gamma_{I}\rangle$ on the fibre
$\mu_{\mathfrak{g}}^{-1}(\gamma_{I})$ and its absolute maximum
$\langle\tau_{C},\gamma_{C}\rangle$ on the fibre
$\mu_{\mathfrak{g}}^{-1}(\gamma_{C})$. By connectedness of
$\mu_{\mathfrak{g}}^{-1}(S)$ and equation (2.8), there must exist a
$\delta_{j}$ such that $\langle\tau_{C},\delta_{j}\rangle>0$.

By the cross-section theorem $\nu_{Y,M}|_{C\cap Y}\simeq(C\cap
Y)\times\mathfrak{g}_{\gamma_{C}}^{\perp}$, where the orthogonal complement
$\mathfrak{g}_{\gamma_{C}}^{\perp}$ is embedded in $TM|_{C\cap Y}$ as the
orbit directions. The weights $\mathcal{R}_{+}^{\tau_{C}}$ which are removed
in (2.5) can be identified with the weights of the $\mathfrak{t}_{C}$-action
on $\nu_{Y,M}|_{C\cap Y}$. With this understanding we have
$\textnormal{wt}(\nu_{C\cap
Y,Y})\subset\textnormal{wt}(\nu_{C})-\mathcal{R}_{+}^{\tau_{C}}$. Thus
$\delta_{j}$ indeed contributes to (2.5), establishing (iii) for this case.
This completes the proof. ∎

######  Corollary 2.3.

Suppose $0\in\Delta$ and let $\mathfrak{p}\subset\Delta$ be as in Theorem 2.2.
If $\xi\in\mathfrak{p}$ is rational and $n_{\xi}\in\mathbb{Z}_{>0}$ is the
least positive integer such that $n_{\xi}\xi\in\Lambda$, then the function

$f_{\xi}\colon\ n_{\xi}\cdot\mathbb{Z}_{>0}\rightarrow\mathbb{Z},\qquad
f_{\xi}(k)=m(k,k\xi)$

is quasi-polynomial. Moreover $m\upharpoonright C_{\mathfrak{p}}$ is the
unique quasi-polynomial function such that $m(k,k\xi)\allowbreak=f_{\xi}(k)$
for all rational, weakly regular values $\xi$ in the relative interior of
$\mathfrak{p}$.

######  Remark 2.4.

A suitable finite collection of the functions $f_{\xi}$ already fully
determines $m\upharpoonright C_{\mathfrak{p}}$.

##  3 Stationary phase calculation

Assume $0\in\Delta$ and let $\mathfrak{p}\subset\Delta$ be as in Theorem 2.2,
so that $m\upharpoonright C_{\mathfrak{p}}$ is quasi-polynomial. By Corollary
2.3, $m\upharpoonright C_{\mathfrak{p}}$ is completely determined by the
collection of quasi-polynomial functions $f_{\xi}(k)=m(k,k\xi)$, for $\xi$
ranging over rational, weakly regular values of $\mu_{\mathfrak{g}}$ lying in
the relative interior of $\mathfrak{p}$. In this section we use the
Berline–Vergne index formula and the stationary phase expansion to compute the
functions $f_{\xi}$, and hence also $m\upharpoonright C_{\mathfrak{p}}$. The
end result will be the formula (1.3) in Theorem 1.2.

Let $t\in T$. By the Berline–Vergne formula, for $X\in\mathfrak{t}$
sufficiently small one has $Q\big{(}k,t{\rm e}^{X}\big{)}=Q_{t}(k,X)$ where

$\displaystyle Q_{t}(k,X):=\int_{M^{t}}\frac{t_{L}^{k}{\rm
e}^{k(\omega+2\pi{\mathrm{i}}\langle\mu,X\rangle)}\textnormal{Td}\big{(}M^{t},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\mathcal{D}_{\mathbb{C}}^{t}\big{(}\nu_{M^{t},M},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}{\rm
e}^{-X}\big{)},$ (3.1)

and $\textnormal{Td}\big{(}M^{t},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}$,
$\mathcal{D}_{\mathbb{C}}^{t}\big{(}\nu_{M^{t},M},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}$
denote equivariant extensions of the usual Chern-Weil forms, closed with
respect to the differential ${\rm d}+2\pi{\mathrm{i}}\iota(X_{M})$, obtained
by replacing curvatures with equivariant curvatures (evaluated at
$\tfrac{2\pi}{{\mathrm{i}}}X$) in the usual formulas (cf. [1] for details,
although note that we are using the topologist’s convention for characteristic
classes).

Let $B_{r}$ denote the ball of radius $r>0$ around the origin in
$\mathfrak{g}/\mathfrak{t}$. Let $\mu_{\mathfrak{g}/\mathfrak{t}}$ denote the
composition of $\mu_{\mathfrak{g}}$ with the quotient map
$\mathfrak{g}\rightarrow\mathfrak{g}/\mathfrak{t}$. Let
$\mathfrak{g}^{t}\subset\mathfrak{g}$ denote the fixed-point set of
$\textnormal{Ad}_{t}$. Then $B_{r}^{t}$ is a neighborhood of $0$ in
$\mathfrak{g}^{t}/\mathfrak{t}$. Recall $\mathfrak{g}^{t}/\mathfrak{t}$,
$\mathfrak{g}/\mathfrak{g}^{t}$ are equipped with complex structures such that
their $+{\mathrm{i}}$-eigenspaces are identified with sums of positive root
spaces. Equip $\mathfrak{g}^{t}/\mathfrak{t}$ with the orientation induced by
the complex structure, and let $\tau_{\mathfrak{g}^{t}/\mathfrak{t}}(X)$ be a
$T$-equivariant Thom form with support contained in $B_{r}^{t}$, closed for
the differential ${\rm d}-\iota(X_{M})$. Consider the $T$-equivariant
differential form on $\mathfrak{g}^{t}/\mathfrak{t}$ (closed for the
differential ${\rm
d}+2\pi{\mathrm{i}}\iota(X_{\mathfrak{g}^{t}/\mathfrak{t}})$) given by

$\textnormal{Ch}^{t}\big{(}{\sf
b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}=\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{g}^{t}}\big{(}1-t^{-1}{\rm
e}^{-X}\big{)}\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}^{t}/\mathfrak{t}}\left(\frac{1-{\rm
e}^{-X}}{X}\right)\tau_{\mathfrak{g}^{t}/\mathfrak{t}}\big{(}\tfrac{2\pi}{{\mathrm{i}}}X\big{)},$

The map $\mu_{\mathfrak{g}/\mathfrak{t}}$ restricts to a map
$M^{t}\rightarrow\mathfrak{g}^{t}/\mathfrak{t}$, which we use to pull back the
form $\textnormal{Ch}^{t}\big{(}{\sf b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}$.

######  Lemma 3.1.

$\displaystyle
Q_{t}(k,X)=\int_{\mu_{\mathfrak{g}/\mathfrak{t}}^{-1}(B_{r}^{t})}\frac{t_{L}^{k}{\rm
e}^{k(\omega+2\pi{\mathrm{i}}\langle\mu,X\rangle)}\textnormal{Td}\big{(}M^{t},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\mathcal{D}_{\mathbb{C}}^{t}\big{(}\nu_{M^{t},M},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}\textnormal{Ch}^{t}\big{(}{\sf
b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}.$ (3.2)

###### Proof.

The pullback of $\tau_{\mathfrak{g}^{t}/\mathfrak{t}}(X)$ to
$0\in\mathfrak{g}^{t}/\mathfrak{t}$ is the equivariant Euler class, which
(since $0$ is just a point) is the function

$\prod_{\alpha\in\mathcal{R}^{\mathfrak{g}^{t}}_{+}}-\langle\alpha,X\rangle=\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}^{t}/\mathfrak{t}}\big{(}\tfrac{{\mathrm{i}}}{2\pi}X\big{)},$

where $\mathcal{R}^{\mathfrak{g}^{t}}_{+}\subset\mathcal{R}_{+}$ is a set of
positive roots for $\mathfrak{g}^{t}$. Note also that $t$ acts trivially on
$\mathfrak{g}^{t}/\mathfrak{t}$, since $\mathfrak{g}^{t}$ is the fixed point
subspace under the adjoint action. It follows that the pullback of
$\textnormal{Ch}^{t}\big{(}{\sf b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}$ to
$0\in\mathfrak{g}^{t}/\mathfrak{t}$ is the function
$\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}{\rm
e}^{-X}\big{)}$. Since pullback to
$\\{0\\}=(\mathfrak{g}^{t}/\mathfrak{t})^{T}$ is injective on equivariant
cohomology classes, $\textnormal{Ch}^{t}\big{(}{\sf
b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}$,
$\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}{\rm
e}^{-X}\big{)}$ determine the same class in $T$-equivariant cohomology of
$\mathfrak{g}^{t}/\mathfrak{t}$. As $M$ is compact, we may make this
replacement in (3.1) without changing the value of the integral. ∎

######  Remark 3.2.

The reason for the notation is that $\textnormal{Ch}^{t}\big{(}{\sf
b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}$ is a representative for the $t$-twisted
Chern character of a Bott element ${\sf b}\in
K_{T}^{0}(\mathfrak{g}/\mathfrak{t})$, which generates the latter as an
$R(T)=K_{T}^{0}(\textnormal{pt})$-module. To be more precise, ${\sf b}$ is the
generator whose pullback to $0\in\mathfrak{g}/\mathfrak{t}$ is
$[\wedge^{\textnormal{ev}}\mathfrak{n}_{-}]-[\wedge^{\textnormal{odd}}\mathfrak{n}_{-}]\in
K^{0}_{T}(\textnormal{pt})$, $\mathfrak{n}_{-}$ being the direct sum of the
negative root spaces.

Since $T$ is compact, there exists a finite set $S\subset T$ and an open cover
$\\{U_{t}\,|\,t\in S\\}$ of $T$ where $U_{t}$ is a small open ball around $t$
in $T$ such that $Q\big{(}k,t{\rm e}^{X}\big{)}=Q_{t}(k,X)$ for $t{\rm
e}^{X}\in U_{t}$. Let $\sigma_{t}$, $t\in S$ be bump functions on
$\mathfrak{t}$ such that $\\{\hat{t}_{\ast}\sigma_{t}\,|\,t\in S\\}$ is a
partition of unity subordinate to the cover, where $\hat{t}$ is the map

$\hat{t}\colon\ \mathfrak{t}\rightarrow T,\qquad X\mapsto t{\rm e}^{X},$

which we may assume restricts to a diffeomorphism of a small ball around
$0\in\mathfrak{t}$ onto $U_{t}$. By equations (3.1) and (3.2)

$Q=\sum_{t\in S}\hat{t}_{\ast}(\sigma_{t}Q_{t}).$

The multiplicity function $m$ is the Fourier transform of $Q$:

$m(k,\lambda)=\sum_{t\in S}\int_{\mathfrak{t}}\sigma_{t}(X)\big{(}t{\rm
e}^{X}\big{)}^{-\lambda}Q_{t}(k,X).$

To do the stationary phase calculation (for $k\rightarrow\infty$) following
the approach outlined at the beginning of this section, we now set
$\lambda=k\xi$ where $\xi\in(\Lambda\otimes\mathbb{Q})\cap\mathfrak{p}$ is a
rational, weakly regular value of $\mu_{\mathfrak{g}}$ contained in the
relative interior of $\mathfrak{p}\subset\Delta$ as in Corollary 2.3, $k\in
n_{\xi}\mathbb{Z}_{>0}$ and $n_{\xi}$ is the least positive integer such that
$n_{\xi}\xi\in\Lambda$. Thus

$\displaystyle m(k,k\xi)=\sum_{t}t^{-k\xi}\\!\\!\int_{\mathfrak{t}}{\rm
d}X\sigma_{t}(X)\\!\\!\int_{\mu_{\mathfrak{g}/\mathfrak{t}}^{-1}(B_{r}^{t})}\frac{t_{L}^{k}\textnormal{Td}\big{(}M^{t},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\mathcal{D}_{\mathbb{C}}^{t}\big{(}\nu_{M^{t},M},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}\textnormal{Ch}^{t}\big{(}{\sf
b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}{\rm
e}^{k(\omega+2\pi{\mathrm{i}}\langle\mu-\xi,X\rangle)}.\\!\\!\\!\\!$ (3.3)

Let $f(m,X)=\langle\mu(m)-\xi,X\rangle$ viewed as a real-valued function on
$\mu_{\mathfrak{g}/\mathfrak{t}}^{-1}\big{(}B_{r}^{t}\big{)}\times\mathfrak{t}$.
According to the principle of stationary phase, we can include a bump function
supported in a small neighborhood of the critical set of $f$ in the integrand
of (3.3), and the error will be $o(k^{-\infty})$. The derivative

${\rm d}_{(m,X_{0})}f=\langle{\rm
d}_{m}\mu,X_{0}\rangle+\langle\mu(m)-\xi,{\rm d}_{X_{0}}X\rangle$

and in particular
$\textnormal{Crit}(f)\subset\mu^{-1}(\xi)\times\mathfrak{t}$. Let $\chi$ be
the pullback by $\mu$ of a bump function in $\mathfrak{t}^{\ast}$ supported in
a small neighborhood of $\xi$. Thus

$\displaystyle m(k,k\xi)\sim\sum_{t}t^{-k\xi}\int_{\mathfrak{t}}{\rm
d}X\sigma_{t}(X)$
$\displaystyle\hphantom{m(k,k\xi)\sim}{}\times\int_{\mu_{\mathfrak{g}/\mathfrak{t}}^{-1}(B_{r}^{t})}\chi\frac{t_{L}^{k}\textnormal{Td}\big{(}M^{t},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\mathcal{D}_{\mathbb{C}}^{t}\big{(}\nu_{M^{t},M},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}\textnormal{Ch}^{t}\big{(}{\sf
b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}{\rm
e}^{k(\omega+2\pi{\mathrm{i}}\langle\mu-\xi,X\rangle)},$ (3.4)

where $\sim$ denotes equality modulo an $o(k^{-\infty})$ error.

Let $Y=\mu_{\mathfrak{g}}^{-1}(\sigma)$ be the cross-section for the principal
face. By the cross-section theorem, a neighborhood $N$ of $Y$ in $M$ is
$G_{\sigma}$-equivariantly diffeomorphic to

$Y\times\mathfrak{g}/\mathfrak{g}_{\sigma},$

where $\mathfrak{g}/\mathfrak{g}_{\sigma}\simeq\mathfrak{g}_{\sigma}^{\perp}$
is embedded in the orbit directions. Since
$\mu^{-1}(\xi)\cap\mu_{\mathfrak{g}}^{-1}(\mathfrak{t}^{\ast})=\mu_{\mathfrak{g}}^{-1}(\xi)\subset
Y$, by taking $r$ and $\textnormal{supp}(\chi)$ sufficiently small, we can
assume that
$\textnormal{supp}(\chi)\cap\mu_{\mathfrak{g}/\mathfrak{t}}^{-1}(B_{r})$ is
contained in a small neighborhood of $\mu_{\mathfrak{g}}^{-1}(\xi)$ where the
local model $Y\times\mathfrak{g}/\mathfrak{g}_{\sigma}$ is valid, and so we
may replace $\mu_{\mathfrak{g}/\mathfrak{t}}^{-1}\big{(}B_{r}^{t}\big{)}$ with
$N^{t}$ in equation (3.4). In the next lemma we use the Thom form to integrate
over the $(\mathfrak{g}/\mathfrak{g}_{\sigma})^{t}$ directions.

######  Lemma 3.3.

$\displaystyle m(k,k\xi)\sim\sum_{t\in S}t^{-k\xi}\int_{\mathfrak{t}}{\rm
d}X\sigma_{t}(X)$
$\displaystyle\hphantom{m(k,k\xi)\sim}{}\times\int_{Y^{t}}\chi\frac{t_{L}^{k}\textnormal{Td}\big{(}Y^{t},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\mathcal{D}_{\mathbb{C}}^{t}\big{(}\nu_{Y^{t},Y},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\rm
e}^{k(\omega+2\pi{\mathrm{i}}\langle\mu-\xi,X\rangle)}\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}_{\sigma}/\mathfrak{t}}\big{(}1-t^{-1}{\rm
e}^{-X}\big{)}.$ (3.5)

###### Proof.

The neighborhood $N^{t}$ of $Y^{t}$ in $M^{t}$ is $T$-equivariantly
diffeomorphic to

$Y^{t}\times(\mathfrak{g}/\mathfrak{g}_{\sigma})^{t}=Y^{t}\times\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t},$

where $\mathfrak{g}_{\sigma}^{t}=(\mathfrak{g}_{\sigma})^{t}$ is the subspace
of $\mathfrak{g}_{\sigma}$ fixed by $t$. Moreover the almost complex structure
on $N^{t}$ is homotopic to a product almost complex structure, where $Y^{t}$
is equipped with an almost complex structure compatible with the symplectic
form in the cross-section, and $\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}$ is
equipped with the almost complex structure whose $+{\mathrm{i}}$-eigenspace is
identified with a sum of positive root spaces. Let

$p\colon\ N^{t}\rightarrow Y^{t}$

denote the projection. For the normal bundle

$\nu_{M^{t},M}|_{N^{t}}\simeq
p^{\ast}\nu_{Y^{t},Y}\oplus(\mathfrak{g}/\mathfrak{g}_{\sigma})/(\mathfrak{g}/\mathfrak{g}_{\sigma})^{t}=p^{\ast}\nu_{Y^{t},Y}\oplus\mathfrak{g}/\big{(}\mathfrak{g}^{t}+\mathfrak{g}_{\sigma}\big{)},$

and again the almost complex structure is homotopic to a product one, using a
compatible almost complex structure on the symplectic vector bundle
$\nu_{Y^{t},Y}$, and an almost complex structure on
$\mathfrak{g}/\big{(}\mathfrak{g}^{t}+\mathfrak{g}_{\sigma}\big{)}$ whose
$+{\mathrm{i}}$-eigenspace is identified with a sum of positive root spaces.
Using the identifications above we obtain, up to equivariantly exact forms:

$\displaystyle\textnormal{Td}\big{(}M^{t},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}|_{N^{t}}=\textnormal{Td}\big{(}Y^{t},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}}\left(\frac{X}{1-{\rm
e}^{-X}}\right),$ (3.6)
$\displaystyle\mathcal{D}_{\mathbb{C}}^{t}\big{(}\nu_{M^{t},M},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}|_{N^{t}}=\mathcal{D}_{\mathbb{C}}^{t}\big{(}\nu_{Y,Y^{t}},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/(\mathfrak{g}^{t}+\mathfrak{g}_{\sigma})}\big{(}1-t^{-1}{\rm
e}^{-X}\big{)}.$

Since $Y^{t}\subset\mu_{\mathfrak{g}}^{-1}(\mathfrak{t}^{\ast})$, the pullback
of the equivariant Thom form $\tau_{\mathfrak{g}^{t}/\mathfrak{t}}(X)$ to
$Y^{t}$ is just the function

$\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}^{t}/\mathfrak{t}}\big{(}\tfrac{{\mathrm{i}}}{2\pi}X\big{)}=\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}}\big{(}\tfrac{{\mathrm{i}}}{2\pi}X\big{)}\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}_{\sigma}^{t}/\mathfrak{t}}\big{(}\tfrac{{\mathrm{i}}}{2\pi}X\big{)}.$

We recognize
$\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}}\big{(}\tfrac{{\mathrm{i}}}{2\pi}X\big{)}$
as the equivariant Euler class of the trivial bundle
$Y^{t}\times\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}$. Thus up to an
equivariantly exact form, we have

$\displaystyle\tau_{\mathfrak{g}^{t}/\mathfrak{t}}(X)=\tau_{p}(X)\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}_{\sigma}^{t}/\mathfrak{t}}\big{(}\tfrac{{\mathrm{i}}}{2\pi}X\big{)},$
(3.7)

where $\tau_{p}(X)$ is an equivariant Thom form for the vector bundle $p\colon
N^{t}=Y^{t}\times\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}\rightarrow Y^{t}$.

We next want to make the replacements (3.6), (3.7) in equation (3.4), and then
use the Thom form $\tau_{p}(X)$ to integrate over the fibres of $p\colon
N^{t}\rightarrow Y^{t}$. In the integral over $N^{t}$ in (3.4), the integrand
has compact support and all terms in the integrand are equivariantly closed
except for the bump function $\chi$. By Stokes’ theorem, replacing a form by a
cohomologous form in the integrand leads to an error term containing ${\rm
d}\chi$; but ${\rm d}\chi$ vanishes near $\mu^{-1}(\xi)$, so the principle of
stationary phase implies the error will be $o(k^{-\infty})$. Let
$\iota_{Y^{t}}\colon Y^{t}\hookrightarrow N^{t}$ denote the inclusion.
Similarly the formula
$p_{\ast}(\tau_{p}(X)\alpha(X))=\iota_{Y^{t}}^{\ast}\alpha(X)$ applies when
$\alpha(X)$ is equivariantly closed. But writing $\chi=1-(1-\chi)$, the
principle of stationary phase again shows that we can make this replacement up
to an $o(k^{-\infty})$ error term.

After making these replacements and integrating over the fibre, the form
$\tau_{p}\big{(}\tfrac{2\pi}{{\mathrm{i}}}X\big{)}$ disappears. There are
various Lie theoretic factors left over:

$\frac{\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{g}^{t}}\big{(}1-t^{-1}{\rm
e}^{-X}\big{)}}{\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/(\mathfrak{g}^{t}+\mathfrak{g}_{\sigma})}\big{(}1-t^{-1}{\rm
e}^{-X}\big{)}}\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}^{t}/\mathfrak{t}}\left(\frac{1-{\rm
e}^{-X}}{X}\right)\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}_{\sigma}^{t}/\mathfrak{t}}(X)\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}}\left(\frac{X}{1-{\rm
e}^{-X}}\right),$

which simplify to
$\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}_{\sigma}/\mathfrak{t}}\big{(}1-t^{-1}{\rm
e}^{-X}\big{)}$ (one uses that $t$ acts trivially on
$\mathfrak{g}_{\sigma}^{t}/\mathfrak{t}$ and that
$\big{(}\mathfrak{g}^{t}+\mathfrak{g}_{\sigma}\big{)}/\mathfrak{g}^{t}\simeq\mathfrak{g}_{\sigma}/\mathfrak{g}_{\sigma}^{t}$).
∎

Choose a complementary subtorus $T_{I}^{\prime}$ so that $T\simeq T_{I}\times
T_{I}^{\prime}$. The quotient map $T\rightarrow T/T_{I}$ induces an
isomorphism of groups $T_{I}^{\prime}\xrightarrow{\sim}T/T_{I}$. By adding
additional points if necessary, we may assume the finite subset $S\subset T$
is a product $S_{I}\times S_{I}^{\prime}$, where $S_{I}\subset T_{I}$,
$S_{I}^{\prime}\subset T_{I}^{\prime}$ and that the image of $S_{I}^{\prime}$
in $T/T_{I}$ contains the set $S_{P}$ from the introduction. Thus we will
write elements of $S$ as products $hg$ with $h\in S_{I}\subset T_{I}$ and
$g\in S_{I}^{\prime}\subset T_{I}^{\prime}$. We may assume the bump function
$\sigma_{t}$ is a product $\sigma_{h}\cdot\sigma_{g}$, where $\sigma_{h}$
(resp. $\sigma_{g}$) is a bump function on $\mathfrak{t}_{I}$ (resp.
$\mathfrak{t}_{I}^{\prime}$), satisfying

$\displaystyle\sum_{h\in S_{I}}\hat{h}_{\ast}\sigma_{h}=1,\qquad\sum_{g\in
S_{I}^{\prime}}\hat{g}_{\ast}\sigma_{g}=1.$ (3.8)

The next lemma gives a further simplification of (3.5).

######  Lemma 3.4.

$\displaystyle m(k,k\xi)\sim\sum_{g\in
S_{I}^{\prime}}g^{-k\xi}\int_{\mathfrak{t}_{I}^{\prime}}{\rm
d}X\sigma_{g}(X)\int_{Y^{g}}\chi\frac{g_{L}^{k}\textnormal{Td}\big{(}Y^{g},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\mathcal{D}_{\mathbb{C}}^{g}\big{(}\nu_{Y^{g},Y},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\rm
e}^{k(\omega+2\pi{\mathrm{i}}\langle\mu-\xi,X\rangle)}.$ (3.9)

###### Proof.

As $T_{I}$ acts trivially on $Y$ and $\mu(Y)\subset I$, the characteristic
forms in (3.5) only depend on the component of $X$ (resp. $t$) in
$\mathfrak{t}_{I}^{\prime}$ (resp. $T_{I}^{\prime}$). Likewise as
$\xi\in(\Lambda\otimes\mathbb{Q})\cap I$, $t^{-k\xi}$ only depends on the
component $g$ of $t$ in $T_{I}^{\prime}$. This means the following expression
can be split off from (3.5) and evaluated separately:

$\displaystyle\sum_{h\in S_{I}}\int_{\mathfrak{t}_{I}}{\rm
d}X\sigma_{h}(X)\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}_{\sigma}/\mathfrak{t}}\big{(}1-h^{-1}g^{-1}{\rm
e}^{-X}\big{)}.$ (3.10)

The determinant is given by a product:

$\prod_{\alpha\in\mathcal{R}_{+}^{\mathfrak{g}_{\sigma}}}\big{(}1-h^{-\alpha}g^{-\alpha}{\rm
e}^{-2\pi{\mathrm{i}}\langle\alpha,X\rangle}\big{)}.$

When the product over $\mathcal{R}_{+}^{\mathfrak{g}_{\sigma}}$ is expanded,
we obtain an alternating sum of terms of the form $h^{-\zeta}g^{-\zeta}{\rm
e}^{-2\pi{\mathrm{i}}\langle\zeta,X\rangle}$, where $\zeta$ is a sum of a
subset of $\mathcal{R}_{+}^{\mathfrak{g}_{\sigma}}$. The elements of
$\mathcal{R}_{+}^{\mathfrak{g}_{\sigma}}$ lie in
$\textnormal{ann}(\mathfrak{z}_{\sigma})$, the annihilator of
$\mathfrak{z}_{\sigma}$ in $\mathfrak{t}^{\ast}$. Since
$\mathfrak{t}^{\ast}=\mathfrak{z}_{\sigma}^{\ast}\oplus\textnormal{ann}(\mathfrak{z}_{\sigma})$
and $I\subset\mathfrak{z}_{\sigma}^{\ast}$, it follows that either $\zeta=0$
or else $\zeta\notin I$.

We claim that if $\zeta\neq 0$, then the corresponding contribution to (3.10)
is $0$. Indeed taking the Fourier transform of the first equation in (3.8), we
find that for any $[\zeta]\in\Lambda/(\Lambda\cap I)$, the weight lattice of
$T_{I}$, we have

$\sum_{h\in S_{I}}h^{-[\zeta]}\int_{\mathfrak{t}_{I}}\sigma_{h}(X){\rm
e}^{-2\pi{\mathrm{i}}\langle[\zeta],X\rangle}{\rm d}X=\delta_{0}([\zeta]),$

where $\delta_{0}$ is the function on $\Lambda/(\Lambda\cap I)$ equal to $1$
at $0$ and $0$ otherwise, obtained by Fourier transform of the constant
function $1$ on $T_{I}$. Thus for $\zeta\in\Lambda$,

$\displaystyle\sum_{h\in
S_{I}}h^{-\zeta}\int_{\mathfrak{t}_{I}}\sigma_{h}(X){\rm
e}^{-2\pi{\mathrm{i}}\langle\zeta,X\rangle}{\rm d}X=\delta_{\Lambda\cap
I}(\zeta),$

where $\delta_{\Lambda\cap I}$ is the function on $\Lambda$ equal to $1$ on
$\Lambda\cap I$ and $0$ otherwise. In particular if $\zeta\notin I$ we see
that the corresponding contribution in (3.10) vanishes.

On the other hand, using equation (3.8), the contribution from $\zeta=0$ to
(3.10) is

$\sum_{h\in S_{I}}\int_{\mathfrak{t}_{I}}{\rm d}X\sigma_{h}(X)=1.$

This yields the expression on the right-hand-side of (3.9). ∎

We can now complete the proof of Theorem 1.2. The fibre
$P=\mu^{-1}(\xi)\subset Y$ is smooth, and the quotient
$\Sigma_{e}:=M_{\xi}=P/G_{\sigma}=P/T_{I}^{\prime}$ is an orbifold
($T_{I}^{\prime}$ acts locally freely on $P$). By the coisotropic embedding
theorem, a neighborhood of $P$ in $Y$ is $T$-equivariantly symplectomorphic to

$P\times B_{I}\subset P\times I,$

where $B_{I}$ is a small ball around $\xi$ in the subspace
$I\subset\mathfrak{t}^{\ast}$, the moment map $\mu$ is projection to the
second factor, and the symplectic form

$\omega|_{P\times B_{I}}=\omega_{\xi}+{\rm
d}\langle\eta-\xi,\theta\rangle=\omega_{\xi}+\langle{\rm
d}\eta,\theta\rangle+\langle\eta-\xi,F_{\theta}\rangle,$

where $\omega_{\xi}$ is the pullback of the symplectic form on the reduced
space $M_{\xi}$, $\eta$ is the variable in $B_{I}$,
$\theta\in\Omega^{1}(P,\mathfrak{t}^{\ast})^{T}$ is a connection on $P$ with
curvature $F_{\theta}={\rm d}\theta$, and here as well as below we have
omitted pullbacks from the notation. A neighborhood of $P^{g}$ in $Y^{g}$ is
$T$-equivariantly symplectomorphic to

$P^{g}\times B_{I}^{g}=P^{g}\times B_{I},$

and $T_{I}^{\prime}$ acts locally freely on $P^{g}$, with the quotient
$\Sigma_{g}=P^{g}/T_{I}^{\prime}$ being an orbifold. On the same neighborhood
we have

$\displaystyle\textnormal{Td}\big{(}Y^{g},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}=\textnormal{pr}_{1}^{\ast}\textnormal{Td}\big{(}P^{g},\tfrac{2\pi}{{\mathrm{i}}}X\big{)},$
$\displaystyle\nu_{Y^{g},Y}=\textnormal{pr}_{1}^{\ast}\nu_{P^{g},P}\quad\Rightarrow\quad\mathcal{D}_{\mathbb{C}}^{g}\big{(}\nu_{Y^{g},Y},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}=\textnormal{pr}_{1}^{\ast}\mathcal{D}_{\mathbb{C}}^{g}\big{(}\nu_{P^{g},P},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}.$

Below we will omit $\textnormal{pr}_{1}^{\ast}$ from the notation.

Take the bump function $\chi$ to have its support contained in the
neighborhood of $P$ where the above local normal forms are valid. We may then
integrate over $I$ instead of $B_{I}$, since $\chi$ vanishes outside of
$P\times B_{I}$ by assumption. On $\textnormal{supp}(\chi)$,

${\rm e}^{k(\omega+2\pi{\mathrm{i}}\langle\mu-\xi,X\rangle)}={\rm
e}^{k(\omega_{\xi}+\langle{\rm
d}\eta,\theta\rangle+\langle\eta-\xi,F_{\theta}\rangle+2\pi{\mathrm{i}}\langle\eta-\xi,X\rangle)}.$

Only the top degree part of ${\rm e}^{k\langle{\rm d}\eta,\theta\rangle}$
contributes to the integral over $I$; this top degree part is
$(-1)^{n(n-1)/2}k^{n}{\rm d}\eta\cdot\Theta$, where $n=\textnormal{dim}(I)$,
${\rm d}\eta=\Pi{\rm d}\eta^{a}$, $\Theta=\Pi\theta_{a}$ in terms of
coordinates on $I$. The sign $(-1)^{n(n-1)/2}$ relates the symplectic and
product orientations for $P^{g}\times I$, so will be absorbed when we use
Fubini’s theorem to write the integral over $P^{g}\times I$ as an iterated
integral. Let $\overline{\chi}(\eta)=\chi(\eta+\xi)$, a bump function on $I$
supported near $0$. Making these substitutions, as well as a change of
variables $\eta\leadsto\eta+\xi$ in the integral over $I$, the asymptotic
expression (3.9) for $m(k,k\xi)$ simplifies to

$\displaystyle k^{n}\sum_{g}g^{-k\xi}\int_{\mathfrak{t}_{I}^{\prime}\times
I}{\rm d}X\,{\rm d}\eta\,{\rm
e}^{2\pi{\mathrm{i}}k\langle\eta,X\rangle}\sigma_{g}(X)\overline{\chi}(\eta)\int_{P^{g}}\Theta\frac{g_{L}^{k}\textnormal{Td}\big{(}P^{g},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\mathcal{D}_{\mathbb{C}}^{g}\big{(}\nu_{P^{g},P},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\rm
e}^{k(\omega_{\xi}+\langle\eta,F_{\theta}\rangle)}.$

We need the following special case of the stationary phase expansion.

######  Proposition 3.5 (stationary phase expansion, cf. [4, Lemma 7.7.3]).

Let $u(X,\eta)$ be a Schwartz function. We have the following asymptotic
expansion in $k$:

$\int_{\mathfrak{t}_{I}^{\prime}\times(\mathfrak{t}_{I}^{\prime})^{\ast}}{\rm
d}X\,{\rm d}\eta\,{\rm
e}^{2\pi{\mathrm{i}}k\langle\eta,X\rangle}u(X,\eta)\sim\frac{1}{k^{n}}\sum_{j=0}^{\infty}\frac{1}{j!}\left(\sum_{a}\frac{{\mathrm{i}}}{2\pi
k}\frac{\partial}{\partial\eta_{a}}\frac{\partial}{\partial
X^{a}}\right)^{j}u(0,0).$

######  Remark 3.6.

To obtain the expression here from the expression appearing in _loc. cit._ ,
one sets $x=(X,\eta)\in\mathbb{R}^{2n}$ and $A(X,\eta)=(\eta,X)$. Note also
that in Hörmander’s notation $D=-{\mathrm{i}}({\rm d}/{\rm d}x)$.

We apply this to the smooth compactly supported function

$u(X,\eta)=\sigma_{g}(X)\overline{\chi}(\eta)\int_{P^{g}}\Theta\frac{g_{L}^{k}\textnormal{Td}\big{(}P^{g},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\mathcal{D}_{\mathbb{C}}^{g}\big{(}\nu_{P^{g},P},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}}{\rm
e}^{k(\omega_{\xi}+\langle\eta,F_{\theta}\rangle)}.$

Although this function depends on $k$, the dependence is quasi-polynomial, and
so the expansion still applies. Since $\sigma_{g}(X)$, $\overline{\chi}(\eta)$
equal $1$ in a neighborhood of $0$, they have no effect on the expansion. The
$\eta$ derivatives $k^{-1}\partial_{\eta_{a}}$ operate only on the factor
${\rm e}^{k\langle\eta,F_{\theta}\rangle}$. The combined effect of the
operator $\Sigma_{a}\,({\mathrm{i}}/2\pi
k)\partial_{\eta_{a}}\partial_{X^{a}}$ is to replace $X$ with
$({\mathrm{i}}/2\pi)F_{\theta}$, yielding the asymptotic expansion

$\displaystyle
m(k,k\xi)\sim\sum_{g}g^{-k\xi}\int_{P^{g}}\Theta\frac{g_{L}^{k}\textnormal{Td}\big{(}P^{g},F_{\theta}\big{)}}{\mathcal{D}_{\mathbb{C}}^{g}(\nu_{P^{g},P},F_{\theta})}{\rm
e}^{k\omega_{\xi}}.$ (3.11)

(By substituting $F_{\theta}$ for $X$ in
$\textnormal{Td}\big{(}P^{g},X\big{)}$,
$\mathcal{D}_{\mathbb{C}}^{g}(\nu_{P^{g},P},X)^{-1}$, we mean to take the
Taylor expansion around $X=0$ and substitute the differential form
$F_{\theta}$.) At this stage we see that the contribution of $g\in
S_{I}^{\prime}$ vanishes unless $P^{g}\neq\varnothing$, so that
$S_{I}^{\prime}=S_{P}$ ($S_{P}$ is as in Theorem 1.2). As the characteristic
forms $\textnormal{Td}\big{(}P^{g},F_{\theta}\big{)}$,
$\mathcal{D}_{\mathbb{C}}^{g}(\nu_{P^{g},P},F_{\theta})$ appear multiplied by
the form $\Theta$, which has top degree in the $T_{I}^{\prime}$ orbit
directions, we can replace these characteristic forms with their horizontal
parts. Substituting $F_{\theta}$ for $X$ and taking the horizontal part is the
definition of the Cartan map $\textnormal{Car}_{\theta}$ for the locally free
action of $T_{I}^{\prime}$ on the space $P^{g}$, hence the result is the
pullback along the map $P^{g}\rightarrow\Sigma_{g}=P^{g}/T_{I}^{\prime}$ of
the form

$\frac{\textnormal{Td}(\Sigma_{g})}{\mathcal{D}_{\mathbb{C}}^{g}(\nu_{\Sigma,\Sigma^{g}})}.$

(See our remarks in the introduction regarding characteristic forms for
orbifolds.) Similarly the $1^{\rm st}$ Chern form $c_{1}(L_{\Sigma})$ is
obtained by applying the Cartan map to the equivariant symplectic form
$\omega_{\mathfrak{t}}(X)=\omega-\langle\mu,X\rangle$, and results in
$c_{1}(L_{\Sigma})=\omega_{\xi}-\langle\xi,F_{\theta}\rangle$. Hence
$\textnormal{Ch}(L_{\Sigma})={\rm e}^{c_{1}(L_{\Sigma})}={\rm
e}^{\omega_{\xi}-\langle\xi,F_{\theta}\rangle}$. The integral over the fibres
of $P^{g}\rightarrow\Sigma_{g}$ then gives $1/d$, where
$d\colon\Sigma=\sqcup\Sigma_{g}\rightarrow\mathbb{Z}$ is the locally constant
function giving the size of the generic stabilizer for the
$T_{I}^{\prime}\simeq T/T_{I}$ action on $\sqcup P^{g}\rightarrow\Sigma$.
Equation (3.11) becomes

$\displaystyle m(k,k\xi)\sim\sum_{g\in
S_{P}}g^{-k\xi}\int_{\Sigma_{g}}\frac{1}{d}\frac{g_{L}^{k}\textnormal{Ch}(L_{\Sigma})^{k}\textnormal{Td}(\Sigma)}{\mathcal{D}_{\mathbb{C}}^{g}(\nu_{\Sigma_{g},\Sigma})}{\rm
e}^{k\langle\xi,F_{\theta}\rangle}.$ (3.12)

By Corollary 2.3, $m(k,k\xi)$ is a quasi-polynomial function of $k$, hence the
asymptotic expansion must be exact, or in other words, ‘$\sim$’ in equation
(3.12) can be replaced with ‘$=$’. Thus setting $\lambda=k\xi$ we have

$\displaystyle m(k,\lambda)=\sum_{g\in
S_{P}}g^{-\lambda}\int_{\Sigma_{g}}\frac{1}{d}\frac{g_{L}^{k}\textnormal{Ch}(L_{\Sigma})^{k}\textnormal{Td}(\Sigma)}{\mathcal{D}_{\mathbb{C}}^{g}(\nu_{\Sigma_{g},\Sigma})}{\rm
e}^{\langle\lambda,F_{\theta}\rangle}.$ (3.13)

The right-hand-side of equation (3.13) is quasi-polynomial in $(k,\lambda)$.
Hence by Corollary 2.3, equation (3.13) holds on _all_ of $C_{\mathfrak{p}}$
(and not only at points $(k,\lambda)$ with $\lambda=k\xi$, $\xi$ a rational,
weakly regular value in the relative interior of $\mathfrak{p}$). This
completes the proof of Theorem 1.2.

### Acknowledgements

I thank M. Vergne and E. Meinrenken for helpful conversations. I thank the
referees for helpful comments and suggestions that improved the article.

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