Coexact spectral gaps of flat bundles on spherical space forms and the McKay distance

Abstract: On a spherical space form $S^3/\Gamma$ with the round metric of radius $R$, let $E_\tau$ be the flat bundle with holonomy a finite-dimensional representation $\tau$ of $\Gamma$, and consider the Hodge Laplacian on coexact $1$-forms valued in $E_\tau$. Its spectral gap is $q_\tau^2/R^2$, where $q_\tau\ge 2$ is the lowest coexact level carrying a constituent of $\tau$. When $-I\in\Gamma$, equivalently when the McKay graph is bipartite, the entry level is determined constituentwise by McKay distance: an irreducible constituent at distance $d$ enters at level $d$ for $d\ge 2$, at level $3$ for $d=1$, and at level $2$ for $d=0$; $q_\tau$ is the minimum of these entry levels over the constituents of $\tau$. In particular, if $d_\tau\ge 2$ then the gap is $d_\tau^2/R^2$. The adjoint bundle of an irreducible flat $\mathrm{SU}(2)$ connection always has $d_\tau\ge 2$, so its gap is the squared adjoint distance. Across the ADE classification of finite subgroups of $\mathrm{SU}(2)$ this gap is $4/R^2$ for every irreducible flat connection, with one exception: the Galois connection on the Poincar'e homology sphere $S^3/2I$ gives $36/R^2$, its adjoint on the short branch of affine $E_8$; here $2I$ is the unique perfect finite subgroup of $\mathrm{SU}(2)$. The ingredients are classical, drawn from the form spectra of Ikeda and Taniguchi, the representation-theoretic descent of Lauret, Miatello and Rossetti, and the McKay correspondence; the content is the uniform expression of the coexact gap through the first coexact occurrence level $q_\tau$, with its McKay-distance form in the bipartite cases, the adjoint as application, and the icosahedral case as the one exception.

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Introduction

Let $\Gamma\subset\mathrm{SU}(2)$ be finite and let $S^3/\Gamma$ carry the round metric of constant curvature $1/R^2$, with $\Gamma$ acting by left multiplication through the fixed inclusion, so the McKay graph is that of the inclusion. A finite-dimensional unitary representation $\tau$ of $\Gamma$ determines a flat bundle $E_\tau$ on $S^3/\Gamma$; the flat adjoint bundle of a flat $\mathrm{SU}(2)$ connection $\rho$ is the case $\tau=\mathrm{Ad}\circ\rho$. Because $S^3$ is $2$-connected, $H^1(S^3/\Gamma;E_\tau)\cong H^1(\Gamma;\tau)$, and since $\Gamma$ is finite and the coefficients have characteristic zero, $H^1(\Gamma;V)=0$ for every finite-dimensional $\Gamma$-module $V$. The twisted harmonic $1$-forms therefore vanish for every $\tau$, so the Hodge Laplacian $\Delta_\tau$ on coexact $E_\tau$-valued $1$-forms is positive, and we study the bottom of its spectrum.

On round $S^3$ of radius $R$ the coexact $1$-forms have eigenvalues $m^2/R^2$ for integers $m\ge 2$, and the level-$m$ space has left content $V_m|_\Gamma$ and $V_{m-2}|_\Gamma$, up to the right-factor multiplicities, with $V_j=\mathrm{Sym}^j\mathbb{C}^2$ [IkedaTaniguchi]. The modes on $S^3/\Gamma$ are the $\Gamma$-invariants $(E_m\otimes\tau)^\Gamma$, and by Peter-Weyl, Frobenius reciprocity, and the self-duality of the $V_j$ these are nonzero exactly when a constituent of $\tau$ occurs in $V_m|_\Gamma$ or $V_{m-2}|_\Gamma$. Section [ref:sec:spectrum] gives this descent, already present for form spectra in Ikeda's work on $p$-spectra [Ikeda1989], with the Dirac analogue due to B"ar [Bar1996] and the representation-equivalence form due to Lauret, Miatello and Rossetti [LauretMiatelloRossetti]; the present use inserts an arbitrary finite-dimensional twist by the diagonal action.

For a flat bundle $E_\tau$, set

$$q_\tau=\min\{\,m\ge 2:\ \mathrm{Hom}_\Gamma(\tau^*,\,V_m|_\Gamma\oplus V_{m-2}|_\Gamma)\ne 0\,\},$$

equivalently, since the $V_j$ are self-dual, the least coexact level at which a constituent of $\tau$ meets the left content. The spectral gap of $\Delta_\tau$ on coexact $1$-forms is $q_\tau^2/R^2$.

When $-I\in\Gamma$, equivalently when the McKay graph is bipartite, the entry level is read constituentwise from the McKay distance: a constituent at distance $d\ge2$ enters at $m=d$, a trivial constituent at $m=2$, a distance-one constituent at $m=3$, and $q_\tau$ is the minimum of these entry levels over the constituents. Thus the value $9/R^2$ at $d_\tau=1$ requires that no distance-two constituent be present. Proposition [ref:prop:gapvalue] proves this and treats the non-bipartite failure.

The flat adjoint bundle is the case $\tau=\mathrm{Ad}\circ\rho$, with $(\mathrm{ad}_\rho)_{\mathbb C}\cong\mathrm{Sym}^2\rho$. For an irreducible connection this representation carries no trivial summand and sits on even-distance nodes, so the adjoint distance is at least two. Proposition [ref:prop:adjoint] then reduces the adjoint coexact gap to a McKay-distance calculation. Section [ref:sec:gap] carries this calculation across the ADE classification: the gap is $4/R^2$ for every irreducible flat connection, with one exception. For the binary icosahedral group $2I$, which is perfect, no one-dimensional character relates the defining connection $Q$ to its Galois conjugate $Q'$; the adjoint $\mathrm{Sym}^2 Q'$ sits at distance six in affine $E_8$, so the Galois connection on the Poincar'e homology sphere $S^3/2I$ has gap $36/R^2$.

The distinguished status of $\Sigma(2,3,5)=S^3/2I$ is familiar in gauge theory: among spherical space forms with nontrivial fundamental group it is the unique integral homology sphere, and in the classical instanton-Floer setting for homology spheres it is the spherical example singled out by Fintushel and Stern [FintushelStern]. Nasatyr computed the Floer indices of Seifert fibred homology spheres as the spectral flow of twisted signature operators with adjoint-bundle coefficients [Nasatyr1992]; Kirk and Klassen developed general spectral-flow techniques for paths of flat connections [KirkKlassen]; and Helle records the character tables, Dynkin diagrams and flat-connection organization for the binary polyhedral spaces [Helle]. On the spectral side, the closest neighbors compute the untwisted form spectrum directly: the Hodge Laplacian on $1$-forms of homogeneous $3$-spheres is treated by Henkel and Lauret [HenkelLauret2026], and the explicit $p$-form spectra of lens spaces by Lauret [LauretLens2018]; neither carries a nonabelian flat twist or a McKay-distance organization.

On the coexact $1$-form summand the operator is the square of the curl-type first-order operator $\ast d_\nabla$, so the gap records the square of the bottom absolute coexact curl eigenvalue rather than spectral flow. In the bipartite principal case $d_\tau\ge 2$, that eigenvalue is $d_\tau/R$. The bottom eigenvalue is read from the classical restricted form spectrum together with the McKay correspondence, so the value $36/R^2$ rests on classical analysis; the spectral-flow and eta-invariant literature does not single it out as a coexact spectral gap.

The objects throughout are flat bundles on spherical space forms carrying their round metrics, and the spectrum studied is that of the Hodge Laplacian on the coexact $1$-form summand. Section [ref:sec:setup] fixes the metric, the bundles, and the operator. Section [ref:sec:spectrum] records the $S^3$ coexact spectrum and the descent to $S^3/\Gamma$. Section [ref:sec:gap] proves the gap for a general flat bundle and derives the adjoint comparison across the classification.

Setup

The space and the McKay graph

Identify $S^3$ with $\mathrm{SU}(2)$ carrying the bi-invariant metric of constant sectional curvature $1/R^2$. Let $\Gamma\subset\mathrm{SU}(2)$ be a finite subgroup acting on $S^3$ by left translation, and write $X=S^3/\Gamma$ for the resulting spherical space form, a closed oriented $3$-manifold of constant curvature $1/R^2$ with $\pi_1(X)=\Gamma$. The finite subgroups are the cyclic groups $\mathbb{Z}_n$, the binary dihedral groups $D_n^{*}$ of order $4n$, and the binary tetrahedral, octahedral, and icosahedral groups $2T,2O,2I$ of orders $24,48,120$.

Let $V_j=\mathrm{Sym}^j\mathbb{C}^2$ be the irreducible $\mathrm{SU}(2)$ representation of dimension $j+1$, so $V_1$ is the defining representation. The McKay graph of $\Gamma$ has the irreducible representations of $\Gamma$ as vertices, the trivial representation as distinguished vertex, and $m_{\sigma\sigma'}=\dim\mathrm{Hom}_\Gamma\bigl(\sigma',\,V_1|_\Gamma\otimes\sigma\bigr)$ edges between $\sigma$ and $\sigma'$. By the McKay correspondence [McKay] it is the affine Dynkin diagram attached to $\Gamma$. The graph is bipartite exactly when $-I\in\Gamma$. If $-I\in\Gamma$, then $-I$ acts by a scalar on each irreducible $\Gamma$-representation, and tensoring with $V_1$ changes this scalar; the two central characters give the bipartition, so the scalar on a vertex at distance $d$ from the trivial vertex is $(-1)^d$. If $-I\notin\Gamma$, then $\Gamma$ is an odd cyclic group, and its McKay graph is the odd cycle $\widetilde A_{|\Gamma|-1}$. Graph distances $d(\sigma)$ from the trivial vertex are taken on the underlying unweighted graph, and for a representation $\tau$ we set

$$d_\tau=\min\{\,d(\sigma):\sigma\text{ a constituent of }\tau\,\}.$$

Flat bundles and the adjoint

A finite-dimensional unitary representation $\tau\colon\Gamma\to\mathrm{U}(V)$ determines a flat Hermitian bundle $E_\tau=(S^3\times V)/\Gamma$ on $X$, with the flat connection $\nabla$ descended from the trivial connection upstairs. Gauge-equivalence classes of flat $\mathrm{SU}(2)$ connections on $X$ are identified with conjugacy classes of homomorphisms $\rho\colon\Gamma\to\mathrm{SU}(2)$; the real adjoint bundle $\mathrm{ad}_\rho$ has fibre $\mathfrak{su}(2)$ and holonomy $\mathrm{Ad}\circ\rho$. We work with the complexification, the flat bundle $E_\tau$ for

$$\tau=(\mathrm{ad}_\rho)_{\mathbb{C}}\cong\mathrm{Sym}^2\rho \quad\text{as a complex }\Gamma\text{-representation,}$$

the isomorphism being the identification of $\mathfrak{su}(2)_{\mathbb{C}}=\mathfrak{sl}(2,\mathbb{C})$ with the spin-$1$ representation $V_2$ of the $\mathrm{SU}(2)$ acting through $\rho$.

The operator and positivity

Let $d_\nabla$ be the flat exterior derivative of $E_\tau$ and $\Delta_\tau=d_\nabla^{*}d_\nabla+d_\nabla d_\nabla^{*}$ the twisted Hodge Laplacian. For the flat unitary connection the Hodge decomposition reads

$$\Omega^1(X;E_\tau)=\operatorname{im}d_\nabla\ \oplus\ \operatorname{im}d_\nabla^{*} \ \oplus\ \mathcal H^1(X;E_\tau),$$

with $\Delta_\tau$ preserving each summand; the coexact $1$-forms are $\operatorname{im}d_\nabla^{*}$. On that summand the curl operator $\ast d_\nabla$ is formally self-adjoint and

$$(\ast d_\nabla)^2=d_\nabla^{*}d_\nabla=\Delta_\tau,$$

so the coexact spectrum of $\Delta_\tau$ is the set of squares of the $\ast d_\nabla$ spectrum, and the bottom coexact eigenvalue is the square of the smallest $|\ast d_\nabla|$.

\begin{lemma}[Vanishing of harmonic $1$-forms] For every finite-dimensional $\tau$, $H^1(X;E_\tau)=0$. Consequently the coexact $1$-forms coincide with the coclosed $1$-forms, and $\Delta_\tau$ is strictly positive on them. \end{lemma}

\begin{proof} By de Rham with local coefficients $\mathcal H^1(X;E_\tau)\cong H^1(X;E_\tau)$, and degree-one cohomology with local coefficients is computed by crossed homomorphisms, hence depends only on $\pi_1 X=\Gamma$; so $H^1(X;E_\tau)\cong H^1(\Gamma;\tau)$. Since $\Gamma$ is finite and the coefficients have characteristic zero, averaging over $\Gamma$ gives $H^1(\Gamma;V)=0$ for every finite-dimensional $\Gamma$-module $V$; hence $H^1(X;E_\tau)=\mathcal H^1(X;E_\tau)=0$. The kernel of $\Delta_\tau$ on $1$-forms is exactly $\mathcal H^1$, so $\Delta_\tau$ is strictly positive on $\Omega^1(X;E_\tau)$, and in particular on the coexact summand; and with $\mathcal H^1=0$ that summand $\operatorname{im}d_\nabla^{*}$ is the full space $\ker d_\nabla^{*}$ of coclosed $1$-forms. \end{proof}

The coexact spectrum and its descent

The coexact spectrum on $S^3$

Write $L^2(S^3)=\bigoplus_{k\ge 0}V_k\boxtimes V_k^{*}$ for the Peter–Weyl decomposition of functions, fixing the convention that $\mathrm{SU}(2)_L$ acts by left translation on the first factor and the commuting right $\mathrm{SU}(2)$-action on the second. By Ikeda and Taniguchi [IkedaTaniguchi], the Hodge Laplacian on coexact $1$-forms on $S^3(R)$ has eigenvalues

$$\lambda_m=\frac{m^2}{R^2},\qquad m\ge 2,$$

with level-$m$ eigenspace

$$E_m=\bigl(V_m\boxtimes V_{m-2}\bigr)\ \oplus\ \bigl(V_{m-2}\boxtimes V_m\bigr)$$

as an $\mathrm{SU}(2)_L\times\mathrm{SU}(2)_R$ representation, the first factor carrying left translation and the second the commuting right action. Each summand is a distinct $\mathrm{SU}(2)_L\times\mathrm{SU}(2)_R$ irreducible, so by Schur the curl $\ast d$ is scalar on each; with $(\ast d)^2=\Delta$ the two summands are the $\pm m/R$ eigenspaces of $\ast d$, up to the orientation convention. The lowest level $m=2$ gives $E_2=(V_2\boxtimes V_0)\oplus(V_0\boxtimes V_2)$ and eigenvalue $4/R^2$.

Descent to $X=S^3/\Gamma$

Fix the associated-bundle convention $\gamma\cdot(x,v)=(\gamma x,\tau(\gamma)v)$ on $E_\tau=(S^3\times V_\tau)/\Gamma$, so sections of $E_\tau$ identify with $\Gamma$-invariant $V_\tau$-valued forms on $S^3$ for the diagonal action, $\Gamma$ acting by left translation on the form and by $\tau$ on $V_\tau$. Left translation commutes with the curl and preserves each $E_m$. Pulling $E_\tau$ back to $S^3$ trivializes it with the product flat connection, so the lifted twisted Laplacian is $\Delta_{S^3}\otimes 1$ on $V_\tau$-valued forms. Thus the level-$m$ twisted eigenspace on $S^3/\Gamma$ is

$$\bigl(E_m\otimes V_\tau\bigr)^{\Gamma},\qquad\text{eigenvalue }\ m^2/R^2 .$$

This is the descent of the $p$-spectrum of a constant-curvature space form, due to Ikeda [Ikeda1989] and given its representation-equivalence form by Lauret, Miatello and Rossetti [LauretMiatelloRossetti], with the coefficient $V_\tau$ inserted by the diagonal action.

Under left translation only the first factor of each piece of $E_m$ is acted on, so the right factors are flat multiplicity spaces:

$$E_m\big|_{\Gamma} =\bigl(V_m\big|_{\Gamma}\bigr)^{\oplus(m-1)} \ \oplus\ \bigl(V_{m-2}\big|_{\Gamma}\bigr)^{\oplus(m+1)} ,$$

the left types $V_m|_\Gamma$ and $V_{m-2}|_\Gamma$ carrying the right-factor multiplicities $\dim V_{m-2}=m-1$ and $\dim V_m=m+1$. The dimensions close, $(m-1)(m+1)+(m+1)(m-1)=2(m^2-1)$, the Ikeda–Taniguchi multiplicity of the level-$m$ coexact eigenvalue. Computing the invariants by restriction and the tensor–hom adjunction,

$$\bigl(E_m\otimes V_\tau\bigr)^{\Gamma} \cong\mathrm{Hom}_\Gamma\!\bigl(\tau^{*},\,E_m|_\Gamma\bigr),$$

nonzero exactly when $\tau^{*}$ shares a constituent with the left content $W_m:=V_m|_\Gamma\oplus V_{m-2}|_\Gamma$. Each $V_j$ is self-dual, so $W_m$ is self-dual as a $\Gamma$-module; hence $\tau^{*}$ meets $W_m$ iff $\tau$ meets $W_m^{*}=W_m$, and the criterion reads

$$\bigl(E_m\otimes V_\tau\bigr)^{\Gamma}\ne 0 \iff \text{a constituent of }\tau\text{ occurs in }W_m .$$

The reduction uses the self-duality of the left content, not of $\tau$, so it holds for arbitrary $\tau$; the adjoint case is self-dual only incidentally.

\begin{proposition}[Twisted coexact spectrum] For $m\ge 2$, the multiplicity of the eigenvalue $m^2/R^2$ in the $E_\tau$-valued coexact $1$-form spectrum on $X$ is

$$\mu_\tau(m) =(m-1)\dim\mathrm{Hom}_\Gamma\!\bigl(\tau^{*},V_m|_\Gamma\bigr) +(m+1)\dim\mathrm{Hom}_\Gamma\!\bigl(\tau^{*},V_{m-2}|_\Gamma\bigr).$$

As a set of eigenvalues the twisted coexact spectrum is

$$\Bigl\{\,m^2/R^2:\ m\ge 2,\ \mathrm{Hom}_\Gamma\!\bigl(\tau^{*},W_m\bigr)\ne 0\,\Bigr\},$$

equivalently those $m$ at which a constituent of $\tau$ occurs in $W_m=V_m|_\Gamma\oplus V_{m-2}|_\Gamma$. Its bottom is $q_\tau^2/R^2$, where

$$q_\tau=\min\bigl\{\,m\ge 2:\ \text{a constituent of }\tau\text{ occurs in }W_m\,\bigr\}.$$

The defining set is nonempty: $V_1|_\Gamma$ is faithful, so by Burnside–Brauer every irreducible representation of $\Gamma$ occurs in some tensor power $V_1^{\otimes k}|_\Gamma$, and Clebsch–Gordan decomposes these tensor powers into symmetric powers $V_j|_\Gamma$. Hence every irreducible occurs in some $V_m|_\Gamma$; the sharp first occurrence, at $m=d(\sigma)$, is proved in Lemma [ref:lem:first-occurrence] below. Thus $q_\tau$ is finite. \end{proposition}

The gap and the ADE classification

First occurrence

Let $A$ be the McKay adjacency matrix, with entries $A_{\sigma\sigma'}=\dim\mathrm{Hom}_\Gamma(\sigma',V_1|_\Gamma\otimes\sigma)$, so that multiplication by $V_1$ in the representation ring acts as $A$. The symmetric powers satisfy $V_{a+1}=V_1V_a-V_{a-1}$, hence $V_a=U_a(A)$ applied to the trivial node, where $U_a$ is the degree-$a$ polynomial of leading coefficient one with $U_a(2\cos\theta)=\sin((a+1)\theta)/\sin\theta$.

\begin{lemma}[First occurrence] For an irreducible $\Gamma$-representation $\sigma$ at McKay distance $d(\sigma)$, the least $a$ with $\sigma\subset V_a|_\Gamma$ is $a=d(\sigma)$. If moreover $-I\in\Gamma$, equivalently the McKay graph is bipartite, then $\sigma\subset V_a|_\Gamma$ forces $a\equiv d(\sigma)\pmod 2$. \end{lemma}

\begin{proof} The multiplicity of $\sigma$ in $V_a|_\Gamma$ is the matrix element $\bigl(U_a(A)\bigr)_{0\sigma}$. Since $U_a$ has degree and parity $a$ and leading coefficient one, this is a combination of $(A^a)_{0\sigma},(A^{a-2})_{0\sigma},\dots$, and $(A^k)_{0\sigma}$ counts length-$k$ walks from $0$ to $\sigma$, which vanishes for $k<d(\sigma)$. Hence the multiplicity vanishes for $a<d(\sigma)$, and at $a=d(\sigma)$ it equals $(A^{d(\sigma)})_{0\sigma}>0$, the lower terms contributing nothing. When $-I\in\Gamma$, the preceding bipartition gives that $-I$ acts on $\sigma$ by $(-1)^{d(\sigma)}$, while it acts on $V_a|_\Gamma$ by $(-1)^a$. Hence occurrence of $\sigma$ in $V_a|_\Gamma$ forces $a\equiv d(\sigma)\pmod 2$. \end{proof}

The gap

\begin{proposition}[Gap from first occurrence] Let $\tau$ be a finite-dimensional $\Gamma$-representation and $d_\tau=\min\{d(\sigma):\sigma\text{ an irreducible constituent of }\tau\}$. For an irreducible constituent $\sigma$, let $e(\sigma)$ be the least $m\ge2$ for which $\sigma$ occurs in $W_m=V_m|_\Gamma\oplus V_{m-2}|_\Gamma$. Then

$$e(\sigma)=2\quad\text{if }d(\sigma)=0,\qquad e(\sigma)=d(\sigma)\quad\text{if }d(\sigma)\ge2,$$

with no hypothesis on $\Gamma$. If $-I\in\Gamma$, then also $e(\sigma)=3$ for $d(\sigma)=1$. Consequently

$$q_\tau=\min_{\sigma\subset\tau} e(\sigma),$$

where the minimum runs over irreducible constituents. In particular, if $d_\tau\ge2$, then $q_\tau=d_\tau$ and the bottom of the twisted coexact spectrum is $d_\tau^2/R^2$. \end{proposition}

\begin{proof} The equality $q_\tau=\min_{\sigma\subset\tau}e(\sigma)$ is just the definition of $q_\tau$. If $d(\sigma)=0$, then $\sigma$ is the trivial representation and occurs in $V_0$, hence in $W_2$, and no level below $2$ is allowed. If $d(\sigma)\ge2$, Lemma [ref:lem:first-occurrence] places $\sigma$ first in $V_{d(\sigma)}|_\Gamma$, so $W_{d(\sigma)}$ contains $\sigma$ through its $V_{d(\sigma)}$ summand. For $m<d(\sigma)$, both $m$ and $m-2$ are below the first occurrence, so $W_m$ contains no copy of $\sigma$. Thus $e(\sigma)=d(\sigma)$. Finally assume $-I\in\Gamma$ and $d(\sigma)=1$. The first occurrence is in $V_1|_\Gamma$, while the parity restriction excludes $\sigma$ from the even levels $V_0|_\Gamma$ and $V_2|_\Gamma$ appearing in $W_2$. Hence the first coexact level is $m=3$, through the $V_{m-2}=V_1$ summand. \end{proof}

The bipartite hypothesis enters only for $d_\tau\le1$. On $\mathbb{Z}_3$, where $-I\notin\Gamma$ and $\widetilde A_2$ is not bipartite, a nontrivial character has $d=1$ but already occurs in $V_2|_{\mathbb{Z}_3}$, so $q=2$ rather than $3$. The adjoint uses only the $d_\tau\ge2$ case.

The adjoint

A flat $\mathrm{SU}(2)$ connection is irreducible precisely when $\rho\colon\Gamma\to\mathrm{SU}(2)$ is an irreducible $2$-dimensional representation. Its complexified adjoint is

$$(\mathrm{ad}_\rho)_{\mathbb C}\cong\mathrm{Sym}^2\rho,$$

a $3$-dimensional representation of $\Gamma$, irreducible for the polyhedral groups $2T,2O,2I$ and reducible for the binary dihedral groups, where $\mathrm{Sym}^2\tau_{2j-1}$ contains a fixed distance-two sign character $\varepsilon_D$.

\begin{proposition}[Adjoint gap] For an irreducible flat $\mathrm{SU}(2)$ connection $\rho$ on $X=S^3/\Gamma$, the bottom of the adjoint coexact spectrum is $d_{\mathrm{Sym}^2\rho}^2/R^2$, and $d_{\mathrm{Sym}^2\rho}\ge 2$. \end{proposition}

\begin{proof} An irreducible $2$-dimensional $\rho$ forces $\Gamma$ nonabelian, hence binary dihedral or polyhedral, so $-I\in\Gamma$. Being central, $-I$ acts on the irreducible $\rho$ by a scalar, necessarily $\pm I$ since $\det\rho=1$, so $\mathrm{Sym}^2\rho(-I)=I$: the adjoint factors through $\Gamma/\{\pm I\}$, and as $-I$ acts on a distance-$d$ node by $(-1)^d$, every constituent lies at even distance. The trivial is not among them: since $\det\rho=1$ the representation $\rho$ is self-dual, so $\mathrm{Hom}_\Gamma(\mathbf1,\rho\otimes\rho)\cong\mathrm{Hom}_\Gamma(\rho^{*},\rho)$ is one-dimensional by Schur, and as $\rho\otimes\rho=\mathrm{Sym}^2\rho\oplus\Lambda^2\rho$ with $\Lambda^2\rho=\det\rho=\mathbf 1$, that unique trivial summand is the determinant, lying in $\Lambda^2$. Even distance and no trivial constituent give $d_{\mathrm{Sym}^2\rho}\ge 2$, and Proposition [ref:prop:gapvalue] returns the bottom $d_{\mathrm{Sym}^2\rho}^2/R^2$. \end{proof}

The ADE table

The distance $d_{\mathrm{Sym}^2\rho}$ is read from the McKay graph; for the binary polyhedral groups the graph and the flat $\mathrm{SU}(2)$ connections are tabulated by Helle [Helle]. Cyclic groups carry no irreducible flat $\mathrm{SU}(2)$ connection, their image being abelian. In every other case the nearest constituent of the adjoint sits at distance two, with a single exception.

The binary dihedral row covers every odd index at once. In $D_n^{*}=\langle a,b\mid a^{2n}=1,\ b^2=a^n,\ bab^{-1}=a^{-1}\rangle$ the $2$-dimensional irreducible $\tau_\ell$ has $\chi_{\tau_\ell}(a^k)=2\cos(\pi\ell k/n)$ and $\chi_{\tau_\ell}(ba^k)=0$. From $(ba^k)^2=a^n$ and $\chi_{\tau_\ell}(a^n)=2(-1)^\ell$,

$$\chi_{\mathrm{Sym}^2\tau_\ell}(a^k)=1+2\cos(2\pi\ell k/n),\qquad \chi_{\mathrm{Sym}^2\tau_\ell}(ba^k)=(-1)^\ell .$$

For the connections $\tau_{2j-1}$, subtracting the distance-two sign character $\varepsilon_D$ ($a\mapsto1,\ b\mapsto-1$) leaves a two-dimensional character, while neither the trivial nor the distance-one character occurs. The nearest constituent is $\varepsilon_D$ at distance two and the gap is $4/R^2$. The two $2O$ connections share an adjoint, since $\varepsilon^2=\mathbf 1$ gives $\mathrm{Sym}^2(Q\otimes\varepsilon)=\mathrm{Sym}^2 Q$, again distance two. The binary icosahedral group is the exception: it is perfect, so it has no $1$-dimensional character to twist $Q$ by, and its only other irreducible $\mathrm{SU}(2)$ connection is the Galois conjugate $Q'$, whose adjoint is a distinct node. The Galois adjoint $\mathrm{Sym}^2 Q'$ sits at distance six, so the Galois connection has gap $36/R^2$. Explicitly, the character of $V_a=\mathrm{Sym}^a\mathbb{C}^2$ is

$$\chi_a(\theta)=\frac{\sin((a+1)\theta)}{\sin\theta},$$

with the limiting values at $\theta=0,\pi$. Evaluating it on the nine conjugacy classes of $2I$ and decomposing against the $2I$ character table (The nine classes have half-angles $0,\pi/5,2\pi/5,\pi/3,\pi/2,2\pi/3,3\pi/5,4\pi/5,\pi$, and the entries are the inner products of $\chi_a$ with the irreducible characters of $2I$ [Helle].) gives

$$\begin{array}{c|c|c|c} a & \dim V_a & V_a|_{2I} & \text{McKay distance(s)}\\ \hline 0 & 1 & \mathbf 1 & 0\\ 1 & 2 & Q & 1\\ 2 & 3 & \mathrm{Sym}^2 Q & 2\\ 3 & 4 & \mathbf 4 & 3\\ 4 & 5 & \mathbf 5 & 4\\ 5 & 6 & \mathbf 6 & 5\\ 6 & 7 & \mathbf 4'\oplus \mathrm{Sym}^2 Q' & 6,6 \end{array}$$

where $\mathbf 4'$ denotes the second $4$-dimensional irreducible. Thus $\mathrm{Sym}^2 Q$ first occurs at level two, and $\mathrm{Sym}^2 Q'$, the Galois conjugate of $\mathrm{Sym}^2 Q$, is absent from $V_a|_{2I}$ for $a<6$ and first occurs in $V_6|_{2I}$.

Every irreducible flat $\mathrm{SU}(2)$ connection on a spherical space form thus has adjoint coexact gap $4/R^2$, with the single exception of the Galois connection on $S^3/2I$, whose gap is $36/R^2=6^2/R^2$. The factor $9$ between them is the square of the distance ratio $6/2$ on the affine $E_8$ graph.

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