text:
\section*{Introduction}

A (Riemannian) orbifold  is  a  versatile generalization  of  a (Riemannian) manifold that  permits  the  presence of well-structured  singular  points. Orbifolds appear in a variety of mathematical areas and have applications in physics, in particular to string theory.

 Orbifolds are locally the orbit spaces of finite effective group actions on $\mathbf R^d$.  Orbifold singularities correspond to orbits with non-trivial isotropy and thus each singularity has an associated ``isotropy type.''   The notions of the Laplace-Beltrami operator and Hodge Laplacian extend to Riemannian orbifolds and many results on the spectral geometry of Riemannian manifolds, such as Weyl asymptotics and Cheeger's inequality, extend to this more general setting.    We will always assume all orbifolds under consideration are connected and compact without boundary.

 Because  the possible presence of singular  points is a defining  characteristic  of  the class of  orbifolds,  the question ``Can one hear the singularities of an orbifold?" is  fundamental  in  the  study  of  the  spectral  geometry  of  orbifolds.    More precisely, one asks:

\begin{enumerate}
\item \label{it:orbimani} Does the spectrum distinguish orbifolds with singularities from manifolds?
\item Does the spectrum detect the topology and geometry of the set of singular points, including the isotropy types of singularities?
\end{enumerate}

Many authors have addressed these questions for the spectrum of the Laplace-Beltrami operator.   However, the question of whether the spectrum of the Laplace-Beltrami operator \emph{always} distinguishes Riemannian orbifolds with singularities from Riemannian manifolds, remains open.    In this article, we focus primarily on question \ref{it:orbimani} and consider whether additional spectral data, specifically the spectra of the Hodge Laplacians on $p$-forms, suffice to detect the presence of singularities.    We will say that two closed Riemannian orbifolds are $p$-\emph{isospectral} if their Hodge Laplacians acting on $p$-forms are isospectral.  In particular, 0-isospectrality means that the Laplace-Beltrami operators are isospectral.

We first construct the fundamental solution of the heat equation and the heat trace for the $p$-spectrum on closed Riemannian orbifolds, analogous to that of the 0-spectrum in \cite{DGGW08} discussed below, and then apply the resulting spectral invariants to question \ref{it:orbimani}.   Our first main result is that the 0-spectrum and 1-spectrum together distinguish orbifolds with a sufficiently large singular set from manifolds.  Orbifolds are stratified spaces in which the regular points form an open dense stratum, and each lower-dimensional stratum is a connected component of the set of singularities of a given isotropy type. By the \emph{codimension} of the singular set, we mean the minimum codimension of the singular strata.

 \begin{thm}\label{thm:main1V4}  The $0$-spectrum and $1$-spectrum together distinguish closed Riemannian orbifolds with singular sets of codimension $\leq 3$  from closed Riemannian manifolds.
 \end{thm}

Consequently:

\begin{cor}\label{thm:lowdim} For $d\leq 3$, the $0$ and $1$-spectra together distinguish singular closed $d$-dimensional Riemannian orbifolds from closed Riemannian manifolds.
\end{cor}

 The case $d=3$ is perhaps of particular interest since 3-dimensional orbifolds play a role in Thurston's Geometrization Conjecture, later proven by Grigori Perel'man.

 We next consider the inverse spectral problem for individual $p$-spectra, obtaining both positive and negative results.   The theorem below refers to the combinatorial Krawtchouk polynomials (see Example~\ref{isotropy2}, Equation~\eqref{krawtpoly}); there is a large literature on the zeros of these polynomials.  Theorem~\ref{thm:krawt} part~\ref{negative} is motivated by and applies the work of Roberto Miatello and Juan Pablo Rossetti \cite{MR01}, where the Krawtchouk polynomials were used to construct $p$-isospectral Bieberbach manifolds.

 \begin{thm}\label{thm:krawt} Denote by ${\mathcal Orb}^d_k$ the class of all closed $d$-dimensional Riemannian orbifolds with singular set of codimension $k$.    Let $p\in \{0,1,2,\dots , d\}$ and let $K_p^d$ be the Krawtchouk polynomial given by Equation~\eqref{krawtpoly}.
\begin{enumerate}
\item\label{volume} If $k$ is odd and $K_p^d(k)\neq 0$, then the $p$-spectrum determines the $(d-k)$-dimensional volume of the singular set of each orbifold in ${\mathcal Orb}^d_k$.   In particular, the $p$-spectrum distinguishes orbifolds in ${\mathcal Orb}^d_k$ from closed Riemannian manifolds.
\item\label{negative} For every $k\in\{1,\dots, d\}$ such that $K_p^d(k)=0$, there
exists an orbifold ${\mathcal{O}}\in {\mathcal Orb}^d_k$ that is $p$-isospectral to a closed Riemannian manifold.

\end{enumerate}
\end{thm}

\begin{remark}\label{rem on thm krawt}~
\begin{enumerate}
\item  In case $p=0$, the Krawtchouk polynomial $K_0^d(1)\equiv 1$, so the first statement of item~\ref{volume} says that, for every closed Riemannian orbifold with singular set of odd codimension, the $0$-spectrum detects the volume of the singular set.    To our knowledge, this statement is new even in the case $p=0$. On the other hand, as mentioned in the review below, the second statement in part~\ref{volume} was previously known in this case.

\item Consider the special case of Riemannian orbifolds for which \emph{all} singular strata have codimension $k=1$.  The underlying space of such an orbifold ${\mathcal{O}}$ is a Riemannian manifold $M$ with boundary; the singular set of the orbifold corresponds to the manifold boundary.   (See Remark~\ref{abs boun}.)    In this case, the $p$-spectrum of the orbifold coincides with the $p$-spectrum of $M$ with absolute boundary conditions (Neumann boundary conditions if $p=0$).   Statement~\ref{volume} of Theorem~\ref{thm:krawt} generalizes the well-known fact that, except when the dimension $d$ is even and $p=\frac{d}{2}$, the $p$-spectrum of a Riemannian manifold determines the volume of the boundary.   (We note that $p=\frac{d}{2}$ is the unique $p$ for which $K_p^d(1)=0$.)

\item As discussed in \cite{GR03} and \cite{GR03-errata},  the middle degree $p=\frac{d}{2}$ Hodge spectrum of even-dimensional Riemannian manifolds, or more generally orbifolds, contains less information than the other spectra due to Hodge duality.   Theorem~\ref{thm:krawt} illustrates another weakness of the middle degree Hodge spectrum: when $p=\frac{d}{2}$, the Krawtchouk polynomial $K_p^d(k)=0$ for every odd integer $k$ in the interval $[1,d-1]$.  (Krawtchouk polynomials for other values of $p$ yield far fewer integer zeros.)  In \cite{GR03}, J. P. Rossetti and the second author gave examples of singular Riemannian orbifolds of even dimension $d$ that are $\frac{d}{2}$-isospectral to Riemannian manifolds.  The construction in Theorem~\ref{thm:krawt}, part~\ref{negative} reproduces some of those examples in this case.

Theorem~\ref{thm:krawt}, part~\ref{negative} also yields examples of Riemannian orbifolds of both even and odd dimensions with singularities that are $p$-isospectral to manifolds for various $p\neq 0$.    To our knowledge, the only previously known examples of such isospectralities were in the special case $p=\frac{d}{2}$.

\end{enumerate}
\end{remark}

We then focus on the Hodge Laplacian on 1-forms, obtaining both positive and negative results.   In particular:

\begin{itemize}
\item We address analogues for the 1-spectrum of those inverse spectral results for the $0$-spectrum appearing in Theorems~\ref{thm:common good} and \ref{thm:dggw5.1} in the review below.  We find that these results remain valid for the 1-spectrum \emph{provided that} one imposes an inequality on the codimension of the singular set in the case Theorem~\ref{thm:common good} and on the codimension of the singular stratum that appears in the statement of Theorem~\ref{thm:dggw5.1}.    Counterexamples constructed using Krawtchouk polynomials show that the conditions imposed are sharp at least for all even-dimensional orbifolds.

\item We construct a flat ``3-pillow'' (an orbifold ${\mathcal{O}}$ whose underlying space is a 2-sphere with three cone points) such that the 1-spectrum of ${\mathcal{O}}$, endowed with a flat Riemannian metric, coincides with the 1-spectrum of a square in $\mathbf R^2$ with absolute boundary conditions.

\end{itemize}

\subsection{Partial review of known results for the Laplace-Beltrami operator.}

The earliest results in the spectral geometry of Riemannian orbifolds are those of Yuan-Jen Chiang \cite{YJC} who established the existence of the Laplace spectrum and heat kernel of a Riemannian orbifold, and Carla Farsi \cite{F01} who extended the Weyl law to the orbifold setting.  Results from a variety of authors soon followed, including several examples of isospectral non-isometric orbifolds with a variety of properties.   For example, Naveed Bari, the last author, and David Webb \cite{BSW06} produced arbitrarily large finite families of mutually strongly isospectral orbifolds with non-isomorphic maximal isotropy groups of the same order.  In particular, they are $p$-isospectral for all $p$.   Shortly afterward, J. P. Rossetti, Dorothee Schueth, and Martin Weilandt \cite{RSW08} produced examples of strongly isospectral orbifolds with maximal isotropy groups of distinct orders.

Emily Dryden and Alexander Strohmaier \cite{DS09} proved that hyperbolic orientable closed 2-orbifolds are 0-isospectral if and only if they have the same geodesic length spectrum and the same number of cone points of each order.  Benjamin Linowitz and Jeffrey Meyer \cite{LM17} obtained interesting results for locally symmetric spaces.  A series of papers extended to the orbifold setting the result of Robert Brooks, Peter Perry, and Peter Petersen \cite{BPP92} that a set $\mathcal I_k(d)$ of 0-isospectral $d$-manifolds, sharing a uniform lower bound $k$ on sectional curvature, contains orbifolds of only finitely many homeomorphism types (diffeomorphism types for $d \neq 4$) .  Specifically, the last author \cite{St05} obtained an upper bound on the order of isotropy groups that can arise in elements of the set $\mathcal I_k(d)$ (expanded to contain orbifolds), the last author and Emily Proctor \cite{PrSt10} showed orbifold diffeomorphism finiteness for $\mathcal I_k(2)$, Proctor \cite{Pr12} obtained orbifold homeomorphism finiteness for $\mathcal I_k(d)$ assuming only isolated singularities, and John Harvey \cite{H16} obtained orbifold homeomorphism finiteness for the full set $\mathcal I_k(d)$. In addition, Farsi, Proctor and Christopher Seaton \cite{FPS14} introduced and studied a stronger notion of isospectrality of orbifolds.

Recall that the trace of the heat kernel of the Laplace-Beltrami operator on a closed $d$-dimensional Riemannian manifold ${M}$ is a primary source of spectral invariants.  Denoting the spectrum by $\{\lambda_j\}_{j=1}^\infty$, the heat trace (on the left below) yields an asymptotic expansion of the following form as $t \downarrow 0^{+}$:
\begin{equation*}
    \sum_{j=0}^\infty e^{-\lambda_j t}
    \sim
    (4\pi t)^{-d/2} \sum_{k=0}^{\infty} a_k({M}) \, t^k.
\end{equation*}
The coefficients $a_k$ are spectral invariants referred to as the \emph{heat invariants}. The first two, for example, give the volume and the total scalar curvature of the Riemannian manifold.

A Riemannian orbifold is said to be \emph{good} if it is a global quotient ${\mathcal{O}}=\Gamma{\backslash} M$ of a Riemannian manifold $M$ by a (possibly infinite) discrete group $\Gamma$ acting effectively and isometrically.  (We say $M$ is a Riemannian cover of ${\mathcal{O}}$ even though the group action need not be free.)   In \cite{D76}, Harold Donnelly obtained an asymptotic expansion of the form
$$
    \sum_{j=0}^\infty e^{-\lambda_j t}
    \sim
(4\pi t)^{-d/2} \sum_{k=0}^{\infty} c_k(\mathcal{{\mathcal{O}}}) \, t^{k/2}$$ for the heat trace of a good Riemannian orbifold by adding contributions from each element $\gamma\in \Gamma$.  Each $c_k({\mathcal{O}})$ is a sum of terms $c_k({\mathcal{O}},\gamma)$, $\gamma\in \Gamma$, which in turn is a sum of integrals over the connected components of the fixed point set of $\gamma$.  The integrands are universal expressions in the germs of the metric and the isometry $\gamma$.

We cite some consequences of Donnelly's heat trace asymptotics for good orbifolds:

\begin{thm}\label{thm:common good} Let ${\mathcal{O}}$ be a closed Riemannian orbifold with singularities and $M$ a closed Riemannian manifold.  If either of the following conditions hold, then ${\mathcal{O}}$ cannot be 0-isospectral to $M$:
\begin{enumerate}
\item\label{sutton}\cite[Theorem 1.2]{S10}.  ${\mathcal{O}}$ and $M$ have 0-isospectral finite Riemannian covers;
\item\label{homog cover} (See \cite{GR03}, \cite{GR03-errata})  ${\mathcal{O}}$ and $M$ are have a common Riemannian cover $\widetilde{M}$ that is a homogenous Riemannian manifold.
\end{enumerate}
\end{thm}

(We note that the article \cite{GR03} by J. P. Rossetti and the second author originally asserted that a closed Riemannian orbifold with singularities and a closed Riemannian manifold cannot be 0-isospectral if they have a common Riemannian cover $\widetilde{M}$.  Recently, a gap was found in the proof unless $\widetilde{M}$ is assumed to be either compact -- so that it is a finite cover of ${\mathcal{O}}$ and $M$ -- or else Riemannian homogeneous.  In the interim, Craig Sutton generalized the case of a common finite Riemannian cover to obtain the first statement of the theorem above.)

In \cite{DGGW08}, Dryden, the second author, Sarah Greenwald and Webb extended the heat trace asymptotics to the case of arbitrary (not necessarily good) closed Riemannian orbifolds expressing the invariants $c_k({\mathcal{O}})$ as sums of contributions from the various ``primary'' strata of the orbifold.   (See Notation and Remarks~\ref{notarem:isomax} part~\ref{it:isomax-primary} for the definition of primary.)   An orbifold ${\mathcal{O}}$ with singularities will always contain primary singular strata; the singular strata of minimal codimension in ${\mathcal{O}}$ are always primary, and any component of the singular set that contains a non-primary stratum must contain at least three primary ones (See Notation and Remarks~\ref{notarem:isomax} parts \ref{it:isomax-mincodim} and \ref{it:isomax-three}.)    As in the good case, the small-time heat trace expansion is of the form $(4\pi t)^{-d/2} \sum_{k=0}^{\infty} c_k(\mathcal{{\mathcal{O}}}) \, t^{k/2}$.    Primary strata of even, respectively odd, codimension in ${\mathcal{O}}$ contribute to coefficients $c_k$ for $k$ even, odd respectively.   As a consequence:

\begin{thm}[{\cite[Theorem 5.1]{DGGW08}}] \label{thm:dggw5.1}
A closed Riemannian orbifold that contains a primary singular stratum of odd codimension cannot be 0-isospectral to any closed Riemannian manifold.
\end{thm}

Sean Richardson and the last author \cite{RS20} proved that an orbifold ${\mathcal{O}}$ is locally orientable if and only if ${\mathcal{O}}$ does not contain any primary singular strata of odd codimension.     They thus concluded that the 0-spectrum determines local orientability of an orbifold.
In addition,
\cite[Theorem 5.15]{DGGW08} gives that the $0$-spectrum distinguishes closed, locally orientable, 2-orbifolds with non-negative Euler characteristic from smooth, oriented, closed surfaces (in fact, a stronger result is proven there, see Remark~\ref{rem:orbisurfaces}).

\bigskip
This paper is organized as follows: Section \ref{sec:background} provides background on Riemannian orbifolds, their singular strata, differential forms and the Hodge Laplacian. In Section \ref{sec:heattrace}, we construct the heat kernel, heat trace, and heat invariants for $p$-forms on orbifolds.  Our construction follows the construction in the manifold case by Gaffney \cite{G58} and Patodi \cite{P71}, uses results of Donnelly and Patodi \cite{DP77} for good orbifolds, and parallels the adaptations in \cite{DGGW08}.

 Section \ref{applications} contains all the inverse spectral results  -- both positive and negative -- discussed above as well as additional results.  Theorem~\ref{thm:main1V4} is proven in Subsection~\ref{ss:codim2}.  The negative results appear in Subsection~\ref{sec.isospec}, where we construct flat Riemannian orbifolds that are $p$-isospectral to Riemannian manifolds.  In particular, part~\ref{negative} of Theorem~\ref{thm:krawt} above appears as Proposition~\ref{it:flat-involution}.  Section~\ref{sec: common cover} addresses positive inverse spectral results for individual $p$-spectra including, in particular, Theorem~\ref{thm:krawt}, part~\ref{volume}, which appears as Theorem~\ref{thm:krawt4}, and the analogues for 1-forms of Theorems~\ref{thm:common good} and \ref{thm:dggw5.1}, which appear jointly as Theorem~\ref{thm:halfdim}.

  In an appendix, we outline the computation of the heat invariants for good orbifolds in \cite{D76} and \cite{DP77}, indicating aspects of the construction that are applicable to more general settings.

\section{Riemannian Orbifold Background}
\label{sec:background}

\subsection{Definitions and Basic Properties}\label{background}
In this section we recall the definition and basic properties of a Riemannian orbifold. We also present some examples of singular strata in orbifolds that will be revisited in later sections. For comprehensive information about orbifolds see the paper by Peter Scott \cite{Scott83}, as well as the texts \cite{Thurston97} by William Thurston and \cite{ALR07} by Alejandro Adem, Johann Leida, and Yongbin Ruan. Here we follow a somewhat abridged form of the presentation given in \cite{G12} by the second author,  and \cite{DGGW08} by Dryden, Gordon, Greenwald, and Webb.

\begin{definition}
\label{defn:ofld}~
Let $X$ be a second countable Hausdorff space.
\begin{enumerate}
\item For a connected open subset $U \subseteq X$, an \emph{orbifold chart} (of dimension $d$) over $U$ is a triple $\cc$ where $\widetilde{U} \subseteq \mathbf R^d$ is a connected open subset, $G_U$ is a finite group acting on $\widetilde{U}$ effectively and by diffeomorphisms, and $\pi_U\colon \widetilde{U} \to X$ is a map inducing a homeomorphism $\widetilde{U}/G \xrightarrow{\cong} U$. The open set $U$ is sometimes referred to as the image of the chart.

\item \label{it:injection} If $U \subseteq V \subseteq X$, an orbifold chart $\cc$ is said to \emph{inject} into an orbifold chart $\cc[V]$ if there exists a smooth embedding $i\colon \widetilde{U} \to \widetilde{V}$ and a monomorphism $\lambda\colon G_U \to G_V$ such that $\pi_U = \pi_V \circ i$ and $i \circ \gamma = \lambda(\gamma) \circ i$ for all $\gamma \in G_U$.

\item \label{it:ro} Orbifold charts $\cc$ and $\cc[V]$ on open sets $U$ and $V$ are said to be \emph{compatible} if for every $x\in U\cap V$, there exists a neighborhood $W\subset U\cap V$ of $x$ that admits an orbifold chart injecting into both $\cc$ and $\cc[V]$.   A $d$-dimensional \emph{orbifold atlas} on $X$ consists of a collection of compatible $d$-dimensional orbifold coordinate charts whose images cover $X$.  An \emph{orbifold} is a second countable Hausdorff topological space together with a maximal $d$-dimensional atlas.

\item A Riemannian structure on an orbifold ${\mathcal{O}}$ is an assignment of a $G_U$-invariant Riemannian metric $\widetilde{U}$ to each orbifold chart $\cc$, compatible in the sense that each injection of charts as in part~\ref{it:injection} is an isometric embedding.

\item If an orbifold is the quotient space of a manifold under the
smooth action of a discrete group it is called a \emph{good} orbifold. Otherwise, it is called a \emph{bad} orbifold.
\end{enumerate}
\end{definition}

\begin{definition}\label{def:iso}
Let $\mathcal O$ be a Riemannian orbifold and let $x \in \mathcal O$.
\begin{enumerate}

\item \label{it:isotropy}A chart $\cc$ about $x$ defines a smooth action of $G_{U}$ on $\widetilde U \subset \mathbb R^d$. Fix a lift $\widetilde x \in \widetilde{U}$ of $x$. The set of $\gamma \in G_U$ such that $\gamma \cdot \widetilde{x}=\widetilde{x}$ is called the \emph{isotropy group} of $x$ denoted ${\operatorname{{Iso}}}(x)$. An element $\gamma \in {\operatorname{{Iso}}}(x)$ defines an invertible linear map $d\gamma_x: T_{\widetilde{x}}\widetilde{U} \to T_{\widetilde{x}}\widetilde{U}$, which gives an injective linear representation of the isotropy group of $x$.    Every finite dimensional linear representation is equivalent to an orthogonal representation, unique up to orthogonal equivalence.    Thus ${\operatorname{{Iso}}}(x)$ can be viewed as a subgroup of the orthogonal group $O(d)$, unique up to conjugacy.

\item \label{it:welldef}  The \emph{isotropy type} of $x$ is the conjugacy class of the isotropy group of $x \in O(d)$. This definition is independent of both the choice of lift of $x$ and the chart.

\item We say $x$ is a \emph{regular point} if it has trivial isotropy type, and a \emph{singular point} otherwise.

\end{enumerate}
\end{definition}

We conclude our discussion of the basic properties of Riemannian orbifolds by describing their singular stratification. Define an equivalence relation on $\mathcal O$ by saying that two points in $\mathcal O$ are \emph{isotropy equivalent} if they have the same isotropy type (this is well defined by Definition~\ref{def:iso} part~\ref{it:welldef}). The connected components of the isotropy equivalence classes of $\mathcal O$ form a smooth stratification of $\mathcal O$. When the corresponding isotropy type is non-trivial these strata are called \emph{singular strata}.  Note that in the literature, the requirement that a singular stratum be connected is sometimes dropped.

\begin{notarem}\label{notarem:isomax}~
\begin{enumerate}
\item \label{it:isomax-primary} Let $N$ be a singular stratum in $\mathcal O$ and let $\operatorname{Iso}(N)$ denote a representative of its isotropy type, unique up to conjugacy. We denote by $\iiso^{\rm max}(N)$ the set of all elements $\gamma \in \operatorname{Iso}(N)$ such that in a coordinate chart $\cc$ over a neighborhood $U$ about a point in $N$, $\pi_U^{-1}(U \cap N)$ is open in the fixed point set of $\gamma$. We say that a stratum $N$ is \emph{primary} if $\iiso^{\rm max}(N)$ is non-empty. One can check that these definitions are independent of the choice of chart and of the choice of subgroup in the isotropy type of $N$.
\item \label{it:isomax-mincodim} Not every singular stratum is primary. (See Example 2.16 in \cite{DGGW17}.) The singular strata of minimal codimension must be primary. (See \cite[Notation and Remarks 2.14]{DGGW17}.)
\item \label{it:isomax-three} A component of the singular set $C$ that contains a non-primary singular stratum $N$ must also contain at least three primary singular strata. To see this consider a small enough neighborhood of $x$ for which we can take a coordinate chart $\cc$ about $x$ with $\widetilde U$ a ball about the origin in $\mathbf R^d$, $\pi_U(0)=x$, and $G_U = \operatorname{Iso}(N)$ acting on $\widetilde U$ by orthogonal transformations. For $\gamma \in \operatorname{Iso}(N)$, $Fix(\gamma)$ will be the intersection of a subspace of $\mathbf R^d$ with $\widetilde U$. Now $\gamma \in \operatorname{Iso}(N)$ must be in $\iiso^{\rm max}(N')$ for some stratum $N' \subset C$. If all nontrivial $\gamma \in \operatorname{Iso}(N)$ were in $\iiso^{\rm max}(N')$, we would have $\operatorname{Iso}(N)=\operatorname{Iso}(N')$ and $N$ and $N'$ wouldn't be distinct strata. Thus there are at least two primary singular strata $N',N'' \subset C$.  Suppose these were the only two. Then for every nontrivial element $\gamma\in \operatorname{Iso}(N)$, at least one of $\operatorname{Iso}(N')$ or $\operatorname{Iso}(N'')$ contains $\gamma$.    Since neither $N'$ nor $N''$ contain all elements of $\operatorname{Iso}(N)$, there are $\gamma_1, \gamma_2 \in \operatorname{Iso}(N)$ for which $\gamma_1 \notin \operatorname{Iso}(N')$ and $\gamma_2 \notin \operatorname{Iso}(N'')$.  Then we must have  $\gamma_1\in \operatorname{Iso}(N'')$ and $\gamma_2\in \operatorname{Iso}(N')$. Now $\gamma_1\gamma_2$ needs to be in one of these two isotropy groups, say $\operatorname{Iso}(N')$. But this implies $\gamma_1 \in \operatorname{Iso}(N')$, a contradiction.
\item We observe that $\iiso^{\rm max}(N)$ consists of all $\gamma\in {{\operatorname{{Iso}}}(N)}$ such that the multiplicity of $1$ as an eigenvalue of the orthogonal representation of $ \gamma$ is precisely the dimension of $N$.
\end{enumerate}
\end{notarem}

\subsection{Examples of Orbifold Singular Strata}

The singular strata described below will be important to the discussion in Section~\ref{applications}.

\begin{ex}
By considering all the finite subgroups of the orthogonal group $O(2)$, we obtain a classification of the isotropy types of singular points that can occur in dimension $2$:
\begin{enumerate}
\item A rotation about the origin in $\mathbf R^2$ through an angle $\frac{2\pi}{k}$, for some integer $k\geq2$, generates a finite cyclic group of order $k$. The image of the origin in the quotient of $\mathbf R^2$ under this action is an isolated singular point called a \emph{cone point} of order $k$.
\item A reflection across a line through the origin generates a cyclic group of order $2$. The fixed points of the reflection correspond to a $1$-dimensional singular stratum in the quotient called a \emph{mirror} or \emph{reflector edge}.
\item A dihedral group generated by a pair of reflections across lines forming an angle $\frac{\pi}{k}$, for $2\leq k\in \mathbb Z^+$, yields a $0$-dimensional stratum in the quotient called a \emph{corner reflector} or \emph{dihedral point}. The corner reflector is not an isolated singular point, rather it is the point
where two mirror edges intersect.
\end{enumerate}
\end{ex}

\begin{ex}
Starting from an orbifold $\mathcal O$ of dimension $\ell$, we can construct orbifolds of higher dimension $d$ by taking a product with a manifold $M$ of dimension $d-\ell$. For example, let $\mathcal O$ be an orbifold with a single cone point $x_0$ and underlying topological space $S^2$, and let $M$ be a closed $(d-2)$-dimensional manifold. The product $\mathcal{O} \times M$ has the structure of a $d$-dimensional orbifold with a unique singular stratum $\{x_0\} \times M$ of codimension $2$.
\end{ex}

\begin{nota}\label{nota:Rstuff} Let $r$ and $s$ be integers with $0 \le r \le d$ and $0 \le 2s \le d$, and $\theta_1, \dots \theta_s \in (0,\pi)$. For a given $\gamma \in O(d)$, the expression $E(\theta_1, \theta_2, \dots, \theta_s;r)$ will be called the \emph{eigenvalue type} of $\gamma$ and indicates that $\gamma$ has eigenvalues $e^{\pm i \theta_j}$ for $1 \le j \le s$, $-1$ with multiplicity $r$, and $1$ with multiplicity $d-2s-r$. We emphasize that the $\theta_j$ are repeated according to multiplicity. When $r=0$ we write $E(\theta_1, \theta_2, \dots, \theta_s;)$ and when $s=0$ we write $E(;r)$. We note that the parity of $r$ determines the parity of the codimension of the fixed point set.
\end{nota}

\begin{remark} We take $\theta_j \neq \pi$ for all $1 \le j \le s$ for the notational convenience of letting $r$ count the total number of eigenvalues equal to $-1$. This convention can be dropped without affecting the results of Section~\ref{applications}.
\end{remark}

\subsection{Differential forms and the Hodge Laplacian}

As the main results of this article concern the spectrum of the Laplacian acting on $p$-forms on a closed Riemannian orbifold, we give a brief description of how to define differential forms on orbifolds.
For details of the constructions we refer to \cite{C19} or \cite{W12}. We take $\mathcal O$ to be a $d$-dimensional orbifold throughout this section.

\begin{defn}\label{def:pform}
\hspace{1in}
\begin{itemize}
\item As in the case of manifolds, a $p$-form $\omega$ on an orbifold ${\mathcal{O}}$ is defined to be a section of the $p$-th exterior power of the cotangent bundle $T^*(\mathcal O)$.   The cotangent bundle and its exterior powers are examples of orbibundles; see, for example, \cite{C19} or \cite{G12} for an expository introduction to orbibundles or the references therein for more in-depth explanations.   We will only use the following:   If $U\subset {\mathcal{O}}$ is an open set on which there exists a coordinate chart $(\widetilde U, G_U, \pi_U)$, then any $p$-form $\omega$ on $U$ corresponds to a $G_U$-invariant $p$-form $\widetilde{\omega}$ on $\widetilde U$, which we call the lift of $\omega$ to ${\widetilde{U}}$. The differential form $\omega$ is said to be of class $C^k$ on $U$ if $\widetilde{\omega}$ is of class $C^k$.  The compatibility condition on overlapping charts in Definition~\ref{defn:ofld} part~\ref{it:ro} leads to a natural compatibility condition on the lifts of $p$-forms.   One can use partitions of unity to construct globally defined $p$-forms from $p$-forms defined on an open covering of ${\mathcal{O}}$.

\item For a Riemannian manifold $M$, recall that the Hodge Laplacian $\Delta^p$ acting on the space of smooth $p$-forms $\Omega^p(M)$ is given by
\begin{align}
\label{defhodge}
\Delta^p:= -(d \delta + \delta d),
\end{align}
where $d$ is the exterior derivative and $\delta$ is the codifferential operator, i.e.\ the formal adjoint of $d$ obtained via the Riemannian metric.  It is straightforward to check that the Hodge Laplacian commutes with isometries.   The notion of Hodge Laplacian extends to Riemannian orbifolds ${\mathcal{O}}$ as follows:   Let $\omega$ be a smooth $p$-form on ${\mathcal{O}}$.  On each coordinate chart $\cc$, the $G$-invariance of the lift $\widetilde{\omega}$ and the fact that the Hodge Laplacian on Riemannian manifolds commutes with isometries together imply that $\Delta^p(\widetilde{\omega})$ is $G_U$-invariant.   We define $\Delta^p(\omega)$ on $U$ by
$$\widetilde{\Delta^p(\omega)}=\Delta^p(\widetilde{\omega}).$$
We can again use the fact that isometries between Riemannian manifolds intertwine the Hodge Laplacians along with the compatibility condition on orbifold charts to conclude that $\Delta^p(\omega)$ is well-defined on ${\mathcal{O}}$.
\end{itemize}
 \end{defn}

\begin{remark}\label{hodgestar-adjoint} The results in the present work apply to both orientable and nonorientable orbifolds and manifolds.  For clarity we note the following:
\begin{itemize}
\item In what follows the notation $d V_{M}$ indicates the Riemannian volume element in the orientable case, and the Riemannian density in the nonorientable case.
\item The codifferential operator on a Riemannian  manifold is a differential operator and can be expressed in terms of the Hodge-$*$ operator in any orientable neighborhood by the expression
\begin{align}
    \label{codifforiented}
\delta = (-1)^{n(p+1)+1}*d*,
\end{align}
which is independent of the choice of orientation.
\end{itemize}
\end{remark}

It is known, see e.g.\ \cite[Theorem 4.8.1]{B99}, that the operator $\Delta^p$ viewed as an unbounded operator acting in $ L_p^2(\mathcal O)$, the space of differential $p$-forms whose components are square-integrable functions, is essentially self-adjoint and has a purely discrete spectrum. By abuse of notation, in what follows we denote its closure by $\Delta^p$, which is then self-adjoint with compact resolvent. The spectral theorem thus applies to
$\Delta^p$ and we denote its eigenvalues by $0\leq\lambda_1 \leq \lambda_2\leq\dots \to +\infty$, with associated smooth eigenforms, denoted by $(\varphi_i)_i$, which form an orthonormal $ L_p^2$-basis.

We conclude this section by reviewing the notion of multi $p$-forms.
\begin{nota}\label{nota:doubleform}~
\begin{enumerate}
\item Recall that if $\pi: B\to M$ and $\pi':B'\to M'$ are vector bundles over manifolds $M$ and $M'$, then the external tensor product $B\boxtimes B'\to M\times M'$ is the vector bundle whose fiber over the point $(m,m')$ is $\pi^{-1}(m)\otimes \pi^{-1}(m')$.    Given a $C^\infty$ manifold $M$, let $M^k$ denote the $k$-fold Cartesian product $M\times \dots\times M$.  Sections of the $k$-fold external tensor product $\boxtimes^k(\wedge^p T^*(M))\to M^k$ are called multi $p$-forms on $M$ of order $k$.    In particular, a multi $p$-form of order $1$ is simply a $p$-form.  Those of order $2$ or $3$ are called double or triple $p$-forms, respectively.
Thus for example, if $(\mathcal U,x)$ and   $(\mathcal V,y)$ are local coordinate charts on $M$ and $F$ is a double $p$-form on $M$, then $F_{|\mathcal U\times \mathcal V}$ can be expressed as $\sum_{I, J} a_{I,J} dx^I \otimes dy^J,
$
where $a_{I,J} \in C^\infty(\mathcal U \times \mathcal V)$.  (Here $I$ and $J$ vary over $p$-tuples $ 1\leq i_1<\dots <i_p\leq d=\dim(M).$)

\item \label{it:gammasigma}  Given smooth manifolds $M$ and $N$ and smooth maps $\gamma: M\to M$ and $\sigma :N\to N$, denote by $\gamma_{1,\cdot}$ and $\sigma_{\cdot,2}$ the maps $M\times N\to M\times N$ given by $(a,b)\mapsto (\gamma(a), b)$ and $(a,b)\mapsto (a, \sigma(b))$.    If $M=N$, we also define $\gamma_{1,2}: M^2\to M^2$ by $(a,b)\mapsto (\gamma(a),\gamma(b))$.    We use analogous notation for maps of higher order Cartesian products.  Thus, for example, the map $\gamma_{1,\cdot,3}: M^3\to M^3$ is given by $(m_1,m_2,m_3)\mapsto (\gamma(m_1), m_2,\gamma(m_3)).$

\item \label{it:multipformofld} Analogous to the case of $p$-forms in Definition~\ref{def:pform}, we can use the notion of orbibundle to extend the definition of multiple $p$-forms directly to the orbifold setting, but we will not need the orbibundle formalism in what follows.   If $U$ and $V$ are open sets in an orbifold ${\mathcal{O}}$ on which there exist orbifold coordinate charts $\cc$ and $(\widetilde{V},G_V, \pi_V)$, then the restriction to $U\times V$ of a double $p$-form $F$ on ${\mathcal{O}}$ lifts to a section $\tilde{F}$ of $\wedge^p T^*({\widetilde{U}})\boxtimes \wedge^p T^*(\widetilde{V})$ that is invariant under pullback by each of the maps $\gamma_{1,\cdot}$ and $\sigma_{\cdot, 2}$ for $\gamma\in G_U$ and $\sigma\in G_V$.    Again, the compatibility condition on overlapping charts in Definition~\ref{defn:ofld} part~\ref{it:ro} of an orbifold leads to a natural compatibility condition on the lifts of double $p$-forms.  One can use partitions of unity to construct double $p$-forms on ${\mathcal{O}}$ from ones on an open covering of ${\mathcal{O}}\times{\mathcal{O}}$.   Multiple $p$-forms of any order are defined similarly.

\end{enumerate}

\end{nota}

Observe that if $F$ and $H$ are multi $p$-forms, say of orders $k$ and $\ell$, respectively, then $F\otimes H$ is a multi $p$-form of order $k+\ell$.

\begin{definition}\label{defcontraction}
Recall that if  $V_1,...,V_k$ are inner product spaces and if $1\leq i <j \leq k$ with $V_i=V_j$, then the contraction
$
C_{i,j}: V_1\otimes...\otimes V_n\to V_1\otimes...\otimes\widehat{V_i}\otimes...\otimes\widehat{V_j}\otimes...\otimes V_k $ is the linear map whose value on decomposable vectors is given by
$$\alpha_1\otimes ...\otimes \alpha_n \mapsto \langle \alpha_i, \alpha_j\rangle \alpha_1\otimes..\otimes \widehat{\alpha_i}\otimes...\otimes \widehat{\alpha_j}\otimes ... \otimes \alpha_k$$ where $\langle\,,\,\rangle$ is the inner product on $V_i$.

In particular, let $(M,g)$ be a Riemannian manifold, $F$ a multi $p$-form of order $k$ on $M$, and suppose $m=(m_1,\dots, m_k)\in M^k$ satisfies $m_i=m_j$.    Then we can use the inner product on $\wedge^pT^*_{m_i}(M)$ to define the contraction $C_{i,j}(F(m)).$

To define the contraction of such a multi $p$-form in the setting of an orbifold ${\mathcal{O}}$ at a point $m=(m_1,\dots, m_k)\in {\mathcal{O}}^k$ with $m_i=m_j$, let $\widetilde{F}$ be a local lift on a neighborhood $U_1\times \dots \times U_k$ as in Notation~\ref{nota:doubleform} part~\ref{it:multipformofld}.   Choose the lift $\widetilde{m}$ of $m$ so that $\widetilde{m}_i=\widetilde{m}_j\in \widetilde{U}_i=\widetilde{U}_j$.  We can then define $C_{ij}(F(m))$  by the condition that it lifts to $C_{ij}(\widetilde{F}(\widetilde{m}))$.   This definition is independent of the choice of $\widetilde{m}$ (subject only to the condition $\widetilde{m}_i=\widetilde{m}_j$), and it is independent of the choice of chart since the compatibility condition on charts respects the Riemannian structure.
\end{definition}

\section{Heat trace for differential forms on orbifolds}
\label{sec:heattrace}

In Subsection~\ref{subsec:kernel} we will construct the fundamental solution of the heat equation for the Hodge Laplacian on arbitrary closed Riemannian orbifolds by first constructing a parametrix.  Our construction follows and generalizes the construction in the manifold case by Gaffney \cite{G58} and Patodi \cite{P71}.  (The assumption of orientability in the work of Gaffney and Patodi is not needed.)  In Subsection~\ref{subsec:Don}, we translate a result of Donnelly into our context and employ it
to develop the small-time asymptotic expansion of the heat trace.   Our arguments are similar to those used in \cite{DGGW08} to construct the heat kernel and heat trace asymptotics for the Laplacian acting on functions on a closed Riemannian orbifold.

\subsection{Heat kernel for $p$-forms on manifolds and orbifolds}\label{subsec:kernel}

\begin{definition}\label{fundsol}
We say that $K^p: \mathbb (0,\infty) \times  {\mathcal{O}} \times  {\mathcal{O}} \to \Lambda^p( T^* {\mathcal{O}}) \otimes \Lambda^p (T^* {\mathcal{O}})$ is a \emph{fundamental solution} or \emph{heat kernel} of the heat equation for $p$-forms if it satisfies
\begin{enumerate}
\item $K^p(t,\cdot,\cdot)$ is a double $p$-form  for any $t\in (0,\infty)$;
\item $K^p$ is continuous in the three variables, $C^1$ in the first variable and $C^2$ in the second variable;
\item $(\partial _t + \Delta^p_x ) K^p(t,x,y)= 0$ for each fixed $y\in {\mathcal{O}}$, where $\Delta^p_x$ is the Hodge Laplacian with respect to the variable $x$;
\item\label{initcond} For every continuous $p$-form $\omega$ on ${\mathcal{O}}$ and for all $x\in {\mathcal{O}}$, we have
$$\lim_{t\to 0^+}\,\int_{\mathcal{O}}\,C_{2,3} K^p(t,x,y)\otimes \omega(y) \, dV_{\mathcal{O}}(y) = \omega(x).$$
\end{enumerate}
(Because the variable $t$ in the preceding definition is a real parameter, in contrast to a space variable, it is not counted when specifying indices in expressions using Notation~\ref{nota:doubleform} part~\ref{it:gammasigma}, or in expressions involving contractions using the notation from Definition~\ref{defcontraction}. Also, the variables over which a contraction is occuring will be repeated.  For example, in part~\ref{initcond} above, $C_{2,3} K^p(t,x,y)\otimes \omega(y)$ indicates contraction in the second and third entries, not counting $t$.)
\end{definition}

The same argument as in the manifold case shows that if the heat kernel exists, then it is unique and given by
\begin{equation}\label{fundsoldvp}
K^p(t,x,y)= \sum_{j= 1}^{\infty} \varphi_j\otimes\varphi_j(x,y) e^{-\lambda_j t}
\end{equation}
where $\{\varphi_j\}_{j=1}^\infty$ is an orthonormal basis of $L_p^2({\mathcal{O}})$ consisting of eigenfunctions and the $\lambda_j$ are the corresponding eigenvalues as above.  In particular, the heat kernel is invariant under any isometry $\gamma$, i.e.,
\begin{equation}(\gamma^*_{1,2} K^p)(t,x,y)=K^p(t,x,y)\end{equation}
in the notation of Notation~\ref{nota:doubleform} part~\ref{it:gammasigma}.

\begin{defn}\label{def.param}
A \emph{parametrix} for the heat operator on $p$-forms on ${\mathcal{O}}$ is a function
$H:(0,\infty)\times{\mathcal{O}}\times{\mathcal{O}}\to \Lambda^p(T^* {\mathcal{O}}) \otimes \Lambda^p(T^*{\mathcal{O}})$  satisfying:
\begin{enumerate}
\item $H(t,\cdot,\cdot)$ is a double $p$-form for each $t\in \mathbb (0,\infty)$;
\item $H$ is  $C^\infty$ on $(0,\infty)\times{\mathcal{O}}\times{\mathcal{O}}$;
\item $(\partial_t + \Delta^p_x) H(t,x,y)$ extends continuously to $[0,\infty)\times{\mathcal{O}}\times{\mathcal{O}}$;
\item For every continuous $p$-form $\omega$ on ${\mathcal{O}}$ and for all $x\in {\mathcal{O}}$, we have $$\lim_{t\to 0^+}\,\int_{\mathcal{O}}\,C_{2,3} H(t,x,y)\otimes\omega(y)\,dV_{\mathcal{O}}(y) = \omega(x).$$
\end{enumerate}
\end{defn}

We first recall the construction of a parametrix by Patodi, see also Gaffney \cite{G58}.

\begin{prop}[{\cite{P71}}]
\label{localparam}
There exist smooth double $p$-forms $u^p_i$ for $i=0,1,2,\dots$ defined on a neighborhood of the diagonal of $M\times M$ in any Riemannian manifold $M$ of dimension $d$ (more precisely, $u^p_i(x,y)$ is well-defined whenever $y$ is in a normal neighborhood about $x$) satisfying the following.

\begin{enumerate}
\item \label{uzero}
If $\{\omega_1,\dots, \omega_r\}$, where  $r=\binom{d}{p}$, is an orthonormal basis of $\Lambda^p\,T^*_x(M)$, then
$$u_0^p(x,x)=\sum_{j=1}^r\, \omega_j\otimes \omega_j.$$
(Note that this expression is independent of the choice of orthonormal basis).   Equivalently under the identification of $\Lambda^p\,T^*_x(M)\otimes \Lambda^p\,T^*_x(M)$ with $\operatorname{End}(\Lambda^p\,T^*_x(M)) $,
$u^p_0(x,x)$ is the identity endomorphism of $\Lambda^p( T_x^*M)$.
\item \label{um} For each $m=1,2,\dots$, we have
$$(\partial_t + \Delta^p_x) H^{(m)}(t,x,y)= (4\pi)^{-d/2}e^{-d^2(x,y)/4t}t^{m-\frac{d}{2}}\Delta^p_x u^p_m(x,y),$$
 where
$$H^{(m)}(t,x,y):= {(4\pi t)^{-d/2}e^{-d^2(x,y)/4t}}{(u^p_0(x,y)+\dots +t^mu^p_m(x,y))},$$
and $d^2(x,y)$ is the square of the Riemannian distance between $x$ and $y$.
In particular, if $\psi:M\times M\to \mathbf R$ is supported on a sufficiently small neighborhood of the diagonal and is identically one on a smaller neighborhood of the diagonal, then  $\psi H^{(m)}$ is a parametrix for the heat operator on $p$-forms when $m>\frac{d}{2}$.

\item \label{traceinv} The $u_i^p$ are the unique double $p$-forms satisfying conditions~\ref{uzero} and \ref{um}.  If $\gamma$ is an isometry of an open set in the domain of $u^p_i $, then $\gamma^*_{1,2}u^p_i=u^p_i$.
\end{enumerate}
\end{prop}

\begin{notarem}\label{notarem:hm} Fix ${\epsilon}>0$ so that each $x\in{\mathcal{O}}$ is the center of a ``convex geodesic ball'' $W$ of radius $\epsilon$.   By this we mean that W admits a coordinate chart $(\widetilde{W},G_W,\pi_W)$, where $\widetilde{W}$ is a convex geodesic ball of radius $\epsilon$ centered at the necessarily unique point ${\widetilde{x}}$ satisfying $\pi_U({\widetilde{x}})=x$.   (In particular, ${\widetilde{x}}$ has isotropy $G_W$.)     Cover ${\mathcal{O}}$ by finitely many such geodesic balls ${W_\a}$ with charts
$({\widetilde{W}_\a}, G_{\alpha},\pi_{\alpha})$, $\alpha=1,\dots, s$.  (Here we write $G_{\alpha}$ for
$G_{{W_\a}}$ and $\pi_{\alpha}$ for $\pi_{W_\a}$.)   Let ${\widetilde{U}_\a}\subset {\widetilde{W}_\a}$,
respectively ${\widetilde{V}_\a}\subset{\widetilde{W}_\a}$, be the concentric geodesic balls of radius $\frac{{\epsilon}}{4}$, respectively
$\frac{{\epsilon}}{2}$, and let ${U_\a}=\pi_{\alpha}({\widetilde{U}_\a})$ and ${V_\a}=\pi_{\alpha}({\widetilde{V}_\a})$.  We may assume that the family of balls
$\{{U_\a}\}_{1\le{\alpha}\le s}$ still covers ${\mathcal{O}}$.

For each ${\alpha}$ and each nonnegative integer $m$, we define  ${\widetilde{H}^{(m)}_\a}:
(0,\infty)\times{\widetilde{W}_\a}\times{\widetilde{W}_\a}$ by
\begin{align}
\label{nota.hm}
{\widetilde{H}^{(m)}_\a}(t,{\widetilde{x}},{\widetilde{y}}):={(4\pi t)^{-d/2}e^{-d(\tx,\ty)^2/4t}}{(u^p_0(\tx,\ty)+\dots +t^mu^p_m(\tx,\ty))},
\end{align}
where the $u^p_i$ are the double $p$-forms defined in Proposition~\ref{localparam}.
Since each ${\gamma}\in{G_\a}$ is an isometry of ${\widetilde{W}_\a}$, we have
$\gamma^*_{1,2}u^p_i=u^p_i$.

It follows that the double $p$-form (depending on the parameter $t$)

$$\sum_{\gamma\in{G_\a}}\gamma^*_{\cdot,2}{\widetilde{H}^{(m)}_\a}$$
is ${G_\a}$-invariant in both ${\widetilde{x}}$ and ${\widetilde{y}}$ and
thus descends to a well-defined function on
$(0,\infty)\times{W_\a}\times{W_\a}$, which we denote by ${H^{(m)}_\a}$.

Let ${\psi_\a}:{\mathcal{O}}\to\mathbf R$ be a $C^\infty$ cut-off function  which is identically one
on ${V_\a}$
and is supported in ${W_\a}$.  Let $\{{\eta_\a} :{\alpha}=1,\dots,s\}$ be a partition of
unity on
${\mathcal{O}}$ with the support of ${\eta_\a}$ contained in  $ \overline{{U_\a}}$.   Define ${L^{(m)}}$ on $(0,\infty)\times{\mathcal{O}}\times{\mathcal{O}}$
by
\begin{equation}\label{eq.hm}
{L^{(m)}}(t,x,y):=\sum_{{\alpha}
=1}^s\,{\psi_\a}(x){\eta_\a}(y){H^{(m)}_\a}(t,x,y).
\end{equation}

\end{notarem}
\begin{prop}\label{prop.param}
${L^{(m)}}$ is a parametrix for the heat kernel on ${\mathcal{O}}$ when
$m>\frac{d}{2}$.

Moreover, the extension of ${\left(\partial_t+\Delta^p_x\right)} {L^{(m)}}(t,x,y)$ to $[0,\infty)\times{\mathcal{O}}\times{\mathcal{O}}$ is of class $C^k$ if $m>\frac{d}{2}+k$ for any $k\in \mathbb N$.

\end{prop}
\begin{proof}
The proof is similar to that carried out in \cite{DGGW08} (see also references therein) for the case of the Laplacian on functions.   The first three properties of a parametrix and the final statement of the proposition are straightforward.     We include here the proof of the last property of a parametrix (the reproducing property).

Let $\omega$ be a continuous $p$-form on ${\mathcal{O}}$.
Let ${\widetilde{\psi}_\a}$ and ${\widetilde{\eta}_\a}$ be the lifts of ${\psi_\a}$ and ${\eta_\a}$ to
${\widetilde{W}_\a}$.
We consider the lifts so as to make use of Proposition~\ref{localparam}
part~\ref{um} and we also use the fact that the properties of a parametrix hold locally (see, \cite[Remark 3.7]{DGGW08}).

Since ${\rm supp}({\eta_\a})\subset\overline{{U_\a}}\subset{W_\a}$, we have
\begin{equation}
    \label{eq.ham}\begin{aligned}
&\pi_{\alpha}^*\int_{\mathcal{O}}{\psi_\a}(x){\eta_\a}(y)C_{2,3}{H^{(m)}_\a}(t,x,y)\otimes \omega(y)\,dy\\&
=\frac{{\psi_\a}(x)}{|G_{\alpha}|}\sum_{{\gamma}\in{G_\a}}\int_{\widetilde{W}_\a}\,{\widetilde{\eta}_\a}({\widetilde{y}})C_{2,3}{\gamma}_{.,2}^*{\widetilde{H}^{(m)}_\a}(t,{\widetilde{x}},{\widetilde{y}})\otimes
\widetilde{\omega}_\alpha({\widetilde{y}}) \, d{\widetilde{y}},
\end{aligned}
\end{equation}
where $\widetilde{\omega}_\alpha$
  is the pull-back of $\omega_{|{\widetilde{W}_\a}}$ to
${\widetilde{W}_\a}$ and ${\widetilde{x}}$ is an arbitrarily chosen point in the preimage of $x$
under the map ${\widetilde{W}_\a}\to{W_\a}$. We change variables in each of the integrals in the right-hand side of
Equation~\eqref{eq.ham}, letting ${\widetilde{u}}={\gamma}^{-1}({\widetilde{y}})$.  Since
${\gamma}$ is an isometry and since ${\widetilde{\eta}_\a}$ and $\widetilde{\omega}_\alpha$
are ${\gamma}$-invariant, each integral in the summand is equal to
\begin{equation}\label{eq.tham}\int_{\widetilde{W}_\a}\,{\widetilde{\eta}_\a}({\widetilde{u}})C_{2,3} {\widetilde{H}^{(m)}_\a}(t,{\widetilde{x}},{\widetilde{u}})\otimes
\widetilde{\omega}_\alpha({\widetilde{u}}) \,d{\widetilde{u}}.\end{equation}
As ${t\to 0^+}$, the
integral \eqref{eq.tham} above converges to
${\widetilde{\eta}_\a}({\widetilde{x}})\widetilde{\omega}_\alpha({\widetilde{x}})={\eta_\a}(x)\pi_{\alpha}^*(\omega(x))$.  Noting
that ${\psi_\a}\equiv 1$ on the support of ${\eta_\a}$, it follows that  both sides of
Equation~\eqref{eq.ham} converge to ${\eta_\a}(x)\pi_{\alpha}^*(\omega(x))$ as ${t\to 0^+}$.    Since both sides of
Equation~\eqref{eq.ham} are identically zero when $x$ lies outside of
${\rm supp}({\psi_\a})\subset {W_\a}$, the left-hand side converges to
${\eta_\a}(x)\pi_{\alpha}^*(\omega(x))$ for each fixed $x\in {\mathcal{O}}$.
Thus
\[
\lim_{t\to 0^+}\int_{\mathcal{O}}{\psi_\a}(x){\eta_\a}(y)C_{2,3} {H^{(m)}_\a}(t,x,y)\otimes \omega(y)\,dy ={\eta_\a}(x)\omega(x),
\]
  and by  Equation~\eqref{eq.hm} we have
\[
\lim_{t\to 0^+}\int_{\mathcal{O}}\,{L^{(m)}}(t,x,y)\omega(y)\,dy=\sum_{\alpha}\,{\eta_\a}(x)\omega(x)=\omega(x). \qedhere
\]
\end{proof}

The construction of the heat kernel from the parametrix ${L^{(m)}}$ then follows exactly as in \cite{G58}.

\begin{nota}\label{conv} ~
\begin{enumerate}
\item For $A$ and $B$ continuous double $p$-form valued functions on $[0,\infty)\times{\mathcal{O}}\times{\mathcal{O}}$, we define the convolution $A*B$ on $(0,\infty)\times{\mathcal{O}}\times{\mathcal{O}}$ as
$$A*B(t,x,y):=\int_0^t\,d\theta\int_{\mathcal{O}}\,C_{2,4} A(\theta,x,z)\otimes B(t-\theta,y,z) dV_{\mathcal O}(z).$$
\item Fix
$m>\frac{d}{2} +2$.  Define $\kappa_0(t,x,y)
:={\left(\frac{\partial}{\partial t}+\Delta_x\right)}{L^{(m)}}(t,x,y)$ and, for $j=1,2,\dots$, set $\kappa_j(t,x,y):=\kappa_0*\kappa_{j-1}$.
\end{enumerate}
\end{nota}

An argument analogous to that in the manifold case \cite{G58} shows that the series
\begin{equation}\label{lem.qm} P_m(t,x,y):=\sum_{j=1}^\infty\,(-1)^{j+1}\kappa_j(t,x,y)\end{equation}
converges uniformly and absolutely on $[0,T]\times{\mathcal{O}}\times{\mathcal{O}}$ for each $T>0$.  Thus $P_m$ is continuous.  Moreover, $P_m$ is of class $C^2$ on ${\R_+}\times{\mathcal{O}}\times{\mathcal{O}}$ and for any $T>0$, there exists a constant $C$ such that
\begin{equation} P_m(t,x,y)\leq Ct^2\end{equation}
on $[0,T]\times{\mathcal{O}}\times{\mathcal{O}}$.
(Our indexing is different from that in \cite{G58}.   Our $\kappa_j$ is Gaffney's $\kappa_{j+1}$.)

We are now able to state the main theorem of this section, which follows in the same way as that in \cite{G58}.
\begin{thm}
\label{lem.conv}  Let $m>\frac{d}{2}+2$. Define  $P_m$ as in Equation~\eqref{lem.qm} and let
$$K^p(t,x,y) :={L^{(m)}}(t,x,y)
+(P_m*{L^{(m)}})(t,x,y).$$
Then $K^p$ is a heat kernel.
Moreover,
$$K^p(t,x,y)={L^{(m)}}(t,x,y) +O(t^{m-\frac{d}{2}+1}), \text{ as } t\to 0^+.$$
\end{thm}

\begin{nota}\label{nota.ha}  Let
$${\widetilde{H}_\a}(t,{\widetilde{x}},{\widetilde{y}})=\sum_{{\gamma}\in{G_\a}}{(4\pi t)^{-d/2}e^{-d(\tx,\g(\ty))^2/4t}}{(\g_{.,2}^*u^p_0(\tx,\ty)+ t\g_{.,2}^*u^p_1(\tx,\ty)+\dots)}.$$
Observe that ${\widetilde{H}_\a}$ is ${G_\a}$-invariant in both ${\widetilde{x}}$ and ${\widetilde{y}}$ and thus is the pull-back
of a double $p$-form valued function, which we denote by ${H_\a}$,
on $(0,\infty)\times{W_\a}\times{W_\a}$.
\end{nota}

As a consequence of Theorem~\ref{lem.conv}, we have:

\begin{thm}
\label{thm.heattrace}  In the notation of Equation\eqref{eq.hm}
and Notation~\ref{nota.ha}, the trace of the heat kernel has an asymptotic expansion as
${t\to 0^+}$ given by
\[
\tr K^p(t):=\int_{\mathcal{O}}\,C_{1,2}K^p(t,x,x)\,dV_{\mathcal O}(x)
\sim\,\sum_{{\alpha}=1}^s\,\int_{\mathcal{O}}\,{\eta_\a}(x)C_{1,2}{H_\a}(t,x,x)\,dV_{\mathcal O}(x).
\]
\end{thm}

\subsection{Heat invariants for $p$-forms on orbifolds}\label{subsec:Don}

We will use Theorem~\ref{thm.heattrace} together with a theorem of Donnelly \cite[Theorem 4.1]{D76} (see also \cite[Theorem 3.1]{DP77}) to compute the small-time asymptotics of the heat trace for the Laplacian on $p$-forms on closed Riemannian orbifolds.
Our presentation is similar to that in \cite{DGGW08} where the case $p=0$ is carried out.

\begin{notarem}\label{nota:don} ~
\begin{enumerate}
\item \label{it:dontriples} Let $\mathcal{T}$ be the set of all triples $(M,\gamma, a)$ where $M$ is a Riemannian manifold, $\gamma$ is an isometry of $M$ and $a$ is in the fixed point set $Fix(\gamma)$ of $\gamma$.   A function $h:\mathcal{T}\to \mathbf R$ is said to satisfy the \emph{locality property} if $h(M,\gamma, a)$ depends only on the germs at $a$ of the Riemannian metric and of $\gamma$.     The function $h$ is said to be \emph{universal} if whenever
$\sigma:M_1\to M_2$ is an isometry between two Riemannian manifolds and $\gamma_i$ is an isometry of $M_i$ with $\gamma_2=\sigma\circ\gamma_1\circ\sigma^{-1}$, then $h(M_1,\gamma_1,a)=h(M_2,\gamma_2,\sigma(a))$ for all $a\in Fix(\gamma)$.

\item We will also consider functions on the set $\mathcal{T}'$ of all $(M, \eta, \gamma, a)$ where $(M,\gamma, a)$ is given as in part~\ref{it:dontriples} and $\eta$ is a $\gamma$-invariant function on $M$.   The locality and universality properties are analogously defined.   (In the definition of locality, $h(M,\eta, \gamma, a)$ may also depend on the germ of $\eta$ at $a$.  In the definition of of universality, we have $h(M_1,\eta_1,\gamma_1,a)=h(M_2,\eta_2,\gamma_2,\sigma(a))$ if $\eta_2\circ\sigma=\eta_1$ and the other conditions in part~\ref{it:dontriples} hold.)

\item Let $M$ be a Riemannian manifold of dimension $d$ and let  $\gamma: M\to  M$ be an isometry.   Then each component of the fixed point set $Fix(\gamma)$ is a totally geodesic, closed submanifold of $M$. (See \cite{K58}.) If $M$ is compact, then $Fix(\gamma)$ has only finitely many components. In any Riemannian manifold, at most one component of $Fix(\gamma)$ can intersect any given geodesically convex ball.  For $a\in Fix(\gamma)$, note that the differential $d\gamma_a: T_a M \to T_a M$ fixes $T_aQ$ and restricts to an isomorphism of $T_a(Q)^\perp$, where $Q$ is the connected component of $Fix(\gamma)$ containing $a$.
In particular, we have $\det({\operatorname{{Id}}}_{\operatorname{codim}(Q)}-A_a)\neq 0$, where $A_a$ is the restriction of the action of $d\gamma_a$ to $(T_aQ)^\perp$ and $\operatorname{codim}(Q)$ denotes the codimension of $Q$.

\item \label{it:don-trace} For $\alpha$ in the orthogonal group $O(d,\mathbf R)$, we denote by $\tr_p(\alpha)$ the trace of the natural action of $\alpha$ on $\wedge^p(\mathbf R^d)$.  For $M$ and $\gamma$ as in part~\ref{it:dontriples}, the Riemannian inner product on $T_a M$ allows us to identify $d\gamma_a$ with an element of  $O(d,\mathbf R)$ and $\tr_p(d\gamma_a)$ coincides with the trace of $\gamma_a^*: \wedge^p(T_a^*M)\to  \wedge^p(T_a^*M)$.

\item \label{it:don-CFix} Let $M$ be a compact Riemannian manifold and  $\gamma: M \to  M$ be an isometry. We denote the set of components of the fixed point set of $\gamma$ by $\mathcal{C}(Fix(\gamma))$.
\end{enumerate}

\end{notarem}

\begin{thm}\label{bkthm}\label{bklocal}
\hspace{1in}$\text{  }$
\newline\begin{enumerate}

\item \label{it:bkthm-global} \cite[Theorem 4.1]{D76}; \cite[Theorem 3.1]{DP77}. In the notation of Notation~\ref{nota:don}, there exist functions $b_k$, $k=0, 1, 2, \dots$, on $\mathcal{T}$ satisfying both the locality and universality properties such that if $M$ is a compact Riemannian manifold and  $\gamma: M \to  M$ is an isometry, then, as $t\rightarrow 0^+$,

\[
\int_{M} C_{1,2} \gamma^*_{\cdot, 2} K^p(t,x,x) \, dV_{M}(x) \sim \sum_{Q\in \mathcal{C}(Fix(\gamma))} (4\pi t)^{-\dim(Q)/2} \sum_{k=0}^{\infty} t^k \int_{Q} b^p_k(\gamma,a) \, dV_Q (a).
\]
(Here we are writing $b_k(\gamma,a)$ for $b_k(M,\gamma, a)$.) Moreover,
\[
b^p_0(\gamma,a)=\frac{\tr_p(d\gamma_a)}{\lvert \det({\operatorname{{Id}}}_{\operatorname{codim}(Q)}-A_a)\rvert}
\]

\item \label{it:bkthmlocal} (Local version.)  There exist functions $c_k$, $k=0, 1, 2, \dots$, on $\mathcal{T}'$ satisfying both the locality and universality properties such that the following condition holds:  If $M$ is an arbitrary  Riemannian manifold, $\gamma$ is an isometry of $M$, and $\eta$ is a $\gamma$-invariant function whose support is compact and geodesically convex, then as $t\to 0^+$, we have
\begin{multline*}
    \int_{M}\,(4\pi t) ^{-d/2} e^{-\frac{ d^2(x, \gamma(x))}{4t}}\eta(x)\left ( \sum_{j=0}^\infty t^j C_{1,2} \gamma^*_{\cdot, 2} u^p_j(x,x)\right) \, dV_{M}(x) \\
\sim \sum_{Q\in \mathcal{C}(Fix(\gamma))} (4\pi t)^{-\dim(Q)/2} \sum_{k=0}^{\infty} t^k \int_{Q} c^p_k(\eta,\gamma,a) \, dV_Q (a).
\end{multline*}
where the $u^p_j$, $j=1,2,\dots$ are the double $p$-forms defined in Proposition~\ref{localparam}.  (We are writing $c_k(\eta,\gamma,a)$ for $c_k(M,\eta, \gamma, a).$   Note that there is at most one component $Q$ of $Fix(\gamma)$ that intersects the support of $\eta$ and thus at most one non-zero term in the sum.)

Moreover, the dependence on $\eta$ is linear.  In particular, if $\eta\equiv 1$ near $a$, then $c_k^p(\eta, \gamma,a)=b_k^p(\gamma,a)$.
\end{enumerate}
\end{thm}

\begin{remark} In the first statement in the previous theorem, we have omitted the hypotheses in \cite{DP77} that $M$ is orientable and that $\gamma$ is orientation-preserving as they are not needed for the proof.

The proof of the second statement is a minor adaptation of Donnelly's proof of the first statement.   In the appendix, we will give an exposition of the proof, making clear that aspects of the statement apply to much more general integrals.
\end{remark}

Recall that for closed Riemannian manifolds, the heat trace has a small-time asymptotic expansion \begin{equation}\label{manifoldasymp}
\tr K^p(t) :=\int_M C_{1,2}K^p(t,m ,m)\,dV_M(m)\sim_{t\to 0^+}
(4\pi t) ^{-d/2} \sum_{i=0}^{\infty} a^p_i(M) t^i,
\end{equation}

where
\[
a^p_i( M):= \int_{ M} C_{1,2} u^p_i(m,m)\, dV(m).
\]
The $a_i^p$ are called the \emph{heat invariants}.

The first two heat invariants for $p$-forms on manifolds are given by
\begin{align}
\label{a10}
    a_0^p({M}) &= \binom{d}{p} {\rm vol} ({M}), \\
    \label{a11}
    a_1^p({M}) &= \left(\frac{1}{6}\binom{d}{p}-\binom{d-2}{p-1}\right) \int_{{M}} \tau (m) \, dV_M(m),
    \end{align}
where $\tau$ is the scalar curvature, and we use the convention that $\binom{m}{n}=0$ whenever $n < 0$. (See, for example, \cite{P70} and references therein.)

We now address the heat invariants for the Hodge Laplacian on closed Riemannian orbifolds.
\begin{nota}\label{invariants} Let ${\mathcal{O}}$ be a closed Riemannian orbifold of dimension $d$.
\begin{enumerate}
\item \label{it:invariants-Ip0t}  For $k=0,1,2,\dots$, define $U_k^p\in C^\infty({\mathcal{O}})$ as follows:    For $x\in {\mathcal{O}}$, choose an orbifold chart $(\widetilde{U},G_U, \pi_U)$ about $x$, let ${\widetilde{x}}$ be any lift of $x$ in $\widetilde{U}$ and set $U_k^p(x):= C_{1,2}u_k^p({\widetilde{x}},{\widetilde{x}})$ where $u_k^p$ is the double $p$-form
defined in Proposition~\ref{localparam}.  This definition is independent of both the choice of chart and the choice of lift ${\widetilde{x}}$ since the $u_k^p$ are local isometry invariants.

Define
\begin{equation}\label{eq:heatinvM}
a_k^p({\mathcal{O}}):=\int_{\mathcal{O}}\, U_k^p(x) \, dV_\mathcal O(x).
\end{equation}
One easily verifies that the resulting expressions for the $a_k^p({\mathcal{O}})$ are identical to those for manifolds.  E.g.,  $a_0^p({{\mathcal{O}}}) = \binom{d}{p} {\rm vol} ({O})$ and $a_1^p({\mathcal{O}})$ is given by Equation~\eqref{a11} with $M$ replaced by ${\mathcal{O}}$.

Set
$$I_0^p(t):=(4\pi t)^{-d/2}\sum_{k=0}^\infty\,a_k^p({\mathcal{O}}) t^k.$$

\item \label{it:invariants-IpNt}  Let $N$ be a stratum of the singular set of ${\mathcal{O}}$.    For $\gamma\in \iiso^{\rm max}(N)$, define $b_k^p(\gamma,\cdot): N\to \mathbf R$ as follows:  For $a\in N$, let $(\tilde{U}, G_U, \pi_U)$ be an orbifold chart about $a$ in ${\mathcal{O}}$ and let $\tilde{a}$ be a lift of $a$ in $\tilde{U}$.   Note that $\tilde{a}$ lies in a stratum $W$ of $\tilde{U}$ and $\gamma$ is naturally identified with an element of $\iiso^{\rm max}(W)$.     We define $$b_k^p(\gamma,a):=b_k^p(\gamma,\tilde{a}),$$ where the right-hand side is defined as in Theorem~\ref{bkthm} part~\ref{it:bkthm-global}.   The universality of the functions $b_k^p :\mathcal{T}\to \mathbf R$ (as stated in Theorem~\ref{bkthm} part~\ref{it:bkthm-global}) implies that  the left-hand side is independent both of the choice of chart and of the choice of lift $\tilde{a}$.   Define $b_k^p(N,\cdot): N\to \mathbf R$ by

$$ b_ k^p(N,a):=\sum_{\gamma\in \iiso^{\rm max}(N)}\, b_k^p(\gamma,a),$$

and set
\begin{align}
\label{nota_heatb}
b_ k^p(N):=\sum_{\gamma\in \iiso^{\rm max}(N)}\,\int_N\, b_k^p(\gamma,a)\,dV_N(a),
\end{align}
where $dV_N$ is the volume element on $N$ for the Riemannian metric on $N$ induced by that of ${\mathcal{O}}$.

Aside: this notation differs slightly from that in \cite{DGGW08} where the case $p=0$ is studied.

Set
$$I_N^p(t):=(4\pi t)^{-\dim(N)/2}\sum_{k=0}^\infty\,b_k^p(N) t^k.$$
Observe that $I_N^p(t)=0$ if $N$ is not a primary stratum.
\end{enumerate}
\end{nota}

\begin{remark}
Since each stratum $N$ consists of points of a fixed isotropy type, the expression for $b_0^p(\gamma, a)$ in Theorem~\ref{bkthm} part~\ref{it:bkthm-global} is independent of $a\in N$, we denote it by $b_0^p(\gamma)$. In the notation of Equation~\eqref{nota_heatb}, we have
\begin{equation}\label{niceb0} b_0^p(N)={\rm vol}(N)\sum_{\gamma\in\iiso^{\rm max}(N)}\,b_0^p(\gamma).
\end{equation}
Note that in Equation~\eqref{niceb0} the volume of $N$ is present instead of the integral over $N$. This is because the invariant $b_0$ does not depend on the curvature (see Appendix).
\end{remark}

\begin{thm}\label{thm.asympt} Let ${\mathcal{O}}$ be a closed $d$-dimensional Riemannian orbifold, let $p\in \{1,\dots, d\}$ and let $0\leq\lambda_1\leq\lambda_2\leq\dots \to +\infty$ be the spectrum of the Hodge Laplacian acting on smooth $p$-forms on ${\mathcal{O}}$.   The heat trace has an asymptotic expansion as $t\to 0^+$ given by
\begin{equation}
\label{eq:traceO}
 \tr K^p(t) \sim_{t\to 0^+}\, I_0^p(t)+\sum_{N\in S({\mathcal{O}})} \, \frac{I_N^p(t)}{|\operatorname{Iso}(N)|},
 \end{equation}
where $S({\mathcal{O}})$ is the set of all singular ${\mathcal{O}}$-strata and where $|\operatorname{Iso}(N)|$ is the order of the isotropy group of $N$.    This asymptotic expansion is of the form
\begin{align}
    \label{heatascoeffc}
(4\pi t)^{-d/2}\sum_{j=0}^\infty\, c^p_j(\mathcal O)\,t^{\frac{j}{2}}
\end{align}
with $c^p_j(\mathcal O)\in\mathbf R$.
\end{thm}
Observe that if there are no singular strata, then Equation~\eqref{eq:traceO} agrees with Equation~\eqref{manifoldasymp}.

\begin{proof} By Theorem~\ref{thm.heattrace}, we can write
\begin{equation}\label{prelim}
\tr K^p(t)
\sim_{{t\to 0^+}}\,\sum_{{\alpha}=1}^s\,\int_{\mathcal{O}}\,{\eta_\a}(x)C_{1,2}{H_\a}(t,x,x)\,dx.
\end{equation}
Recalling the notation introduced in Equation~\eqref{nota.hm}, Equation~\eqref{prelim}, we have
\begin{equation}\label{start}
\tr K^p(t) \sim_{{t\to 0^+}}\sum_{{\alpha}=1}^s\frac{1}{|G_{\alpha}|}\sum_{{\gamma}\in{G_\a}}L(t,\alpha,\gamma)
\end{equation}
where
\begin{equation}\label{start2}
L(t,\alpha,\gamma):= \int_{{\widetilde{U}_\a}}(4\pi t)^{-d/2}e^{-d^2({\widetilde{x}},{\gamma}({\widetilde{x}}))/4t}{\widetilde{\eta}_\a}({\widetilde{x}}) \notag\,
 C_{1,2}\Big({\gamma}_{.,2}^*u^p_0({\widetilde{x}},{\widetilde{x}})+ t{\gamma}_{.,2}^*u^p_1({\widetilde{x}},{\widetilde{x}})+\dots\Big) dV_{{\widetilde{U}_\a}}({\widetilde{x}}).
\end{equation}

We will group together various terms in this double sum.   First consider the identity element $1_{\alpha}$ of each $G_{\alpha}$.   By Notation~\ref{invariants} part~\ref{it:invariants-Ip0t}, we have
$$\sum_{{\alpha}=1}^s\frac{1}{|G_{\alpha}|}L(t,{\alpha}, 1_{\alpha})= (4\pi t)^{-d/2}\int_{\mathcal{O}} (U_0^p(x) + tU_1^p(x) +\dots) \,dV_{\mathcal{O}}(x)   =I_0^p(t).$$

Next let
\begin{equation}\label{I'} I'(t):=\sum_{{\alpha}=1}^s\frac{1}{|G_{\alpha}|}\sum_{1_{\alpha}\neq{\gamma}\in{G_\a}}L(t,\alpha,\gamma).
  \end{equation}
  It remains to show that
  \begin{equation}\label{toshow}I'(t)=\sum _{N\in S({\mathcal{O}})}\frac{I_N^p(t)}{|\operatorname{Iso}(N)|}.\end{equation}
We will apply Theorem~\ref{bklocal} to each term in $I'(t)$.   For each ${\alpha}$ and each nontrivial element $\gamma\in G_{\alpha}$, Theorem~\ref{bklocal} (with ${\widetilde{U}_\a}$ playing the role of $M$ in Theorem~\ref{bklocal} part~\ref{it:bkthmlocal}) expresses $L(t,{\alpha}, \gamma)$  as a sum of integrals over the various components of the fixed point set of $\gamma$ in ${\widetilde{U}_\a}$.   Each such component is a union of strata in ${\widetilde{U}_\a}$.  Since the union of those strata $W$ in ${\widetilde{U}_\a}$ for which $\gamma\in \iiso^{\rm max}(W)$ is an open subset of ${\operatorname{{Fix}}}(\gamma)$ of full measure, we can instead add up the integrals over such strata.  Letting $S({\widetilde{U}_\a})$ denote the singular strata of ${\widetilde{U}_\a}$ and applying Theorem~\ref{bklocal}, we have
\begin{equation}
\label{ip}
I'(t)=\sum_{{\alpha}=1}^s \sum_{W\in S({\widetilde{U}_\a})}\,\sum_{{\gamma}\in\iiso^{\rm max}(W)}\,\frac{1}{|G_{\alpha}|}(4\pi t)^{-\dim(W)/2}\sum_{k=0}^{\infty}\,t^k\int_W\,c_k^p({\widetilde{\eta}_\a}, \gamma, \tilde{a})\, dV_W(\tilde{a}).
\end{equation}

 Let $N\in S({\mathcal{O}})$.
For $a\in N$, set
$$c_k^p({\eta_\a},\gamma, a)= c_k^p({\widetilde{\eta}_\a}, \gamma, \tilde{a})$$
 where $\tilde{a}$ is any lift of $a$ in ${\widetilde{U}_\a}$.   This is well-defined since any two lifts differ by an element of $G_{\alpha}$ and ${\widetilde{\eta}_\a}$ is $G_{\alpha}$-invariant.  Moreover, Theorem~\ref{bklocal} part~\ref{it:bkthmlocal} implies that
\begin{equation}\label{sumck}\sum_{{\alpha}=1}^s c_k^p({\eta_\a}, \gamma, a)=b_k^p(\gamma, a).\end{equation}

For each $\alpha$ and each $W\in S({\widetilde{U}_\a})$, there exists $N\in S({\mathcal{O}})$ such that $\pi_{\alpha}$ maps $W$ isometrically onto $N\cap U_{\alpha}$.  Let
\begin{equation}\label{san}S({\widetilde{U}_\a};N)=\{W\in S({\widetilde{U}_\a}): \pi_{\alpha}(W)=N\cap U_{\alpha}\} \end{equation}
 and observe that
\begin{equation}\label{osan}|S({\widetilde{U}_\a}; N)|= \frac{|G_{\alpha}|}{|{\operatorname{{Iso}}}(N)|}.\end{equation}
For $W\in S({\widetilde{U}_\a};N)$, we identify $\iiso^{\rm max}(N)$ with $\iiso^{\rm max}(W)$.   Observe that
\begin{equation}\label{WN}\int_W\,c_k^p({\widetilde{\eta}_\a}, \gamma, \tilde{a})\, dV_W(\tilde{a}) =\int _{N\cap U_{\alpha}}\,c_k^p({\eta_\a},\gamma, a)\, dV_N(a).\end{equation}
Since $\dim(W)=\dim(N)$, Equations~\eqref{ip}, \eqref{sumck}, \eqref{san}, \eqref{osan} and \eqref{WN}  yield
\begin{align}\label{inp}
I'(t)&=\sum_{N\in S({\mathcal{O}})}\sum_{{\alpha}=1}^s \frac{|G_{\alpha}|}{|{\operatorname{{Iso}}}(N)|}\sum_{{\gamma}\in\iiso^{\rm max}(N)}\,\frac{1}{|G_{\alpha}|}(4\pi t)^{-\dim(N)/2}\sum_{k=0}^\infty\,t^k\int _{N\cap U_{\alpha}}\,c_k^p({\eta_\a},\gamma, a) dV_N(a)\\
&=\sum_{N\in S({\mathcal{O}})}\frac{1}{|{\operatorname{{Iso}}}(N)|}\sum_{{\gamma}\in\iiso^{\rm max}(N)}\,(4\pi t)^{-\dim(N)/2}\sum_{k=0}^\infty\,t^k\int_N\,b_k^p(\gamma, a) \, dV_N(a)=\sum_{N\in S({\mathcal{O}})} \, \frac{I_N^p(t)}{|\operatorname{Iso}(N)|}.
\end{align}
  The theorem follows.
\end{proof}

\begin{remark}\label{abs boun}
Let ${\mathcal{O}}$ be a closed Riemannian orbifold all of whose singular strata $N$ have co-dimension one; i.e., the singular strata are mirrors.   Every such stratum is necessarily totally geodesic in ${\mathcal{O}}$.  (Indeed, about any point $a$ in $N$, there exists a coordinate chart $({\widetilde{U}}, G_U,\pi_U)$ where $G_U$ is generated by a reflection $\tau$, and the Riemannian metric on $U=\pi({\widetilde{U}})$ arises from a $\tau$-invariant metric on ${\widetilde{U}}$.    The fixed point set of any involutive isometry is necessarily totally geodesic.)   The underlying topological space $\underline{{\mathcal{O}}}$ has the structure of a smooth Riemannian manifold each of whose boundary components is totally geodesic.

The orbifold ${\mathcal{O}}$ is good in this case, as we may double ${\mathcal{O}}$ over its reflectors
to obtain a smooth Riemannian manifold $M$ admitting a reflection symmetry $\tau$ so that ${\mathcal{O}}=\langle\tau\rangle\backslash M$.   Let $P:M\to \underline{{\mathcal{O}}}$ be the projection.  Identify the fixed point set of $\tau$ with $N$.  Let $\omega$ be a smooth $p$-form on the orbifold ${\mathcal{O}}$.  Then $\widetilde{\omega}:=P^*\omega$ is $\tau$ invariant.   Let $\nu$ be a unit normal vector field along $N$ in $M$ and let $j:N\to M$ be the inclusion.   The $\tau$-invariance of $\widetilde{\omega}$ is equivalent to the conditions $j^*\iota_\nu\widetilde{\omega}=j^*\iota_\nu d\widetilde{\omega}=0$.   Using the same notation $\nu$ for the unit normal $P_*\nu$ along $N$ in ${\mathcal{O}}$ and now letting $j:N\to {\mathcal{O}}$ be the inclusion, this says
\[
j^*\iota_\nu \omega=j^*\iota_\nu d\omega=0,
\]
and these are precisely the absolute boundary conditions for the manifold $\underline{{\mathcal{O}}}$.  The spectrum of the Hodge Laplacian on $p$-forms on ${\mathcal{O}}$ thus coincides with the spectrum of the Hodge Laplacian on $p$-forms on $\underline{{\mathcal{O}}}$ with absolute boundary conditions. It is then straightforward to prove that the spectral invariants $b^p_k(N)$ computed here for the orbifold agree with the familiar contributions to the heat trace asymptotics arising from the boundary of $\underline{{\mathcal{O}}}$ as obtained in \cite[(2) Theorem 3.2]{P03} (see also \cite[Theorem 1.2]{BG90}).

\end{remark}

\section{Applications}\label{applications}

In this section we use the asymptotic results derived in the previous section to prove our main theorems.
We first compute the invariant $b_0^p$ for various types of singular strata $N$ in a $d$-dimensional closed orbifold $\mathcal O$. We then prove an inverse spectral result involving both the 0-spectrum and 1-spectrum.
Finally, we construct examples of orbifolds that are $p$-isospectral to manifolds (for various $p \geq 1$) and obtain some inverse spectral results for the $p$-spectrum alone.

\subsection{Computation of $b_0^p(N)$ for various strata $N$}\label{sec.computation}

\begin{prop}\label{b01general}  Let $N$ be a singular stratum of codimension $k$ in the $d$-dimensional closed orbifold ${\mathcal{O}}$. Suppose $\gamma \in\iiso^{\rm max}(N)$ has eigenvalue type $E(\theta_1, \theta_2, \dots, \theta_s;r)$, using Notation~\ref{nota:Rstuff}.

Then
\begin{equation}\label{eq.b01}b_0^1(\gamma)=\left( d-k-r+\sum_{j=1}^s\,2\cos(\theta_j)\right) \left( 2^{-k}\prod_{j=1}^s\,\csc^2(\theta_j/2)\right) .\end{equation}
Here we use the convention that when $s=0$, we have $\prod_{j=1}^s\,\csc^2(\theta_j/2)=1$.
\end{prop}

\begin{proof} We apply Theorem~\ref{bkthm}.
The expression in the first set of large parentheses in Equation~(\ref{eq.b01}) is the trace of $\gamma$.    The expression in the second set of large parentheses equals $\displaystyle\frac{1}{|\det({\operatorname{{Id}}}_{\operatorname{codim}(N)}-A)|}$, where (taking note of Notation~\ref{invariants} and Notation and Remarks \ref{nota:don}), we are writing $A$ to denote the expression $A_a$ in Theorem~\ref{bkthm} part~\ref{it:bkthm-global}.
\end{proof}

\begin{ex}\label{isotropy2} ~
Suppose $N$ is a singular stratum of codimension $k$ with isotropy group of order $2$.  The generator $\gamma$ of ${\operatorname{{Iso}}}(N)$ must have eigenvalue type $E(;k)$, and thus
\begin{equation}\label{Krawt_trace}\tr_p(\gamma)=\sum_{j=0}^p\,(-1)^j\,\binom{k}{j}\binom{d-k}{p-j}\end{equation}
with the understanding that $\binom{m}{n}=0$ when $n>m$.     The expression on the right-hand side of Equation~\eqref{Krawt_trace} is the value at $k$ of the
so-called (binary) Krawtchouk polynomial of degree $p$ given by
\begin{equation}\label{krawtpoly}K^d_p(x)=\sum_{j=0}^p\,(-1)^j\,\binom{x}{j}\binom{d-x}{p-j}.\end{equation}
(The authors learned of the Krawtchouk polynomials through the work of Miatello and Rossetti \cite{MR01}, where these polynomials arise in the computation of the spectrum of $p$-forms on Bieberbach manifolds with holonomy $\mathbf Z_2^k$.)

By Theorem~\ref{bkthm} part~\ref{it:bkthm-global} and Notation~\ref{invariants} part~\ref{it:invariants-IpNt}, we have
\begin{equation}
\label{Krawt}b_0^p(N) =\frac{{\rm vol}(N)}{2^k}K_p^d(k).
\end{equation}

In particular, $b_0^p(N)$ vanishes if and only if the codimension $k$ of $N$ is a zero of the Krawtchouk polynomial $K_p^d$.   There is a large literature on the zeros of Krawtchouk polynomials.  We include a few elementary observations here:

\begin{itemize}
\item $K^d_0(k)=1$ and $K^d_d(k) = (-1)^k$.
\item $K^d_1(k)=0$ if and only if $k=\frac{d}{2}$.
\item $K^d_2(k)=0$ if and only if the dimension $d=
n^2$ is a perfect square and $k=\frac{n(n\pm 1)}{2}$.  (For later use, we observe that if $4|n$, then both zeros $\frac{n(n\pm 1)}{2}$ are even.)
\item When $d$ is even and $p=\frac{d}{2}$, we have $K^d_p(k)=0$ for all odd $k$.
\item When $d$ is even and $p$ is odd, we have $K^d_p(\frac{d}{2})=0$.
\end{itemize}

\end{ex}

\begin{ex}\label{isotropy3} Suppose $N$ is a singular stratum of codimension $k$ with isotropy group ${{\operatorname{{Iso}}}(N)}$ of order $3$.  Then ${{\operatorname{{Iso}}}(N)}$ is necessarily cyclic, generated by an element $\gamma$ with eigenvalue type $E(\theta_1, \theta_2, \dots, \theta_s;)$ where $k=2s$ and $\theta_j \in \{\frac{2\pi}{3},\frac{4\pi}{3}\}$ for $1 \le j \le s$. We have $\gamma \in \iiso^{\rm max}(N)$ as $\gamma$ has eigenvalue $1$ with multiplicity $d-k$. As $\gamma^2$ satisfies the same conditions, we conclude $\iiso^{\rm max}(N)=\{\gamma, \gamma^2\}$. Using the fact that $\cos(\theta_j)=-\frac{1}{2}$ and $\csc^2(\theta_j/2)=4/3$ for all $j$, we have
$$b_0^1(\gamma)=b_0^1(\gamma^2)=(d-k-s)2^{-k}(4/3)^s=\frac{d-3s}{3^s}.$$
Thus $$b_0^1(N)=\frac{2(d-3s)}{3^s}{\rm vol}(N).$$

Thus $b_0^1(N)=0$ if and only if $k=\frac{2}{3}d$.    (In particular, we must have $3|d$ in this case.)   When $d=3$ and $k=2$, we will construct an example of a flat orbifold that is 1-isospectral to a flat torus.  (See Example \ref{it:flat-trianglat}).
 \end{ex}

We next consider strata of low codimension.
If $N$ is a stratum of codimension one, then the isotropy necessarily has order 2 so this case is covered by Example~\ref{isotropy2}.

\begin{prop}
\label{b01codim2}
Let $N$ be a stratum of codimension two with cyclic  isotropy group of order $m$.  Then,
\begin{equation}\label{eq.codim2} b_0^1(N)=\left( (d-2)\frac{m^2 - 1}{12} + \frac{m^2 - 6m + 5}{6}\right) {\rm vol}(N).
\end{equation}

\end{prop}
In order to prove Proposition~\ref{b01codim2}, we need an intermediary result.

\begin{lemma}
\label{lem:trigsum2}

For $2\leq m \in \mathbb{Z}$, we have

\begin{equation*}\sum_{j=1}^{m -1} \csc^2(\pi j/ m)
  = \frac{m^2 - 1}{3},
\end{equation*}
and
\begin{equation*}\sum_{j=1}^{m -1} \cos (2\pi j/ m)\csc^2 (\pi j/ m)
  = \frac{m^2 - 6m + 5}{3}.
\end{equation*}

\end{lemma}

\begin{proof}

These formulas are proved using the calculus of residues.   See \cite[Lemma 5.4]{DGGW08} for the first formula.
For the second, by \cite[Equation (2.3)]{CS12}, we have
  \begin{align*}
    S_3(m,1,1) &= \sum_{j=1}^{m-1} \cos (2\pi j/ m) \csc^2 (\pi j/ m) \\
    &= 2 \sum_{\alpha =0}^{1} \binom{2}{2\alpha} B_{2\alpha}(1/m) B_{2-2\alpha}^{(2)}(1) m^{2\alpha} \\
    &= 2B_0(1/m)B_2^{(2)}(1) + 2B_2(1/m)B_0^{(2)}(1)m^2
    = \frac{m^2 - 6m + 5}{3},
  \end{align*}
where $B_{n}^{(m)}(x)$ are the Bernoulli polynomials of order $m$ and degree $n$ in $x$
(see \cite{CS12} and references therein).
\end{proof}

\begin{proof}[Proof of Proposition~\ref{b01codim2}] Let $m$ be the order of ${{\operatorname{{Iso}}}(N)}$.  If $m=2$, then the expression \eqref{eq.codim2} in Proposition \ref{b01codim2} agrees with that in Example~\ref{isotropy2}, so for notational convenience we assume $m\geq 3$.  In this case ${{\operatorname{{Iso}}}(N)}$ is generated by an element $\gamma$ with eigenvalue type $E(2\pi/m;)$. All elements of ${{\operatorname{{Iso}}}(N)}$ other than the identity lie in $\iiso^{\rm max}(N)$, and the proposition follows from Notation~\ref{invariants} part~\ref{it:invariants-IpNt}, Proposition~\ref{b01general} and Lemma~\ref{lem:trigsum2}.
\end{proof}

\begin{ex}\label{compound rotation} Generalizing Proposition~\ref{b01codim2}, we can compute $b_0^1(N)$ for strata for which ${{\operatorname{{Iso}}}(N)}$ is cyclic and
\begin{itemize}
\item when $m>2$, the generator has eigenvalue type $E(2\pi/m, \dots, 2\pi/m;)$ with $2\pi/m$ repeated $s$ times, or
\item when $m=2$, the generator has eigenvalue type $E(;r)$ with $r$ even.
\end{itemize}

Consider the case that $N$ has codimension $4$ in ${\mathcal{O}}$. By \cite[Equation~(5.2) and Corollary~5.4]{BY02}, we have
\begin{equation}\label{trigsum4}
    \sum_{j=1}^{m-1} \cos (2\pi j/ m) \csc^4(\pi j/ m) = \frac{m^4 -20m^2 +19}{45}.
\end{equation}

From \cite[Equation~(5.2) and Corollary~5.2]{BY02}, we have that
\begin{equation}\label{trigsum4'}
    \sum_{j=1}^{m -1} \frac{d-4}{\sin^4(\pi j/ m)}
    = \frac{(d-4) (m^4 +10m^2 -11)}{45}.
\end{equation}

Thus by Proposition~\ref{b01general},
\begin{equation*}
    b_0^1(N) =
   {\rm vol}(N) \frac{4(m^4 -20m^2 +19)+(d-4) (m^4 +10m^2 -11)}{720}.
\end{equation*}

By applying Example~\ref{isotropy2}, one can confirm the same expression is valid when $m=2$.

Reference \cite{BY02} also contains formulas analogous to Equations~(\ref{trigsum4}) and (\ref{trigsum4'}) with the power $4$ replaced by the power $2s$ for arbitrary positive $s$, thus yielding expressions for $b_0^1(N)$ for rotation strata of constant rotation angle and higher codimension.

\end{ex}

\subsection{The proof of Theorem~\ref{thm:main1V4}}\label{ss:codim2} The goal of this subsection is to prove Theorem~\ref{thm:main1V4} from the Introduction. Let ${\mathcal{O}}$ be a $d$-dimensional orbifold.

\begin{proof}[Proof of Theorem~\ref{thm:main1V4}]
If ${\mathcal{O}}$ contains a stratum $S$ of codimension three, then either $S$ is primary or else ${\mathcal{O}}$ contains a singular stratum of lower codimension.   In the first case, Theorem~\ref{thm:dggw5.1} implies that the 0-spectrum alone suffices to distinguish ${\mathcal{O}}$ from any Riemannian  manifold.   Thus we may assume that ${\mathcal{O}}$ contains a stratum of codimension one or two.

If ${\mathcal{O}}$ contains a stratum of codimension one, then the theorem is immediate from Theorem~\ref{thm:dggw5.1}.   Thus we may assume that there are no such strata.   It then follows that every stratum $N$ of codimension two is primary and moreover that its isotropy group is cyclic.  Indeed, the isotropy group $\operatorname{Iso}(N)$ is isomorphic to a subgroup of the orthogonal group $O(d)$ of the form $A\times  I_{d-2}$ where $I_{d-2}$ is the $(d-2)\times (d-2)$ identity matrix and $A$ is a finite  subgroup of $O(2)$ with the property that no element of $A$ other than the identity fixes any non-trivial vector.  Every such subgroup $A$ is a cyclic subgroup of $SO(2)$.

In the following, we use the notation introduced in Equation~\eqref{heatascoeffc}.
If $M$ is any Riemannian manifold of dimension $d$, then its heat invariants $c_2^0(M)=a_1^0(M)$ and $c_2^1(M)=a_2^1(M)$ satisfy
$$
c_2^1(M)=\frac{d-6}{6} \int_M \tau \, dV_M=(d-6)c_2^0(M).
$$
Thus it suffices to show that the orbifold ${\mathcal{O}}$ satisfies
\begin{equation}\label{d-6}
c_2^1({\mathcal{O}})>(d-6)c_2^0({\mathcal{O}}).
\end{equation}
We will show that
\begin{equation}\label{greater d-6}
b_0^1(N)>(d-6)b_0^0(N)
\end{equation}
for every stratum of codimension two in ${\mathcal{O}}$.   Equation~(\ref{d-6}) will then follow from Theorem~\ref{thm.asympt}.

Let $N$ be a stratum of codimension two with (necessarily cyclic) isotropy group of order $m$. The computation of $b_0^0(N)$ is analogous to the case of cone points in dimension 2, carried out in \cite[Proposition 5.5]{DGGW08}, yielding
$$b_ 0^0(N)=\frac{(m^2-1)}{12} {\rm vol}(N).$$
Using Proposition~\ref{b01codim2}, we have
$$
b_0^1(N)=\left((d-2)\frac{m^2-1}{12}+\frac{m^2-6m+5}{6}\right){\rm vol}(N).
$$
Thus, noting that $m\geq 2$, we have
$$b_0^1(N)-(d-6)b_0^0(N)=\frac{3m^2 - 6m + 3}{6}>0.$$
This proves Equation~(\ref{greater d-6}) and the theorem follows.
\end{proof}

\begin{remark}\label{rem:orbisurfaces} It is shown in \cite[Theorem 5.15]{DGGW08} that the $0$-spectrum alone suffices to distinguish singular, closed, locally orientable Riemannian orbisurfaces with nonnegative Euler characteristic from smooth, oriented, closed Riemannian surfaces. In fact, it is shown there that the degree zero term in the small-time heat trace asymptotics for functions gives rise to a complete topological invariant for orbisurfaces satisfying these constraints. In the previous theorem, by also appealing to the $1$-spectrum, we do not require local orientability or nonnegative Euler characteristic to distinguish Riemannian orbisurfaces from closed Riemannian surfaces.
\end{remark}

\subsection{Negative inverse spectral results for the $p$-spectra}\label{sec.isospec}

The article \cite{GR03} contains examples of orbifolds of even dimension $d=2m$ that are $m$-isospectral to manifolds. Here, we will construct examples of flat orbifolds that are $p$-isospectral to manifolds for various values of $p$.

\begin{nota}\label{nota:flat} Every closed flat orbifold or manifold is of the form ${\mathcal{O}}=\Sigma{\backslash} \mathbf R^d$ where $\Sigma$ is a discrete subgroup of the Euclidean motion group $I(\mathbf R^d)=\mathbf R^d\rtimes O(d)$.  (Note that ${\mathcal{O}}$ is a manifold, i.e., $\Sigma$ acts freely on $\mathbf R^d$, if and only if $\Sigma$ is a Bieberbach group). The restriction of the projection $I(\mathbf R^d)\to O(d)$ to $\Sigma$ has finite image $F$ and kernel a lattice $\Lambda$ of rank $d$.   We will refer to $\Lambda$ as the translation lattice of $\Sigma$. The group $F$ is the holonomy group of ${\mathcal{O}}$.   For each $\gamma\in F$, there exists $a=a(\gamma)\in \mathbf R^d$, unique modulo $\Lambda$, such that $\gamma\circ L_a\in \Sigma$, where $L_a$ denotes translation by $a$.   Let $\Lambda^*$ denote the lattice dual to $\Lambda$.  For $\mu\geq 0$ and $\gamma\in F$, set
$$e_{\mu,\Sigma}(\gamma)=\sum_{v\in \Lambda^*, \|v\|=\mu,\gamma(v)=v}e^{2\pi i v\cdot a}.$$

\end{nota}

In the notation of Notation~\ref{nota:flat}, let $T$ be the torus $\Lambda{\backslash}\mathbf R^d$.    Let $\eta =\sum_J \, f_J\,dx^J$ be the pullback to $\mathbf R^n$ of a $p$-form on ${\mathcal{O}}$.  (Here $J$ varies over all multi-indices $1\leq j_1<\dots<j_p\leq d$, and the functions $f_J$ are $\Sigma$-invariant.)  We have
$$\Delta^p(\eta)=\sum_J \,\Delta^0( f_J)\,dx^J.$$
Thus every element of the $p$-spectrum of ${\mathcal{O}}$ occurs as an eigenvalue in the 0-spectrum of the torus $T$; i.e., it is of the form $4\pi^2\|\mu\|^2$ for some $\mu$ in the dual lattice $\Lambda^*$ of $\Lambda$.   The following result of Miatello and Rossetti, while originally stated in the context of flat manifolds, is also valid for orbifolds.

\begin{prop}[{\cite[Theorem 3.1]{MR01}}]
\label{flatspeccomp}
We use the notation of Notation~\ref{nota:flat}.
\begin{enumerate}
\item For $\mu\geq 0$, the multiplicity $m_{p,\mu}(\Sigma)$ of $\mu$ in the $p$-spectrum of ${\mathcal{O}}=\Sigma{\backslash}\mathbf R^d$ is given by
$$m_{p,\mu}(\Sigma)=\frac{1}{|F|}\sum_{\gamma\in F}\, \tr_p(\gamma) e_{\mu,\Sigma}(\gamma)=\frac{1}{|F|}\binom{d}{p}+\frac{1}{|F|}\sum_{1\neq\gamma\in F}\, \tr_p(\gamma) e_{\mu,\Sigma}(\gamma).$$
\item \label{it:flatspeccomp-isospectral} Thus if ${\mathcal{O}}'=\Sigma'{\backslash} \mathbf R^d$, where $\Sigma'$ has the same translation lattice $\Lambda'=\Lambda$, and if there is a bijection $\gamma\mapsto\gamma'$ between the holonomy groups of ${\mathcal{O}}$ and ${\mathcal{O}}'$ such that $$\tr_p(\gamma) e_{\mu,\Sigma}(\gamma)=\tr_p(\gamma') e_{\mu,\Sigma'}(\gamma')$$
for all $\gamma\in F$, then ${\mathcal{O}}$ and ${\mathcal{O}}'$ are $p$-isospectral.

\end{enumerate}
\end{prop}

\begin{cor}\label{isosp condition}
Suppose ${\mathcal{O}}=\Sigma{\backslash} \mathbf R^d$ and ${\mathcal{O}}'=\Sigma'{\backslash} \mathbf R^d$ where $\Sigma$ and $\Sigma'$ have the same translation lattice.   If $|F|=|F'|$ and if $\tr_p(\gamma)=0=\tr_p(\gamma')$ for all $1\neq \gamma\in F$ and $1\neq \gamma'\in F'$, then ${\mathcal{O}}$ and ${\mathcal{O}}'$ are $p$-isospectral.

\end{cor}

\begin{ex}\label{it:flat-trianglat} Let $\Lambda$ be the lattice in $\mathbf R^3$ generated by $\{(1,0,0), (\frac{1}{2},\frac{\sqrt{3}}{2},0), (0,0,1)\}$.   Let $\gamma\in I(\mathbf R^3)$ be rotation through angle $\frac{2\pi}{3}$ about the $x_3$-axis and let $F$ be the cyclic group generated by $\gamma$.  Observe that $F$ stabilizes
$\Lambda$.  Let $\Sigma$ be the subgroup of $I(\mathbf R^3)$ generated by $\Lambda$ and $\gamma$, and let $\Sigma'$ be generated by $\Lambda$ and $\gamma\circ L_a$ where $a=(0,0,\frac{1}{3})$.  Corollary~\ref{isosp condition} and the computation in Example~\ref{isotropy3} imply that the orbifold $\Sigma{\backslash} \mathbf R^3$ and the manifold $\Sigma'{\backslash} \mathbf R^3$ are 1-isospectral.

\end{ex}

\begin{prop}\label{it:flat-involution}~
Let $d\geq 2$ and suppose that $p,k\in \{0, \dots, d\}$ satisfy $K_p^d(k)=0$ where $K_p^d$ is the Krawtchouk polynomial given by Equation~\eqref{krawtpoly}.   Then there exists a $d$-dimensional Bieberbach manifold $M_k$ and a closed flat orbifold ${\mathcal{O}}_k$ of dimension $d$ such that:
\begin{enumerate}
\item The singular set of ${\mathcal{O}}_k$ consists precisely of $2^k$ singular strata each of which has codimension $k$.
\item ${\mathcal{O}}_k$ and $M_k$ are $p$-isospectral.
\end{enumerate}
Moreover, if $k'\in \{0, \dots, d\}$ is another zero of $K_p^d$, then the collection $\{M_k, M_{k'}, {\mathcal{O}}_k, {\mathcal{O}}_{k'}\}$ are all mutually $p$-isospectral.
\end{prop}

\begin{proof}    Let $\gamma_k:\mathbf R^{d}\to\mathbf R^{d}$ be given by
$$\gamma_k(x_1,\dots, x_d)=(-x_1,\dots, -x_k, x_{k+1},\dots, x_d),$$
let $a\in (\frac{1}{2}\mathbf Z)^d$ with at least one of the last $d-k$ entries of $a$ equal to $\frac{1}{2}$, and let $\rho_k=\gamma_k\circ L_a$.   Let $\Sigma_k$, respectively $\Sigma'_k$ be the discrete subgroup of the Euclidean motion group $I(\mathbf R^d)$ generated by $\mathbf Z^d$ together with $\gamma_k$, respectively $\rho_k$.   Set ${\mathcal{O}}_k:=\Sigma{\backslash} \mathbf R^d$ and $M_k:= \Sigma'{\backslash} \mathbf R^d$.   The torus $T=\mathbf Z^d{\backslash}  \mathbf R^d$ is a two-fold cover  of each of ${\mathcal{O}}_k$ and $M_k$.  The map $\rho_k$ induces a fixed-point free involution of $T$, so $M_k$ is a manifold.    In contrast, the involution of $T$ induced by $\gamma_k$ fixes all points each of whose first $k$ coordinates lie in $\{0, \frac{1}{2}\}$.    Thus ${\mathcal{O}}_k$ contains $2^k$ singular strata each of codimension $k$.  In the notation of Notation~\ref{nota:flat}, the holonomy groups of both ${\mathcal{O}}_k$ and $M_k$ have order 2 with generator $\gamma_k$.   Since $\tr_p(\gamma_k)=K^d_p(k)=0$ (as seen in Example~\ref{isotropy2}), Corollary~\ref{isosp condition} implies that ${\mathcal{O}}_k$ and $M_k$ are $p$-isospectral .  Moreover, the corollary also implies that their $p$-spectra are independent of the choice of the zero $k$ of $K^d_p$.

\end{proof}

\begin{ex}\label{ex:d/2}
As noted in Example~\ref{isotropy2}, when $d$ is even and $k=\frac{d}{2}$, we have $K^d_p(k)=0$ for all odd $p$.  Thus in every even dimension $d$, there exists a Bieberbach manifold $M$ and a flat $d$-dimensional orbifold with singular set of codimension $\frac{d}{2}$ such that $M$ and ${\mathcal{O}}$ are $p$-isospectral for all odd $p$.
\end{ex}

The fact that the orbifolds and manifolds in Example~\ref{ex:d/2} are $\frac{d}{2}$-isospectral was proven by other methods in \cite{GR03}, Theorem 3.2(i).   As noted there, in the special case that $d=2$, the resulting manifold in Example~\ref{ex:d/2}  is a Klein bottle, while the orbifold ${\mathcal{O}}$ is a cylinder, and these were also shown to be 1-isospectral to a M\"obius strip.  As in Remark~\ref{abs boun}, ${\mathcal{O}}$ can be viewed either as an orbifold or as a manifold with boundary with absolute boundary conditions.   In \cite{GR03}, it was also asserted that these surfaces are $1$-isospectral to a flat $3$-pillow (i.e., a flat Riemannian orbifold whose underlying space is a sphere with three cone points); however as shown in the errata to \cite{GR03}, that assertion was incorrect.

In the example below, we will use the observation in Notation~\ref{abs boun} to construct a flat $3$-pillow that is $1$-isospectral to a square with absolute boundary conditions.

\begin{ex}\label{ex:isospecorb}

Let $\Lambda=\mathbf Z^2$.  Let $F<O(2)$ be the cyclic group of order $4$ generated by rotation $\gamma$ through angle $\frac{\pi}{2}$ about the origin, let $\Sigma=\mathbf Z^2\rtimes F$, and let ${\mathcal{O}}=\Sigma{\backslash}\mathbf R^2$.
Observe that $\tr_1(\gamma)=\tr_1(\gamma^3)=0$ and that  $\gamma^2=-{\operatorname{{Id}}}$.

Let $F'<O(2)$ be the Klein 4-group generated by the reflections $\alpha$ and $\beta$ across the $x$ and $y$-axes, let $\Sigma'=\mathbf Z^2\rtimes F'$, and let ${\mathcal{O}}'=\Sigma'{\backslash}\mathbf R^2$.  Since $\tr_1(\alpha)=\tr_1(\beta)=0$ and $\alpha\circ\beta =-{\operatorname{{Id}}} =\gamma^2$, Proposition~\ref{flatspeccomp} part~\ref{it:flatspeccomp-isospectral} implies that ${\mathcal{O}}$ and ${\mathcal{O}}'$ are $1$-isospectral.

 The underlying space of the orbifold ${\mathcal{O}}'$ is a square and the
 $1$-spectrum of ${\mathcal{O}}'$ as an orbifold coincides with the $1$-spectrum of the square with absolute boundary conditions.  (As an orbifold, the four edges are reflectors and the four corners are dihedral points.)    On the other hand, ${\mathcal{O}}$ is a $3$-pillow. Indeed, the underlying topological space of ${\mathcal{O}}$ is a sphere.    The singular set of ${\mathcal{O}}$ consists only of cone points: two cone points of order $4$ and one cone point of order $2$.   (The cone points of order $4$ correspond to the $\Sigma$-orbits of the points $(0,0)$ and $(\frac{1}{2},\frac{1}{2})$ in $\mathbf R^2$, while the cone point of order $2$ corresponds to the $\Sigma$-orbit of $(\frac{1}{2},0)$, which is the same as the $\Sigma$-orbit of the point $(0,\frac{1}{2})$.)

\end{ex}

\subsection{Positive inverse spectral results for the $p$-spectra}\label{sec: common cover}

We continue to assume that ${\mathcal{O}}$ is a $d$-dimensional closed Riemannian orbifold.

\begin{nota}\label{parity}
We will say that a singular stratum of ${\mathcal{O}}$ has \emph{positive, respectively negative, parity} if its codimension in ${\mathcal{O}}$ is even, respectively odd.  For $\epsilon\in\{\pm\}$, let $k_\epsilon$ be the minimum codimension of the primary singular strata (if any) of ${\mathcal{O}}$ of parity $\epsilon$, and let $S_\epsilon({\mathcal{O}})$ be the collection of all primary strata of codimension $k_\epsilon$.  Set
$$B_\epsilon^p({\mathcal{O}}) =\sum_{N\in S_\epsilon({\mathcal{O}})}\,\frac{1}{|{\operatorname{{Iso}}}(N)|}\,b_0^p(N).$$
If there are no singular strata of parity $\epsilon$, then we set $B_\epsilon^p({\mathcal{O}})=0.$
\end{nota}

\begin{prop}\label{prop: odd par}~
\begin{enumerate}
\item \label{it:oddpar-1} $B_-^p({\mathcal{O}})$ is an invariant of the $p$-spectrum.   In particular, if $B_-^p({\mathcal{O}})\neq 0$, then ${\mathcal{O}}$ cannot be $p$-isospectral to a Riemannian manifold.
\item \label{it:oddpar-2}  If $B_+^p({\mathcal{O}})\neq 0$, then ${\mathcal{O}}$ cannot be $p$-isospectral to any closed Riemannian manifold
$M$ such that $M$ and ${\mathcal{O}}$ have either finite $p$-isospectral Riemannian covers or isometric (possibly infinite) homogeneous
Riemannian covers.
\end{enumerate}
\end{prop}

\begin{proof}~
\begin{enumerate}
\item The fact that $k_-$ is the minimal codimension of the odd primary singular strata implies that $B_-^p({\mathcal{O}})$ is precisely the term of degree $t^{-(d-k_-)/2}$ in the heat trace expansion for $\Delta^p$ on ${\mathcal{O}}$ and thus is a spectral invariant.   The second statement is immediate.

\item The proof is analogous to the proofs of the analogous statements for the 0-spectrum cited in the introduction.  We give a summary.  It suffices to show that if $M$ and ${\mathcal{O}}$ are isospectral, then each of the two conditions on $M$ and ${\mathcal{O}}$ implies that $a_j^p({\mathcal{O}})=a_j^p(M)$ for all $j$.   (See Notation~\ref{invariants} part~\ref{it:invariants-Ip0t}.)   Writing $k_+=2\ell$, one then notes that the coefficient of $t^{\ell-d/2}$ in the heat trace for the $p$-Laplacian on ${\mathcal{O}}$ is $a_\ell^p({\mathcal{O}})+B_+^p({\mathcal{O}})$ whereas the corresponding coefficient for $M$ is $a_\ell^p(M)$, contradicting the $p$-isospectrality of ${\mathcal{O}}$ and $M$.

First consider the case that there exist finite Riemannian covers $M^*$ of ${\mathcal{O}}$ and $M^{**}$ of $M$ that are $p$-isospectral.  Suppose that $M$ and ${\mathcal{O}}$ are $p$-isospectral.  Since the $p$-spectrum determines the volume, we have ${\rm vol}(M)={\rm vol}({\mathcal{O}})$ and also ${\rm vol}(M^{**})={\rm vol}(M^*)$.   Thus the two coverings are of the same order $r$.    Using Notation~\ref{invariants} part~\ref{it:invariants-Ip0t}, we then have
$$a_j^p({\mathcal{O}}) =\frac{1}{r}a_j^p(M^*)=\frac{1}{r}a_j^p(M^{**})= a_j(M)$$
for all $j$.

Next suppose that ${\mathcal{O}}$ and $M$ have a common homogeneous Riemannian cover $\tilde{M}$.  Write $U_j^p(M;\cdot)$ and $U_j^p({\mathcal{O}},\cdot)$ for the functions on $M$ and ${\mathcal{O}}$ in Equation~\eqref{eq:heatinvM} of Notation and Remarks \ref{nota:don} whose integrals yield $a_j^p(M)$ and $a_j^p({\mathcal{O}})$.   One uses homogeneity of the cover and locality and universality of the $U_j^p$ to see that $U_j^p(M;\cdot)$ and $U_j^p({\mathcal{O}},\cdot)$ are constant functions with the same constant value $U_j$.  If $M$ and ${\mathcal{O}}$ are isospectral, then again they have the same volume $V$, so $a_j^p(M)=U_jV=a_j^p({\mathcal{O}})$.
\qedhere
\end{enumerate}
\end{proof}

\begin{remark}\label{rem:p=0}
In case $p=0$, one has $b_0^0(N) > 0$ for every primary singular stratum.   Thus  the condition $B_-^0({\mathcal{O}})\neq 0$, respectively $B_+^0({\mathcal{O}})\neq 0$, is equivalent to the condition that ${\mathcal{O}}$ contains at least one primary singular stratum of odd, respectively even, codimension.    Hence when $p=0$, the first statement of the proposition is the weak analogue of
Theorem~\ref{thm:dggw5.1} and Theorem~\ref{thm:common good} part~\ref{homog cover}.

\end{remark}

\begin{defn}\label{dim of sing set} We define the \emph{codimension} of the singular set to be the minimum codimension of the singular strata.
\end{defn}

\begin{thm}\label{thm:krawt4} Denote by ${\mathcal Orb}^d_k$ the class of all closed $d$-dimensional Riemannian orbifolds with singular set of codimension $k$.    Assume that $k$ is odd.  Let $p\in \{0,1,2,\dots , d\}$ and let $K_p^d$ be the Krawtchouk polynomial given by Equation~\eqref{krawtpoly}. If $K_p^d(k)\neq 0$, then the $p$-spectrum determines the $(d-k)$-dimensional volume of the singular set of elements of ${\mathcal Orb}^d_k$.   In particular, the $p$-spectrum distinguishes elements of orbifolds in ${\mathcal Orb}^d_k$ from Riemannian manifolds.
\end{thm}

\begin{proof} Let $N \subset \mathcal O \in {\mathcal Orb}^d_k$ be a singular stratum of codimension $k$.  View ${\operatorname{{Iso}}}(N)$ as a subgroup of the orthogonal group $O(d)$.  All non-trivial elements of ${\operatorname{{Iso}}}(N)$ must lie in $\iiso^{\rm max}(N)$ and must have the same 1-eigenspace; otherwise there would exist a stratum of smaller codimension.  Thus  ${\operatorname{{Iso}}}(N)$ must be of the form
${\operatorname{{Iso}}}(N)=\Gamma\times \{{\operatorname{{Id}}}_{d-k}\}$ where $\Gamma$ is a subgroup of the orthogonal group $O(k)$ that acts freely on the sphere $S^{k-1}$.   Since $k$ is odd and the only finite group acting freely on even-dimensional spheres has order 2, $\Gamma$ and thus also ${\operatorname{{Iso}}}(N)$ must have order 2.

Using Equation~\eqref{Krawt} and Notation~\ref{parity} we have,
\begin{equation*} B_-^p(\mathcal O)=\frac{K_p^d(k)}{2^{k+1}}\sum_{N \in S_-(\mathcal O)}\,{\rm vol}(N).
\end{equation*}
As shown in Proposition~\ref{prop: odd par} part~\ref{it:oddpar-1}, $B_-^p({\mathcal{O}})$ is a spectral invariant, thus the proof is complete.
\end{proof}

Thus for example (under the assumption that $k$ is odd), Theorem~\ref{thm:krawt} along with the bulleted items in Example~\ref{isotropy2} imply:
\begin{itemize}
\item An element of ${\mathcal Orb}^d_k$ can be 1-isospectral to a Riemannian manifold only if $d$ is even and $k=\frac{d}{2}$.
\item Assume $d$ is not a perfect square.   Then no closed Riemannian $d$-orbifold with singular set of odd codimension can be 2-isospectral to a Riemannian manifold.    (The same statement holds if $d$ is of the form $d=4m^2$ for some integer $m$.)
\end{itemize}

We next focus on the $1$-spectrum.

\begin{thm}\label{thm:halfdim}
Let $\mathcal O$ be a closed Riemannian orbifold of dimension $d$.
\begin{enumerate}

\item \label{it:halfdim-covers} If $\mathcal O$ contains at least one singular stratum of codimension $k<\frac{d}{2}$, then ${\mathcal{O}}$ cannot be $1$-isospectral to any closed Riemannian manifold
$M$ such that $M$ and ${\mathcal{O}}$ have either finite $1$-isospectral Riemannian covers or isometric infinite homogeneous
Riemannian covers.

\item \label{it:halfdim-primary}  If $\mathcal O$ contains at least one primary singular stratum of odd codimension $k<\frac{d}{2}$, then $\mathcal O$ cannot be 1-isospectral to any closed Riemannian manifold.

\item \label{it:halfdim-deven} For $d$ even, both statements remain true when $k=\frac{d}{2}$ provided at least one stratum of codimension $k$ has isotropy group of order at least three.
\end{enumerate}

\end{thm}

\begin{remark}\label{rem: sharp}

Proposition~\ref{it:flat-involution} shows that the hypothesis on isotropy order in Theorem~\ref{thm:halfdim} part~\ref{it:halfdim-deven} cannot be removed .

When $d=3$, Theorem~\ref{thm:halfdim} part~\ref{it:halfdim-covers} is sharp:  Example~\ref{it:flat-trianglat} shows that the conclusion fails when $k=2=\frac{d+1}{2}$.

\end{remark}

\begin{proof}  Parts \ref{it:halfdim-covers} and \ref{it:halfdim-primary} follow from Proposition \ref{prop: odd par} along with the following two observations:
\begin{enumerate}
\item Since any singular stratum of minimum codimension is necessarily primary,  the hypothesis of part~\ref{it:halfdim-covers} implies that $k_\epsilon <\frac{d}{2}$ for at least one $\epsilon\in\{\pm\}$, while the hypothesis of part~\ref{it:halfdim-primary} says that $k_-<\frac{d}{2}$.

\item If $N$ is any primary singular stratum of codimension $k$ and if $\gamma\in\iiso^{\rm max}(N)$, then 1 is an eigenvalue of $\gamma$ with multiplicity $d-k$ while all other eigenvalues lie in $[-1,1)$.  Thus if $k<\frac{d}{2}$, then $\tr(\gamma)>0$ for all $\gamma\in\iiso^{\rm max}(N)$, so $b_0^1(N) > 0$.   In particular, if $k_\epsilon(\mathcal O)<\frac{d}{2}$, then $B_\epsilon(\mathcal O)>0$.
\end{enumerate}

To prove part~\ref{it:halfdim-deven},
suppose that $k_\epsilon=\frac{d}{2}$ and let $N$ be a primary stratum of codimension $k=\frac{d}{2}$.  If $N$ has isotropy order 2, then the unique $\gamma\in \iiso^{\rm max}(N)$ has eigenvalues $1$ and $-1$, each with multiplicity $k$, and thus $\tr(\gamma)=0$.   Hence $b_0^1(N)=0$.  If $N$ has higher order isotropy, then every $\gamma\in \iiso^{\rm max}(N)$ has at least two eigenvalues of absolute value strictly less than 1, while 1 occurs as an eigenvalue with multiplicity $k$, so $\tr(\gamma)>0$.   Thus $b_0^1(N)>0$.   Hence the presence of any primary strata with isotropy order greater than 2 implies that $B_\epsilon(\mathcal O)>0$, so we can again apply Proposition \ref{prop: odd par}.
\end{proof}

\section{Appendix}
\label{app}

We outline the proof of Theorem~\ref{bkthm}.    We are closely following the proof of \cite[Theorem 4.1]{D76} but expressing it in a more general context.

Let $M$ be an arbitrary Riemannian manifold and $\gamma$ an isometry of $M$. Let $d(x,y)$ denote the induced distance function between $x$ and $y$ in $M$.  Outside of a small tubular neighborhood of ${\operatorname{{Fix}}}(\gamma)$, we have that  $(4\pi t) ^{-d/2}e^{-\frac{ d^2(x, \gamma(x))}{4t}}\to 0$ uniformly as $t\to 0^+$.  Thus for any compactly supported smooth function $f$ on $M$, we have
\begin{align*}
&\int_{M}  \,(4\pi t) ^{-d/2}e^{-\frac{ d^2(x, \gamma(x))}{4t}}f(x) \,dV_{ M}(x)
\\
&\qquad = \sum_{Q\in \mathcal{C}(Fix(\gamma))} \int_{U_Q} \,(4\pi t) ^{-d/2}e^{-\frac{ d^2(x, \gamma(x))}{4t}}f(x) \,dV_{ M}(x)
+ O(e^{-c/t}),
\end{align*}
where $U_Q$ is a small tubular neighborhood of $Q$, say of radius $r$, and $c>0$ is a constant, where, as in Notation and Remarks \ref{nota:don} part~\ref{it:don-CFix}, $\mathcal C({\operatorname{{Fix}}}(\gamma))$ denotes the set of connected components of ${\operatorname{{Fix}}}(\gamma).$

We will later specialize to the choices of $f$ that arise in Theorem~\ref{bkthm}, but for now we work in the general setting.

As in \cite[Theorem 4.1]{D76}, we analyze each term in the sum above individually and define
\begin{equation*}
I_Q(f):= \int_{U_Q} \,(4\pi t) ^{-d/2}e^{-\frac{ d^2(x, \gamma(x))}{4t}}f(x) \,dV_{ M}(x).
\end{equation*}

 Let $W_Q\subset TM$ be the normal disk bundle of $Q$ of radius $r$ and let $\varphi: W_Q\to Q$ be the bundle projection.   We use the diffeomorphism $W_Q\to U_Q$ given by ${\rm{x}}\mapsto\exp_{\varphi({\rm{x}})}{\rm{x}}$, where $\exp$ is the Riemannian exponential map of $M$, to express $I_Q(f)$ as
$$
I_Q(f)= \int_Q\int_{\varphi^{-1}(a)} (4\pi t) ^{-d/2} e^{-\frac{ d^2(x, \gamma(x))}{4t}}f(x)\,\psi({\rm{x}}) \,d{\rm{x}} \,dV_{Q}(a)
$$

Here $d{\rm{x}}$ is the Euclidean volume element defined by the Riemannian structure on $(T_aQ)^\perp$, and for ${\rm{x}}\in \varphi^{-1}(a)$, we are writing $x=\exp_a({\rm{x}})$.    The function $\psi$ arising in this change of variables depends only on the curvature and its covariant derivatives and, as shown in \cite[Equation~(2.6)]{D76}, satisfies
\begin{equation}\label{psi}\psi({\rm{x}})=1-\frac{1}{2}R_{i\alpha j\alpha}{\rm{x}}^i{\rm{x}}^j-\frac{1}{6}R_{ikjk}{\rm{x}}^i{\rm{x}}^j+O({\rm{x}}^3) \end{equation}
where the ${\rm{x}}^i$ are the coordinates of ${\rm{x}}$ with respect to an orthonormal basis of $(T_{a}Q)^\perp$ and where the components of the curvature tensor are evaluated at the point $a$.

Letting $\bar{{\rm{x}}}={\rm{x}}-d\gamma_a({\rm{x}})$, Donnelly shows that $d^2(x, \gamma(x))= (\bar{{\rm{x}}})^2 +O(\bar{{\rm{x}}})^3$ and then applies a version of the Morse Lemma, see \cite[Lemma A.1]{D76}, to find new coordinates ${\rm y}$ so that
\begin{equation*}
d^2(x, \gamma(x))=\sum_{i=1}^s{\rm y}^2_i=\|{\rm y}\|^2,
\end{equation*}
Making the two changes of variables ${\rm{x}}\to \bar{{\rm{x}}}\to {\rm y}$ on $(T_aQ)^\perp$, we have
$$d{\rm{x}} =|\det(B(a))|\,d\bar{{\rm{x}}}=|\det(B(a))||J(\bar{{\rm{x}}},{\rm y})|d{\rm y}$$
where
\begin{equation}\label{B} B(a)= ({\operatorname{{Id}}}_{\operatorname{codim}(Q)}-A_a)^{-1}.\end{equation}
 Here $A_a$ is the restriction of $d\gamma_a$ to $(T_aQ)^\perp$ and $|J(\bar{{\rm{x}}},{\rm y})|$ denotes the absolute value of the Jacobian determinant for the second change of variables.
 We are viewing $x=\exp({\rm{x}})$, ${\rm{x}}$ and $\bar{{\rm{x}}}$ as functions of the new variable ${\rm y}$.    Donnelly shows that $|J(\bar{{\rm{x}}},{\rm y})|$ has a Taylor expansion in ${\rm y}$ whose coefficients depend only on the values at $a$ of $B$ and of the curvature tensor $R$ of $M$ and its covariant derivatives:
 \begin{equation}\label{J}
 |J(\bar{{\rm{x}}},{\rm y})|=1+\frac{1}{6}(R_{ikih}B_{ks}B_{ht} +R_{iksh}B_{ki}B_{ht}+R_{ikth}B_{ks}B_{hi}){\rm y}^s{\rm y}^t +O({\rm y}^3).
 \end{equation}

Letting $H_a(f):(T_aQ)^\perp\to \mathbf R$ be given by
\[
H_a(f)({\rm y}):= f(x) |\det(B(a))|\,| J(\overline{{\rm{x}}}, {\rm y})| \,\psi({\rm{x}})
\]

we have
\begin{equation}\label{iqf} I_Q(f)=\int_Q\,\int_{\varphi^{-1}(a)}\,(4\pi t)^{-d/2}e^{-\|{\rm y}\|^2/4t}\,H_a(f)({\rm y})\,d{\rm y}\,dV_{Q}(a).\end{equation}

We expand $H_a(f)$ into its Maclaurin series.  Because of symmetry, only the terms of even degree contribute to the integral $I_Q(f)$.   Thus denoting by $[H_a(f)]_j({\rm y})$ the homogeneous component of degree $2j$ in the Maclaurin series of $H_a(f)$ and then making the change of variable ${\rm z}=\frac{{\rm y}}{\sqrt{t}}$, we have
\begin{multline*}\label{expand} I_Q(f)=(4\pi t)^{-\dim(Q)/2}\int_Q\,\int_{\mathbf R^{\operatorname{codim}(Q)}}\,(4\pi )^{-(\operatorname{codim}(Q))/2}e^{-\|{\rm z}\|^2/4}\sum_{k=0}^L\,t^k [H_a(f)]_k({\rm z})\,d{\rm z}\,dV_Q(a) \\+O(t^{L+1}).
\end{multline*}
(Here we are using the fact that $\varphi^{-1}(a)$ is a disk of radius $\frac{r}{\sqrt{t}}$ in ${\rm z}$, which can be replaced by $\mathbf R^{\operatorname{codim}(Q)}$ without changing the asymptotics.)

Proceeding exactly as in the proof of \cite[Theorem 4.1]{D76}, we complete the integration by doing an iterated integral over the ${\rm z}$ variables, noting that odd powers of any of the coordinates ${\rm z}_i$ lead to integrals of the form $\int_\mathbb{R}\,{\rm z}_i^{2k+1}e^{-({\rm z}_i^2)/4}\,d{\rm z}_i$ which are equal to zero, while the contributions from the terms of the form $\int_\mathbb{R}{\rm z}_i^{2m}e^{-({\rm z}_i^2)/4}\,d{\rm z}_i$ can be computed by using the classical formula
\begin{equation*}
\int_{\mathbb R}x^{2m}e^{-x^2}\,dx=\frac{1\cdot3\cdot5\cdots (2m-1)}{2^{m+1}}\sqrt{\pi}.
\end{equation*}
We are left with
\begin{equation*}
I_Q(f)=(4\pi t)^{-\dim(Q)/2}\sum_{k=0}^L\frac{t^k}{k!}\int_Q\, \Box^k_{{\rm y}}(H_a(f))(0)\,dV_Q(a) +O(t^{L+1}),
\end{equation*}
where, following the notation of Donnelly in \cite[Theorem 4.1]{D76}, we denote
\[
\Box_{\rm y}:=\sum_{i=1}^s\frac{\partial^2}{\partial {\rm y}_i^2}.
\]

Thus we conclude:

\begin{prop}\label{f}  For any compactly supported smooth function $f$ on $M$, we have
\begin{multline*}
\int_{M}  \,(4\pi t) ^{-d/2}e^{-\frac{ d^2(x, \gamma(x))}{4t}}f(x) \,dV_{ M}(x)
\\
= \sum_{Q\in \mathcal{C}(Fix(\gamma))} (4\pi t)^{-\dim(Q)/2}\sum_{k=0}^L\frac{t^k}{k!}\int_Q\, \Box^k_{{\rm y}}\left(H_a(f)\right)(0)\,dV_Q(a) +O(t^{L+1})\end{multline*}
where $H_a(f)=\frac{f(x)\,| J(\overline{{\rm{x}}}, {\rm y})| \,\psi({\rm{x}})}{|\det({\operatorname{{Id}}}_{\operatorname{codim}(Q)}-A_a)|}$ with $\psi(x)$ and $| J(\overline{{\rm{x}}}, {\rm y})|$ given as in Equations~\eqref{psi} and \eqref{J} and with $A_a$ being the restriction of $d\gamma_a$ to $(T_aQ)^\perp$.
\end{prop}
\medskip

To complete the proof of Theorem~\ref{bkthm} part~\ref{it:bkthm-global}, write
\begin{equation}\label{fjp}f_j^p(x)=C_{1,2} \gamma^*_{\cdot, 2} u^p_j(x,x).\end{equation}
Then

\begin{multline*}\int_{M} C_{1,2} \gamma^*_{\cdot, 2} K^p(t,x,x) dV_{M}(x) = \sum_{j=0}^L\, t^j\int_{M}\,(4\pi t) ^{-d/2} e^{-\frac{ d^2(x, \gamma(x))}{4t}}f_j^p(x)dV_{M}(x)+O(t^{L+1})\\
= \sum_{Q\in \mathcal{C}(Fix(\gamma))}\,(4\pi t)^{-\dim(Q)/2}\sum_{k=0}^{L} t^k\,\int_{Q} b^p_k(\gamma,a) dV_Q (a) +O(t^{L+1})
\end{multline*}
where
$$b^p_k(\gamma,a)=\sum_{j=0}^k\,\frac{1}{j!}\Box^j_{{\rm y}}\left(H_a(f^p_{k-j})\right)(0)$$
with $f^p_{k-j}$ defined as in Equation~\eqref{fjp}.

In particular, when $k=0$, we have
\begin{equation}\label{bop}
b^p_0(\gamma,a)= H_a(f_0^p)(0)=\frac{C_{1,2}\gamma^*_{\cdot, 2} u^p_0(a,a)}{\lvert \det({\operatorname{{Id}}}_{\operatorname{codim}(Q)}-A_a)\rvert} = \frac{\tr_p(d\gamma_a)}{\lvert \det({\operatorname{{Id}}}_{\operatorname{codim}(Q)}-A_a)\rvert}
\end{equation}
where $\tr_p(d\gamma_a)$ is defined as in Notation and Remarks \ref{nota:don} part~\ref{it:don-trace}.  (The second equality follows from the fact that $u_0^p(a,a)$ is the identity transformation of $\Lambda^p (T^*_a M)$
as noted in Proposition~\ref{localparam} part~\ref{uzero}.)

Theorem~\ref{bkthm} part~\ref{it:bkthmlocal} also follows with
$$c^p_k(\eta,\gamma,a)=\sum_{j=0}^k\,\frac{1}{j!}\Box^j_{{\rm y}}\left(H_a(\eta f^p_{k-j})\right)(0).$$

\textbf{Funding}: This work was supported by the Banff International Research Station and Casa Matem\'atica Oaxaca [19w5115]; the Association for Women in Mathematics National Science Foundation ADVANCE grant [1500481]; and the Swiss Mathematical Society [SMS Travel Grant] for the travel of KG. IMS was supported by the Leverhulme Trust grant RPG-2019-055.

\textbf{Acknowledgments}:
We thank Carla Farsi for a helpful discussion of orbifold orientability.  We also thank the Banff International Research Station and Casa Matem\'atica Oaxaca for initiating our collaboration by hosting the 2019 Women in Geometry Workshop. In addition, we thank the Association for Women in Mathematics and the organizers of the Women In Geometry Workshop for supporting this collaboration. IMS thanks Prof. Jacek Brodzki for providing her with excellent conditions to work.

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