$B\Phi^{r,P}_{Y,g}(A_{0},\ldots,A_{m-1})=\sum_{i=0}^{m-1}(-1)^{m-1)i}\Phi^{r,P}_{Y,g}(1,A_{i},\ldots,A_{m-1},A_{0},\ldots,A_{i-1}),$

where $1$ is the delta function at the unit. Ignoring the sign, we can write
the $i$’th term of this expression as

$\displaystyle\int_{h\in M/Z_{M}(x)}\int_{KN}\int_{G^{\times m}}\sum_{\tau\in
S_{m}}\mathrm{sgn}(\tau)\cdot
H_{\tau(1)}(g_{1}...g_{m}k)H_{\tau(2)}(g_{2}...g_{m}k)\dots
H_{\tau(m)}(g_{m}k)$
$\displaystyle\delta_{e}\big{(}khgh^{-1}nk^{-1}(g_{1}\dots
g_{m})^{-1}\big{)}^{b}\operatorname{Tr}\Big{(}A_{i}(g_{1})\dots\circ
A_{m}(g_{m-i+1})\circ A_{0}(g_{m-i})\circ A_{i-1}(g_{m})\Big{)}dg_{1}\cdots
dg_{m}dkdndh$ $\displaystyle=\int_{h\in M/Z_{M}(x)}\int_{KN}\int_{G^{\times
m}}\sum_{\tau\in S_{m}}\mathrm{sgn}(\tau)\cdot
H_{\tau(1)}(khgh^{-1}n)H_{\tau(2)}(g_{2}...g_{m}k)\dots H_{\tau(m)}(g_{m},k)$
${}^{b}\operatorname{Tr}\Big{(}A_{i}(khgh^{-1}nk(g_{2}\cdots
g_{m})^{-1})\dots\circ A_{m}(g_{m-i+1})\circ A_{0}(g_{m-i})\circ
A_{i-1}(g_{m})\Big{)}dg_{2}\cdots dg_{m}dkdndh$ $\displaystyle=0,$

because $H(khgh^{-1}n)=0$. This shows that $B\Phi^{r,P}_{Y,g}=0$ and
injectivity of $I^{*}$ show that $(b+B)\sigma^{P}_{\partial Y,g}=0$. ∎

Notation: we shall often omit the parabolic subgroup $P$ from the notation,
thus denoting by $\Phi_{g}$ the cyclic cocycle on $\mathcal{L}_{s}(G)$, by
$\Phi_{Y,g}$ the cyclic cocycle on $\mathcal{L}^{c}_{G}(Y)$ and by
$(\Phi^{r}_{Y,g},\sigma_{\partial Y,g})$ the relative cyclic cocycle for
${{}^{b}\mathcal{L}}^{c}_{G}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{c}_{G,\mathbb{R}}(\operatorname{cyl}(\partial
Y))$.

##  7\. Toward a general higher APS index formula

######  7.1 Proposition.

Assume that $G$ is a connected, linear real reductive group. Then:

  * 1

the cyclic cocycle $\Phi^{P}_{Y,g}$ extends continuously from
$\mathcal{L}^{c}_{G}(Y)$ to $\mathcal{L}^{\infty}_{G,s}(Y)$;

  * 2

the relative cyclic cocycle $(\Phi^{r,P}_{Y,g},\sigma_{\partial Y,g})$ extends
continuously from the pair
${{}^{b}\mathcal{L}}^{c}_{G}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{c}_{G,\mathbb{R}}(\operatorname{cyl}(\partial
Y))$ to the pair
${{}^{b}\mathcal{L}}^{\infty}_{G,s}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y))$.

###### Proof.

The proof of the first statement is analogous to the considerations in [hst]
depending crucially on the inequality proved in [st, Thm A.5]:

$\left|\Phi^{P}_{g}(f_{0},\ldots,f_{m})\right|\leq
C\nu_{d_{0}+T_{0}+1}(f_{0})\cdots\nu_{d_{0}+T_{0}+1}(f_{m}),\qquad
f_{0},\ldots,f_{m}\in\mathcal{L}_{s}(G),$

where

$\nu_{t}(f):=\sup_{g\in G}|(1+||g||)^{t}\Xi(g)^{-1}f(g)|,$

and $T_{0}$ and $d_{0}$ as in [st]. Given $A\in\mathcal{L}^{c}_{G}(Y)$, we
introduce the norm

$|||A|||_{t}:=\sup_{g\in G}|(1+||g||)^{t}\Xi(g)^{-1}||A(g)||_{b},$

where$||P||_{b}^{2}:=\|\chi
P\|^{2}_{1}+\|\phi[\mathcal{V},P]\|^{2}_{1}+\|[\mathcal{V},P]\|^{2}_{1}+\|[\phi,P]\|^{2}+\|P\|^{2}$
for $P\in{}^{b}\Psi^{-\infty,\epsilon}(S)+\Psi^{-\infty,\epsilon}(S)$ (c.f.
[PP2, Definition 7.3]). Because $\mathcal{L}^{\infty}_{G,s}(Y)$ lies inside
the norm-completion of $\mathcal{L}^{c}_{G}(Y)$ with respect to
$|||~{}|||_{s}$, it suffices to estimate the cocycle $\Phi^{P}_{Y,g}$ in one
of these norms. For this we use the inequality

$|{}^{b}\operatorname{Tr}(P_{0}P_{1}\cdots P_{k})|\leq
C||P_{0}||_{b}\cdots||P_{k}||_{b},$

c.f. [GMPi, Lemma 6.4.] to obtain

$\left|\Phi^{P}_{Y,g}(A_{0},\ldots,A_{m})\right|\leq
C|||A_{0}|||_{d_{0}+T_{0}+1}\cdots|||A_{m}|||_{d_{0}+T_{0}+1}.$

This proves the first claim. For the second claim we proceed as in [PP2,
Proposition 7.12]: we use the usual trace inequality
$|\operatorname{Tr}(AB)|\leq||A||_{1}||B||_{1}$ together with the estimate
$|H_{i}(gk)|\leq C_{i}L(g)$ of [st, Proposition A.2] to find

$\displaystyle|$ $\displaystyle\sigma^{P}_{\partial Y,g}(B_{0},...,B_{m+1})|$
$\displaystyle\leq C\int_{h\in
M/Z_{M}(g)}\int_{KN}\int_{G^{\times{(m+1)}}}\tilde{f}_{0}(khgh^{-1}nk^{-1}(g_{1}\dots
g_{m}g_{m+1})^{-1},\lambda)\tilde{f}_{1}(g_{1},\lambda)\cdots$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\tilde{f}_{m+1}(g_{m+1},\lambda)dg_{1}\cdots
dg_{m+1}dkdndhd\lambda,$

where

$\displaystyle\tilde{f}_{0}(g,\lambda)$
$\displaystyle:=||\hat{B}_{0}(g,\lambda)||_{1},$
$\displaystyle\tilde{f}_{i}(g,\lambda)$
$\displaystyle:=||\hat{B}_{i}(g,\lambda)||_{1}(1+L(g))^{i},\qquad
i=1,\ldots,m,$ $\displaystyle\tilde{f}_{m+1}(g,\lambda)$
$\displaystyle:=||\frac{\partial\hat{B}_{m+1}(g,\lambda)}{\partial\lambda}||_{1}$

By continuity of the map $||~{}||_{1}:\mathcal{L}^{\infty}_{G,s}(\partial
Y)\to\mathcal{L}_{s}(G)$, c.f. §4.1, we see that
$\tilde{f}_{j}\in\mathcal{L}_{s}(G)$, for all $j=1,\ldots,m+1$. We can
therefore rewrite the right hand side of the equality above as

$\int_{h\in
M/Z_{M}(g)}\int_{KN}\int_{\mathbb{R}}F(khgh^{-1}nk^{-1},\lambda)dkdndhd\lambda,$

with $F:=\tilde{f}_{0}*\ldots*\tilde{f}_{m+1}$. Convergence of this integral
now follows as in [st, Theorem A.5]. ∎

######  7.2 Definition.

Let $p_{t}=V(tD_{\text{cyl}})$ and
$c_{m}=(-1)^{\frac{m}{2}}\frac{m!}{(\frac{m}{2})!}$. Fix a cuspidal parabolic
subgroup $P=MAN<G$ and $g\in M$ a semisimple element. Let

$\eta^{P}_{g}(t):=2c_{m}\sum_{i=0}^{m}\sigma^{P}_{\partial
Y,g}(p_{t},...,[\dot{p}_{t},p_{t}],...,p_{t})$

We define the higher eta invariant associated to $\Phi^{P}_{g}$ and the
boundary operator $D_{\partial}$ as

(7.3) $\eta_{g}^{P}(D_{\partial}):=\lim_{\epsilon\downarrow
0}\int_{\epsilon}^{1/\epsilon}\eta^{P}_{g}(t)dt\equiv\lim_{\epsilon\downarrow
0}\int_{\epsilon}^{1/\epsilon}2c_{m}\sum_{i=0}^{m}\sigma^{P}_{\partial
Y,g}(p_{t},...,[\dot{p}_{t},p_{t}],...,p_{t})dt$

if this limit exists.

######  7.4 Theorem.

Let $s\in(0,1]$. For the index pairing
$\langle\operatorname{Ind}_{\infty}(D),[\Phi_{Y,g}]$ the following formula
holds:

$c_{m}\langle\operatorname{Ind}_{\infty}(D),[\Phi^{P}_{Y,g}]\rangle=\Phi^{r,P}_{Y,g}(V(sD),\dots,V(sD))-\frac{1}{2}\int_{s}^{\infty}\eta^{P}_{g}(t)dt$

where part of the statement is that the $t$-integral converges at $+\infty$.

###### Proof.

One first establishes the equality

(7.5)
$\langle\operatorname{Ind}_{\infty}(D),[\Phi^{P}_{Y,g}]\rangle=\langle\operatorname{Ind}_{\infty}(D,D_{\partial}),[\Phi^{r,P}_{Y,g},\sigma^{P}_{\partial
Y,g}]\rangle\,,$

exactly as in [moriyoshi-piazza, Theorem 9.7] and [PP2, Theorem 7.17]. By
definition of relative pairing and by the very definition of our relative
index class, that is

$\operatorname{Ind}_{\infty}(D,D_{\partial}):=[V(D),e_{1},q_{t}]\,,\;\;t\in[1,+\infty]\,,\;\;\text{
with
}q_{t}:=\begin{cases}V(tD_{\operatorname{cyl}})\;\;\quad\text{if}\;\;\;t\in[1,+\infty)\\\
e_{1}\;\;\;\;\;\;\;\;\;\;\;\;\;\,\text{ if }\;\;t=\infty\end{cases}$

we then obtain from (7.5) the following formula

$c_{m}\langle\operatorname{Ind}_{\infty}(D),[\Phi^{P}_{Y,g}]\rangle=\Phi^{r,P}_{Y,g}(V(D),\dots,V(D))-\frac{1}{2}\int_{1}^{\infty}\eta^{P}_{g}(t)dt$

The formula we want to prove is obtained by rescaling $D$ to $sD$. ∎

######  7.6 Remark.

We would like to take the limit as $s\downarrow 0$ in Theorem 7.4 and obtain
directly a higher delocalized APS index formula as the sum of a geometric term
and the higher delocalized eta invariant. Unfortunately at the moment it is
quite unclear how to study the limit as $s\downarrow 0$ of
$\Phi^{r,P}_{Y,g}(V(sD),\dots,V(sD))$ which is why in the next section we give
a treatment of the higher APS index formula corresponding to $\Phi^{P}_{g}$
through reduction, as in the closed case treated by Hochs, Song and Tang in
[hst]. Notice that even for a cocompact $G$-proper manifold without boundary
$X$ it is a difficult problem to study the limit

$\lim_{s\to 0}\Phi^{P}_{X,g}(V(sD),\dots,V(sD))$

with $V(D)$ the (symmetrized) Connes-Moscovici projector.

##  8\. Reduction

Suppose that $Y$ is a smooth $G$-proper manifold with boundary. Let $P=MAN$ be
a cuspidal parabolic subgroup. Since the nilpotent subgroup $N$ acts freely on
$X$, we can consider $Y_{MA}:=Y/N$ which is a smooth manifold with a proper
$MA$-action. In addition, the abelian group $A$ also acts freely on $Y_{MA}$,
we define

$Y_{M}=Y/(AN)=Y_{MA}/A$

which is a smooth manifold with a proper $M$-action. The following equation of
the index pairings was proved [hst, Proposition 4.8] for a smooth $G$-proper
manifold $X$ without boundary,

$\langle\Phi_{X,g},\operatorname{Ind}_{\infty}(D)\rangle=\langle\Phi_{X_{MA},g},\operatorname{Ind}_{\infty}(D_{X_{MA}})\rangle\,.$

Moreover, if the metric is slice compatible (Definition 3.5), we have [hst,
Lemma 5.3]

$\langle\Phi_{X_{MA},g},\operatorname{Ind}_{\infty}(D_{X_{MA}})\rangle=\langle\Phi_{X_{M},g},\operatorname{Ind}_{\infty}(D_{X_{M}})\rangle.$

These two equations together give a cohomological formula for
$\langle\Phi_{X,g},\operatorname{Ind}_{\infty}(D)\rangle$, with geometric
information coming from $X_{M,g}$. In this section, we generalize the above
computation for $\langle\Phi_{X,g},\operatorname{Ind}_{\infty}(D)\rangle$ to a
$G$-proper manifold $Y$ with boundary. In particular, we will need to study
the invertibility of the Dirac operators on $\partial(Y/N)$ and
$\partial(Y_{M})$.

###  8.1. First reduction for manifolds without boundary

Recall that

$\mathcal{L}^{\infty}_{G,s}(X,E)\cong\left(\mathcal{L}_{s}(G)\hat{\otimes}\Psi^{-\infty}(S,E|_{S})\right)^{K\times
K},\quad\mathcal{L}^{\infty}_{G,s}(X_{MA},E|_{X_{MA}})\cong\left(\mathcal{L}_{s}(MA)\hat{\otimes}\Psi^{-\infty}(S,E|_{S})\right)^{(K\cap
M)\times(K\cap M)}.$

######  8.1 Definition.

For any $f\in\mathcal{L}_{s}(G)$, Harish-Chandra [Harish-Chandra-dis] defines
a map $f^{N}$ on $MA$ by

$f(ma)\colon=\int_{N}f(man)dn.$

Then for any $k\in\mathcal{L}^{\infty}_{G,s}(X,E)$, and for all $m\in M,a\in
A,y,y^{\prime}\in S$, put

(8.2)
$k^{N}(ma,y,y^{\prime}):=\int_{N}k(nma,y,y^{\prime})dn\in\text{Hom}\left(E|_{y},E|_{y^{\prime}}\right).$

######  8.3 Lemma.

The map defined in (8.2) is a multiplicative map:

$\phi^{N}\colon{\mathcal{L}}^{\infty}_{G,s}(X,E)\ni k\longrightarrow
k^{N}\in{\mathcal{L}}^{\infty}_{G,s}(X_{MA},E|_{X_{MA}})$

Moreover, the map $\phi^{N}$ is continuous with respect to Fréchet topologies.

###### Proof.

Harish-Chandra proved in [Harish-Chandra-dis][Lemma 22] that (8.2) defines a
continuous linear map

$\mathcal{L}_{s}(G)\ni k\longrightarrow k^{N}\in\mathcal{L}_{s}(MA).$

It remains to show that it is multiplicative. Suppose that

$\kappa_{1},\kappa_{2}\in{\mathcal{L}}^{\infty}_{G,s}(X,E).$

Because Lafforgue’s Schwartz algebra is closed under convolution,

$\kappa_{1}^{N}\star\kappa_{2}^{N}\in{\mathcal{L}}^{\infty}_{G,s}(X_{MA},E|_{X_{MA}}).$

By definition and the Iwasawa decomposition $G=KNMA$,

(8.4) $\displaystyle\left(\kappa_{1}\star\kappa_{2}\right)^{N}(ma)=$
$\displaystyle\int_{G}\int_{N}\kappa_{1}(nmag^{\prime-1})\kappa_{2}(g^{\prime})\;dndg^{\prime}$
$\displaystyle=$
$\displaystyle\int_{K}\int_{M}\int_{A}\int_{N}\int_{N}\kappa_{1}(nmaa^{\prime-1}m^{\prime-1}n^{\prime-1}k^{\prime-1})\kappa_{2}(k^{\prime}n^{\prime}m^{\prime}a^{\prime})\;dk^{\prime}dm^{\prime}da^{\prime}dn^{\prime}dn$

Since the kernels $\kappa_{i},i=1,2$ are $K\times K$-equivariant, we have

$\kappa_{1}(k^{\prime}gk)=k^{\prime}\cdot\kappa_{1}(g)\cdot k,$

where the $k,k^{\prime}$ on the right-hand side denotes the $K$-action on
$\Psi^{-\infty}(S)$. A similar equation holds for $\kappa_{2}$. The last
equation in (8.4) becomes

$\int_{M}\int_{A}\int_{N}\int_{N}\kappa_{1}(nmaa^{\prime-1}m^{\prime-1}n^{\prime-1})\kappa_{2}(n^{\prime}m^{\prime}a^{\prime})\;dm^{\prime}da^{\prime}dn^{\prime}dn$

Recall that $MA$ normalizes $N$, that is,

$nmaa^{\prime-1}m^{\prime-1}n^{\prime-1}=n^{\prime\prime}maa^{\prime-1}m^{\prime-1}$

for some $n^{\prime\prime}\in N$. We conclude that

$\displaystyle\left(\kappa_{1}\star\kappa_{2}\right)^{N}(ma)=$
$\displaystyle\int_{M}\int_{A}\int_{N}\int_{N}\kappa_{1}(n^{\prime\prime}maa^{\prime-1}m^{\prime-1})\kappa_{2}(n^{\prime}m^{\prime}a^{\prime})\;dm^{\prime}da^{\prime}dn^{\prime}dn^{\prime\prime}$
$\displaystyle=$
$\displaystyle\int_{M}\int_{A}\kappa_{1}^{N}(maa^{\prime-1}m^{\prime-1})\kappa_{2}^{N}(m^{\prime}a^{\prime})\;dm^{\prime}da^{\prime}$
$\displaystyle=$
$\displaystyle\left(\kappa_{1}^{N}\star\kappa_{2}^{N}\right)(ma).$

∎

######  8.5 Remark.

By Lemma 8.3, the map $\phi^{N}$ induces a map

$\phi^{N}_{*}\colon K^{*}\left(\mathcal{L}^{\infty}_{G,s}(X,E)\right)\cong
K^{*}\left(C^{*}_{r}(G)\right)\to
K^{*}\left(\mathcal{L}^{\infty}_{G,s}(X_{MA},E|_{X_{MA}})\right)\cong
K^{*}\left(C^{*}_{r}(MA)\right).$

We consider $C^{\infty}(X,E)$ and $C^{\infty}(X,E)^{N,c}$, the smooth
$N$-invariant sections of $E$ with compact support in $X/N\cong X_{MA}$. Let
$D$ be a Dirac operator on $X$ and $k_{t}\in{\mathcal{L}}^{\infty}_{G,s}(X,E)$
be the Schwartz kernel of $\exp(-tD^{2})$. Since
$k_{t}^{N}\in{\mathcal{L}}^{\infty}_{G,s}(X_{MA},E|_{X_{MA}})$, we can regard
it as an operator on $C^{\infty}(X,E)^{N,c}$.

######  8.6 Lemma.

For all $\sigma\in C^{\infty}_{c}(X,E)$ and $s\in C^{\infty}(X,E)^{N,c}$ we
have

$(k_{t}^{*}\sigma,s)_{L^{2}(X,E)}=(\sigma,k_{t}^{N}s)_{L^{2}(X,E)},$

where $k_{t}^{*}$ is the adjoint kernel of $k_{t}$. Moreover,

$\frac{d}{dt}\left(\sigma,k_{t}^{N}s\right)_{L^{2}(X,E)}=\left(\sigma,-D^{2}\exp(-tD^{2})s\right)_{L^{2}(X,E)}$

###### Proof.

This lemma is proved in [hst, Lemma 4.1 and Lemma 4.2]. We have included it
here for the reader’s convenience. ∎

######  8.7 Lemma.

Let $D_{X_{MA}}$ be the Dirac operator on $X_{MA}$ induced from $D$. Then the
Schwartz kernel of $\exp(-tD_{X_{MA}}^{2})$ is $k^{N}_{t}$.

###### Proof.

Since $X_{MA}=X/N$, we can identify $C^{\infty}(X,E)^{N,c}\cong
C^{\infty}(X_{MA},E|_{X_{MA}})$. Under such an identification, the restriction
$D$ on $N$-invariant sections equals $D_{X_{MA}}$. By Lemma 8.6, we have that

$\frac{d}{dt}\left(\sigma,k_{t}^{N}s\right)_{L^{2}(X,E)}=\left(\sigma,-D_{X_{MA}}^{2}k_{t}^{N}s\right)_{L^{2}(X,E)}$

and

$\lim_{t\downarrow
0}\left(\sigma,k_{t}^{N}s\right)_{L^{2}(X,E)}=\left(\sigma,s\right)_{L^{2}(X,E)}.$

Using the uniqueness of the heat equation with usual initial data, we conclude
that $k_{t}^{N}$ is the Schwartz kernel of $\exp(-tD_{X_{MA}}^{2})$. ∎

######  8.8 Remark.

Lemma 8.7 is stated in [hst, Lemma 4.5] for the wrong operator $D_{X_{MA}}$
introduced in [hst, Eq. (4.10)]. It is the operator $D_{X_{MA}}$ introduced in
Lemma 8.7 that carries the right property for the development. It is not hard
to see that $D_{X_{MA}}$ is a twisted Dirac operator on $X_{MA}$.

######  8.9 Remark.

As in [hst, Lemma 4.6], one can similarly show that
$V(tD)^{N}=V(tD_{X_{MA}})$, where $V$ denotes the Connes-Moscovici projection.

######  8.10 Lemma.

If the Dirac operator $D$ on $X$ is invertible, then the corresponding Dirac
operator $D_{X_{MA}}$ is invertible as well.

###### Proof.

Suppose that

$\text{Spec}(D)\cap(-\delta,\delta)=\emptyset$

for some $\delta>0$. We have shown in Part II (we should move the proof to
Part I later) that the Schwartz kernel

$e^{\frac{t\delta}{2}}\cdot k_{t}\in\mathcal{L}^{\infty}_{G,s}(X,E)$

converges to zero in Frechet topology as $t\to\infty$. Now the key observation
is that the map

$\mathcal{L}^{\infty}_{G,s}(X,E)\ni k_{t}\longrightarrow
k_{t}^{N}\in\mathcal{L}^{\infty}_{G,s}(X_{MA},E|_{X_{MA}})$

is continuous by Lemma 8.3. Thus, $k_{t}^{N}$, which is the Schwartz kernel of
$e^{-tD_{X_{MA}}^{2}}$ by Lemma 8.7, converges to zero faster than
$e^{\frac{-t\delta}{2}}$ as $t\to\infty$. It follows that the operator
$e^{\frac{t\delta}{2}-tD_{X_{MA}}^{2}}$ is uniformly bounded. Thus, we can
find a constant $C$ such that for $t\gg 0$, and any $s\in
L^{2}(X_{MA},E|_{X_{MA}})$,

(8.11) $\langle e^{-tD_{X_{MA}}^{2}}s,s\rangle\leq
e^{-\frac{t\delta}{3}}\cdot\|s\|^{2}.$

We denote by $E_{\lambda}$ the spectrum measure associated to the self-adjoint
operator $D_{X_{MA}}^{2}$, that is

$E_{\lambda}\colon\lambda\to\text{projection on
}L^{2}(X_{MA},E|_{X_{MA}}),\quad\lambda\in\text{Spec}(D_{X_{MA}}^{2})\subseteq[0,+\infty).$

For any $s\in L^{2}(X_{MA},E|_{X_{MA}})$,

$\displaystyle\langle e^{-tD_{X_{MA}}^{2}}s,s\rangle=$
$\displaystyle\int_{0}^{\infty}e^{-t\lambda}\;d\langle E_{\lambda}s,s\rangle$
$\displaystyle\geq\int_{0}^{\frac{\delta}{4}}e^{-t\lambda}\;d\langle
E_{\lambda}s,s\rangle\geq
e^{-\frac{t\delta}{4}}\cdot\int_{0}^{\frac{\delta}{4}}\;d\langle
E_{\lambda}s,s\rangle$

If
$\text{Spec}(D_{X_{MA}}^{2})\cap\left[0,\frac{\delta}{4}\right)\neq\emptyset$,
then

$P\colon=\int_{0}^{\frac{\delta}{4}}\;dE_{\lambda}$

defines a non-zero projection on $L^{2}(X_{MA},E|_{X_{MA}})$. If we take
$s_{0}\in\text{Image}\left(P\right)$, then

$\langle e^{-tD_{X_{MA}}^{2}}s_{0},s_{0}\rangle\geq
e^{-\frac{t\delta}{4}}\cdot\|s_{0}\|^{2},$

which contradicts to (8.11). This completes the proof. ∎

###  8.2. First reduction for manifold with boundary

Suppose that $Y$ is a smooth $G$-proper manifold with boundary, denoted
$\partial Y$. Let

$g\in\mathcal{L}^{\infty}_{G,s}(Y,E)\,;\quad
k\in{{}^{b}\mathcal{L}}^{\infty}_{G,s}(Y,E)\,;\quad
k_{\mathbb{R}}\in{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y),p^{*}E_{\partial Y}),$

where for the sake of clarity we have now included the bundles in the
notation; $p:\mathbb{R}\times\partial Y\to\partial Y$ is the obvious
projection. We see these kernels as functions on $G$ with values in
pseudodifferential operators on the slice $S$ satisfying a $K\times
K$-equivariance. We can then define

$g^{N}\in\mathcal{L}^{\infty}_{G,s}(Y_{MA},E|_{Y_{MA}})\,;\quad
k^{N}\in{{}^{b}\mathcal{L}}^{\infty}_{G,s}(Y_{MA},E|_{Y_{MA}})\,;\quad
k^{N}_{\mathbb{R}}\in{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y_{MA}),p^{*}E|_{\partial Y_{MA}}).$

These are functions on $MA$ with values in psudodifferential operators on the
slice $Z$ satisfying a $(K\cap M)\times(K\cap M)$-equivariance. See [hst],
Section 4.1. We can of course extend this map to $M_{n\times
n}(\mathcal{L}^{\infty}_{G,s}(Y,E))$, $M_{n\times
n}({{}^{b}\mathcal{L}}^{\infty}_{G,s}(Y,E))$ and $M_{n\times
n}({}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y),p^{*}E_{\partial Y}))$. By an argument similar to that given in Lemma 8.3,
we have that the following three maps

  1. (1)

${\mathcal{L}}^{\infty}_{G,s}(Y,E)\ni g\longrightarrow
g^{N}\in{\mathcal{L}}^{\infty}_{G,s}(Y_{MA},E|_{Y_{MA}})$

  2. (2)

${{}^{b}\mathcal{L}}^{\infty}_{G,s}(Y,E)\ni k\longrightarrow
k^{N}\in{{}^{b}\mathcal{L}}^{\infty}_{G,s}(Y_{MA},E|_{Y_{MA}})$

  3. (3)

${}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y),p^{*}E_{\partial Y})\ni k_{\mathbb{R}}\longrightarrow
k_{\mathbb{R}}^{N}\in{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y_{MA}),p^{*}E|_{\partial Y_{MA}})$

are all multiplicative maps and continuous with respect to Fréchet topologies.
As before, we have the following result:

######  8.12 Proposition.

We have that

  * •

$k^{N}_{t}$ is equal to the Schwartz kernel of $\exp(-tD^{2}_{Y_{MA}})$; an
analogous result holds for $k^{N}_{\mathbb{R}}$.

  * •

$V(tD)^{N}=V(tD_{Y_{MA}})$.

  * •

If the boundary operator of $D$ is $L^{2}$-invertible, then
$({}^{b}V(tD))^{N}={}^{b}V(D_{Y_{MA}})$.

We shall now consider various index classes. We consider $Y_{MA}$ and $Y_{M}$.
The manifold $Y_{MA}$ is a product, i.e.

$Y_{MA}=Y_{M}\times A,$

where $M$ acts properly on $Y_{M}$ and trivially on $A$, and $A$ acts properly
and freely on $A$ and trivially on $Y_{M}$.

######  8.13 Proposition.

Assume that the metric on $Y$ is slice compatible and that the boundary
operator of $D$ is $L^{2}$-invertible. Then the boundary operator of
$D_{Y_{MA}}$ and of $D_{Y_{M}}$ are also $L^{2}$-invertible. Consequently
there are well defined index classes:

(8.14) $\operatorname{Ind}_{\infty}(D_{Y_{MA}})\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y_{MA},E|_{Y_{MA}}))\,,\quad\operatorname{Ind}_{\infty}(D_{Y_{M}})\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y_{M},E_{M}))$

with $E_{M}$ as in (9.2) below.

###### Proof.

If $D_{\partial Y}$ is $L^{2}$-invertible, then we can use Lemma 8.10 in order
to see directly that the boundary operator of $D_{Y_{MA}}$ is also
$L^{2}$-invertible. Next, as the metric is slice compatible, we have that the
induced metric on $Y_{MA}=Y_{M}\times A$ is a product metric. It follows that
the Dirac operator $D_{Y_{MA}}$ decomposes as

(8.15) $D_{Y_{MA}}=D_{Y_{M}}\hat{\otimes}1+1\hat{\otimes}D_{A},$

where we use the graded tensor products. This means that

$D^{2}_{Y_{MA}}=D^{2}_{Y_{M}}\otimes 1+1\otimes D^{2}_{A}$

As $D^{2}_{Y_{MA}}$ is $L^{2}$-invertible and $D^{2}_{A}$ is not
$L^{2}$-invertible ($A$ is isomorphic to $\mathbb{R}^{n}$), we see that
$D^{2}_{Y_{M}}$ and thus $D_{Y_{M}}$ must be $L^{2}$-invertible. ∎

Consider now the cyclic cocycle $\Phi_{MA,g}$ on $\mathcal{L}_{s}(MA)$, see
[hst, Section 3.1], and the associated cyclic cocycle on
$\mathcal{L}^{\infty}_{G,s}(Y_{MA},E|_{Y_{MA}})$, denoted $\Phi_{Y_{MA},g}$.
Consider the index class $\operatorname{Ind}_{\infty}(D)\equiv[{}^{b}V(D)]$
and the index class
$\operatorname{Ind}_{\infty}(D_{Y_{MA}})\equiv[{}^{b}V(D_{Y_{MA}})]$, which is
well defined because of Proposition 8.13.

######  8.16 Proposition.

The following equality holds

(8.17)
$\langle\Phi_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle=\langle\Phi_{Y_{MA},g},\operatorname{Ind}_{\infty}(D_{Y_{MA}})\rangle\,.$

###### Proof.

We must prove that

(8.18)
$\langle\Phi_{Y,g},[{}^{b}V(D)]\rangle=\langle\Phi_{Y_{MA},g},[{}^{b}V(D_{Y_{MA}})]\rangle\,.$

This follows combining [hst, Proposition 3.2], which clearly holds for the
algebra of residual operators $\mathcal{L}^{\infty}_{G,s}(Y,E)$, and
Proposition 8.12 above. ∎

###  8.3. Second reduction

The Lie group $M$ is of equal rank and acts properly on $Y_{M}$. Consider the
orbital integral $\tau^{M}_{g}$ on $\mathcal{L}_{s}(M)$ associated to a
semisimple element $g$ in $M$. As in subsection 4.2, we can associated the
orbital integral $\tau^{M}_{g}$ a $0$-cocycle on $Y_{M}$, denoted
$\Phi_{Y_{M},g}$. More precisely, $\Phi_{Y_{M},g}$ is a cyclic 0-cocycle on
the algebra $\mathcal{L}^{\infty}_{G,s}(Y_{M},E_{M})$, with $E_{M}$ as in
(9.2).

######  8.19 Proposition.

If the metric on $Y$ is slice compatible and the boundary operator on $Y$ is
$L^{2}$-invertible, then the following equality holds:

(8.20)
$\langle\Phi_{Y_{MA},g},\operatorname{Ind}_{\infty}(D_{Y_{MA}})\rangle=\langle\Phi_{Y_{M},g},\operatorname{Ind}_{\infty}(D_{Y_{M}})\rangle$

###### Proof.

We prove this identity in two steps.

Step I: Recall that $\operatorname{Ind}(D_{Y_{MA}})$ is an $K$-theory element
of the algebra $\mathcal{L}_{MA,s}^{\infty}(Y_{MA},E_{MA})$. Recall that the
manifold $Y_{MA}$ is a product, i.e. $Y_{MA}=Y_{M}\times A$, where $M$ acts
properly on $Y_{M}$ and trivially on $A$, and $A$ acts properly and freely on
$A$ and trivially on $Y_{M}$. With this decomposition, we can write the
algebra $\mathcal{L}_{MA,s}^{\infty}(Y_{MA},E_{MA})$ as follows,

(8.21)
$\mathcal{L}_{MA,s}^{\infty}(Y_{MA},E_{MA})=\mathcal{L}_{M,s}^{\infty}(Y_{M},E_{M})\hat{\otimes}\mathcal{L}_{s}(A,\operatorname{End}(\mathcal{S}_{A})),$

where $\mathcal{S}_{A}$ is the space of spinors on $\mathfrak{a}$, the Lie
algebra of $A$, and $\operatorname{End}(\mathcal{S}_{A})$ is the algebra of
endomorphisms on $\mathcal{S}_{A}$. Now, we consider the index elements
$\operatorname{Ind}_{\infty}(D_{Y_{MA}})$,
$\operatorname{Ind}_{\infty}(D_{Y_{M}})$, and
$\operatorname{Ind}_{\infty}(D_{A})$, i.e.

$\operatorname{Ind}_{\infty}(D_{Y_{MA}})\in
K_{*}(\mathcal{L}_{MA,s}^{\infty}(Y_{MA},E_{MA})),\
\operatorname{Ind}_{\infty}(D_{Y_{M}})\in
K_{*}(\mathcal{L}^{\infty}_{M,s}(Y_{M},E_{M})),\
\operatorname{Ind}_{\infty}(D_{A})\in
K_{*}(\mathcal{L}_{s}(A,\operatorname{End}(\mathcal{S}_{A}))).$

We notice that the decomposition Eq. (8.21) of
$\mathcal{L}_{MA,s}^{\infty}(Y_{MA},E_{MA})$ defines an element

$\operatorname{Ind}_{\infty}(D_{Y_{M}})\otimes\operatorname{Ind}_{\infty}(D_{A})$

through the external product

$K_{*}(\mathcal{L}^{\infty}_{M,s}(Y_{M},E_{M}))\otimes
K_{*}(\mathcal{L}_{s}(A,\operatorname{End}(\mathcal{S}_{A})))\to
K_{*}(\mathcal{L}^{\infty}_{MA,s}(Y_{MA},E_{MA})).$

We claim that

(8.22)
$\operatorname{Ind}_{\infty}(D_{Y_{MA}})=\operatorname{Ind}_{\infty}(D_{Y_{M}})\otimes\operatorname{Ind}_{\infty}(D_{A})\in
K_{*}(\mathcal{L}^{\infty}_{MA,s}(Y_{MA},E_{MA})).$

To prove this formula we argue as follows.
Recall that $\mathcal{L}^{\infty}_{MA,s}(Y_{MA},E_{MA})$ (respectively
$\mathcal{L}^{\infty}_{M,s}(Y_{M},E_{M})$ and
$\mathcal{L}_{s}(A,\operatorname{End}(\mathcal{S}_{A}))$) is a dense
subalgebra of the Roe algebra $C^{*}(Y_{MA},E_{MA})^{MA}$ (respectively
$C^{*}(Y_{M},E_{M})^{M}$ and
$C^{*}_{r}(A,\operatorname{End}(\mathcal{S}_{A}))$) closed under holomorphic
functional calculus. The image of the smooth index classes in the K-theory of
the respective Roe $C^{*}$-algebras define the (isomorphic) index classes

$\operatorname{Ind}(D_{Y_{MA}})\in
K_{*}(C^{*}(Y_{MA},E_{MA})^{MA})\,,\;\;\;\operatorname{Ind}(D_{Y_{M}})\in
K_{*}(C^{*}(Y_{M},E_{M})^{M}),\ \operatorname{Ind}(D_{A})\in
K_{*}(C^{*}_{r}(A,\operatorname{End}(\mathcal{S}_{A}))\,.$

Furthermore, the Roe algebra $C^{*}(Y_{MA},E_{MA})^{MA}$ (respectively
$C^{*}(Y_{M},E_{M})^{M}$ and
$C^{*}_{r}(A,\operatorname{End}(\mathcal{S}_{A}))$) is strongly Morita
equivalent to $C^{*}_{r}(MA)$ (respectively $C^{*}_{r}(M)$ and $C^{*}_{r}(A)$)
and as explained in [PP2] we have explicit representatives of the (strongly)
Morita equivalent index classes

(8.23) $\operatorname{Ind}_{C^{*}_{r}(MA)}(D_{Y_{MA}})\in
K_{*}(C^{*}_{r}(MA))\,,\quad\operatorname{Ind}_{C^{*}_{r}(M)}(D_{Y_{M}})\in
K_{*}(C^{*}_{r}(M))\,\,\text{ and
}\,\,\operatorname{Ind}_{C^{*}_{r}(A)}(D_{A})\in K_{*}(C^{*}_{r}(A)).$

Indeed, these index classes are defined in terms of our operators acting on
suitable $C^{*}_{r}H$-modules, with $H$ one of the above 3 groups. Now,
following [Wu, LP-JFA] and [Wahl-product] we can also express these index
classes in terms of unbounded KK-classes associated to a Atiyah-Patodi-Singer
boundary condition, denoted here

(8.24) $\begin{split}\operatorname{Ind}^{{\rm
APS}}_{C^{*}_{r}(MA)}(D_{Y_{MA}})&\in KK_{*}(\mathbb{C},C^{*}_{r}(MA))\ ,\ \
\operatorname{Ind}^{{\rm APS}}_{C^{*}_{r}(M)}(D_{Y_{M}})\in
KK_{*}(\mathbb{C},C^{*}_{r}(M)),\\\ &\operatorname{Ind}^{{\rm
APS}}_{C^{*}_{r}(A)}(D_{A})\in KK_{*}(\mathbb{C},C^{*}_{r}(A)).\end{split}$

Notice that our $b$-index classes do not define KK-elements (the resolvent is
not $C^{*}$-compact); this is why we need to pass to the APS-index classes
defined through the well-known boundary condition. Using formula (8.15) of the
Dirac operator $D_{Y_{MA}}$ and proceeding exactly as in [Wahl-product,
Theorem 2.2] we can prove the following identity

$\operatorname{Ind}^{{\rm
APS}}_{C^{*}_{r}(MA)}(D_{Y_{MA}})=[\operatorname{Ind}^{{\rm
APS}}_{C^{*}_{r}(M)}(D_{Y_{M}})\otimes\operatorname{Ind}^{{\rm
APS}}_{C^{*}_{r}(A)}(D_{A})]\in KK_{*}(\mathbb{C},C^{*}_{r}(MA))\equiv
KK_{*}(C^{*}_{r}(MA))$

Following the stated isomorphisms of $K$-theory groups and the compatibility
of the various index classes we obtain finally

$\operatorname{Ind}_{\infty}(D_{Y_{MA}})=\operatorname{Ind}_{\infty}(D_{Y_{M}})\otimes\operatorname{Ind}_{\infty}(D_{A})\in
K_{*}(\mathcal{L}^{\infty}_{MA,s}(Y_{MA},E_{MA})).$

This is precisely the claim we wanted to prove.

Step II: Using the product structure (8.21) of the algebra
$\mathcal{L}^{\infty}_{MA,s}(Y_{MA},E_{MA})$, we compute the cocycle
$\Phi_{Y_{MA},g}$ as follows. For any $f_{i}\otimes
g_{i}\in\mathcal{L}^{\infty}_{M,s}(Y_{MA})\otimes\mathcal{L}_{s}(A,\operatorname{End}(\mathcal{S}_{A}))$,

(8.25) $\displaystyle\Phi_{Y_{MA},g}(f_{0}\otimes g_{0},\dots f_{n}\otimes
g_{n})$ $\displaystyle=$
$\displaystyle\int_{M/Z_{M,g}}\int_{(MA)^{n}}\text{det}(a_{1},\dots,a_{n})\cdot\operatorname{Tr}\Big{(}f_{0}(hgh^{-1}(m_{1}\dots
m_{n})^{-1})g_{0}((a_{1}\cdot\dots\cdot a_{n})^{-1})$ $\displaystyle
f_{1}(m_{1})g_{1}(a_{1})\cdot\dots\cdot
f_{n}(m_{n})g_{n}(a_{n})\Big{)}\;dhda_{1}\dots da_{n}dm_{1}\dots dm_{n}$
$\displaystyle=$
$\displaystyle\left(\int_{M/Z_{M,g}}\int_{(M)^{n}}\operatorname{Tr}\Big{(}f_{0}(hgh^{-1}(m_{1}\dots
m_{n})^{-1})f_{1}(m_{1})\cdot\dots\cdot f_{n}(m_{n})\Big{)}\;dhdm_{1}\dots
dm_{n}\right)$ $\displaystyle\times$
$\displaystyle\left(\int_{(A)^{n}}\text{det}(a_{1},\dots,a_{n})\cdot\operatorname{Tr}\Big{(}g_{0}((a_{1}\cdot\dots\cdot
a_{n})^{-1})g_{1}(a_{1})\cdot\dots\cdot g_{n}(a_{n})\Big{)}\;da_{1}\dots
da_{n}\right).$

In the above equation, we denote

$\widetilde{\Phi}_{Y_{M},g}(f_{0},\dots,f_{n})=\int_{M/Z_{M,g}}\int_{(M)^{n}}\operatorname{Tr}\Big{(}f_{0}(hgh^{-1}(m_{1}\dots
m_{n})^{-1})f_{1}(m_{1})\cdot\dots\cdot f_{n}(m_{n})\Big{)}\;dhdm_{1}\dots
dm_{n}$

and

$\Phi_{A,e}(g_{0},\dots,g_{n})=\int_{(A)^{n}}\text{det}(a_{1},\dots,a_{n})\cdot\operatorname{Tr}\Big{(}g_{0}((a_{1}\cdot\dots\cdot
a_{n})^{-1})g_{1}(a_{1})\cdot\dots\cdot g_{n}(a_{n})\Big{)}\;da_{1}\dots
da_{n}$

By (8.22) and (8.25), we conclude that

$\langle\Phi_{Y_{MA},g},\operatorname{Ind}_{\infty}(D_{Y_{MA}})\rangle=\langle\widetilde{\Phi}_{Y_{M},g},\operatorname{Ind}_{\infty}(D_{Y_{M}})\rangle\cdot\langle\Phi_{A,e},\operatorname{Ind}_{\infty}(D_{A})\rangle.$

By a special case [PPT, Theorem 4.6] (the case for G = A), we know that

$\langle\Phi_{A,e},\operatorname{Ind}_{\infty}(D_{A})\rangle=1.$

On the other hand, we can directly check that if $f\star f=f$, then

$\displaystyle\widetilde{\Phi}_{Y_{M},g}(f,\dots,f)=$
$\displaystyle\int_{M/Z_{M,g}}\int_{M^{\times
n}}\operatorname{Tr}\Big{(}f(hgh^{-1}(m_{1}\dots
m_{n})^{-1})f(m_{1})\cdot\dots\cdot f(m_{n})\Big{)}\;dhdm_{1}\dots dm_{n}$
$\displaystyle=$
$\displaystyle\int_{M/Z_{M,g}}\operatorname{Tr}\big{(}f(hgh^{-1})\big{)}dh=\Phi_{Y_{M},g}(f).$

Thus,

$\langle\widetilde{\Phi}_{Y_{M},g},\operatorname{Ind}_{\infty}(D_{Y_{M}})\rangle=\langle\Phi_{Y_{M},g},\operatorname{Ind}_{\infty}(D_{Y_{M}})\rangle.$

This completes the proof. ∎

##  9\. An index theorem for higher orbital integral through reduction

We shall now put things together and give a formula for the higher delocalized
Atiyah-Patodi-Singer index
$\langle\Phi^{P}_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle$.

Let $G$ be connected, linear real reductive, $P=MAN$ a cuspidal parabolic
subgroup and $g\in M$ a semisimple element. Let $(Y_{0},\mathbf{h}_{0})$ be a
cocompact $G$-proper manifold with boundary. We fix a slice $Z_{0}$ (and $Z$)
for the $G$ action on $Y_{0}$ (and $Y$), so that

$Y_{0}\cong G\times_{K}Z_{0},\quad Y\cong G\times_{K}Z.$

with $Z_{0}$ a smooth compact manifold with boundary. We assume that
$D_{\partial Y}$ is $L^{2}$-invertible. Consider $Y/AN$, an $M$-proper
manifold. Recall that we have the following $K\cap M$-invariant decomposition

$\displaystyle\mathfrak{p}=$
$\displaystyle(\mathfrak{p}\cap\mathfrak{m})\oplus\mathfrak{a}\oplus\left(\left(\mathfrak{p}\cap\mathfrak{m}\right)\oplus\mathfrak{a}\right)^{\perp}$
$\displaystyle\cong$
$\displaystyle(\mathfrak{p}\cap\mathfrak{m})\oplus\mathfrak{a}\oplus\left(\mathfrak{k}/(\mathfrak{k}\cap\mathfrak{m})\right).$

Accordingly, the spinor bundle

$S_{\mathfrak{p}}\cong S_{\mathfrak{p}\cap\mathfrak{m}}\otimes
S_{\mathfrak{a}}\otimes S_{\mathfrak{k}/(\mathfrak{k}\cap\mathfrak{m})},$

where $S_{\mathfrak{a}}$ is a vector space of dimension
$2^{\lceil\frac{\dim\mathfrak{a}}{2}\rceil}$ on which $M\cap K$ acts
trivially. The following facts will play a central role in our study:

  1. (1)

We realize $Y/N$ in terms of a slice:

$Y_{MA}\colon=Y/N\cong AM\times_{K\cap M}Z.$

Restricting to $Y_{MA}$,

$E_{MA}\colon=E|_{Y_{MA}}\cong AM\times_{M\cap
K}\left(S_{\mathfrak{p}\cap\mathfrak{m}}\otimes S_{\mathfrak{a}}\otimes
S_{\mathfrak{k}/(\mathfrak{k}\cap\mathfrak{m})}\otimes E_{Z}\right).$

For convenience, we introduce the following

(9.1) $\widetilde{E_{Z}}:=E_{Z}\otimes
S_{\mathfrak{k}/(\mathfrak{k}\cap\mathfrak{m})}.$

Thus

(9.2) $E_{M}=M\times_{M\cap K}\left(S_{\mathfrak{p}\cap\mathfrak{m}}\otimes
S_{\mathfrak{k}/(\mathfrak{k}\cap\mathfrak{m})}\otimes E_{Z}\right)\cong
M\times_{M\cap
K}\left(S_{\mathfrak{p}\cap\mathfrak{m}}\otimes\widetilde{E_{Z}}\right).$

By the assumption (3.3) on the dimension of $Y$, we can see that $Y_{M}$ is
even dimensional because both $Z$ and $M/M\cap K$ are even dimensional. The
quotient Dirac operator $D_{Y_{M}}$ on $Y_{M}$ acts on

$L^{2}(Y_{M},E_{M})\cong\left[L^{2}(M)\otimes
S_{\mathfrak{p}\cap\mathfrak{m}}\otimes
L^{2}(Z,\widetilde{E_{Z}})\right]^{K\cap M}.$

We have assumed that $E$ is a $G$-equivariant twisted spinor bundle on $Y$.
Accordingly, $E_{M}$ is an $M$-equivariant twisted spinor bundle on $Y_{M}$,
and we can write

(9.3) $E_{M}=\mathcal{E}_{M}\otimes W_{M}.$

where $\mathcal{E}_{M}$ is the spinor bundle associated to the induced Spinc-
structure on $Y_{M}$ and $W_{M}=M\times_{M\cap M}\widetilde{E}_{Z}$. It is
important to point out that since

$S_{\mathfrak{k}/(\mathfrak{k}\cap\mathfrak{m})}=S^{+}_{\mathfrak{k}/(\mathfrak{k}\cap\mathfrak{m})}\oplus
S^{-}_{\mathfrak{k}/(\mathfrak{k}\cap\mathfrak{m})}$

has a $\mathbb{Z}_{2}$-grading, the auxiliary $M$-equivariant vector $W_{M}$
is equipped with a $\mathbb{Z}_{2}$-grading as well, denoted by

(9.4) $W_{M}=W_{M}^{+}\oplus W_{M}^{-}$

######  9.5 Remark.

Unless $P$ is maximal cuspidal parabolic, $K\cap M$ has a lower rank than $K$.
In this case, there exists a $K\cap M$-equivariant isomorphism $\lambda$
defined by the action of an element in a Cartan subgroup of $K$ but not in
$K\cap M$,

$\lambda:S^{+}_{\mathfrak{k}/(\mathfrak{k}\cap\mathfrak{m})}\cong
S^{-}_{\mathfrak{k}/(\mathfrak{k}\cap\mathfrak{m})}.$

Such an isomorphism $\lambda$ is compatible with the respective connections
and metrics. In the constructions in (9.1), (9.2) and (9.3), the tensor
products are all graded and grading compatible. Hence, $\lambda$ gives an
$M$-equivariant isomorphism between the two vector bundles

$W_{M}^{+}\cong W_{M}^{-},$

compatible with the respective connections and metrics.

######  9.6 Remark.

As $W_{M}$ in Equation (9.4) is $\mathbb{Z}_{2}$-graded, the vector bundle
$E_{M}$ in Equation (9.3) can be written as follows with respect to the
gradings on $\mathcal{E}_{M}$ and $W_{M}$,

$E_{M}=\big{(}\mathcal{E}_{M}^{+}\oplus\mathcal{E}_{M}^{-}\big{)}\otimes
W_{M}^{+}\oplus\big{(}\mathcal{E}_{M}^{+}\oplus\mathcal{E}_{M}^{-}\big{)}\otimes
W^{-}_{M}.$

Accordingly, the $\mathbb{Z}_{2}$-grading on $E_{M}$ gives the following
decomposition,

(9.7) $E_{M}=E_{M}^{+}\oplus E_{M}^{-},\qquad
E_{M}^{+}=\mathcal{E}_{M}^{+}\otimes W^{+}_{M}\oplus\mathcal{E}_{M}^{-}\otimes
W^{-}_{M},\qquad E_{M}^{-}=\mathcal{E}_{M}^{-}\otimes
W^{+}_{M}\oplus\mathcal{E}_{M}^{+}\otimes W_{M}^{-}.$

Given a linear operator $T$ on $E_{M}$, we write $T$ into a $4\times 4$ block
matrix,

(9.8)
$T=\left[\begin{array}[]{cccc}T^{++}_{++}&T^{++}_{-+}&T^{++}_{+-}&T^{++}_{--}\\\
T^{-+}_{++}&T^{-+}_{-+}&T^{-+}_{+-}&T^{-+}_{--}\\\
T^{+-}_{++}&T^{+-}_{-+}&T^{+-}_{+-}&T^{+-}_{--}\\\
T^{--}_{++}&T^{--}_{-+}&T^{--}_{+-}&T^{--}_{--}\\\ \end{array}\right],$

where $T^{\alpha\beta}_{\gamma\delta}$ is a linear operator from
$\mathcal{E}_{M}^{\gamma}\otimes W_{M}^{\delta}$ to
$\mathcal{E}_{M}^{\alpha}\otimes W^{\beta}_{M}$, for
$\alpha,\beta,\gamma,\delta=\pm$. The $\mathbb{Z}_{2}$-grading, c.f. Equation
(9.7), on $E_{M}$ introduces a supertrace on $T$ as follows,

(9.9)
$\begin{split}\operatorname{Str}(T):=&\operatorname{tr}(T^{++}_{++})+\operatorname{tr}(T^{--}_{--})-\big{[}\operatorname{tr}(T^{-+}_{-+})+\operatorname{tr}(T^{+-}_{+-})\big{]}\\\
=&\operatorname{tr}(T^{++}_{++})-\operatorname{tr}(T^{-+}_{-+})-\big{[}\operatorname{tr}(T^{+-}_{+-})-\operatorname{tr}(T^{--}_{--})\big{]}.\end{split}$

The following theorem is one of the main results of this paper:

######  9.10 Theorem.

Suppose that the metric on $Y$ is slice compatible. Assume that $D_{\partial
Y}$ is $L^{2}$-invertible and consider the higher index
$\langle\Phi^{P}_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle$. The following
formula holds:

(9.11)
$\langle\Phi^{P}_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle=\int_{(Y_{0}/AN)^{g}}c^{g}_{(Y_{0}/AN)^{g}}{\rm
AS}(Y_{0}/AN)_{g}-\frac{1}{2}\eta_{g}(D_{\partial Y_{M}})$

with

$\eta_{g}(D_{\partial
Y_{M}})=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{\partial
Y_{M}}_{g}(D_{\partial Y_{M}}\exp(-tD^{2}_{\partial
Y_{M}})\frac{dt}{\sqrt{t}}.$

Recall that $c^{g}_{(Y_{0}/AN)^{g}}$ is a compactly supported smooth cutoff
function on $(Y_{0}/AN)^{g}$ associated to the $Z_{M,g}$ action on
$(Y_{0}/AN)^{g}$.

To explain the right-hand side of (1.11) more explicitly, we fix the following
data.

  * •

In $(\ref{eq WM})$, we constructed a $M$-equivariant, $\mathbb{Z}_{2}$-graded
vector bundle

$W_{M}=W_{M}^{+}\oplus W_{M}^{-}.$

We define

$R^{W^{\pm}_{M}}=\text{the curvature form of the Hermitian connection on
}W_{M}^{\pm}.$

  * •

$R^{\mathcal{N}}$, the curvature form associated to the Hermitian connection
on $\mathcal{N}_{(Y_{0}/AN)^{g}}\otimes\mathbb{C}$
($\mathcal{N}_{(Y_{0}/AN)^{g}}$ is the normal bundle of the $g$-fixed point
submanifold $(Y_{0}/AN)^{g}$ in $Y_{0}/AN$);

  * •

$R^{L}$, the curvature form associated to the Hermitian connection on
$L_{\operatorname{det}}|_{(Y_{0}/AN)^{g}}$ ($L_{\operatorname{det}}$ is the
determinant line bundle of the Spinc-structure on $Y_{0}/AN$ and
$L_{\operatorname{det}}|_{(Y_{0}/AN)^{g}}$ is its restriction to
$(Y_{0}/AN)^{g}$);

  * •

$R_{(Y_{0}/AN)^{g}}$, the Riemannian curvature form associated to the Levi-
Civita connection on the tangent bundle of $(Y_{0}/AN)^{g}$;

  * •

The form ${\rm AS}(Y_{0}/AN)_{g}$ is then given by the following expression:

(9.12) $\frac{\widehat{A}\left(\frac{R_{(Y_{0}/AN)^{g}}}{2\pi
i}\right)\left[\operatorname{tr}\left(g\exp\left(\frac{R^{W^{+}_{M}}}{2\pi
i}\right)\right)-\operatorname{tr}\left(g\exp\left(\frac{R^{W^{-}_{M}}}{2\pi
i}\right)\right)\right]\exp(\operatorname{tr}(\frac{R^{L}}{2\pi
i}))}{\operatorname{det}\left(1-g\exp(-\frac{R^{\mathcal{N}}}{2\pi
i})\right)^{\frac{1}{2}}}$

  * •

Since the auxiliary vector bundle $W_{M}$ on $Y_{M}$ is
$\mathbb{Z}_{2}$-graded, the Dirac operator $D_{Y_{M}}$ is equipped with a
$\mathbb{Z}_{2}$-grading as well. Following the decomposition (9.7) and (9.8),
we have the decomposition for $D_{Y_{M}}$,

$\begin{split}&D_{Y_{M}}=\left[\begin{array}[]{cccc}0&D^{+}_{Y_{M},+}&0&0\\\
D^{-}_{Y_{M},+}&0&0&0\\\ 0&0&0&D^{+}_{Y_{M},-}\\\ 0&0&D^{-}_{Y_{M},-}&0\\\
\end{array}\right]=\left[\begin{array}[]{cc}D_{Y_{M},+}&0\\\
0&D_{Y_{M},-}\end{array}\right]\\\
&D_{Y_{M},+}=\left[\begin{array}[]{cc}0&D^{+}_{Y_{M},+}\\\
D^{-}_{Y_{M},+}&0\end{array}\right],\qquad
D_{Y_{M},-}=\left[\begin{array}[]{cc}0&D^{+}_{Y_{M},-}\\\
D^{-}_{Y_{M},-}&0\end{array}\right],\end{split}$

where

$D^{\alpha}_{Y_{M},\beta}\colon
L^{2}\left(Y_{M},\mathcal{E}^{\alpha}_{Y_{M}}\otimes W^{\beta}_{M}\right)\to
L^{2}\left(Y_{M},\mathcal{E}^{-\alpha}_{Y_{M}}\otimes
W^{\beta}_{M}\right),\quad\alpha,\beta=\pm.$

Accordingly, since $\partial Y_{M}$ is odd dimensional, the Dirac operator
$D_{\partial Y_{M}}$ is a self-adjoint even differential operator on the
$\mathbb{Z}_{2}$-graded spinor bundle $E_{\partial Y_{M}}$, which is defined
as follows,

$E_{\partial Y_{M}}:=\mathcal{E}_{\partial Y_{M}}\otimes W_{M}|_{\partial
Y_{M}},\qquad E_{\partial Y_{M}}^{+}:=\mathcal{E}_{\partial Y_{M}}\otimes
W^{+}_{M}|_{\partial Y_{M}},\qquad E^{-}_{\partial
Y_{M}}:=\mathcal{E}_{\partial Y_{M}}\otimes W^{-}_{M}|_{\partial Y_{M}}.$

We can write $D_{\partial Y_{M}}$ into a $2\times 2$ block diagonal matrix
with respect to the grading on $E|_{\partial Y_{M}}$,

$\left[\begin{array}[]{cc}D_{\partial Y_{M},+}&0\\\ 0&D_{\partial Y_{M},-}\\\
\end{array}\right],$

where $D_{\partial Y_{M},+}$ (and $D_{\partial Y_{M},-}$) is the Dirac
operator on $E_{\partial Y_{M}}^{+}$ (and $E_{\partial Y_{M}}^{-}$). Following
the definition of the supertrace (9.9) and the $\mathbb{Z}_{2}$-grading of
$D_{\partial Y_{M}}$, we obtain the following expression for the delocalized
eta invariant of $D_{\partial Y_{M}}$:

(9.13) $\eta_{g}(D_{\partial Y_{M}})=\eta_{g}(D_{\partial
Y_{M},+})-\eta_{g}(D_{\partial Y_{M},-}).$

This point is slightly different from the 0-degree pairing case, where the
operator $D_{\partial Y_{M}}$ is not graded.

###### Proof.

Denote $\Phi^{P}_{Y,g}$ briefly by $\Phi_{Y,g}$. By (8.17) we know that

$\langle\Phi_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle=\langle\Phi_{Y_{MA},g},\operatorname{Ind}_{\infty}(D_{Y_{MA}})\rangle\,.$

On the other hand, by equation (8.20) we have that

$\langle\Phi_{Y_{MA},g},\operatorname{Ind}_{\infty}(D_{Y_{MA}})\rangle=\langle\Phi_{Y_{M},g},\operatorname{Ind}_{\infty}(D_{Y_{M}})\rangle$

so that

$\langle\Phi_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle=\langle\Phi_{Y_{M},g},\operatorname{Ind}_{\infty}(D_{Y_{M}})\rangle$

It suffices to apply now the 0-degree delocalized APS index theorem, Theorem
4.39, to the right hand side of the above equation and recall that $Y/AN$ is
diffeomorphic to $Y_{M}:=M\times_{K\cap M}Z$ and that is obtained by addition
of a cylindrical end to $Y_{0}/AN$. ∎

When $P$ is not a maximal parabolic subgroup, we have the following vanishing
result:

######  9.14 Remark.

If $P$ is not a maximal parabolic subgroup, then $\Phi^{P}_{Y,g}$ is a trivial
class in cyclic cohomology and

$\langle\Phi^{P}_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle=0.$

For the right-hand side of Equation (9.11), we know from Remark 9.5 that

$W_{M}^{+}\cong W_{M}^{-}$

compatible with the respective connections and metrics. Thus, $D_{\partial
Y_{M},+}$ is unitary equivalent to $D_{\partial Y_{M},-}$ under the above
isomorphism. By using the expression of $\eta_{g}(D_{\partial Y_{M}})$ as
$\eta_{g}(D_{\partial Y_{M},+})-\eta_{g}(D_{\partial Y_{M},-})$, we obtain the
following

$\eta_{g}(D_{\partial Y_{M}})=0\,.$

Moreover, the integral

$\int_{(Y_{0}/AN)^{g}}c^{g}_{(Y_{0}/AN)^{g}}{\rm AS}(Y_{0}/AN)_{g}=0$

as follows from the fact that

$\operatorname{tr}\left(g\exp\left(\frac{R^{W^{+}_{M}}}{2\pi
i}\right)\right)-\operatorname{tr}\left(g\exp\left(\frac{R^{W^{-}_{M}}}{2\pi
i}\right)\right)=0,$

because of the the isomorphism $W^{+}_{M}\cong W^{-}_{M}$.

##  10\. Numeric rho invariants on $G$-proper manifolds

In this section we shall introduce (higher) rho numbers associated to positive
scalar curvature (psc) metrics.

###  10.1. Rho numbers associated to delocalized 0-cocycles

We consider a closed $G$-proper manifold $X$ without boundary, $G$ connected,
linear real reductive, $g\in G$ a semisimple element, $D_{X}$ a
$G$-equivariant $L^{2}$-invertible Dirac operator defined in (3.2). We know
that the following integral is convergent:

$\eta_{g}(D_{X}):=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{X}_{g}(D_{X}\exp(-tD^{2}_{X})\frac{dt}{\sqrt{t}}.$

######  10.1 Definition.

Let $X$ be $G$-equivariantly spin and $D_{X}\equiv D_{\mathbf{h}}$, the spin
Dirac operator associated to a $G$-equivariant PSC metric $\mathbf{h}$. We
know that $D_{\mathbf{h}}$ is $L^{2}$-invertible. We define

$\rho_{g}(\mathbf{h}):=\eta_{g}(D_{\mathbf{h}})\,.$

###  10.2. Rho numbers associated to higher delocalized cocycles

We can generalize the example of the previous subsection and define rho
numbers associated the higher cocycles $\Phi^{P}_{g}$.

######  10.2 Definition.

Let $P=MNA$ be a cuspidal parabolic subgroup, $g\in M$ as above and consider
$\Phi_{g}^{P}$ and $\Phi_{X,g}^{P}$ (we recall that $X$ is without boundary).
Assume that $\mathbf{h}$ is a $G$-invariant PSC metric on $X$. In this case we
also assume that $\mathbf{h}$ is slice-compatible. Then

(10.3) $\rho_{g}^{P}(\mathbf{h}):=\eta_{g}(D_{X_{M}})$

(with $X_{M}$ the reduced manifold associated to $X$) is well defined.

Notice that as $D_{X}$ is invertible, we have that $D_{X_{M}}$ is also
invertible, see Proposition 8.13.

###  10.3. Bordism properties

The APS index theorems proved in this article can be used in order to study
the bordism properties of these rho invariants.

We assume, unless otherwise stated that we are on a $G$-proper manifold which
is endowed with a $G$-invariant metric and a $G$-equivariant spin structure.
If needed, we shall assume slice-compatibility of the metric and of the spin
structure.

######  10.4 Definition.

Consider a cocompact proper $G$-manifold $(W,\mathbf{h})$, possibly with
boundary, endowed with an equivariant spin structure. Let $S$ be the spinor
bundle.Let $g\in G$ be semisimple. We shall say that $g$ is geometrically-
simple on $W$ if

$\int_{W^{g}}c^{g}{\rm AS}_{g}(W,S)=0\,.$

with ${\rm AS}_{g}(W,S)$ the usual integrand appearing in the 0-degree
delocalized APS-index theorem.

######  10.5 Example.

The following proposition provides many examples of geometrically-simple
elements $g$ on an arbitrary $G$-proper manifold $W$.

######  10.6 Proposition.

If $g$ is non-elliptic, that is, does not conjugate to a compact element, then
every element of the conjugacy class $C(g):=\\{hgh^{-1}|h\in G\\}$ in $G$ does
not have any fixed point on $W$.

###### Proof.

Without loss of generality, we assume that $W^{g}\neq\emptyset$. Take

$x=(g_{1},s)\in W=G\times_{K}S$

such that

$gx=(gg_{1},s)=(g_{1},s)=x.$

As $(g_{1},s)$ and $(gg_{1},s)$ correspond to the same point in
$G\times_{K}S$, we can find some $k\in K$ such that

$gg_{1}=g_{1}k,\quad ks=s.$

Thus, $g_{1}^{-1}gg_{1}\in K$, that is, $g$ has to be elliptic, which
contradicts to our assumption on $g$. ∎

######  10.7 Definition.

Let $(Y,\mathbf{h}_{0})$ and $(Y,\mathbf{h}_{1})$ be two psc metrics. We say
that they are $G$-concordant if there exists a $G$-invariant metric
$\mathbf{h}$ on $Y\times[0,1]$ which is of psc, product-type near the boundary
and restricts to $\mathbf{h}_{0}$ at $Y\times\\{0\\}$ and to $\mathbf{h}_{1}$
at $Y\times\\{1\\}$. If $\mathbf{h}_{0}$ and $\mathbf{h}_{1}$ are slice
compatible, we also require the metric $\mathbf{h}$ on $Y\times[0,1]$ to be
slice compatible.

######  10.8 Definition.

Let $(Y_{0},\mathbf{h}_{0})$ and $(Y_{1},\mathbf{h}_{1})$ be two $G$-proper
manifolds with psc metrics. We shall say that they are $G$-psc-bordant if
there exists a $G$-manifold with boundary $W$ with a $G$-invariant metric
$\mathbf{h}$ such that:
(i) $\partial W=Y_{0}\sqcup Y_{1}$;
(ii) $\mathbf{h}$ is product-type near the boundary and of psc;
(iii) $\mathbf{h}$ restricts to $\mathbf{h}_{0}\sqcup\mathbf{h}_{1}$ on
$\partial W=Y_{0}\sqcup Y_{1}$.

If $\mathbf{h}_{0}$ and $\mathbf{h}_{1}$ are slice compatible, we also require
the metric $\mathbf{h}$ to be slice compatible.

######  10.9 Proposition.

1] Assume that the psc metrics $\mathbf{h}_{0}$ and $\mathbf{h}_{1}$ on $Y$
are $G$-concordant. Assume that $g$ is geometrically-simple on $Y\times[0,1]$,
Then $\rho_{g}(\mathbf{h}_{0})=\rho_{g}(\mathbf{h}_{1})$.
2] Let $P=MAN<G$ be a cuspidal parabolic subgroup and let $g\in M$ be a
semisimple element. Assume that the slice-compatible psc metrics
$\mathbf{h}_{0}$ and $\mathbf{h}_{1}$ on $Y$ are $G$-concordant. Assume that
$g$ is geometrically-simple on $Y_{M}\times[0,1]$. Then
$\rho^{P}_{g}(\mathbf{h}_{0})=\rho^{P}_{g}(\mathbf{h}_{1})$.
3] Let $(Y_{0},\mathbf{h}_{0})$ and $(Y_{1},\mathbf{h}_{1})$ be $G$-psc-
bordant through $(W,h)$. Assume that $g$ is geometrically-simple on $W$. Then
$\rho_{g}(\mathbf{h}_{0})=\rho_{g}(\mathbf{h}_{1})$.

###### Proof.

In both cases the proof is an immediate consequence of the geometric-
simplicity of $g\in G$ and the relevant delocalized APS index theorems. ∎

######  10.10 Remark.

Examples of geometrically-simple $g$ as in Proposition 10.9 are given by
elements in the conjugacy class of a non-elliptic element, c.f. Proposition
10.6. Notice that this applies to any bordism entering into Definition 10.7
and Definition 10.8. Thus for such $g$ our (higher) rho invariants are in fact
concordance and bordism invariants.

######  10.11 Remark.

It is a challenge to understand whether these results can be employed in
studying the spaces $\mathcal{R}^{+}_{G,{\rm slice}}(Y)$ of slice-compatible
$G$-invariant metrics of psc and the larger space $\mathcal{R}^{+}_{G}(Y)$ of
$G$-invariant metrics of psc (if non-empty). It would be also interesting to
understand the relationship between the following 3 spaces, especially from
the point of view of homotopy theory:
(i) $\mathcal{R}^{+}_{G,{\rm slice}}(Y)$;
(ii) $\mathcal{R}^{+}_{G}(Y)$;
(iii) $\mathcal{R}^{+}_{K}(S)$, the space of $K$-invariant metrics of psc on
the slice $S$.
Clearly there is an inclusion $\mathcal{R}^{+}_{G,{\rm
slice}}(Y)\hookrightarrow\mathcal{R}^{+}_{G}(Y)$; there is also a natural map
$\mathcal{R}^{+}_{K}(S)\to\mathcal{R}^{+}_{G,{\rm slice}}(Y)$, see [MGW]. For
all these questions it would be interesting to develop a $G$-equivariant
Stolz’ sequence and investigate its basic properties. We leave this task to
future research.

######  10.12 Remark.

If $Y_{1}$ and $Y_{2}$ are two oriented $G$-proper manifolds and
$\mathbf{f}:Y_{1}\to Y_{2}$ is an oriented $G$-equivariant homotopy
equivalence then we can also define rho invariants associated to $f$ and to a
semisimple element $g$:

$\rho_{g}(\mathbf{f}):=\eta_{g}(D^{{\rm sign}}_{X}+A({\mathbf{f}}))$

with $D^{{\rm sign}}_{X}$ the signature operator on $X=Y_{1}\cup(-Y_{2})$ and
$A({\mathbf{f}})$ the Hilsum-Skandalis perturbation, as developed by Fukumoto
in the $G$-proper context [Fukumoto]. Notice however that both the definition
of this invariant and the proof of the associated APS index theorem are not a
routine extension of the case treated here. Especially, the large time
behaviour of a perturbed heat kernel and the proof of the analogue of
Proposition 4.34 are particularly intricate. We plan to report on our results
in this direction in a separate article.

##  Appendix A Dirac operator on $G$

In this appendix, we discuss the difference between the Dirac operator, Eq.
(3.2), considered in this article and the split Dirac operator operator, Eq.
(3.6), on the manifold $G$ with the free proper left $G$-action.

Suppose that $G$ is a connected reductive Lie group with maximal compact
subgroup $K$. We denote by $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ the
Cartan decomposition of the Lie algebra and
$\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}\otimes\mathbb{C}$ its
complexification. Let $B$ be a non-degenerate bilinear symmetric form on
$\mathfrak{g}$, which is invariant under the adjoint action of $G$ on
$\mathfrak{g}$. We assume $B$ to be negative on $\mathfrak{p}$, and positive
on $\mathfrak{k}$. Let $S_{\mathfrak{g}}$ and $S_{\mathfrak{k}}$ be the
irreducible $\mathbb{Z}_{2}$-graded spin representation for
$\text{Cliff}(\mathfrak{g}_{\mathbb{C}})$ and
$\text{Cliff}(\mathfrak{k}_{\mathbb{C}})$ respectively. Then we have the
following decomposition

$S_{\mathfrak{g}}\cong S_{\mathfrak{k}}\otimes S_{\mathfrak{p}}.$

The adjoint actions on $\mathfrak{g}$ induce maps

$\displaystyle\text{ad}^{\mathfrak{g}}\colon$
$\displaystyle\mathfrak{g}_{\mathbb{C}}\to\operatorname{End}(S_{\mathfrak{g}})\cong\text{Cliff}(\mathfrak{g}_{\mathbb{C}})$
$\displaystyle\text{ad}^{\mathfrak{k}}\colon$
$\displaystyle\mathfrak{k}_{\mathbb{C}}\to\operatorname{End}(S_{\mathfrak{k}})\cong\text{Cliff}(\mathfrak{k}_{\mathbb{C}})$

defined by the following formulas: for any $X\in\mathfrak{g}_{\mathbb{C}}$,

$\text{ad}^{\mathfrak{g}}(X)\colon=\frac{1}{4}\sum_{i}^{\dim\mathfrak{g}}c\left([X,e_{i}]\right)\cdot
c(e_{i}),\quad\text{ad}^{\mathfrak{k}}(X)\colon=\frac{1}{4}\sum_{i}^{\dim\mathfrak{k}}c\left([X,e_{i}]\right)\cdot
c(e_{i})$

where

$e_{1},\cdots e_{\text{dim}\mathfrak{k}},\quad\text{and}\quad
e_{\text{dim}\mathfrak{k}+1},\cdots,e_{\text{dim}\mathfrak{g}}$

are the orthonormal bases for $\mathfrak{k}$ and $\mathfrak{p}$ respectively.
We have the following relation

$\text{ad}^{\mathfrak{g}}(X)=\text{ad}^{\mathfrak{k}}(X)+\text{ad}^{\mathfrak{p}}(X),\quad
X\in\mathfrak{k}_{\mathbb{C}},$

where

$\text{ad}^{\mathfrak{p}}(X)=-\frac{1}{4}\sum_{i,j=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}B(X,[e_{i},e_{j}])\cdot
c(e_{i})\cdot c(e_{j})\in\text{Cliff}(\mathfrak{p}),\quad
X\in\mathfrak{k}_{\mathbb{C}}$

By left-trivialization, we can identify the vector field $\mathfrak{X}(G)$
with $C^{\infty}(G)\otimes\mathfrak{g}$. In particular, for any
$X\in\mathfrak{g}$, we can identify it with a left-invariant vector field on
$G$. We define a $K$-invariant inner product on $\mathfrak{g}$ by

$\langle\ ,\ \rangle=-B|_{\mathfrak{k}}\oplus+B|_{\mathfrak{p}}.$

Our computation below is similar to those in [MR3052646, Chapter 9], while the
author considers the pseudo Riemannian metric B, introduced above, instead of
a Riemannian metric.

By the Koszul formula, the Levi-Civita connection is determined by

$\langle\nabla_{X}Y,Z\rangle=\frac{1}{2}\left(\langle[X,Y],Z\rangle-\langle[Y,Z],X\rangle+\langle[Z,X],Y\rangle\right),\quad
X,Y,Z\in\mathfrak{g}.$

More general, for any vector field $V$ on $G$, it can be written as

$V=\sum_{i=1}^{\dim\mathfrak{g}}f_{i}\cdot V_{i}$

where $f_{i}\in C^{\infty}(G)$ and $\\{V_{i}\\}$ is a basis for
$\mathfrak{g}$. Then

$\nabla_{X}V=\sum_{i=1}^{\dim\mathfrak{g}}X(f_{i})\cdot
V_{i}+\sum_{i=1}^{\dim\mathfrak{g}}f_{i}\cdot\nabla_{X}V_{i}.$

If $X,Y\in\mathfrak{p}$ and $Z\in\mathfrak{k}$, then

$\langle[Y,Z],X\rangle=-B([Y,Z],X)=B(Z,[Y,X])=\langle
Z,[Y,X]\rangle=-\langle[X,Y],Z\rangle$

and

$\langle[Z,X],Y\rangle=-B([Z,X],Y)=-B(Z,[X,Y])=-\langle[X,Y],Z\rangle.$

Thus,

$\nabla_{X}Y=\frac{1}{2}[X,Y].$

Similarly, one can check the following:

  * •

$\nabla_{X}Y=\frac{1}{2}[X,Y],\quad X,Y\in\mathfrak{k}$.

  * •

$\nabla_{X}Y=\frac{3}{2}[X,Y],\quad X\in\mathfrak{k},Y\in\mathfrak{p}$.

  * •

$\nabla_{X}Y=-\frac{1}{2}[X,Y],\quad X\in\mathfrak{p},Y\in\mathfrak{k}$.

Therefore, the induced Clifford connection $\nabla^{S_{\mathfrak{g}}}$ acts on
left $G$-invariant section of $S_{\mathfrak{g}}$ by the following formula:

$\nabla^{S_{\mathfrak{g}}}_{X}=\frac{1}{8}\sum_{i=1}^{\dim\mathfrak{k}}c\left([X,e_{i}]\right)\cdot
c(e_{i})+\frac{3}{8}\sum_{i=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}c\left([X,e_{i}]\right)\cdot
c(e_{i}),\quad X\in\mathfrak{k}$

and

$\nabla^{S_{\mathfrak{g}}}_{X}=-\frac{1}{8}\sum_{i=1}^{\dim\mathfrak{k}}c\left([X,e_{i}]\right)\cdot
c(e_{i})+\frac{1}{8}\sum_{i=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}c\left([X,e_{i}]\right)\cdot
c(e_{i}),\quad X\in\mathfrak{p}.$

Note that

$e_{1},\cdots
e_{\text{dim}\mathfrak{k}},\quad\text{and}\quad\sqrt{-1}e_{\text{dim}\mathfrak{k}+1},\cdots,\sqrt{-1}e_{\text{dim}\mathfrak{g}}$

give an orthonormal basis for $\mathfrak{g}_{\mathbb{C}}$ with respect to the
inner product $\langle\ ,\ ,\rangle$. The corresponding Dirac operator on
$L^{2}(G,S_{\mathfrak{g}})$ is given by

$\displaystyle
D_{G}=\sum_{i=1}^{\dim\mathfrak{k}}c(e_{i})\nabla^{S_{\mathfrak{g}}}_{e_{i}}-\sum_{i=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}c(e_{i})\nabla^{S_{\mathfrak{g}}}_{e_{i}}.$

As an element in
$\mathcal{U}(\mathfrak{g}_{\mathbb{C}})\otimes\text{Cliff}(\mathfrak{g}_{\mathbb{C}})$,

(A.1) $\displaystyle D_{G}=$
$\displaystyle\sum_{i=1}^{\dim\mathfrak{k}}e_{i}\otimes
c(e_{i})-\sum_{i=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}e_{i}\otimes c(e_{i})$
$\displaystyle+$
$\displaystyle\frac{1}{8}\sum_{j=1}^{\dim\mathfrak{k}}\sum_{i=1}^{\dim\mathfrak{k}}c\left([e_{j},e_{i}]\right)\cdot
c(e_{i})c(e_{j})+\frac{3}{8}\sum_{j=1}^{\dim\mathfrak{k}}\sum_{i=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}c\left([e_{j},e_{i}]\right)\cdot
c(e_{i})c(e_{j})$ $\displaystyle-$
$\displaystyle\frac{1}{8}\sum_{j=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}\sum_{i=1}^{\dim\mathfrak{k}}c\left([e_{j},e_{i}]\right)\cdot
c(e_{i})c(e_{j})+\frac{1}{8}\sum_{j=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}\sum_{i=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}c\left([e_{j},e_{i}]\right)\cdot
c(e_{i})c(e_{j})$

Let us introduce a _cubic term_

$\phi^{\mathfrak{k}}=\frac{1}{4}\sum_{j=1}^{\dim\mathfrak{k}}\sum_{i=1}^{\dim\mathfrak{k}}c\left([e_{j},e_{i}]\right)\cdot
c(e_{i})c(e_{j})=\sum_{i=1}^{\dim\mathfrak{k}}\text{ad}^{\mathfrak{k}}(e_{j})\cdot
c(e_{j})$

and a _torsion term_ by

$\Omega=\frac{1}{4}\sum_{j=1}^{\dim\mathfrak{k}}\sum_{i=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}c\left([e_{j},e_{i}]\right)\cdot
c(e_{i})c(e_{j})=\sum_{i=1}^{\dim\mathfrak{k}}\text{ad}^{\mathfrak{p}}(e_{j})\cdot
c(e_{j})$

By straightforward computation, we check that

$\frac{1}{4}\sum_{j=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}\sum_{i=1}^{\dim\mathfrak{k}}c\left([e_{j},e_{i}]\right)\cdot
c(e_{i})c(e_{j})=\Omega$

and

$\displaystyle\frac{1}{4}\sum_{j=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}\sum_{i=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}c\left([e_{j},e_{i}]\right)\cdot
c(e_{i})c(e_{j})=\Omega.$

Thus, (A.1) can be simplified as follow

$D_{G}=\left(\sum_{i=1}^{\dim\mathfrak{k}}e_{i}\otimes
c(e_{i})+\frac{1}{2}\phi^{\mathfrak{k}}\right)-\sum_{i=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}e_{i}\otimes
c(e_{i})+\frac{3}{2}\Omega.$

If we identify

$L^{2}(G,S_{\mathfrak{g}})\cong\left[L^{2}(G)\otimes S_{\mathfrak{p}}\otimes
L^{2}(K,S_{\mathfrak{k}})\right]^{K},$

the split Dirac operator is given by

$\displaystyle D_{G,\text{split}}=$ $\displaystyle D_{G,K}\otimes 1+1\otimes
D_{K}$ $\displaystyle=$
$\displaystyle-\sum_{i=\dim\mathfrak{k}+1}^{\dim\mathfrak{g}}e_{i}\otimes
c(e_{i})+\left(\sum_{i=1}^{\dim\mathfrak{k}}e_{i}\otimes
c(e_{i})+\frac{1}{2}\phi^{\mathfrak{k}}\right)$

Therefore, the difference between the Dirac operator associated to the
Clifford connection and split Dirac operator is given by the torsion term
$\frac{3}{2}\cdot\Omega$ (see also [BGV, Theorem 10.38]).

## References