$\displaystyle=\frac{1}{4}\left(-2\chi(X)-3\sigma(X)-2\right)\mod\mathbb{Z}.$

as $\mathrm{gr}(X^{*},\mathfrak{s}_{X},[\mathfrak{b}])$ is an integer (it is
the expected dimension of a moduli space) and $c_{1}(\mathfrak{s}_{X})$ is a
torsion class, and the claim follows.

Therefore, we only need to approximate $\eta_{\mathrm{Dir}}$ with a good
error. The spectral picture can be found in Figure 6. Here, we computed the
spin length spectrum up to cutoff $R=7.5$ using the algorithm described in
Section 2.

Figure 6. The function $J_{\mathrm{SW},\mathfrak{s}_{0}}(t)$, computed with
cutoff $R=7.5$.

For a spin structure, the Dirac operator is quaternionic and therefore all
eigenspaces have even multiplicity. We can therefore conclude that
$|s_{1}|\geq 1.45$. We compute an approximation to the eta invariant using
$G(x)=e^{-x^{2}/2}$ and $L=1.7$ by truncating the geometric sum at $R=7.5$ and
considering the whole spectral side as an error term. We obtain

$\displaystyle\eta_{\mathrm{Dir}}$
$\displaystyle\approx-\frac{1}{\pi}\sum_{\ell(\gamma)\leq
7.5}\ell(\gamma_{0})\cdot\frac{2\sin(\mathrm{hol}(\tilde{\gamma}))\cdot{\cos(\varphi_{\tilde{\gamma}})}}{|1-e^{\ell+i\theta}|\cdot|1-e^{-(\ell+i\theta)}|}\cdot\frac{1}{\ell(\gamma)}\cdot
e^{-\frac{\ell^{2}}{2}\cdot(1.7)^{2}}$ $\displaystyle=0.641083369621\dots$

and therefore the approximate value

$-2h(\mathsf{SW},\mathfrak{s}_{0})\approx 0.968861147398778$

This shows $h(\mathsf{SW},\mathfrak{s}_{0})=-1/2$ provided we can bound the
error term for $\eta_{\mathrm{Dir}}$ effectively. Again, the error from
truncating the sum on the geometric side is bounded above by $0.0376$ by
Proposition 9.4. For the spectral side, we bound the number of spectral
parameters as in the proof of Proposition 10.1 using the even spinor trace
formula for the function

(25)
$\cos(Tx)\cdot\left(\frac{1}{2\delta}\mathbf{1}_{[-\delta,\delta]}\right)^{\ast
6}\text{ where }\delta=7.5/6$

(whose geometric side we can compute as it is supported in $[-7.5,7.5]$). We
get that there are at most $8$ spectral parameters in $[1.45,2]$, and each
contributes at most

$\frac{1}{\pi}\int_{1.7\cdot 1.45}^{\infty}\sqrt{2\pi}e^{-t^{2}/2}dt<0.0108.$

Furthermore, there are at most $12$ parameters in $[2,3]$ and $31$ parameters
in $[3,4]$; larger parameters are again negligible. Putting everything
together, we see that the error is bounded above by $0.1564$. Therefore, we
have

$-2h(\mathsf{SW},\mathfrak{s}_{0})\in[0.812,1.126]$

and the result in Theorem 0.2 follows because $1$ is the only number of the
form $n/4$ in that interval.

###  10.3. Conclusion of the proof.

We now complete the proof of Theorem 0.2. The arguments in this section will
be quite ad hoc because the spinc structures have a smaller spectral gap than
the spin case, and in particular the error bounds we will be able to provide
will not be as good as in the spin case. Of course, one could in principle
obtain better bounds by computing larger portions of the length spectrum, but
this is a quite challenging task from the computational viewpoint (compare
with the discussion in [28]).

A corollary of the computation of the previous section is the simply connected
spin manifold $(X,\mathfrak{s}_{X})$ bounding $\mathsf{SW}$ we fixed satisfies

$\frac{1}{4}\left(-2\chi(X)-3\sigma(X)-2\right)\in\mathbb{Z}$

As $H^{3}(X,\partial X;\mathbb{Z})=H_{1}(X,\mathbb{Z})=0$, the restriction map

$H^{2}(X,\mathbb{Z})\rightarrow H^{2}(\partial X,\mathbb{Z})$

is surjective; therefore every spinc structure
$\mathfrak{s}=\mathfrak{s}_{0}+x$ on $\mathsf{SW}$ extends to a spinc
structure $\tilde{\mathfrak{s}}=\mathfrak{s}_{X}+\tilde{x}$ on $X$. As
$c_{1}(\tilde{\mathfrak{s}})$ equals $2\tilde{x}$ up to torsion elements, we
have

$\displaystyle\eta_{\mathrm{sign}}/4+\eta_{\mathrm{Dir}}$
$\displaystyle=\frac{1}{4}(c_{1}^{2}(\tilde{\mathfrak{s}})-2\chi(X)-3\sigma(X)-2)\mod\mathbb{Z}$
$\displaystyle=\tilde{x}^{2}\mod\mathbb{Z}.$

As $H_{1}(X,\mathbb{Z})=0$, we have
$\tilde{x}^{2}=\mathrm{lk}(x,x)\mod\mathbb{Z}$ hence we conclude,

$-2h(\mathsf{SW},\mathfrak{s})=\mathrm{lk}(x,x)\mod\mathbb{Z}.$

By the computations of Proposition 8 we know exactly the number of spinc which
are not spin and for which the self-linking number equals a given value in
$\left(\frac{1}{5}\mathbb{Z}\right)/\mathbb{Z}$. Also, we know from Corollary
8.2 that the isometry group acts transitively on the set of non-spin spinc
structures with fixed self-linking number, so we expect a priori to only have
$5$ possible spectral pictures for the remaining spinc structures. This is
indeed the case, see Figure 7.

(a) Class $2$: $30$ elements.

(b) Class $3$: $30$ elements

(c) Class $4$: $24$ elements.

(d) Class $5$: $20$ elements

(e) Class $6$: $20$ elements

Figure 7. Pictures for $\mathsf{SW}$

Comparing the number of spinc structures with a given linking form, we
conclude the following.

######  Lemma 10.2.

The following statements hold:

  1. (1)

for the spinc structures in Class $2$ and $3$, $-2h$ has fractional part $1/5$
and $-1/5$, or vice versa;

  2. (2)

for the spinc structures in Class $4$, $-2h$ has fractional part $0$;

  3. (3)

for the spinc structures in Class $5$ and $6$, $-2h$ has fractional part $2/5$
and $-2/5$, or vice versa.

As in the case of the spin structure, we compute an approximation to the Dirac
eta invariant using $G(x)=e^{-x^{2}/2}$ and $L=1.7$ by truncating the
geometric sum at $R=7.5$ and considering the whole spectral side as an error
term. The results for the approximated value of
$-2h(Y,\mathfrak{s})=\eta_{\mathrm{sign}}/4+\eta_{\mathrm{Dir}}$ are as in the
following table.

Spinc class | approximate value of $\eta_{\mathrm{sign}}/4+\eta_{\mathrm{Dir}}$
---|---
2 | 0.292743626654778…
3 | $-0.133807427751222\dots$
4 | 0.645708045190778…
5 | 0.456382720492778…
6 | 0.542040336591778…

We then again need to bound errors, as in the case of the spin structure. The
error from truncating the sum on the geometric side is bounded above by
$0.0376$, see Proposition 9.4. For the spectral side, using the even spinor
trace formula for the function (25) we can provide decent bounds on spectral
parameters on any given interval. Using this, we can conclude; each of cases
requires a slightly different argument, and we treat them separately.

###### Proof of Theorem 0.2 for Classes $2$, $3$ and $4$..

We begin with Class $4$, which is the most involved because it has the
smallest spectral gap. Using our refined estimates on the number of spectral
parameters as in the previous sections using the test functions (25), we see
that there are at most $1$ eigenvalue in $[0.545,1.05]$, $2$ in $[1.05,1.39]$,
$4$ in $[1.39,1.85]$, $7$ in $[1.85,2.2]$, $15$ in $[2.2,3]$, $30$ in $[3,4]$,
for a total error of $0.6279$. We therefore conclude

$-2h\in[0.017,1.274]$

so that is $-2h=1$ in this case, as it is an integer by Lemma 10.2.

For Classes $2$ and $3$ the gap is wider, so it is easier to get good bounds.
For Class $2$ we can obtain for example

$-2h\in[-0.197,0.784].$

Because the fractional part is either $-1/5$ or $1/5$, we obtain that
$-2h=1/5$. This in turn implies that for Class $3$, $-2h$ has fractional part
$-1/5$; as we obtain with the same approach

$-2h\in[-0.614,0.347],$

we conclude that $-2h=-1/5$ in this case. ∎

The case of classes $5$ and $6$, which have fractional part $2/5$ and $-2/5$
or vice versa, is more delicate because for Class $5$ we can only prove a very
small spectral gap. We first obtain more information on the fractional parts
as follows.

######  Lemma 10.3.

For class $5$, $-2h$ has fractional part $2/5$, while for Class $6$ it has
fractional part $3/5$.

###### Proof.

The output of our computations described in Section 2 provides the description
of a spinc structure as a homomorphism

$\varphi:{\mathbb{Z}/5\mathbb{Z}}\oplus{\mathbb{Z}/5\mathbb{Z}}\oplus{\mathbb{Z}/5\mathbb{Z}}\rightarrow\mathbb{Q}/\mathbb{Z},$

where the natural basis is different from the one used in Section 8. It is
easy to check that the linking form in this new basis is given by

$\frac{1}{5}\begin{bmatrix}0&2&2\\\ 2&0&0\\\ 2&0&1\\\ \end{bmatrix}.$

Indeed, this is the unique symmetric matrix that takes the correct values for
the spinc structures of class $2,3$ and $4$. The lemma is then proved by
plugging in the explicit values for classes $5$ and $6$. ∎

###### Proof of Theorem 0.2 for Class $6$..

For Class $6$, we compute using again the test functions (25) that there are
at most $4$ parameters in $[1,1.5]$, $5$ in $[1.5,2]$, $16$ in $[2,3]$ and
$30$ in $[3,4]$. We therefore have a total error of at most $0.459$, and
therefore

$-2h\in[0.083,1.002]$

hence by the previous Lemma we have $-2h=3/5$. ∎

Finally, to deal with the case of Class $5$, we need some additional
information about the small eigenvalues.

######  Lemma 10.4.

For spinc class $5$, the smallest eigenvalue $s_{1}$ satisfies

(26) $|s_{1}|\in[0.0408361,0.4077692]$

and has multiplicity $1$. Furthermore, $|s_{2}|>1.55$.

###### Proof.

Using the same method as above involving the test functions (25) we conclude
that there are at most two eigenvalues in this interval.

To prove that there is exactly one, we apply the trace formula to the Gaussian
$e^{-(x/1.7)^{2}/2}$ by truncating the sum over geodesics at $R=7.5$. The
computations of the error bound involves the exact same quantities as in the
proof of Proposition 9.4, and we obtain the following estimate for the sum
over the spectrum:

$\sum_{s_{n}}e^{-(1.7\cdot s_{n})^{2}/2}\in[0.9678,1.0789].$

Now, for $t\in[0.0408361,0.4077692]$ we have $e^{-(1.7\cdot
t)^{2}/2}\in[0.786417,0.9976]$ and the spectral picture shows that eigenvalues
which are not in this interval are $\geq 1.55$. The same trace formula
argument used seversal times above to prove upper bounds on spectral density
shows that there are at most $6$ parameters in $[1.55,2]$, $16$ in $[2,3]$,
$30$ in $[3,4]$, from which we see that the contribution of the eigenvalues
$\geq 1.55$ has to be at most $0.2432$. Using the fact that there are at most
two eigenvalues in the interval, we conclude by direct inspection that there
has to be exactly one. ∎

###### Proof of Theorem 0.2 for Class $5$..

For our purposes the main complication arises from the fact that it is hard to
determine the sign of the small eigenvalue using the trace formula. This is
because in order to study the sign we need to use odd test functions, and
these will be small near zero. We will therefore study the two possibilities
separately. We approximate the contribution of the small eigenvalue with the
average $0.7157945$ of

$\displaystyle\frac{1}{\pi}\int_{1.7\cdot
0.04}^{\infty}\sqrt{2\pi}e^{-t^{2}/2}dt$ $\displaystyle\approx 0.945786$
$\displaystyle\frac{1}{\pi}\int_{1.7\cdot
0.41}^{\infty}\sqrt{2\pi}e^{-t^{2}/2}dt$ $\displaystyle\approx 0.485803$

which introduces an error of at most $0.229992$. The total error is computed
to be then at most $0.3289$ (luckily, $|s_{2}|$ is large). If the small
eigenvalue $s_{1}$ were negative we would have

$-2h\in[-0.588,0.069]$

which does not contain any number with fractional part $2/5$. Therefore
$s_{1}>0$, and we have

$-2h\in[0.8432,1.5011]$

from which we conclude $-2h=7/5$. ∎

##  Appendix A Generalities on the group $G$ and the trace formula

All the trace formulas we will need in the present paper are obtained by
specializing the general trace formula for a cocompact torsion-free lattice
$\widetilde{\Gamma}$ in the Lie group

$G=\\{g\in\mathrm{GL}_{2}(\mathbb{C}):|\det(g)|=1\\}.$

The strategy of the proof follows very closely that of [28, Appendix $B$],
where we studied the case of $\mathrm{PGL}_{2}(\mathbb{C})$:

  1. (1)

express the very general trace formula in terms of irreducible representations
and geometric data;

  2. (2)

understand the representation theoretic incarnation of the differential
operators under consideration;

  3. (3)

choose suitable test functions to isolate the relevant representations.

In [28, Appendix $B$], we tackled the first step for
$\mathrm{PGL}_{2}(\mathbb{C})$: we derived a general trace formula for
cocompact lattices in the group $\mathrm{PGL}_{2}(\mathbb{C})$ (which was
called $G$ in the reference). So as not to repeat ourselves, we simply state
the trace formula for $G$ and highlight the differences between it and the
very similar trace formula from [28].

######  Remark A.1.

As it will be clear from the discussion, this approach involving $G$ is only
strictly necessary when dealing with spinors; the trace formulas for functions
and forms, both even and odd, can be derived from the general trace formula
for $\mathrm{PGL}_{2}(\mathbb{C})$.

###  A.1. Notation for subgroups of $G$.

We will work with the maximal torus $T\subset G$ of diagonal matrices; we will
use the parametrizations

(27) $T=\left\\{\left(\begin{array}[]{cc}e^{u/2}e^{i\alpha}&0\\\
0&e^{-u/2}e^{i\beta}\end{array}\right):u\in\mathbb{R},\alpha,\beta\in\mathbb{R}/2\pi\mathbb{Z}\right\\}$

and

(28) $T=\left\\{\left(\begin{array}[]{cc}e^{i\phi}e^{v+i\theta}&0\\\
0&e^{i\phi}e^{-v-i\theta}\end{array}\right)v\in\mathbb{R},\phi,\theta\in\mathbb{R}/2\pi\mathbb{Z}\right\\}.$

The center $Z$ of $G$ is the diagonal copy of $U_{1}$. We will equip $T$ with
the Haar measure

(29) $dt=du\frac{d\alpha}{2\pi}\frac{d\beta}{2\pi}.$

The Weyl group $W$ has two elements, and is generated by
$(u,\alpha,\beta)\mapsto(-u,\beta,\alpha)$.

######  Remark A.2.

Using this notation, the complex length of the image in
$\mathrm{PGL}_{2}(\mathbb{C})$ of an element of $T$ is $u+i2\theta$.

We will denote by $B$ the subgroup of upper triangular matrices; the modular
function is then

$\displaystyle\delta:B$ $\displaystyle\rightarrow\mathbb{R}^{>0}$
$\displaystyle\left(\begin{array}[]{cc}t_{1}&\ast\\\
0&t_{2}\end{array}\right)$ $\displaystyle\mapsto|t_{1}/t_{2}|^{2}$

Finally, $K=U_{2}$ is the maximal compact subgroup; we can identify

$G/K=\mathrm{PGL}_{2}(\mathbb{C})/PU_{2}=\mathbb{H}^{3}$

using the upper half-space model
$\mathbb{H}^{3}\equiv\mathbb{C}\times\mathbb{R}_{>0}$. The tangent space at
$(0,1)$ is then identified with $\mathfrak{p}_{0}=i\mathfrak{su}(2)$. We will
also denote $\mathfrak{p}=\mathfrak{p}_{0}\otimes\mathbb{C}$ for its
complexification.

###  A.2. Unitary representations of $G$

Every irreducible unitary representation of $G$ is isomorphic to one of the
following:

  * (a)

the one dimensional representation with character $\mathrm{det}^{k}$ for some
$k\in\mathbb{Z}$.

  * (b)

the representation $\pi_{s_{1},s_{2},n_{1},n_{2}}$ with
$s_{1}=ir_{1},s_{2}=ir_{2}$ purely imaginary and $n_{1},n_{2}\in\mathbb{Z},$
defined as follows. Consider the character

$\displaystyle\chi_{s_{1},s_{2},n_{1},n_{2}}:B$
$\displaystyle\rightarrow\mathbb{C}^{*}$
$\displaystyle\left(\begin{array}[]{cc}t_{1}&\ast\\\
0&t_{2}\end{array}\right)$
$\displaystyle\mapsto|t_{1}|^{s_{1}}\cdot|t_{2}|^{s_{2}}\cdot\left(\frac{t_{1}}{|t_{1}|}\right)^{n_{1}}\cdot\left(\frac{t_{2}}{|t_{2}|}\right)^{n_{2}}.$

We then define $\pi_{s_{1},s_{2},n_{1},n_{2}}$ to be the representation
induced on $G$ by $\delta^{1/2}\chi_{s_{1},s_{2},n_{1},n_{2}}$. Because
$|t_{1}|\cdot|t_{2}|=1,$ the representation $\pi_{s_{1},s_{2},n_{1},n_{2}}$
only depends on $s_{1}-s_{2},n_{1},n_{2}.$

  * (c)

complementary series representations, which are abstractly isomorphic to
$\pi_{s_{1},s_{2},0,0}$ where the difference $s_{1}-s_{2}$ is real and lies in
the interval $[-1,1].$

Again, there are some coincidences among these representations.

######  Lemma A.1.

We have $\pi_{s_{1},s_{2},n_{1},n_{2}}\cong\pi_{t_{1},t_{2},m_{1},m_{2}}$ iff

$(t_{1},t_{2},m_{1},m_{2})=(s_{1},s_{2},n_{1},n_{2})\text{ or
}(s_{2},s_{1},n_{2},n_{1})\in\frac{i\mathbb{R}\times i\mathbb{R}}{\langle
i(1,1)\rangle}\times\mathbb{Z}\times\mathbb{Z}.$

###### Proof.

The proof is very similar to the case of $\mathrm{PGL}_{2}(\mathbb{C})$
covered in [28, Appendix $B$]. For every smooth compactly supported test
function $f$ on $G,$

$\mathrm{trace}(\pi_{s_{1},s_{2},n_{1},n_{2}})=\widehat{Sf}(\chi_{s_{1},s_{2},n_{1},n_{2}}^{-1}),$

where we view $Sf$ as a $W$-invariant function on the diagonal torus $T.$ By
the main theorem of Bouaziz [9], the Satake transform $f\mapsto Sf$ maps onto
the $W$-invariant compactly supported smooth function on $T.$ Therefore, two
representations $\pi_{s_{1},s_{2},n_{1},n_{2}}$ and
$\pi_{t_{1},t_{2},m_{1},m_{2}}$ are isomorphic iff

$\widehat{F}(\chi_{s_{1},s_{2},n_{1},n_{2}})=\widehat{F}(\chi_{t_{1},t_{2},m_{1},m_{2}})$

for all smooth, compactly supported functions $F$ on $T$ which is
$W$-invariant, i.e.

$F(u,\alpha,\beta)=F(-u,\beta,\alpha).$

The function $F$ is $W$-invariant iff $\widehat{F}(s,n,m)=\widehat{F}(-s,m,n)$
(the forward implication is immediate from the definition, and the reverse
implication follows from Fourier inversion). Since

$\displaystyle\widehat{F}(\chi_{s_{1},s_{2},n_{1},n_{2}}^{-1})$
$\displaystyle=\int F(t)\chi_{s_{1},s_{2},n_{1},n_{2}}(t)dt$
$\displaystyle=\int F(u,\alpha,\beta)\cdot e^{u/2\cdot(s_{1}-s_{2})}\cdot
e^{in_{1}\alpha}\cdot
e^{in_{2}\beta}du\frac{d\alpha}{2\pi}\frac{d\beta}{2\pi}$
$\displaystyle=\widehat{F}(-(s_{1}-s_{2}),-n_{1},-n_{2}),$

the result follows. ∎

###  A.3. The trace formula for $G$

We have the following general result.

######  Proposition A.2.

Let $F$ be any smooth, compactly supported, $W$-invariant function on the
diagonal torus $T$. Let $\widetilde{\Gamma}\subset G$ be a cocompact torsion-
free lattice. The identity

$\displaystyle{}\sum_{s_{1}-s_{2},n,m}m_{\widetilde{\Gamma}}(\pi_{s_{1},s_{2},n,m})\cdot\widehat{F}(\chi_{s_{1},s_{2},n,m}^{-1})+\sum_{k}\frac{1}{|W|}\int_{T}|D(t^{-1})|^{1/2}\cdot\det(t)^{k}\cdot
F(t)dt$
$\displaystyle=-{\frac{1}{8\pi}}\cdot\mathrm{vol}(Y)\cdot\sum_{{z\in\widetilde{\Gamma}\cap
Z}}\left(\frac{d^{2}}{dv^{2}}+\frac{d^{2}}{d\theta^{2}}\right)F|_{t=z}+\sum_{[\gamma]\neq
1}\ell(\gamma_{0})\cdot|D(t_{\gamma}^{-1})|^{-1/2}\cdot F(t_{\gamma}).$

holds. Here the second derivatives in $v$ and $\theta$ are taken with respect
to the parametrization (28) of $T$, and $|W|=2$.

This is in fact a generalization of the trace formula from [28], which can be
recovered by using test functions on the diagonal torus $T$ invariant by the
center $Z$ of $G.$ The proof of this result is essentially identical to the
one of the formula in [28], so we will focus on pointing out similarities and
differences between Proposition A.2 and the trace formula from our first
paper.

  * •

The “identity contribution” to the trace formula has a contribution for each
element of $\widetilde{\Gamma}\cap Z.$ The group $Z$ is compact so this sum is
finite. In our case of interest, $\widetilde{\Gamma}$ is the lift of a lattice
$\Gamma\subset\mathrm{PGL}_{2}(\mathbb{C}),$ and $\widetilde{\Gamma}\cap
Z=\\{1\\}$.

  * •

Recall that the function $F(u,\alpha,\beta)$ is $W$-invariant iff

$F(u,\alpha,\beta)=F(-u,\beta,\alpha).$

In particular, we will use $F(u,\alpha,\beta)=K_{\pm}(\alpha,\beta)\cdot
H_{\pm}(u),$ where $K_{\pm}$ are symmetric and antisymmetric respectively and
$H_{\pm}$ are even and odd respectively, define $W$-invariant functions on
$T.$

  * •

The universal constant $-\frac{1}{8\pi}$ can be determined in several ways;
for example, by specializing this formula in the case of the even trace
formula for coexact $1$-forms and comparing it to the one from [28].

##  Appendix B The odd signature and Dirac operators via representation theory

We need to specialize the general trace formula in Proposition A.2 to our
cases of interest, namely to isolate the eigenvalue spectrum of the odd
signature and Dirac operators. In [28], we showed how to interpret the Hodge
Laplacian and its spectrum in terms of the representation theory of
$\mathrm{PGL}_{2}$; in particular, we identified coexact $1$-eigenforms in
terms of isotypic vectors in the irreducible representations, and the
corresponding eigenvalue using the Casimir eigenvalue. We will do the same
here, where we consider the maximal compact subgroup $K=U_{2}$. In particular
we show the following two results.666Recall that our convention is that
$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\quad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$ forms a positively oriented basis of
$\mathfrak{p}_{0}=i\mathfrak{su}(2)$.

######  Proposition B.1.

Eigenforms of $\ast d$ correspond to $\mathfrak{p}$-isotypic vectors where
$\mathfrak{p}=\mathfrak{p}_{0}\otimes\mathbb{C}$. An irreducible unitary
representation of $G$ contains a $\mathfrak{p}$-isotypic vector if and only if
it is isomorphic to $\pi_{s_{1},s_{2},1,-1}$ with $s_{1},s_{2}$ purely
imaginary. Furthermore, each copy of $\pi_{s_{1},s_{2},1,-1}\subset
L^{2}(\widetilde{\Gamma}\backslash G)$ contains exactly a one dimensional
eigenspace of $\ast d$, spanned by an eigenform with eigenvalue
$i(s_{1}-s_{2})/2$.

Recall that a $\mathfrak{p}$-isotypic vector in a $G$-representation $\pi$ is
a non-zero element of $\mathrm{Hom}_{K}(\mathfrak{p},\pi)$. Notice that as the
Hodge Laplacian on coexact $1$-forms is the square of $\ast d$, this result
implies (after changing the parametrization) those in [28]; in that case we
computed the corresponding eigenvalue via Kuga’s lemma. Notice that while this
result can be proved directly using the group $\mathrm{PGL}_{2}$, the
following one relies essentially on the larger group $G$.

######  Proposition B.2.

Eigenspinors correspond to $S^{\vee}$-isotypic vectors where $S^{\vee}$ is the
dual of the standard representation of $K=U_{2}$ on $\mathbb{C}^{2}$. An
irreducible unitary representation of $G$ contains an $S^{\vee}$-isotypic
vector if and only if it is isomorphic to $\pi_{s_{1},s_{2},-1,0}$ with
$s_{1},s_{2}$ purely imaginary. Furthermore, each copy of
$\pi_{s_{1},s_{2},-1,0}\subset L^{2}(\widetilde{\Gamma}\backslash G)$ contains
exactly a one dimensional eigenspace of the Dirac operator, spanned by an
eigenspinor with eigenvalue $i(s_{1}-s_{2})/2$.

The present Appendix is dedicated to the proof of these results. The new main
ingredient is to describe the Dirac and $\ast d$ operators in terms of
representation theory, which we do in Subsections B.1, B.2 and B.3. We then
identify the relevant representations in Subsection B.4, and the corresponding
eigenvalues in Subsection B.5.

###  B.1. Generalities on associated bundles and connections for the principal
$K$-bundle $G\rightarrow G/K$

####  B.1.1. The invariant connection on $G\rightarrow G/K$

We follow the notation of Subsection A.1. Note that $G\xrightarrow{\pi}G/K$
(with $K$ acting on the right by multiplication) is a principal $K$-bundle.
The assignment

$H_{g}:=(L_{g})_{\ast}\mathfrak{p}_{0}\subset T_{g}(G)$

defines a subbundle $H\subset TG.$ Observe that

  1. (1)

The projection $H_{g}\rightarrow T_{g}(G)\xrightarrow{\pi_{\ast}}T_{gK}(G/K)$
is an isomorphism.

  2. (2)

The subbundle $H$ is right $K$-invariant:

$\displaystyle(R_{k})_{\ast}H_{g}$
$\displaystyle=(R_{k})_{\ast}(L_{g})_{\ast}\mathfrak{p}_{0}$
$\displaystyle=(L_{g})_{\ast}(R_{k})_{\ast}\mathfrak{p}_{0}\hskip
56.9055pt\text{because left and right multiplications commute}$
$\displaystyle=\left[(L_{g})_{\ast}(L_{k})_{\ast}\right]\left[(L_{k})_{\ast}^{-1}(R_{k})_{\ast}\right]\mathfrak{p}_{0}$
$\displaystyle=(L_{g}L_{k})_{\ast}(\text{conjugation by
}k^{-1})_{\ast}\mathfrak{p}_{0}$
$\displaystyle=(L_{gk})_{\ast}\mathrm{Ad}(k^{-1})\mathfrak{p}_{0}$
$\displaystyle=(L_{gk})_{\ast}\mathfrak{p}_{0}\hskip 79.6678pt\text{because
}\mathfrak{p}_{0}\text{ is invariant under }\mathrm{Ad}(K)$

  3. (3)

The subbundle $H$ is left $G$-invariant:
$(L_{h})_{\ast}H_{g}=(L_{h})_{\ast}(L_{g})_{\ast}\mathfrak{p}_{0}=H_{hg}.$

Items (1) and (2) imply that $H$ defines a connection on $G\rightarrow G/K.$
Item (3) implies that $H$ is left $G$-invariant.

####  B.1.2. Associated bundles and covariant derivatives

Let $\rho:K\rightarrow\mathrm{GL}(V)$ be a representation of $K.$ The
associated vector bundle $\mathcal{F}_{V}:=G\times_{\rho}V\rightarrow G/K$ has
sections

$\Gamma(\mathcal{F}_{V})=\\{s:G\rightarrow V:s(gk)=\rho(k)^{-1}s(g)\\}.$

The covariant derivative $\nabla^{V}$ on $\mathcal{F}_{V}$ associated to the
connection $H\subset TG$ is given by

$\displaystyle\nabla^{V}_{X}:\Gamma(\mathcal{F}_{V})$
$\displaystyle\rightarrow\Gamma(\mathcal{F}_{V})$ $\displaystyle s$
$\displaystyle\mapsto ds(\widetilde{X}),$

where $\widetilde{X}$ denotes the horizontal lift of $X$ to $TG$ via the
invariant connection $H$ from §B.1.1. Via the trivialization

$\displaystyle\mathfrak{p}_{0}\times G$ $\displaystyle\rightarrow H$
$\displaystyle(X,g)$ $\displaystyle\mapsto(L_{g})_{\ast}X$

from §B.1.1, we may regard
$\nabla^{V}s\in\Gamma\left(\mathcal{F}_{V}\otimes\Omega^{1}(G/K)\right),$
which is the bundle associated to the representation
$V\otimes\mathfrak{p}_{0}^{\vee}\simeq\mathrm{Hom}(\mathfrak{p}_{0},V)$ of
$K,$ as the function

(30) $\displaystyle\nabla^{V}s:G$
$\displaystyle\rightarrow\mathrm{Hom}(\mathfrak{p}_{0},V)$ $\displaystyle g$
$\displaystyle\mapsto\left(X\mapsto ds((L_{g})_{\ast}X)=X(s\circ
L_{g})\right).$

Let $r$ denote the representation

$\displaystyle r:K$
$\displaystyle\rightarrow\mathrm{End}\left(\mathrm{Hom}(\mathfrak{p}_{0},V)\right)$
$\displaystyle k$ $\displaystyle\mapsto\left(T\mapsto\rho(k)\circ
T\circ\mathrm{Ad}(k)^{-1}\right)$

Note that $\nabla^{V}s,$ as defined by formula (30), satisfies the
$K$-equivariance property

$\left(\nabla^{V}s\right)(gk)=r(k)^{-1}\cdot\left(\nabla^{V}s\right)(g).$

####  B.1.3. Dual picture for sections of $\mathcal{F}_{V}$ and the covariant
derivative $\nabla^{V}$

For our purposes it will be convenient to rephrase everything in terms of a
dual picture, which is more manifestly representation theoretic. There is a
canonical isomorphism

$\displaystyle\Phi^{V}:C^{\infty}(G,V)$
$\displaystyle\xrightarrow{\sim}\mathrm{Hom}(V^{\vee},C^{\infty}(G))$ (31)
$\displaystyle s$
$\displaystyle\mapsto\left(\;\ell\mapsto(g\mapsto\ell(s(g)))\;\right)$

The inverse mapping $(\Phi^{V})^{-1}$ is most easily expressed using a basis
$\\{v_{i}\\}$ for $V$ and its corresponding dual basis $\\{v_{i}^{\vee}\\}$
for $V^{\vee}$:

$\displaystyle(\Phi^{V})^{-1}:\mathrm{Hom}(V^{\vee},C^{\infty}(G))$
$\displaystyle\xrightarrow{\sim}C^{\infty}(G,V)$ (32) $\displaystyle T$
$\displaystyle\mapsto\left(g\mapsto\sum_{i}T(v_{i}^{\vee})(g)\cdot
v_{i}\right)$

The map $\Phi:=\Phi^{V}$ satisfies the following equivariance properties:

  1. (1)

$\Phi(L_{g}s)=L_{g}\Phi(s)$ and $\Phi(R_{g}s)=R_{g}\Phi(s).$ In
$C^{\infty}(G,V)$,

$L_{g}s(x)=s(gx),R_{g}s(x)=s(xg)$

and on $\mathrm{Hom}(V^{\vee},C^{\infty}(G)),$ the corresponding actions are

$V^{\vee}\longrightarrow C^{\infty}(G)\xrightarrow{\text{left or right
translation by }g}C^{\infty}(G).$

  2. (2)

$\Phi$ carries the subspace $\Gamma(\mathcal{F}_{V})=\\{s:G\rightarrow
S:s(gk)=\rho(k)^{-1}s(g)\\}$ to the subspace
$\mathrm{Hom}_{K}(V^{\vee},C^{\infty}(G)),$ where $K$ acts by right
translation on $C^{\infty}(G)$ and by $\rho^{\vee}$ on $V^{\vee}.$

  3. (3)

Upon combining (30) with the explicit formula for $\Phi^{-1}$ from (B.1.3), we
see that $\Phi$ intertwines the covariant derivative $\nabla^{V}$ with
operator

$\displaystyle\Theta^{V}:\mathrm{Hom}(V^{\vee},C^{\infty}(G))$
$\displaystyle\rightarrow\mathrm{Hom}((V\otimes\mathfrak{p}_{0}^{\vee})^{\vee},C^{\infty}(G))$
$\displaystyle T$ $\displaystyle\mapsto\left(\ell\otimes
X\mapsto\left(g\mapsto\sum_{i}\frac{d}{dt}|_{t=0}\left[T(v_{i}^{\vee})(ge^{tX})\right]\cdot\ell(v_{i})\right)\right),$

where $\ell\otimes X\in
V^{\vee}\otimes\mathfrak{p}_{0}\cong(V\otimes\mathfrak{p}_{0}^{\vee})^{\vee},$
i.e. there is a commutative diagram

$\begin{CD}C^{\infty}(G,V)@>{\nabla^{V}}>{}>C^{\infty}(G,V\otimes\mathfrak{p}_{0}^{\vee})\\\
@V{\Phi}V{}V@V{}V{\Phi}V\\\
\mathrm{Hom}(V^{\vee},C^{\infty}(G))@>{\Theta^{V}}>{}>\mathrm{Hom}(V^{\vee}\otimes\mathfrak{p}_{0},C^{\infty}(G)).\end{CD}$

The above restricts to a commutative diagram

$\begin{CD}\Gamma(\mathcal{F}_{V})@>{\nabla^{V}}>{}>\Gamma(\mathcal{F}_{V\otimes\mathfrak{p}_{0}^{\vee}})\\\
@V{\Phi}V{}V@V{}V{\Phi}V\\\
\mathrm{Hom}_{K}(V^{\vee},C^{\infty}(G))@>{\Theta^{V}}>{}>\mathrm{Hom}_{K}(V^{\vee}\otimes\mathfrak{p}_{0},C^{\infty}(G)).\end{CD}$

  4. (4)

$K$-equivariant maps $\alpha:V\rightarrow W$ induce maps
$\Gamma(\mathcal{F}_{V})\rightarrow\Gamma(\mathcal{F}_{W})$ (by post-
composition with $\alpha$) and on
$\mathrm{Hom}_{K}(V^{\vee},C^{\infty}(G))\rightarrow\mathrm{Hom}_{K}(W^{\vee},C^{\infty}(G))$
(by pre-composition with $\alpha^{\vee}$). These induced maps commute with the
isomorphism $\Phi.$

####  B.1.4. Going automorphic

Everything in §B.1.1, §B.1.2 and §B.1.3 is left invariant. So the connections,
vector bundles, covariant derivatives and bundle maps defined thus far all
descend to $\widetilde{\Gamma}\backslash G/K.$ We denote descended objects
with an $\overline{\bullet}$, e.g. for the descended vector bundle
$\overline{\mathcal{F}}_{V}$ over $\widetilde{\Gamma}\backslash G/K$

$\Gamma(\overline{\mathcal{F}}_{V})=\\{s:\widetilde{\Gamma}\backslash
G\rightarrow S:s(\widetilde{\Gamma}gk)=\rho(k)^{-1}s(\widetilde{\Gamma}g)\\}$

which is identified with, via the isomorphism $\Phi^{V},$ with
$\mathrm{Hom}_{K}(V^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash G)).$

The descended covariant derivative $\overline{\nabla}^{V}$ and its incanation
$\overline{\Theta}^{V}$ in the dual picture admit the “same formulas mod
$\widetilde{\Gamma}$”. In particular,

$\displaystyle\overline{\Theta}^{V}:\mathrm{Hom}_{K}(V^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G))$
$\displaystyle\rightarrow\mathrm{Hom}_{K}((V\otimes\mathfrak{p}_{0}^{\vee})^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G))$ $\displaystyle T$ $\displaystyle\mapsto\left(\ell\otimes
X\mapsto\left(g\mapsto\sum_{i}\frac{d}{dt}|_{t=0}\left[T(v_{i}^{\vee})(ge^{tX})\right]\cdot\ell(v_{i})\right)\right),$

and there is a commutative diagram

$\begin{CD}\Gamma(\overline{\mathcal{F}}_{V})@>{\overline{\nabla}^{V}}>{}>\Gamma(\overline{\mathcal{F}}_{V\otimes\mathfrak{p}_{0}^{\vee}})\\\
@V{\overline{\Phi}}V{}V@V{}V{\overline{\Phi}}V\\\
\mathrm{Hom}_{K}(V^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G))@>{\overline{\Theta}^{V}}>{}>\mathrm{Hom}_{K}(V^{\vee}\otimes\mathfrak{p}_{0},C^{\infty}(\widetilde{\Gamma}\backslash
G)).\end{CD}$

Given a $K$-equivariant map $\alpha:V\rightarrow W,$ we denote the associated
bundle map (obtained by post-composition by $\alpha$ on
$\Gamma(\overline{\mathcal{F}}_{V})$ or by precomposition by $\alpha^{\vee}$
in the dual picture) by $\overline{\alpha}.$ We also have commutative diagrams

$\begin{CD}\Gamma(\overline{\mathcal{F}}_{V})@>{\overline{\alpha}}>{}>\Gamma(\overline{\mathcal{F}}_{V\otimes\mathfrak{p}^{\vee}_{0}})\\\
@V{\overline{\Phi}}V{}V@V{}V{\overline{\Phi}}V\\\
\mathrm{Hom}_{K}(V^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G))@>{\overline{\alpha}}>{}>\mathrm{Hom}_{K}(V^{\vee}\otimes\mathfrak{p}_{0},C^{\infty}(\widetilde{\Gamma}\backslash
G)).\end{CD}$

Because $\overline{\nabla}^{V}$ is built from the derivative of the right
translation action of $G$ on $C^{\infty}(\widetilde{\Gamma}\backslash G),$ the
operators $\overline{\Theta}^{V}$ are amenable to representation-by-
representation analysis. Likewise, the bundle maps $\overline{\alpha}$ are
amenable to representation-by-representation analysis. For irreducible
(unitary) representations $\pi$ of $G,$ denote by $\pi^{\infty}\subset\pi$ the
dense subspace of smooth elements. We define the following $\pi$-isotypic
versions of the operators $\overline{\Theta}^{V}$ and $\overline{\alpha}$:

$\displaystyle\overline{\Theta}^{V}_{\pi}:\mathrm{Hom}_{K}(V^{\vee},\pi^{\infty})$
$\displaystyle\rightarrow\mathrm{Hom}_{K}((V\otimes\mathfrak{p}_{0}^{\vee})^{\vee},\pi^{\infty})$
(33) $\displaystyle T$ $\displaystyle\mapsto\left(\ell\otimes
X\mapsto\sum_{i}\left(X\cdot T(v_{i}^{\vee})\right)\cdot\ell(v_{i})\right),$

and

$\displaystyle\overline{\alpha}_{\pi}:\mathrm{Hom}_{K}(V^{\vee},\pi^{\infty})$
$\displaystyle\rightarrow\mathrm{Hom}_{k}(W^{\vee},\pi^{\infty})$
$\displaystyle T$ $\displaystyle\mapsto T\circ\alpha^{\vee}.$

Given a subrepresentation $\pi\subset L^{2}(\widetilde{\Gamma}\backslash G),$
the above $\pi$-isotypic operators fit into commutative diagrams

(34)
$\begin{CD}\mathrm{Hom}_{K}(V^{\vee},\pi^{\infty})@>{\overline{\Theta}_{\pi}^{V}}>{}>\mathrm{Hom}_{K}((V\otimes\mathfrak{p}_{0}^{\vee})^{\vee},\pi^{\infty})\\\
@V{\iota}V{}V@V{}V{\iota}V\\\
\mathrm{Hom}_{K}(V^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G))@>{\overline{\Theta}^{V}}>{}>\mathrm{Hom}_{K}((V\otimes\mathfrak{p}_{0}^{\vee})^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G))\end{CD}$

and

(35)
$\begin{CD}\mathrm{Hom}_{K}(V^{\vee},\pi^{\infty})@>{\overline{\alpha}_{\pi}}>{}>\mathrm{Hom}_{K}(W^{\vee},\pi^{\infty})\\\
@V{\iota}V{}V@V{}V{\iota}V\\\
\mathrm{Hom}_{K}(V^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G))@>{\overline{\alpha}}>{}>\mathrm{Hom}_{K}(W^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G)),\end{CD}$

where the vertical maps $\iota$ in both diagrams are induced by the inclusion
$\pi\subset L^{2}(\widetilde{\Gamma}\backslash G).$

###  B.2. Automorphic incarnation of the spinor bundle and Dirac operator

Denote as before by $S$ be the standard representation of $\mathrm{U}_{2}$ on
$\mathbb{C}^{2}$.

######  Definition B.1.

Let $\widetilde{\Gamma}\subset G$ be a lattice lifting the closed hyperbolic
3-manifold lattice $\Gamma\subset\mathrm{PGL}_{2}(\mathbb{C}).$ The spinor
bundle over
$\Gamma\backslash\mathbb{H}^{3}=\widetilde{\Gamma}\backslash\mathbb{H}^{3}=\widetilde{\Gamma}\backslash
G/K$ associated to the lift $\widetilde{\Gamma}\subset G$ of $\Gamma$ is the
vector bundle $\overline{\mathcal{F}}_{S}.$ It comes equipped with the
covariant derivative $\overline{\nabla}^{S}.$

####  B.2.1. Clifford multiplication on $\overline{\mathcal{F}}_{S}$

The linear map $C:S\otimes\mathfrak{p}_{0}\rightarrow S$ given by $2i$ times
matrix multiplication is $\mathrm{U}_{2}$-equivariant:

$\displaystyle C((\mathrm{std}\otimes\mathrm{Ad})(k)(s\otimes X))$
$\displaystyle=C(\mathrm{std}(k)s\otimes\mathrm{Ad}(k)X)$
$\displaystyle={2i\cdot}(kXk^{-1})(ks)$
$\displaystyle=k\left({2i\cdot}Xs\right)$
$\displaystyle=\mathrm{std}(k)C(s\otimes X).$

In what follows, we will make use of the (physicists’) Pauli matrices

(36) $\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\quad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$

######  Remark B.1.

In the hyperbolic metric, the Pauli matrices
$\sigma_{1},\sigma_{2},\sigma_{3}$ are orthogonal to each other, but have norm
$2$. To see why the latter holds, notice that

$e^{t\sigma_{3}}=\left(\begin{array}[]{cc}e^{t}&0\\\
0&e^{-t}\end{array}\right).$

In the upper half-space model, this maps $(0,0,1)$ to $(0,0,e^{2t}),$ and the
geodesic segment connecting them has length $2t.$ On the other hand, it has
length $||\sigma_{3}||\cdot t$, so $||\sigma_{3}||=2$. So the multiplier $2$
in the above definition of $C$ guarantees that Clifford multiplication is
isometric.

######  Remark B.2.

The above Clifford multiplication is compatible with the orientation
$\sigma_{1}\wedge\sigma_{2}\wedge\sigma_{3}$ in the sense that

$C(\sigma_{1})C(\sigma_{2})C(\sigma_{3})=8$

is positive.

The hyperbolic metric on $\mathfrak{p}_{0}$ defines a $K$-equivariant
isomorphism $\iota:\mathfrak{p}_{0}^{\vee}\rightarrow\mathfrak{p}_{0}.$ We
define $C^{\prime}$ to be the composition of $C$ with $\iota$:

$C^{\prime}=S\otimes\mathfrak{p}^{\vee}_{0}\xrightarrow{1\otimes\iota}S\otimes\mathfrak{p}_{0}\xrightarrow{C}S.$

We will refer to the associated map on sections

$\Gamma(\overline{\mathcal{F}}_{S\otimes\mathfrak{p}_{0}^{\vee}})\xrightarrow{\overline{C}^{\prime}}\Gamma(\overline{\mathcal{F}}_{S})$

as _Clifford multiplication_.

####  B.2.2. The Dirac operator on $\Gamma(\overline{\mathcal{F}}_{S})$

######  Definition B.2.

The Dirac operator $\overline{D}$ on $\Gamma(\overline{\mathcal{F}}_{S})$ is
defined to be the composition

$\Gamma(\overline{\mathcal{F}}_{S})\xrightarrow{\overline{\nabla}^{S}}\Gamma(\overline{\mathcal{F}}_{S\otimes\mathfrak{p}_{0}^{\vee}})\xrightarrow{C^{\prime}}\Gamma(\overline{\mathcal{F}}_{S}).$

Let $X_{1},X_{2},X_{3}$ be an oriented orthonormal basis for
$\mathfrak{p}_{0}.$ The canonical isomorphism
$\mathrm{Hom}(\mathfrak{p}_{0},S)\simeq\mathfrak{p}_{0}^{\vee}\otimes S$ can
be expressed as

$T\mapsto\sum_{i=1}^{3}\langle\bullet,X_{i}\rangle\otimes T(X_{i}),$

where $X_{i}$ is any orthonormal basis for $\mathfrak{p}.$ By the discussion
from §B.1.2 and §B.2.1, the Dirac operator is given by the formula

$\displaystyle\overline{D}:\Gamma(\overline{\mathcal{F}}_{S})$
$\displaystyle\rightarrow\Gamma(\overline{\mathcal{F}}_{S})$
$\displaystyle(g\mapsto s(g))$
$\displaystyle\mapsto\left(g\mapsto\sum_{i=1}^{3}{2i\cdot}m_{X_{i}}X_{i}(s\circ
L_{g})\right)$
$\displaystyle=\left(g\mapsto\sum_{i=1}^{3}{2i\cdot}m_{X_{i}}\frac{d}{dt}|_{t=0}s(ge^{tX_{i}})\right).$

where $m_{X_{i}}$ denotes matrix multiplication by
$X_{i}\in\mathfrak{p}_{0}=i\mathfrak{su}(2)$.

In the dual picture,
$\overline{D}=\overline{C}^{\prime}\circ\overline{\Theta}^{S}.$ Unravelling
the formula for $\overline{\Theta}^{S}$ from (B.1.4), we find that
$\overline{D}$ equals

$\displaystyle\overline{D}$
$\displaystyle:\mathrm{Hom}_{K}(S^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G))\rightarrow\mathrm{Hom}_{K}(S^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G))$ $\displaystyle T$
$\displaystyle\mapsto\sum_{i=1}^{3}\left(S^{\vee}\xrightarrow{{2i\cdot}m_{X_{i}}^{\vee}}S^{\vee}\xrightarrow{T}C^{\infty}(\widetilde{\Gamma}\backslash
G)\xrightarrow{f\mapsto\left(g\mapsto\frac{d}{dt}|_{t=0}f(ge^{tX_{i}})\right)}C^{\infty}(\widetilde{\Gamma}\backslash
G)\right).$

####  B.2.3. Representation-by-representation analysis of the Dirac operator

For every irreducible representation $\pi$ of $G$, we define the
$\pi$-isotypic Dirac operators

$\overline{D}_{\pi}:=\overline{C}_{\pi}^{\prime}\circ\overline{\Theta}_{\pi}^{S}.$

Suppose $\pi\subset L^{2}(\widetilde{\Gamma}\backslash G)$ is a
subrepresentation. By the commutativity of the diagrams (34) and (35), we
obtain the commutative diagram

(37)
$\begin{CD}\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})@>{\overline{D}_{\pi}}>{}>\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})\\\
@V{\iota}V{}V@V{}V{\iota}V\\\
\mathrm{Hom}_{K}(S^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G))@>{\overline{D}}>{}>\mathrm{Hom}_{K}(S^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash
G)),\end{CD}$

where the vertical maps $\iota$ in both diagrams are induced by the inclusion
$\pi\subset L^{2}(\widetilde{\Gamma}\backslash G).$

We will show in Proposition B.5 that if $\pi$ is an irreducible (unitary)
representation of $G$ and $\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})$ is non-
zero, then it is $1$-dimensional. For all such representations $\pi,$
$\overline{D}_{\pi}$ thus necessarily acts on
$\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})$ by some scalar
$\lambda_{\mathrm{Dirac},\pi}$ (which will be later computed in Proposition
B.7). Thus, $\iota\left(\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})\right)$
corresponds to an eigenline for the Dirac operator $\overline{D}$ with
eigenvalue $\lambda_{\mathrm{Dirac},\pi}.$ In particular, the natural
decomposition

$L^{2}(\overline{\mathcal{F}}_{S})=\bigoplus_{\pi\in\widehat{G}}m_{\tilde{\Gamma}}(\pi)\cdot\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty}),$

which is the analogue of the Matsushima decomposition we used in [28],
corresponds to the eigenspace decomposition of the Dirac operator.

###  B.3. Automorphic incarnation of $q$-form bundles and the $\ast d$
operator

Reprise all notation from §B.2. As in [28], in this setup it is more
convenient to consider the operators on complex valued forms; to this effect,
we will consider the $K$-representation
$\mathfrak{p}=\mathfrak{p}_{0}\otimes\mathbb{C}$. In particular, the vector
bundle $\overline{\mathcal{F}}_{\wedge^{q}\mathfrak{p}^{\vee}}$ is naturally
identified with the vector bundle of (complex-valued) $q$-forms over
$\widetilde{\Gamma}\backslash\mathbb{H}^{3}$.

####  B.3.1. Hodge star on
$\overline{\mathcal{F}}_{\wedge^{q}\mathfrak{p}^{\vee}}$

Consider the Hodge star operator

$\ast:\wedge^{q}\mathfrak{p}^{\vee}\rightarrow\wedge^{3-q}\mathfrak{p}^{\vee}.$

The action of $\mathrm{U}_{2}$ on $\mathfrak{p}^{\vee}$ factors through
$\mathrm{PU}_{2},$ the group of orientation preserving isometries of
$\mathbb{H}^{3}=G/K$ stabilizing $eK.$ Thus, the actions of $K$ on
$\wedge^{q}\mathfrak{p}^{\vee}$ and $\wedge^{3-q}\mathfrak{p}^{\vee}$ commute
with $\ast.$ It follows immediately:

######  Lemma B.3.

The associated map on sections
$\Gamma(\overline{\mathcal{F}}_{\wedge^{q}\mathfrak{p}^{\vee}})\xrightarrow{\overline{\ast}}\Gamma(\overline{\mathcal{F}}_{\wedge^{3-q}\mathfrak{p}^{\vee}})$
is identified with Hodge star for the hyperbolic metric on
$\widetilde{\Gamma}\backslash\mathbb{H}^{3}.$

####  B.3.2. Exterior derivative

Let $\overline{\nabla}:=\overline{\nabla}^{\wedge^{q}\mathfrak{p}^{\vee}}$ be
the covariant derivative on
$\overline{\mathcal{F}}_{\wedge^{q}\mathfrak{p}^{\vee}}$ induced by the
invariant connection on $G\rightarrow G/K$ described in §B.1.2. Let
$\overline{\Theta}:=\overline{\Theta}^{\wedge^{q}\mathfrak{p}^{\vee}}$ be the
corresponding operator in the dual picture, described in §B.1.3. Composing
with the wedge product map,
$\wedge:\mathfrak{p}^{\vee}\otimes\wedge^{q}\mathfrak{p}^{\vee}\rightarrow\wedge^{q+1}\mathfrak{p}^{\vee},$
which is $K$-equivariant, recovers the exterior derivative:

$d=\Gamma\left(\overline{\mathcal{F}}_{\wedge^{q}\mathfrak{p}^{\vee}}\right)\xrightarrow{\overline{\nabla}}\Gamma\left(\overline{\mathcal{F}}_{\mathfrak{p}^{\vee}\otimes\wedge^{q}\mathfrak{p}^{\vee}}\right)\xrightarrow{\overline{\wedge}}\Gamma\left(\overline{\mathcal{F}}_{\wedge^{q+1}\mathfrak{p}^{\vee}}\right).$

In the dual picture, applying formula (B.1.4) for
$V=\wedge^{q}\mathfrak{p}^{\vee}$ yields the following formula for
$d=\overline{\wedge}\circ\overline{\Theta}^{\wedge^{q}\mathfrak{p}^{\vee}}$:

$\displaystyle
d:\mathrm{Hom}_{K}(\wedge^{q}\mathfrak{p},C^{\infty}(\widetilde{\Gamma}\backslash
G))$
$\displaystyle\rightarrow\mathrm{Hom}_{K}(\wedge^{q+1}\mathfrak{p},C^{\infty}(\widetilde{\Gamma}\backslash
G))$ (38) $\displaystyle\omega$
$\displaystyle\mapsto\left(X_{0}\wedge\cdots\wedge
X_{q}\mapsto\sum_{i}X_{i}\cdot\omega(X_{0}\wedge\cdots\wedge\widehat{X_{i}}\wedge\cdots\wedge
X_{q})\right).$

See [8, Chapter 1, §1]. Implicitly in (B.3.2), we have used the canonical
duality between $\wedge^{q}\mathfrak{p}^{\vee}$ and $\wedge^{q}\mathfrak{p}.$

####  B.3.3. Representation-by-representation analysis of $\ast d$

For every irreducible representation $\pi$ of $G$, we define the
$\pi$-isotypic $\ast d$ operators

$(\ast
d)_{\pi}:=\overline{\ast}_{\pi}\circ\ast\overline{\wedge}_{\pi}\circ\overline{\Theta}_{\pi}^{\wedge^{1}\mathfrak{p}^{\vee}}.$

Suppose $\pi\subset L^{2}(\widetilde{\Gamma}\backslash G)$ is a
subrepresentation. By the commutativity of the diagrams (34) and (35), it
follows that the diagram

(39) $\begin{CD}\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi^{\infty})@>{(\ast
d)_{\pi}}>{}>\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi^{\infty})\\\
@V{\iota}V{}V@V{}V{\iota}V\\\
\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},C^{\infty}(\widetilde{\Gamma}\backslash
G))@>{\ast
d}>{}>\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},C^{\infty}(\widetilde{\Gamma}\backslash
G)),\end{CD}$

where the vertical maps $\iota$ in both diagrams are induced by the inclusion
$\pi\subset L^{2}(\widetilde{\Gamma}\backslash G).$

Similarly to the case of the Dirac operator, we will show in Proposition B.6
that if $\pi$ is an irreducible (unitary) representation of $G$ and
$\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi^{\infty})$ is non-zero, then it
is 1-dimensional. For all such representations $\pi,$ $(\ast d)_{\pi}$ thus
necessarily acts on $\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi^{\infty})$ by
some scalar $\lambda_{\ast d,\pi}$ (which will be computed in Proposition
B.8). Thus,
$\iota\left(\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi^{\infty})\right)$ is
an eigenline for the odd signature operator with eigenvalue $\lambda_{\ast
d,\pi}.$ In particular, the natural decomposition

$L^{2}(\Omega^{1})=\bigoplus_{\pi\in\widehat{G}}m_{\tilde{\Gamma}}(\pi)\cdot\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi^{\infty}),$

which is the essentially the Matsushima decomposition we used in [28],
corresponds to the eigenspace decomposition of odd signature operator. Notice
that of course closed forms are in the kernel of $\ast d$; as in [28], we will
identify exactly which representations correspond to coexact forms.

###  B.4. Frobenius reciprocity and $K$-isotypic vectors for induced
representations

Let $M=K\cap B$ be the subgroup of diagonal unitary matrices. Let
$\chi_{n_{1},n_{2}}$ denote the character of $M$ given by

$\displaystyle\chi_{n_{1},n_{2}}:M$
$\displaystyle\rightarrow\mathbb{C}^{\times}$
$\displaystyle\left(\begin{array}[]{cc}t_{1}&0\\\ 0&t_{2}\end{array}\right)$
$\displaystyle\rightarrow t_{1}^{n_{1}}\cdot t_{2}^{n_{2}},$

which is obtained by restricting the family of characters
$\chi_{s_{1},s_{2},n_{1},n_{2}}$ of $B$.

Suppose $\pi\subset L^{2}(\widetilde{\Gamma}\backslash G)$ is a
subrepresentation isomorphic to $\pi_{s_{1},s_{2},n_{1},n_{2}}.$ The following
Lemma will allow us to characterize whether $\pi$ contributes to the spinor
(resp. $\ast d$) eigenvalue spectrum. And if $\pi$ contributes, it will allow
us to write down explicit vectors in $\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})$
(resp. $\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})$) whose images in
$\mathrm{Hom}_{K}(S^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash G))$ (resp.
in $\mathrm{Hom}_{K}(S^{\vee},C^{\infty}(\widetilde{\Gamma}\backslash G))$)
are Dirac eigenspinors (resp. $\ast d$ eigen 1-forms), per the discussion from
§B.2 (resp. §B.3).

######  Lemma B.4.

Let $(V,\rho)$ be a finite dimensional representation of $K.$ The the map

$\displaystyle\mathrm{Hom}_{M}(V^{\vee},\chi_{n_{1},n_{2}})$
$\displaystyle\rightarrow\mathrm{Hom}_{K}(V^{\vee},\pi_{s_{1},s_{2},n_{1},n_{2}})$
$\displaystyle T$
$\displaystyle\mapsto\left(\ell\mapsto\chi_{s_{1},s_{2},n_{1},n_{2}}(b)\delta(b)^{1/2}\cdot
T(\rho^{\vee}(k)\ell)\right),$

where $\delta$ is the modular function, is an isomorphism.

In particular, this reduces the problem of determining $V^{\vee}$-isotypic
vectors to the (essentially trivial) problem of understanding of $V^{\vee}$ as
a representation of $M$.

###### Proof.

This is proved in [28, Appendix $B$], to which we refer for a more detail
discussion (and unraveling of the formulas). The isomorphism is the composite
of the Frobenius reciprocity isomorphism

$\displaystyle\mathrm{Hom}_{M}(V^{\vee},\chi_{n_{1},n_{2}})$
$\displaystyle\rightarrow\mathrm{Hom}_{K}(V^{\vee},\mathrm{Ind}_{M}^{K}\chi_{n_{1},n_{2}})$
$\displaystyle T$ $\displaystyle\mapsto\left(\ell\mapsto
f_{\ell}(k):=T(\rho^{\vee}(k)\ell)\right)$

and the map

$\displaystyle\mathrm{Hom}_{K}(V^{\vee},\mathrm{Ind}_{M}^{K}\chi_{n_{1},n_{2}})$
$\displaystyle\rightarrow\mathrm{Hom}_{K}(V^{\vee},\pi_{s_{1},s_{2},n_{1}.n_{2}})$
$\displaystyle S$ $\displaystyle\mapsto\left(\ell\mapsto
g_{\ell}(bk):=\chi_{s_{1},s_{2},n_{1},n_{2}}(b)\delta(b)^{1/2}\cdot[S(\ell)](k)\right).$

The latter is an isomorphism because $G=BK,M=B\cap K,$ and
$\chi_{s_{1},s_{2},n_{1},n_{2}}\cdot\delta^{1/2}$ restricted to $M$ equals
$\chi_{n_{1},n_{2}}.$ ∎

####  B.4.1. Representations contributing to the spinor spectrum

We need to apply Lemma B.4 to the representation $S^{\vee}$, where again
$S=\mathbb{C}^{2}$ is the standard representation of $\mathrm{U}_{2}$.

######  Proposition B.5.

Let $\pi$ be an irreducible unitary representation of $G.$ Then
$\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})$ is either 0 or 1 dimensional. It is
non-zero (and hence 1-dimensional) if and only if $\pi$ is isomorphic to
$\pi_{s_{1},s_{2},-1,0}$ (or, equivalently $\pi_{s_{2},s_{1},0,-1}$), in which
case a basis vector for $\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})$ is given by

$\ell\mapsto
f_{\ell}(bk):=\chi_{s_{1},s_{2},-1,0}(b)\delta(b)^{1/2}\cdot[\mathrm{std}^{\vee}(k)\ell](e_{1}).$

###### Proof.

We easily check that $\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})=0$ if $\pi$ is
isomorphic to some power of the determinant. So we focus on the case
$\pi\cong\pi_{s_{1},s_{2},n_{1},n_{2}}.$

By Lemma B.4, $\mathrm{Hom}_{K}(S^{\vee},\pi^{\infty})$ is naturally
isomorphic to $\mathrm{Hom}_{M}(S^{\vee},\chi_{n_{1},n_{2}}).$ Since
$S^{\vee}|_{M}$ is isomorphic to $\chi_{-1,0}\oplus\chi_{0,-1},$ if follows
that

  * •

$\mathrm{Hom}_{K}(S^{\vee},\pi_{s_{1},s_{2},n_{1},n_{2}}^{\infty})$ is non-
zero iff $(n_{1},n_{2})=(-1,0)$ or $(0,-1).$ If non-zero, it is exactly
1-dimensional.

  * •

The intertwining map

$\ell\mapsto\ell(e_{1})$

is a basis vector for the 1-dimensional space
$\mathrm{Hom}_{M}(S^{\vee},\chi_{-1,0}).$ Its image under the isomorphism from
Lemma B.4, namely

$\ell\mapsto
f_{\ell}(bk):=\chi_{s_{1},s_{2},-1,0}(b)\delta(b)^{1/2}\cdot[\mathrm{std}^{\vee}(k)\ell](e_{1}),$

thus defines a basis vector for the 1-dimensional space
$\mathrm{Hom}_{K}(S^{\vee},\pi_{s_{1},s_{2},-1,0}).$

This proves the Lemma. ∎

####  B.4.2. Representations contributing to the coclosed 1-form spectrum

We need to apply Lemma B.4 to the representation
$\wedge^{1}\mathfrak{p}=\mathfrak{p}$, where again
$\mathfrak{p}=\mathfrak{p}_{0}\otimes\mathbb{C}$ and
$\mathfrak{p}_{0}=i\mathfrak{su}(2)$ is acted on by $K$ via the adjoint
representation. Denoting again the Pauli matrices in (36) by $\sigma_{i}$, we
have that

(40) $w_{1}=\sigma_{1}+i\sigma_{2},\quad w_{0}=\sigma_{3},\quad
w_{-1}=\sigma_{1}-i\sigma_{2},$

are the $M$-weight vectors for $\wedge^{1}\mathfrak{p}$ of respective weights
$(1,-1),(0,0)$ and $(-1,1)$. We denote the corresponding dual basis by
$w_{1}^{\vee},w_{0}^{\vee},w_{-1}^{\vee}.$

######  Proposition B.6.

Let $\pi$ be an irreducible unitary representation of $G.$ Then
$\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi^{\infty})$ is either 0 or 1
dimensional. It is non-zero (and hence 1-dimensional) if and only if $\pi$ is
isomorphic to $\pi_{s_{1},s_{2},1,-1}$ (or equivalently
$\pi_{s_{2},s_{1},-1,1}$) or $\pi_{s_{1},s_{2},0,0}$; only the representations
of the form $\pi_{s_{1},s_{2},1,-1}$ contribute to the coclosed 1-form
spectrum, in which case a basis vector for
$\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi^{\infty})$ is given by

$v\mapsto f_{v}(bk):=\chi_{s_{1},s_{2},1,-1}(b)\delta(b)^{1/2}\cdot
w_{1}^{\vee}\left(\mathrm{Ad}(k)v\right).$

###### Proof.

We easily check that $\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi^{\infty})=0$
if $\pi$ is isomorphic to some power of the determinant. So we focus on the
case $\pi\cong\pi_{s_{1},s_{2},n_{1},n_{2}}.$

By Lemma B.4, $\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi^{\infty})$ is
naturally isomorphic to
$\mathrm{Hom}_{M}(\wedge^{1}\mathfrak{p},\chi_{n_{1},n_{2}}).$ Since
$\wedge^{1}\mathfrak{p}|_{M}$ is isomorphic to
$\chi_{1,-1}\oplus\chi_{0,0}\oplus\chi_{-1,1},$ if follows that

  * •

$\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi_{s_{1},s_{2},n_{1},n_{2}}^{\infty})$
is non-zero iff $(n_{1},n_{2})=(1,-1),(0,0),$ or $(-1,1).$ If non-zero, it is
exactly 1-dimensional. Furthermore, as argued in [28, Lemma B.11], the
representations $\pi_{s_{1},s_{2},1,-1}$ and $\pi_{s_{1},s_{2},-1,1},$
isomorphic respectively to $\pi_{s_{1}-s_{2},\pm 1}$ in the notation from [28,
Appendix B], only contribute to the coclosed 1-form spectrum while the
representations $\pi_{s_{1},s_{2},0,0},$ isomorphic to $\pi_{s_{1}-s_{2},0}$
in the notation from [28, Appendix B], only contribute to the closed 1-form
spectrum.

  * •

The intertwining map $v\mapsto w_{1}^{\vee}(v)$ is a basis vector for the
1-dimensional space $\mathrm{Hom}_{M}(\wedge^{1}\mathfrak{p},\chi_{1,-1}).$
Its image under the isomorphism from Lemma B.4, namely

$\ell\mapsto f_{v}(bk):=\chi_{s_{1},s_{2},1,-1}(b)\delta(b)^{1/2}\cdot
w_{1}^{\vee}\left(\mathrm{Ad}(k)v\right),$

thus defines a basis vector for the 1-dimensional space
$\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi_{s_{1},s_{2},1,-1}).$

This proves the Lemma. ∎

###  B.5. Representation-by-representation calculation of Dirac and $\ast d$
eigenvalues

We conclude the proofs on the main results of this Appendix by computing the
eigenvalues $\lambda_{\ast d,\pi}$ and $\lambda_{\mathrm{Dirac},\pi}$ of the
eigenforms/eigenspinors corresponding to isotypic vectors.

######  Remark B.3.

While in [28, Appendix $B$] we performed the computations in the natural
Killing metric, and then normalized the result to obtain the answer for the
hyperbolic metric, we will now work directly with the hyperbolic metric.

######  Proposition B.7 (Calculation of $\lambda_{\mathrm{Dirac},\pi}$).

Let $\pi=\pi_{s_{1},s_{2},-1,0}.$ Then

$\lambda_{\mathrm{Dirac},\pi}={i\cdot\left(\frac{s_{1}-s_{2}}{2}\right)}.$

###### Proof.

The space $\mathrm{Hom}_{T\cap K}\left(S^{\vee},\chi_{-1,0}\right)$ has basis
$\ell\mapsto\ell(e_{1}).$ Thus according to Lemma B.4, the unique (up to
scaling) element $T$ of
$\mathrm{Hom}_{K}\left(S^{\vee},\pi_{s_{1},s_{2},-1,0}\right)$ equals

$\displaystyle T(\ell)=f_{\ell}:G$ $\displaystyle\rightarrow\mathbb{C}$
$\displaystyle f_{\ell}(bk)$
$\displaystyle=\delta^{1/2}(b)\chi_{s_{1},s_{2},0,-1}(b)\cdot\left[\mathrm{std}^{\vee}(k)\ell\right](e_{1}).$

Suppose $\overline{D}_{\pi_{s_{1},s_{2},-1,0}}T=cT.$ Then, as
$T(e_{1}^{\vee})(1)=1$, we have

$\displaystyle\lambda_{\mathrm{Dirac},\pi}$ $\displaystyle=c$
$\displaystyle=cT(e_{1}^{\vee})(1)$
$\displaystyle=(\overline{D}_{\pi_{s_{1},s_{2},-1,0}}T)(e_{1}^{\vee})(1),$

so we will evaluate the latter next. We calculate:

$\displaystyle m_{\sigma_{1}}^{\vee}:e_{1}^{\vee}$ $\displaystyle\mapsto
e_{2}^{\vee},$ $\displaystyle e_{2}^{\vee}$ $\displaystyle\mapsto
e_{1}^{\vee}$ $\displaystyle m_{\sigma_{2}}^{\vee}:e_{1}^{\vee}$
$\displaystyle\mapsto-ie_{2}^{\vee},$ $\displaystyle e_{2}^{\vee}$
$\displaystyle\mapsto ie_{1}^{\vee}$ $\displaystyle
m_{\sigma_{3}}^{\vee}:e_{1}^{\vee}$ $\displaystyle\mapsto e_{1}^{\vee},$
$\displaystyle e_{2}^{\vee}$ $\displaystyle\mapsto-e_{2}^{\vee}$

Therefore we have the explicit expression

$\displaystyle\overline{D}_{\pi_{s_{1},s_{2},-1,0}}T(e_{1}^{\vee})$
$\displaystyle=\frac{2i}{2^{2}}\cdot\left(\sigma_{1}\cdot
Tm_{\sigma_{1}}^{\vee}(e_{1}^{\vee})+\sigma_{2}\cdot
Tm_{\sigma_{2}}^{\vee}(e_{1}^{\vee})+\sigma_{3}\cdot
Tm_{\sigma_{3}}^{\vee}(e_{1}^{\vee})\right)$
$\displaystyle=\frac{i}{2}\left(\sigma_{1}\cdot
f_{e_{2}^{\vee}}+\sigma_{2}\cdot f_{-ie_{2}^{\vee}}+\sigma_{3}\cdot
f_{e_{1}^{\vee}}\right),$

where the factor $2^{2}$ comes the fact that the Pauli matrices have norm $2$
in the hyperbolic metric, see Remark B.1. To evaluate the above expression at
1, we express each Pauli matrix in the form $K_{i}+B_{i}$ where
$K_{i}\in\mathfrak{k}$ and $B_{i}\in\mathfrak{b}$ (here $\mathfrak{k}$ and
$\mathfrak{b}$ are the Lie algebras of $K$ and $B$ respectively):

$\displaystyle\sigma_{1}$ $\displaystyle=\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right)+2\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)=:K_{1}+B_{1}$ $\displaystyle\sigma_{2}$
$\displaystyle=\left(\begin{array}[]{cc}0&i\\\
i&0\end{array}\right)-2\left(\begin{array}[]{cc}0&i\\\
0&0\end{array}\right)=:K_{2}+B_{2}$ $\displaystyle\sigma_{3}$
$\displaystyle=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right)+\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)=:K_{3}+B_{3}$

Notice that $B_{1},B_{2}\in\mathfrak{n}$, the Lie algebra of the group of
unipotent matrices $N$. Because the functions $f_{\ell}$ are invariant under
left translation by elements of $N,$

$\displaystyle\overline{D}_{\pi_{s_{1},s_{2},-1,0}}T(e_{1}^{\vee})$
$\displaystyle=\frac{i}{2}\cdot\left(\sigma_{1}\cdot
f_{e_{2}^{\vee}}+\sigma_{2}\cdot f_{-ie_{2}^{\vee}}+\sigma_{3}\cdot
f_{e_{1}^{\vee}}\right)$ $\displaystyle=\frac{i}{2}\cdot\left(K_{1}\cdot
f_{e_{2}^{\vee}}+K_{2}\cdot f_{-ie_{2}^{\vee}}+B_{3}\cdot
f_{e_{1}^{\vee}}\right).$

We calculate each summand, evaluated at 1, individually:

$\displaystyle(K_{1}\cdot f_{e_{2}^{\vee}})(1)$
$\displaystyle=1\cdot\frac{d}{dt}|_{t=0}[e_{2}^{\vee}](e^{-tK_{1}}e_{1})$
$\displaystyle=[e_{2}^{\vee}](-K_{1}e_{1})$ $\displaystyle=-1,$
$\displaystyle(K_{2}\cdot f_{-ie_{2}^{\vee}})(1)$
$\displaystyle=1\cdot\frac{d}{dt}|_{t=0}[-ie_{2}^{\vee}](e^{-tK_{2}}e_{1})$
$\displaystyle=[-ie_{2}^{\vee}](-K_{2}e_{1})$
$\displaystyle=[-ie_{2}^{\vee}](-ie_{2})$ $\displaystyle=-1,$
$\displaystyle(B_{3}\cdot f_{e_{1}^{\vee}})(1)$
$\displaystyle=\frac{d}{dt}|_{t=0}\delta^{1/2}\left(\begin{array}[]{cc}e^{t}&0\\\
0&e^{-t}\end{array}\right)\cdot\chi_{s_{1},s_{2},-1,0}\left(\begin{array}[]{cc}e^{t}&0\\\
0&e^{-t}\end{array}\right)\cdot[e_{1}^{\vee}](e_{1})$
$\displaystyle=\frac{d}{dt}|_{t=0}e^{2t}\cdot e^{t\cdot(s_{1}-s_{2})}\cdot 1$
$\displaystyle=s_{1}-s_{2}+2.$

Summing these contributions, $\overline{D}_{\pi_{s_{1},s_{2},-1,0}}$ acts on
$\mathrm{Hom}_{K}(S^{\vee},\pi_{s_{1},s_{2},-1,0})$ by the scalar

$\displaystyle c$
$\displaystyle=\overline{D}_{\pi_{s_{1},s_{2},-1,0}}T(e_{1}^{\vee})(1)$
$\displaystyle=\frac{i}{2}\cdot\left((-1)+(-1)+(s_{1}-s_{2}+2)\right)$ (41)
$\displaystyle={i\cdot\left(\frac{s_{1}-s_{2}}{2}\right)}$

and the proof is completed. ∎

We conclude by computing $\lambda_{\ast d,\pi}$, obtaining the desired
refinement of the result of [28].

######  Proposition B.8 (Calculation of $\lambda_{\ast d,\pi}$).

Let $\pi=\pi_{s_{1},s_{2},1,-1}.$ Then

$\lambda_{\ast d,\pi}={i\cdot\left(\frac{s_{1}-s_{2}}{2}\right)}.$

###### Proof.

We will use the notation from (40). According to Proposition B.6, the unique
(up to scaling) element $T$ of
$\mathrm{Hom}_{K}\left(\wedge^{1}\mathfrak{p},\pi_{s_{1},s_{2},1,-1}\right)$
equals

$\displaystyle T(v)=f_{v}:G$ $\displaystyle\rightarrow\mathbb{C}$
$\displaystyle f_{v}(bk)$
$\displaystyle=\delta^{1/2}(b)\chi_{s_{1},s_{2},1,-1}(b)\cdot
w_{1}^{\vee}\left(\mathrm{Ad}(k)v\right).$

Recall the $\pi$-isotypic version $(\ast d)_{\pi}$ of $\ast d$ from §B.3.3.
Because $\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi_{s_{1},s_{2},1,-1})$ is
1-dimensional, $(\ast d)_{\pi}$ acts on it by a scalar. We next calculate the
eigenvalue of $(\ast d)_{\pi}$ acting on
$\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi)$ for
$\pi=\pi_{s_{1},s_{2},1,-1}.$ To lighten notation, we refer to
$d_{\pi},\ast_{\pi}$ and $(\ast d)_{\pi}$ simply by $d,\ast,$ and $\ast d$
respectively. Suppose $(\ast d)T=cT.$ Then

$\displaystyle\lambda_{\ast d,\pi}$ $\displaystyle=c$
$\displaystyle=cT(w_{1})(1)$ $\displaystyle=((\ast d)T)(w_{1})(1),$

so we will evaluate the latter next. Observe that:

$\displaystyle dT(\sigma_{1}\wedge\sigma_{2})$ $\displaystyle=\sigma_{1}\cdot
T(\sigma_{2})-\sigma_{2}\cdot T(\sigma_{1})$ $\displaystyle
dT(\sigma_{2}\wedge\sigma_{3})$ $\displaystyle=\sigma_{2}\cdot
T(\sigma_{3})-\sigma_{3}\cdot T(\sigma_{2})$ $\displaystyle
dT(\sigma_{3}\wedge\sigma_{1})$ $\displaystyle=\sigma_{3}\cdot
T(\sigma_{1})-\sigma_{1}\cdot T(\sigma_{3}),$

implying that

$\displaystyle 2\cdot\ast dT(\sigma_{3})$
$\displaystyle=dT(\sigma_{1}\wedge\sigma_{2})=\sigma_{1}\cdot
T(\sigma_{2})-\sigma_{2}\cdot T(\sigma_{1})$ $\displaystyle 2\cdot\ast
dT(\sigma_{1})$ $\displaystyle=dT(\sigma_{2}\wedge\sigma_{3})=\sigma_{2}\cdot
T(\sigma_{3})-\sigma_{3}\cdot T(\sigma_{2})$ $\displaystyle 2\cdot\ast
dT(\sigma_{2})$ $\displaystyle=dT(\sigma_{3}\wedge\sigma_{1})=\sigma_{3}\cdot
T(\sigma_{1})-\sigma_{1}\cdot T(\sigma_{3}),$

where the factor of $2$ appears because the norm of the Pauli matrices in the
hyperbolic metric is $2$, see Remark B.1. It thus follows:

$\displaystyle 2\cdot\ast dT(w_{1})$ $\displaystyle=2\cdot\left(\ast
dT(\sigma_{1})+i\ast dT(\sigma_{2})\right)$ $\displaystyle=\sigma_{2}\cdot
T(\sigma_{3})-\sigma_{3}\cdot T(\sigma_{2})+i\left(\sigma_{3}\cdot
T(\sigma_{1})-\sigma_{1}\cdot T(\sigma_{3})\right).$

To evaluate the above expression at 1, recall the decomposition of the Pauli
matrices $\sigma_{i}=K_{i}+B_{i}$ introduced in the proof of Proposition B.7.
Again, the functions $f_{v}$ are invariant under left translation by elements
of $N$, and we therefore compute

$\displaystyle 2\cdot\ast dT(w_{1})$ $\displaystyle=\sigma_{2}\cdot
T(\sigma_{3})-\sigma_{3}\cdot T(\sigma_{2})+i\left(\sigma_{3}\cdot
T(\sigma_{1})-\sigma_{1}\cdot T(\sigma_{3})\right)$ $\displaystyle=K_{2}\cdot
f_{\sigma_{3}}-B_{3}\cdot f_{\sigma_{2}}+i\left(B_{3}\cdot
f_{\sigma_{1}}-K_{1}\cdot f_{\sigma_{3}}\right).$

We calculate each summand, evaluated at 1, individually. The terms involving
$B_{3}$ are

$\displaystyle(B_{3}\cdot f_{\sigma_{2}})(1)$
$\displaystyle=\frac{d}{dt}|_{t=0}\delta^{1/2}\left(\begin{array}[]{cc}e^{t}&0\\\
0&e^{-t}\end{array}\right)\cdot\chi_{s_{1},s_{2},1,-1}\left(\begin{array}[]{cc}e^{t}&0\\\
0&e^{-t}\end{array}\right)\cdot w_{1}^{\vee}(\sigma_{2})$
$\displaystyle=\frac{d}{dt}|_{t=0}e^{2t}\cdot
e^{t\cdot(s_{1}-s_{2})}\cdot\frac{1}{2}\left(-i\right)$
$\displaystyle=(2+s_{1}-s_{2})\cdot\frac{1}{2}\left(-i\right),$
$\displaystyle(B_{3}\cdot f_{\sigma_{1}})(1)$
$\displaystyle=\frac{d}{dt}|_{t=0}\delta^{1/2}\left(\begin{array}[]{cc}e^{t}&0\\\
0&e^{-t}\end{array}\right)\cdot\chi_{s_{1},s_{2},1,-1}\left(\begin{array}[]{cc}e^{t}&0\\\
0&e^{-t}\end{array}\right)\cdot w_{1}^{\vee}(\sigma_{1})$
$\displaystyle=\frac{d}{dt}|_{t=0}e^{2t}\cdot
e^{t\cdot(s_{1}-s_{2})}\cdot\frac{1}{2}$
$\displaystyle=(2+s_{1}-s_{2})\cdot\frac{1}{2},$

while those involving the $K_{i}$ are

$\displaystyle(K_{2}\cdot f_{\sigma_{3}})(1)$
$\displaystyle=1\cdot\frac{d}{dt}\lvert_{t=0}w_{1}^{\vee}(\mathrm{Ad}(e^{tK_{2}})\sigma_{3})$
$\displaystyle=w_{1}^{\vee}([K_{2},\sigma_{3}])$
$\displaystyle=w_{1}^{\vee}(2\sigma_{2})$ $\displaystyle=-i,$
$\displaystyle(K_{1}\cdot f_{\sigma_{3}})(1)$
$\displaystyle=1\cdot\frac{d}{dt}|_{t=0}w_{1}^{\vee}(Ad(e^{tK_{1}})\sigma_{3})$
$\displaystyle=w_{1}^{\vee}([K_{1},\sigma_{3}])$
$\displaystyle=w_{1}^{\vee}(2\sigma_{1})$ $\displaystyle=1.$

Summing these contributions, $\ast d$ acts on
$\mathrm{Hom}_{K}(\wedge^{1}\mathfrak{p},\pi_{s_{1},s_{2},1,-1})$ by the
scalar

$\displaystyle c$ $\displaystyle=\ast dT(w_{1})(1)$
$\displaystyle=\frac{1}{2}\cdot\left(-i-(2+s_{1}-s_{2})\cdot\frac{1}{2}\left(-i\right)+i\left((2+s_{1}-s_{2})\cdot\frac{1}{2}-1\right)\right)$
$\displaystyle={i\cdot\left(\frac{s_{1}-s_{2}}{2}\right)}$

and the proof is concluded. ∎

##  Appendix C Proof of the explicit trace formulas

We will now prove the results in Section 3, starting with the more involved
case of spinors in Theorem 3.3.

###  C.1. Isolating the spinor spectrum

We discuss suitable choices of test function to specialize the general trace
formula for $G$ in Proposition A.2. According to Proposition B.5, the
irreducible unitary representations of $G$ contributing to the spinor spectrum
are precisely those isomorphic to $\pi_{s_{1},s_{2},-1,0}$ or
$\pi_{s_{1},s_{2},0,-1}$ for purely imaginary $s_{1},s_{2}.$ To isolate the
latter representations, we use the test functions of the form

$F\left(\begin{array}[]{cc}e^{u/2}e^{i\alpha}&0\\\
0&e^{-u/2}e^{i\beta}\end{array}\right)=\begin{cases}H_{+}(u)\left(e^{i\alpha}+e^{i\beta}\right)&H_{+}\text{
even}\\\ H_{-}(u)\left(e^{i\alpha}-e^{i\beta}\right)&H_{-}\text{
odd}\end{cases}$

We now discuss how each term in the trace formula simplifies when choosing
test functions of the latter shapes.

####  C.1.1. Regular geometric terms for spinors

For our normalization of Haar measure (29), the covolume of the centralizer of
$\gamma$ equals $\ell(\gamma_{0}).$ If its conjugacy class in $G$ is
represented by

$\left(\begin{array}[]{cc}e^{\ell/2}e^{i\theta}e^{i\phi}&0\\\
0&e^{-\ell/2}e^{-i\theta}e^{i\phi}\end{array}\right),$

then its complex length equals $\ell+i2\theta$ and its corresponding summand
on the geometric side of the trace formula from Proposition A.2 equals

$\displaystyle=\ell_{0}\cdot\frac{1}{|1-e^{\ell+i2\theta}|\cdot|1-e^{-\ell-i2\theta}|}\cdot
H_{\pm}(\ell)\cdot\left(e^{i(\theta+\phi)}\pm e^{i(-\theta+\phi)}\right).$

Here we use that the Weyl discriminant is the same as in the trace formula
from [28]:

$|D(t_{\gamma})|^{1/2}=\frac{1}{|1-e^{\mathbb{C}\ell(\overline{\gamma})}|\cdot|1-e^{-\mathbb{C}\ell(\overline{\gamma})}|},$

where $\overline{\gamma}$ denotes the image of $\gamma$ in
$\mathrm{PGL}_{2}(\mathbb{C}).$ In particular, it is independent of $\phi.$

####  C.1.2. Contribution of central elements for spinors

Let $z\in Z\subset G$ be central, so

$z=\left(\begin{array}[]{cc}e^{i\phi}&0\\\ 0&e^{i\phi}\end{array}\right)$

for some $\phi\in\mathbb{R}/2\pi\mathbb{Z}$. Recalling that for the two
different parametrizations of $T$ we have $u=2v$, the contribution of the
conjugacy class of $z$ in the trace formula of Proposition A.2 is then

$\displaystyle-\frac{1}{8\pi}\cdot\mathrm{vol}(Y)\cdot\left(\frac{\partial^{2}}{\partial
v^{2}}+\frac{\partial^{2}}{\partial\theta^{2}}\right)|_{t=z}F$
$\displaystyle=$
$\displaystyle-\frac{1}{8\pi}\cdot\mathrm{vol}(Y)\cdot\left(\frac{\partial^{2}}{\partial
v^{2}}+\frac{\partial^{2}}{\partial\theta^{2}}\right)|_{(v,\theta)=(0,0)}H_{+}(2v)(e^{i(\theta+\phi)}+e^{i(-\theta+\phi)})$
$\displaystyle=$ $\displaystyle\frac{1}{8\pi}\cdot\mathrm{vol}(Y)\cdot
2e^{i\phi}\cdot\left(H_{+}(0)-4H_{+}^{\prime\prime}(0)\right)$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi}\cdot\mathrm{vol}(Y)\cdot
e^{i\phi}\cdot\left(H_{+}(0)-4H_{+}^{\prime\prime}(0)\right).$

In our case, we will be interested in lifts of torsion-free lattices
$\Gamma\subset\mathrm{PSL}_{2}(\mathbb{C})$, so that we will only need to
consider the case $z=1$, corresponding to $\phi=0$.

####  C.1.3. Regular spectral terms for spinors

Let $s_{1}=ir_{1},s_{2}=ir_{2}\in i\mathbb{R}.$ The summand on the spectral
side of the trace formula from Proposition A.2 corresponding to
$\pi_{s_{1},s_{2},n_{1},n_{2}}$ then equals

$\displaystyle\widehat{F}(\chi^{-1}_{s_{1},s_{2},n_{1},n_{2}})$
$\displaystyle=\int_{T}F(t)\chi_{s_{1},s_{2},n_{1},n_{2}}(t)dt$
$\displaystyle=\int\int\int H_{\pm}(u)\cdot\left(e^{i\alpha}\pm
e^{i\beta}\right)\cdot e^{u/2\cdot s_{1}}e^{-u/2\cdot s_{2}}\cdot
e^{in_{1}\alpha}\cdot
e^{in_{2}\beta}du\frac{d\alpha}{2\pi}\frac{d\beta}{2\pi}.$

In the case of $H_{+}\cdot(e^{i\alpha}+e^{i\beta}),$ the latter equals

$\widehat{H_{+}}\left(\frac{r_{2}-r_{1}}{2}\right)\cdot\begin{cases}+1&\text{
if }(n_{1},n_{2})=(-1,0)\text{ or }(0,-1)\\\ 0&\text{ otherwise}\end{cases}.$

while in the case of $H_{-}\cdot(e^{i\alpha}-e^{i\beta}),$ it equals

$\widehat{H_{-}}\left(\frac{r_{2}-r_{1}}{2}\right)\cdot\begin{cases}+1&\text{
if }(n_{1},n_{2})=(-1,0)\\\ -1&\text{ if }(n_{1},n_{2})=(0,-1)\\\ 0&\text{
otherwise.}\end{cases}.$

####  C.1.4. Contribution of $\det^{k}$ to the spectral side

For every $k\in\mathbb{Z}$, the contribution of $\det^{k}$ to the trace
formula for the test function $H_{\pm}(u)\cdot(e^{i\alpha}\pm e^{i\beta})$
equals

$\displaystyle\frac{1}{|W|}\int_{T}|D(t^{-1})|^{1/2}\cdot\det(t)^{k}\cdot
F(t)dt$
$\displaystyle=\frac{1}{|W|}\int\int\int\left(e^{u}+e^{-u}-\left(e^{i(\alpha-\beta)}+e^{i(\beta-\alpha)}\right)\right)\cdot
e^{k\cdot i(\alpha+\beta)}\cdot H_{\pm}(u)\cdot(e^{i\alpha}\pm
e^{i\beta})\;\frac{d\alpha}{2\pi}\frac{d\beta}{2\pi}du$ $\displaystyle=0,$

since the integral of every non-trivial character over the torus
$\mathrm{U}_{1}\times\mathrm{U}_{1}$ equals 0.

###  C.2. Trace formula for spinors

We are finally ready to prove the trace formulas that appear in Theorem 3.3.

####  C.2.1. Even spinor trace formula

Let $H_{+}$ be a smooth, compactly supported, even test function on
$\mathbb{R}.$ Let $\widetilde{\Gamma}$ be the lift to $G$ of the closed
hyperbolic 3-manifold group $\Gamma\subset\mathrm{PGL}_{2}(\mathbb{C})$, (so
that $\tilde{\Gamma}\cap Z=\\{1\\}$). For $\gamma\in\widetilde{\Gamma},$
suppose that $\gamma$ is conjugate to

$\left(\begin{array}[]{cc}z_{1}&0\\\
0&z_{2}\end{array}\right)=\left(\begin{array}[]{cc}e^{u/2}e^{i\alpha}&0\\\
0&e^{-u}e^{i\beta}\end{array}\right)=\left(\begin{array}[]{cc}e^{u/2}e^{i\theta}e^{i\phi}&0\\\
0&e^{-u/2}e^{-i\theta}e^{i\phi}\end{array}\right).$

Let $z=z_{1}/z_{2}.$ Let $\ell_{0}$ denote the length of the primitive closed
geodesic some multiple of which corresponds to $\gamma.$ Unwinding all terms
in the trace formula from Proposition A.2 as in §C.1 yields the following
trace formula:

$\displaystyle\sum
m_{\widetilde{\Gamma}}(\pi_{s_{1},s_{2},-1,0})\cdot\widehat{H_{+}}\left(\frac{r_{2}-r_{1}}{2}\right)$
$\displaystyle={\frac{1}{4\pi}}\cdot\mathrm{vol}(Y)\cdot\left(H_{+}(0)-4H_{+}^{\prime\prime}(0)\right)$
$\displaystyle+\sum_{[\gamma]}\ell_{0}\cdot\frac{1}{|1-z|\cdot|1-z^{-1}|}\cdot\left(e^{i\alpha}+e^{i\beta}\right)\cdot
H_{+}(u),$

where the LHS is summed over all isomorphism classes of irreducible unitary
representations of $G$ isomorphic to some $\pi_{s_{1},s_{2},-1,0}.$ Since
$H_{+}$ is even, the left side is real. So taking real parts:

$\displaystyle\sum
m_{\widetilde{\Gamma}}(\pi_{s_{1},s_{2},-1,0})\cdot\widehat{H_{+}}\left(\frac{r_{2}-r_{1}}{2}\right)=$
$\displaystyle=$
$\displaystyle{\frac{1}{4\pi}}\cdot\mathrm{vol}(Y)\cdot\left(H_{+}(0)-4H_{+}^{\prime\prime}(0)\right)+\sum_{[\gamma]}\ell_{0}\cdot\frac{1}{|1-z|\cdot|1-z^{-1}|}\cdot\left(\cos\alpha+\cos\beta\right)\cdot
H_{+}(u)$ (42) $\displaystyle=$
$\displaystyle{\frac{1}{4\pi}}\cdot\mathrm{vol}(Y)\cdot\left(H_{+}(0)-4H_{+}^{\prime\prime}(0)\right)+\sum_{[\gamma]}\ell_{0}\cdot\frac{1}{|1-z|\cdot|1-z^{-1}|}\cdot\left(2\cos\theta\cos\phi\right)\cdot
H_{+}(u).$

Finally, by Proposition B.7 the Dirac eigenvalue on
$\mathrm{Hom}_{K}(S^{\vee},\pi_{s_{1},s_{2},0,-1})$, with orientation
$\sigma_{1}\wedge\sigma_{2}\wedge\sigma_{3},$ equals
$i\left(\frac{s_{1}-s_{2}}{2}\right)=\frac{r_{2}-r_{1}}{2}.$ Thus, by
Proposition B.5 we can reexpress the trace formula from (C.2.1) geometrically
as

$\displaystyle\frac{1}{2}\cdot\sum_{\text{Dirac eigenvalues for
}\widetilde{\Gamma}}m_{\widetilde{\Gamma}}(\lambda)\cdot\widehat{H_{+}}\left(\lambda\right)=$
$\displaystyle=\frac{1}{2\pi}\cdot\mathrm{vol}(Y)\cdot\left(\frac{1}{4}\cdot
H_{+}(0)-H_{+}^{\prime\prime}(0)\right)+\sum_{[\gamma]}\ell_{0}\cdot\frac{1}{|1-z|\cdot|1-z^{-1}|}\cdot\left(\cos\theta\cos\phi\right)\cdot
H_{+}(u).$

This is the even trace formula in Theorem 3.3.

####  C.2.2. Odd spinor trace formula: representation theoretic form

Let $H_{-}$ be a smooth, compactly supported, odd test function on
$\mathbb{R}.$ Using the notation of the previous subsection, unwinding all
terms in the trace formula from Proposition A.2 as in §C.1 yields the
following trace formula:

$\displaystyle\sum
m_{\widetilde{\Gamma}}(\pi_{s_{1},s_{2},-1,0})\cdot\widehat{H_{-}}\left(\frac{r_{2}-r_{1}}{2}\right)$
$\displaystyle=-\sum_{[\gamma]}\ell_{0}\cdot\frac{1}{|1-z|\cdot|1-z^{-1}|}\cdot\left(e^{i\alpha}-e^{i\beta}\right)\cdot
H_{-}(u)$

where the LHS is summed over all isomorphism classes of irreducible unitary
representations of $G$ isomorphic to some $\pi_{s_{1},s_{2},-1,0}.$ Since
$H_{-}$ is odd, its Fourier transform is purely imaginary. So looking
imaginary parts:

$\displaystyle\sum
m_{\widetilde{\Gamma}}(\pi_{s_{1},s_{2},-1,0})\cdot\widehat{H_{-}}\left(\frac{r_{2}-r_{1}}{2}\right)$
$\displaystyle=i\sum_{[\gamma]}\ell_{0}\cdot\frac{1}{|1-z|\cdot|1-z^{-1}|}\cdot\left(\sin\alpha-\sin\beta\right)\cdot
H_{-}(u)$ (43)
$\displaystyle=i\sum_{[\gamma]}\ell_{0}\cdot\frac{1}{|1-z|\cdot|1-z^{-1}|}\cdot\left(2\sin\theta\cos\phi\right)\cdot
H_{-}(u).$

By (B.7), the Dirac eigenvalue on
$\mathrm{Hom}_{K}(S^{\vee},\pi_{s_{1},s_{2},-1,0})$, with orientation
$\sigma_{1}\wedge\sigma_{2}\wedge\sigma_{3},$ equals
$i\left(\frac{s_{1}-s_{2}}{2}\right)=\frac{r_{2}-r_{1}}{2}.$ Thus by
Proposition B.5, we can reexpress the above trace formula from (C.2.2)
geometrically:

$\displaystyle\frac{1}{2}\sum_{\text{Dirac eigenvalues for
}\widetilde{\Gamma}}m_{\widetilde{\Gamma}}(\lambda)\cdot\widehat{H_{-}}\left(\lambda\right)$
(44)
$\displaystyle=i\sum_{[\gamma]}\ell_{0}\cdot\frac{1}{|1-z|\cdot|1-z^{-1}|}\cdot\left(\sin\theta\cos\phi\right)\cdot
H_{-}(u).$

This is the odd trace formula in Theorem 3.3; let us point out again that in
(C.2.2), the Dirac operator is taken relative to the orientation
$\sigma_{1}\wedge\sigma_{2}\wedge\sigma_{3}$ on $\mathbb{H}^{3}.$

###  C.3. The trace formula for coexact 1-forms

Specializing the trace formula to isolate coexact 1-forms is more
straightforward than specializing the trace formula to isolate spinors as in
§C.2. We content ourselves with highlighting the main differences between
specializing to 1-forms versus specializing to spinors.

  * •

Irreducible unitary subrepresentations of $L^{2}(\widetilde{\Gamma}\backslash
G)$ contributing to the coclosed 1-form spectrum are precisely those
isomorphic to $\pi_{s_{1},s_{2},1,-1}$ and $\pi_{s_{1},s_{2},-1,1}$ for
$s_{1},s_{2}\in i\mathbb{R}.$ We isolate those representations using the test
functions $H_{\pm}(u)\cdot\left(e^{i(\alpha-\beta)}\pm
e^{i(\beta-\alpha)}\right),$ where $H_{+}$ is even and $H_{-}$ is odd.

  * •

For the latter test functions $H_{\pm}(u)\cdot\left(e^{i(\alpha-\beta)}\pm
e^{i(\beta-\alpha)}\right),$ the summand corresponding to the representation
$\pi_{s_{1},s_{2},1,-1}$ equals
$\widehat{H_{\pm}}\left(\frac{r_{2}-r_{1}}{2}\right).$ By Proposition B.8,
$\frac{r_{2}-r_{1}}{2}$ is the eigenvalue of $\ast d$ acting on the
$\wedge^{1}\mathfrak{p}$ isotypic vector of $\pi$ (for the orientation
$\sigma_{1}\wedge\sigma_{2}\wedge\sigma_{3}$ on $\mathbb{H}^{3}$). For a
subrepresentation $\pi\subset L^{2}(\widetilde{\Gamma}\backslash G)$
isomorphic to $\pi_{s_{1},s_{2},1,-1},$ the latter equals the eigenvalue of
$\ast d$ acting on the $\wedge^{1}\mathfrak{p}$-isotypic vectors of
$\pi_{s_{1},s_{2},1,-1}.$

  * •

The contribution of $\det^{k}$ to the trace formula for 1-forms is non-trivial
only if $k=0.$ If $k=0,$ the contribution equals 0 for test functions
$H_{-}(u)\cdot\left(e^{i(\alpha-\beta)}-e^{i(\beta-\alpha)}\right)$ for odd
$H_{-}$ and equals $-\widehat{H_{+}}(0)$ for the test function

$H_{+}(u)\cdot\left(e^{i(\alpha-\beta)}+e^{i(\beta-\alpha)}\right)=2H_{+}(2v)\cos(2\theta)$

with $H_{+}$ even.

  * •

In the even case for the test function $2H_{+}(2v)\cos(2\theta)$ the identity
contribution is $\frac{1}{2\pi}\cdot\mathrm{vol}(Y)\cdot
2(H_{+}(0)-H_{+}^{\prime\prime}(0))$.

From this, one readily obtains the formulas for coexact $1$-forms in Theorem
3.2.

##  Appendix D Admissibility of Gaussian functions

Gaussian functions were convenient to use at several points in our arguments.
The next result proves that functions of sufficiently fast decay, e.g.
Gaussians, are admissible for use in the trace formula.

######  Proposition D.1.

Let $f$ be an even or odd smooth function on $\mathbb{R}.$ Suppose that

$\sum_{n=0}^{\infty}e^{n}\cdot\sup_{x\in[n,n+1]}|f(x)|<\infty.$

Then both the geometric and spectral sides of the spinor trace formula for
closed hyperbolic 3-manifolds converge absolutely for the test function $f$
and they are equal. The same statement holds for the 1-form trace formula
applied to even or odd test functions and the 0-form trace formula applied to
even test functions $f.$

###### Proof.

We focus on the spinor case (arguments in the other cases being identical).
Convergence of the spectral side of the trace formula for $f$ follows because
$\widehat{f}$ is Schwartz and the number of spectral parameters in
$[\nu,\nu+1]$ is quadratic in $\nu$ (see Section 5)

Convergence of the geometric side of the trace formula follows by our
hypothesis. Indeed,

$\displaystyle\sum_{\gamma}\left|\ell(\gamma_{0})\cdot\frac{(\cos\text{ or
}\sin)\theta\cos\phi}{|1-e^{\mathbb{C}\ell(\gamma)}|\cdot|1-e^{-\mathbb{C}\ell(\gamma)}|}\cdot
f(\ell(\gamma))\right|$
$\displaystyle\leq\sum_{n=0}^{\infty}\left(\sum_{\ell(\gamma)\in[n,n+1]}\ell(\gamma_{0})\right)\cdot\frac{C}{e^{n}}\cdot\sup_{x\in[n,n+1]}|f(x)|$
$\displaystyle\leq\sum_{n=0}^{\infty}D\cdot
e^{2n}\cdot\frac{C}{e^{n}}\cdot\sup_{x\in[n,n+1]}|f(x)|$
$\displaystyle=CD\cdot\sum_{n=0}^{\infty}e^{n}\cdot\sup_{x\in[n,n+1]}|f(x)|$
(45) $\displaystyle<\infty.$

Above, we have used that $\sum_{\ell(\gamma)\in[L,L+1]}\ell(\gamma_{0})$ has
order of magnitude $e^{2L}$ (cf. the discussion in Section 9) and that
$|1-e^{\mathbb{C}\ell(\gamma)}|\cdot|1-e^{-\mathbb{C}\ell(\gamma)}|$ has order
of magnitude $e^{\ell(\gamma)}$ for $\ell(\gamma)$ large.

The trace formula for $f$ will be a consequence of the limit of the trace
formulas applied to the test functions $f(x)\cdot b(x/R),$ where $b$ is a
smooth even, compactly supported bump function with $\widehat{b}$ everywhere
positive and $b(0)=1.$ As $R\rightarrow\infty,$ the geometric and spectral
sides of the trace formula for $f(x)\cdot b(x/R)$ respectively converge
pointwise to the geometric and spectral sides of the trace formula for $f$;
this is immediate for the geometric side and it follows on the spectral side
because $f(x)\cdot b(x/R)$ has Fourier transform $\left(\widehat{f}\ast
g_{R}\right)(t),$ where

$g_{R}(t):=\frac{1}{2\pi}\cdot R\cdot\widehat{b}(Rt)$

is an approximate identity (i.e. it is everywhere positive of total mass 1 and
concentrated increasingly near the origin as $R\to\infty$). Here we use that
for our definition of Fourier transform $\widehat{f\cdot
g}=\frac{1}{2\pi}\widehat{f}{\ast}\widehat{g}$. It therefore suffices to prove
that the tails of the geometric and spectral sides of the trace formula,
applied to the test function $f(x)\cdot b(x/R),$ approach 0 uniformly as
$R\rightarrow\infty.$

Uniform smallness of the tail on the geometric side follows by our hypothesis.
Indeed,

$\displaystyle\sum_{\ell(\gamma)\geq
k}\left|\ell(\gamma_{0})\cdot\frac{(\cos\text{ or
}\sin)\theta\cos\phi}{|1-e^{\mathbb{C}\ell(\gamma)}|\cdot|1-e^{-\mathbb{C}\ell(\gamma)}|}\cdot
f(\ell(\gamma))\right|$ $\displaystyle\leq
CD\cdot\sum_{n=k}^{\infty}e^{n}\cdot\sup_{x\in[n,n+1]}|f(x)b(x/R)|\hskip
14.22636pt\text{exactly as in \eqref{geometricsideconverges}}$
$\displaystyle\leq
CD\cdot||b||_{\infty}\cdot\sum_{n=k}^{\infty}e^{n}\cdot\sup_{x\in[n,n+1]}|f(x)|,$

which converges uniformly to 0 as $k\to\infty,$ being the tail of one fixed
convergent sum (independent of $R$).

Uniform smallness of the tail on the spectral side follows by the following
estimate:

$\displaystyle\widehat{f}\ast g_{R}(t)$
$\displaystyle=\int_{\mathbb{R}}\widehat{f}(t-y)g_{R}(y)dy$
$\displaystyle=\int_{y\in[-1/2|t|,+1/2|t|]}\widehat{f}(t-y)g_{R}(y)dy+\int_{y\notin[-1/2|t|,+1/2|t|]}\widehat{f}(t-y)g_{R}(y)dy$
$\displaystyle=O\left(\sup_{y\in[1/2|t|,3/2|t|]}|\widehat{f}(y)|\right)+O\left(||\widehat{f}||_{\infty}\cdot\int_{y\notin[-1/2R|t|,+1/2R|t|]}\widehat{b}(z)dz\right).$

Since $\widehat{b}$ and $\widehat{f}$ are Schwartz, for every $p>0$ there is
some constant $C_{p}$ for which the latter is bounded above by $C_{p}\langle
t\rangle^{-p}$ uniformly in $R.$ In particular, since the number of spectral
parameters in $[\nu,\nu+1]$ grows quadratically with $\nu,$ taking any $p>3,$
the tail of the spectral side of the trace formula converges to 0 uniformly in
$R.$ ∎

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