$\mathfrak{O}^{\prime}\neq\mathfrak{O}$. From the formula (6.13),
$\hat{\delta}_{i}$ is a multiple of the characteristic function of
$-p_{F}^{-2i}Y+M_{2}(P_{F}^{-i})$ and from Lemma 6.11 the nilpotent elements
there form the $K_{i}$-orbit of $p_{F}^{-2i}Y$. It follows that
$I_{\mathfrak{O}^{\prime}}(\hat{\delta}_{i})=0$ if
$\mathfrak{O}^{\prime}\neq\mathfrak{O}$. That
$I_{\mathfrak{O}}(\hat{\delta}_{i})=1$ is proved exactly as in the proof of
Lemma 7 in [Varma14].

#### 6.2.4.

For a locally profinite space $X$, $x\in X$, and a field $C$, two linear forms
$f,f^{\prime}$ on $C_{c}^{\infty}(V;C)$ for some open neighbourhood $V$ of $x$
in $X$, are called equivalent if their restrictions to $C_{c}^{\infty}(W;C)$
for some open neighbourhood $W$ of $x$ contained in $V$ are equal. The
equivalence class of $f$ is called its germ $\tilde{f}$ at $x$. Denote
$\mathfrak{G}_{x}(X)$ the space of the germs at $x$.

For a locally profinite space $X^{\prime}$, an open subset $W$ in $X$ and an
open subset $W^{\prime}$ in $X^{\prime}$, an homeomorphism $j:W\to W^{\prime}$
gives by functoriality an isomorphism $C_{c}^{\infty}(W^{\prime};C)\to
C_{c}^{\infty}(W;C)$ and an isomorphism
$\mathfrak{G}_{j(x)}(X^{\prime})\to\mathfrak{G}_{x}(X)$ from the space of the
germs of $X^{\prime}$ at $j(x)$ to the space of the germs of $X$ at $x\in W$.

The nilpotent orbital integrals $\mathcal{F}_{\mathfrak{O}}:\varphi\mapsto
I_{\mathfrak{O}}(\hat{\varphi})$ for $\varphi\in
C_{c}^{\infty}(\operatorname{Lie}H;\mathbb{C})$, and the nilpotent $H$-orbits
$\mathfrak{O}$ in $\operatorname{Lie}(H)$, are linearly independent
$H$-equivariant linear forms on
$C_{c}^{\infty}(\operatorname{Lie}H;\mathbb{C})$. They form a basis of a
$\mathbb{Z}$-module $I_{H}$ with rank $1+2^{d}$ (6.3). For each
$H$-equivariant open neighborhood $V$ of $0$ in $\operatorname{Lie}H$, the
$\mathcal{F}_{\mathfrak{O}}$ remain independent as linear forms on
$C_{c}^{\infty}(V;\mathbb{C})$. The germs $\tilde{\mathcal{F}}_{\mathfrak{O}}$
form a basis of the $\mathbb{Z}$-module $\tilde{I}_{H}$ of germs of elements
of $I_{H}$. Denote by $I_{H}^{Wh}$ the $\mathbb{Z}$-submodule of $I_{H}$ of
basis $\mathcal{F}_{\mathfrak{O}}$ for $\mathfrak{O}\neq 0$.

Theorems 6.6 and 6.9 say that the germ at $1$ of the trace of an irreducible
complex smooth representation $\pi$ of $H$ identifies via the map $X\to 1+X$
with the germ at $0$ of a unique element
$T_{\pi}=c_{0}(\pi)\mathcal{F}_{0}+T_{\pi}^{Wh}$ where
$c_{0}(\pi)\in\mathbb{Q}$, and $T_{\pi}^{Wh}\in I_{H}^{Wh}$ is determined by
the non-degenerate Whittaker models of $\pi$. Note that $T_{\pi}^{Wh}=0$ if
and only if $\dim_{\mathbb{C}}\pi=1$.

Denote by $T_{H}^{Wh}$ the $\mathbb{Z}$-submodule of $I_{H}^{Wh}$ generated by
respectively the $T_{\pi}^{Wh}$, for all irreducible complex smooth
representations $\pi$ of $H$. Write $\tilde{I}_{H}^{Wh},\tilde{T}_{H}^{Wh}$
for the space of germs at $0$ of $I_{H}^{Wh},T_{H}^{Wh}$.

######  Theorem 6.14.

We have $\tilde{T}_{H}=\tilde{I}_{H}$ when $d=0,1$.

The $\mathbb{Z}$-submodule $\tilde{T}_{H}^{Wh}$ is a submodule of
$\tilde{I}_{H}^{Wh}$ of finite index. The exponent of
$\tilde{I}_{H}^{Wh}/\tilde{T}_{H}^{Wh}$ is $2^{d-2}$ when $d\geq 2$.

###### Proof.

When $d=0$, then $I_{H}$ has $\mathbb{Z}$-rank $2$, and the germs of the
traces of the trivial representation $1$ and of the Steinberg representation
$\operatorname{st}_{H}$ form a $\mathbb{Z}$-basis
$\\{\tilde{\operatorname{tr}}_{1},\tilde{\operatorname{tr}}_{\operatorname{st}_{H}}\\}$
of $\tilde{I}_{H}$.

When $d=1$, then $I_{H}$ has $\mathbb{Z}$-rank $3$, $\det H=N_{E/F}(E^{*})$
for a quadratic separable extension $E/F$ (Remark 6.3), the principal series
$(i_{B}^{G}\eta_{E})|_{H}$ is semi-simple of length $2$ and multiplicity free
(Lemma 2.3 and footnote in the proof of Proposition 4.27), and the germs of
the traces of the trivial representation $1$ and of the two components
$\pi_{E}^{+},\pi_{E}^{-}$ of $(i_{B}^{G}\eta_{E})|_{H}$ form a
$\mathbb{Z}$-basis
$\\{\tilde{\operatorname{tr}}_{1},\tilde{\operatorname{tr}}_{\pi_{E}^{+}},\tilde{\operatorname{tr}}_{\pi_{E}^{-}}\\}$
of $\tilde{I}_{H}$.

When $d\geq 2$, the theorem follows from Lemma 6.4. ∎

Theorem 6.14 can be equally well expressed in terms of the Grothendieck group
$\operatorname{Gr}_{R}(H)$. This is under this form that the theorem extends
to $R$-representations. For an open compact subgroup $K$ of $H$, and $\pi$ a
finite length smooth complex representation $\pi$ of $H$, $\pi|_{K}$ is semi-
simple wifh finite multiplicities, and is determined by the restriction of the
trace of $\pi$ to $C^{\infty}_{c}(K,\mathbb{C})$.

######  Corollary 6.15.

There are $2^{d}$ virtual finite length smooth complex representations
$\pi_{1},\ldots,\pi_{2^{d}}$ of $H$ with the following property: for any
finite length smooth complex representation $\pi$ of $H$, there are unique
integers $a_{0}(\pi),a_{1}(\pi),\ldots,a_{2^{d}}(\pi)$, such that on some
compact open subgroup $K=K(\pi)$ of $H$,

$\pi\simeq a_{0}(\pi)1+\sum_{i=1}^{2^{d}}a_{i}(\pi)\pi_{i}.$

###### Proof.

By Theorem 6.14, the $\mathbb{Z}$-module $\tilde{T}_{H}^{Wh}$ has a basis
$\\{\tilde{T}_{\pi_{1}}^{Wh},\ldots,\tilde{T}_{\pi_{2^{d}}}^{Wh}\\}$ where
$\pi_{1},\ldots,\pi_{2^{d}}$ are virtual finite length smooth representations
of $H$. By Theorem 6.6, for any finite length smooth representation $\pi$ of
$H$ there exist a unique rational number $a_{0}(\pi)$ and unique integers
$a_{1}(\pi),\ldots,a_{2^{d}}(\pi)$, such that

$\operatorname{tr}_{\pi}=a_{0}(\pi)\operatorname{tr}_{1}+\sum_{i=1}^{2^{d}}a_{i}(\pi)\operatorname{tr}_{\pi_{i}}$

on restriction to $C^{\infty}_{c}(K(\pi),\mathbb{C})$ for some compact open
subgroup $K(\pi)$ of $H$. As
$a_{0}(\pi)=\dim_{\mathbb{C}}\pi^{K(\pi)}-\sum_{i=1}^{2^{d}}a_{i}(\pi)\dim_{\mathbb{C}}\pi_{i}^{K(\pi)},$
we see that $a_{0}(\pi)$ is an integer. Equivalently, on restriction to
$K(\pi)$,

$\pi\simeq a_{0}(\pi)1+\sum_{i=1}^{2^{d}}a_{i}(\pi)\pi_{i}.$

∎

#### 6.2.5.

This has consequences for the representations of $G^{\prime}$.

An irreducible complex representation of $G^{\prime}$ extends to
$ZG^{\prime}$, and we can apply Theorem 6.6 to $H=ZG^{\prime}$ when
$\operatorname{char}_{F}\neq 2$. When $p$ is odd, there is an unique
$L$-packet $\tau_{1},\tau_{2},\tau_{3},\tau_{4}$ of $G^{\prime}$ with $4$
elements (Proposition 4.23). One can enumerate the $4$ non-trivial nilpotent
$G^{\prime}$-orbits $\mathfrak{O}_{1},\ldots,\mathfrak{O}_{4}$ such that
$c_{\mathfrak{O}_{i}}(\tau_{j})=\begin{cases}1\ \text{if}\ i=j\cr 0\
\text{if}\ i\neq j\end{cases}$. For $i=1,\ldots,4$ we choose a lower
triangular element $Y_{i}\in\mathfrak{O}_{i}$.

######  Theorem 6.16.

($p$ odd, $R=\mathbb{C}$) Let $\pi$ be a finite length smooth complex
representation of $G^{\prime}$. On restriction to a small enough compact open
subgroup $K(\pi)$ of $G^{\prime}$, we have

(6.14) $\pi\simeq
a_{0}(\pi)1+\sum_{i=1}^{4}c_{\mathfrak{O}_{i}}(\pi)\tau_{i},\ \
c_{\mathfrak{O}_{i}}(\pi)=\dim_{\mathbb{C}}W_{Y_{i}}(\pi),$

where
$a_{0}(\pi)=\dim_{\mathbb{C}}\pi^{K(\pi)}-\sum_{i=1}^{4}c_{\mathfrak{O}_{i}}(\pi)\dim_{\mathbb{C}}\tau_{i}^{K(\pi)}$.
The constant term in Theorem (6.6) is

$c_{0}(\pi)=a_{0}(\pi)-(\sum_{i=1}^{4}c_{\mathfrak{O}_{i}}(\pi))/2.$

The constant term $c_{0}(\pi)$ can be computed using (6.7) and (6.8).

######  Remark 6.17.

When $\operatorname{char}_{F}=0,p$ odd and $R=\mathbb{C}$, the theorem was
known ([Assem94] and the last section of [Nevins23]).

#### 6.2.6.

For any $p$, let $\pi$ be an irreducible smooth complex representation of
$G^{\prime}$ and $r$ the cardinality of the $L$-packet of $\pi$.

For any $L$-packet $\\{\tau_{1},\tau_{2},\tau_{3},\tau_{4}\\}$ of size $4$,
there exist integers $a_{0},b_{0}$ such that on a small enough compact open
subgroup $K$ of $G^{\prime}$ we have

(6.15) $\operatorname{ind}_{B^{\prime}}^{G^{\prime}}1\simeq
b_{0}T_{1}+\sum_{i=1}^{4}\tau_{i},\ \ \ \text{and if $r=1$, \ }\ \pi\simeq
a_{0}T_{1}+\sum_{i=1}^{4}\tau_{i}.$

If $r=2$, then $\det(G_{\pi})=N_{E/F}(E^{*}/F)$ for a quadratic separable
extension $E/F$. Choose a bi-quadratic separable extension of $F$ containing
$E$. There exist $\tau_{1}$ and $\tau_{2}$ in the associated $L$-packet of
size $4$ (Proposition 4.23) and an integer $a_{0}$ such that on a small enough
compact open subgroup $K$ of $G^{\prime}$ we have

(6.16) $\ \pi\simeq a_{0}T_{1}+\sum_{i=1}^{2}\tau_{i}.$

Therefore, when $R=\mathbb{C}$ we have:

######  Theorem 6.18.

Let $\pi$ be an irreducible smooth $R$-representation of $G^{\prime}$. There
are an integer $a_{0}$ and irreducible smooth $R$-representations
$\\{\tau_{1},\tau_{2},\tau_{3},\tau_{4}\\}$ of $G^{\prime}$ forming an
$L$-packet, such that on a small enough compact open subgroup $K$ of
$G^{\prime}$ we have

$\pi\simeq a_{0}1+\sum_{i=1}^{4/r}\tau_{i},$

where $r$ is the cardinality of the $L$-packet containing $\pi$.

#### 6.2.7.

Let us prove Theorem 6.18 for any $R$.

Put $R_{c}=\mathbb{Q}^{ac}$ when $\operatorname{char}_{R}=0$ and
$R_{c}=\mathbb{F}_{\ell}^{ac}$ when $\operatorname{char}_{R}=\ell>0$.

a) We show first that Theorem 1.8 for $R_{c}$ extends to $R$. A cuspidal
$R$-representation of $G^{\prime}$ is the scalar extension
$\pi_{R}=R\otimes_{R_{c}}\pi$ to $R$ of a cuspidal $R_{c}$-representation
$\pi$ of $G^{\prime}$ [Vignéras96] and the $L$-packets of size $4$ are
cuspidal. The scalar extension from $R_{c}$ to $R$ respects irreducibility,
identifies the $L$-packets of size $4$ over $R_{c}$ with those over $R$ and
sends the $L$-packets of size $r$ over $R_{c}$ to $L$-packets of size $r$ over
$R$. Theorem 1.8 for $R_{c}$-representations imply Theorem 1.8 extends for
$R$-representations which are scalar extensions of $R_{c}$-representations:

$\pi\simeq a_{0}1+\sum_{i=1}^{4/r}\tau_{i}\ \ \text{implies by scalar
extension}\ \ \pi_{R}\simeq a_{0}1+\sum_{i=1}^{4/r}\tau_{i,R}.$

The only irreducible smooth $R$-representations ot $G^{\prime}$ which are not
scalar extensions of $R_{c}$-representations, are principal series
$i_{B^{\prime}}^{G^{\prime}}(\eta)$. But

(6.17)
$i_{B^{\prime}}^{G^{\prime}}(\eta)\simeq\operatorname{ind}_{B^{\prime}}^{G^{\prime}}(1)\
\ \text{on some small open compact subgroup $K$ of $G^{\prime}$},$

and we have (6.15) for the $R_{c}$-representation
$\operatorname{ind}_{B^{\prime}}^{G^{\prime}}(1)$. Therefore, for any
$L$-packet $\\{\tau_{1,R},\tau_{2,R},\tau_{3,R},\tau_{4,R}\\}$ of size $4$,
there is an integer $a_{0}$ such that

$\operatorname{ind}_{B^{\prime}}^{G^{\prime}}(1)\simeq
a_{0}1+\sum_{i=1}^{4}\tau_{i,R}\ \ \ \text{on some small open compact subgroup
$K$ of $G^{\prime}$}.$

b) Theorem 6.18 for $\mathbb{C}$ extends to $\mathbb{Q}^{ac}$ because the
scalar extension from $\mathbb{Q}^{ac}$ to $\mathbb{C}$ respects
irreducibility, representations in an $L$-packet of size $4$ are cuspidal, and
cuspidal representations complex representations of $G^{\prime}$ are defined
over $\mathbb{Q}^{ac}$.

c) Via an isomorphism $\mathbb{C}\simeq\mathbb{Q}_{\ell}^{ac}$, Theorem 1.8
for $\mathbb{C}$ extends to $\mathbb{Q}_{\ell}^{ac}$. Theorem 6.18 for
$\mathbb{Q}_{\ell}^{ac}$ extends to $\mathbb{F}_{\ell}^{ac}$-representations.
Indeed, from Proposition 4.31 an irreducible smooth
$\mathbb{F}_{\ell}^{ac}$-representation $\pi$ of $G^{\prime}$ in an $L$-packet
of size $r$, lifts to an integral irreducible smooth
$\mathbb{Q}_{\ell}^{ac}$-representation $\tilde{\pi}$ of $G^{\prime}$ in an
$L$-packet of size $r$ (Proposition 1.6). From Theorem 6.18 for
$\mathbb{Q}_{\ell}^{ac}$, there is an $L$-packet
$\\{\tilde{\tau}_{1},\tilde{\tau}_{2},\tilde{\tau}_{3},\tilde{\tau}_{4}\\}$ of
irreducible smooth $\mathbb{Q}_{\ell}^{ac}$-representations of $G^{\prime}$
and an integer $a_{0},$ such that on a small enough compact open subgroup $K$
of $G^{\prime}$ we have

$\tilde{\pi}\simeq a_{0}1+\sum_{i=1}^{4/r}\tilde{\tau}_{i}.\ \ \text{By
reduction modulo $\ell$}\ \ \pi\simeq a_{0}1+\sum_{i=1}^{4/r}\tau_{i},$

where the reduction $\\{\tau_{1},\tau_{2},\tau_{3},\tau_{4}\\}$ modulo $\ell$
of $\\{\tilde{\tau}_{1},\tilde{\tau}_{2},\tilde{\tau}_{3},\tilde{\tau}_{4}\\}$
forms an $L$-packet of irreducible smooth
$\mathbb{F}_{\ell}^{ac}$-representations of $G^{\prime}$. This ends the proof
of Theorem 1.8.

######  Remark 6.19.

From the proof, the recipes giving (6.15) and (6.16) remain valid for $R$.

#### 6.2.8.

Assume $p$ odd and $R=\mathbb{C}$. Let $x$ be a vertex of the Bruhat-Tits
building of $G^{\prime}$. In [Nevins23], are defined admissible complex
representations $\tau_{x,1},\ldots\tau_{x,5}$ of the maximal open compact
subgroup $G^{\prime}_{x}$ fixing $x$ such that the following is true. Let
$\pi$ be an irreducible smooth complex representation of $G^{\prime}$ of depth
$r_{\pi}$ in the sense of Moy-Prasad. Then, there are integers
$a_{\pi,1},\ldots,a_{\pi,5}$ such that on restriction to
$G^{\prime}_{x,r_{\pi+}}$,

$\pi\simeq\sum_{i=1}^{5}a_{\pi,i}\tau_{x,i}.$

Now allow any $R$ with $\operatorname{char}_{R}\neq p$ and $p$ odd. The
representations $\tau_{x,i}$ of Nevins transfered to $\mathbb{Q}_{\ell}^{ac}$
are integral, defined over $\mathbb{Q}^{ac}$ and can be transfered to
$R$-representations $\tau_{x,i,R}$. The proof in §6.2.7 applies and shows that
the above result is also valid over $R$ with
$\tau_{x,1,R},\ldots\tau_{x,5,R}$.

##  7\. Appendix - The finite group $SL_{2}(\mathbb{F}_{q})$

Let $k$ be a finite field of characteristic $p$ with $q$ elements. In this
appendix we classify irreducible representations of $G=GL_{2}(k)$ and of
$G^{\prime}=SL_{2}(k)$ over an algebraically closed field $R$ of
characteristic $0$ or $\ell>0,\ \ell\neq p$. We could use [Bonnafé11] for
$\operatorname{char}_{R}\neq 2$ and [Kleshchev-Tiep09] for any $R$, but we
prefer using the same methods as in the main text.

Note that the irreducible $R$-representations of the finite groups $G$ and
$G^{\prime}$ are defined over the algebraic closure $R_{c}$ of the prime
field, and we can freely pass from $R$ to any other algebraically closed field
of the same characteristic. Thus it is enough to consider the cases where
$R=\mathbb{C}$ or $R=\mathbb{F}_{\ell}^{ac}$. We aim also to prove the
following theorem.

######  Theorem 7.1.

Any irreducible $\mathbb{F}_{\ell}^{ac}$ representation $\sigma$ of
$GL_{2}(k)$ is the reduction modulo $\ell$ of a
$\mathbb{Q}_{\ell}^{ac}$-representation $\tilde{\sigma}$ of $GL_{2}(k)$ such
that $\tilde{\sigma}|_{SL_{2}(k)}$ and $\sigma|_{SL_{2}(k)}$ have the same
length.

Any irreducible $\mathbb{F}_{\ell}^{ac}$-representation of $SL_{2}(k)$ is the
reduction modulo $\ell$ of a $\mathbb{Q}_{\ell}^{ac}$-representation of
$SL_{2}(k)$.

Write $Z$ for the centre of $G$, $B$ for the upper triangular subgroup of $G$,
and $U$ for its unipotent radical. Let us first recall the known
classification of the $R$-representations of $G$ ([Bushnell-Henniart02] for
$R=\mathbb{C}$ and [Vignéras88] for $R=\mathbb{F}_{\ell}^{ac}$).

The parabolically induced representation $\operatorname{ind}_{B}^{G}(1)$
realised by the space of constant functions on $B\backslash G$, contains the
trivial character. It also has the trivial character as a quotient, given by
the functional $\lambda$ which sums the values of functions on $B\backslash
G$. The map from the trivial subrepresentation to the trivial quotient is
multiplication by $q+1$, so is an isomorphism if $\ell$ does not divide $q+1$,
and is $0$ otherwise. In the first case the quotient
$\operatorname{St}=\operatorname{ind}_{B}^{G}(1)/1$ is irreducible, in the
second case $Ker(\lambda)/1$ is a cuspidal but not supercuspidal
representation $\sigma_{0}$ of $G$.

The irreducible (classes of) $R$-representations $\sigma$ of $G$ are :

1) The characters $\chi\circ\det$ where $\chi$ is an $R$-character of $k^{*}$.

2) When $q+1\neq 0$ in $R$, the twists
$(\chi\circ\det)\otimes\operatorname{St}$ of $\operatorname{St}$ by the
$R$-characters $\chi\circ\det$ of $G$.

2’) When $q+1=0$ in $R$, the twists $(\chi\circ\det)\otimes\sigma_{0}$ of
$\sigma_{0}$ by the $R$-characters $\chi\circ\det$ of $G$.

3) The irreducible principal series
$\operatorname{ind}_{B}^{G}(\chi_{1}\otimes\chi_{2})$, where $\chi_{1}$ and
$\chi_{2}$ are two distinct $R$-characters of $k^{*}$.

4) The supercuspidal representations $\sigma(\theta)$, where $\theta$ is an
$R$-character of $k_{2}^{*}$, $\theta\neq\theta^{q}$, where $k_{2}/k$ is a
quadratic extension.

The only isomorphisms between those representations are given by exchanging
$\chi_{1}$ and $\chi_{2}$ in 3), $\theta$ and $\theta^{q}$ in 4).

Twisting by an $R$-character $\chi\circ\det$ of $G$ has the obvious effect,
for example sending $\theta$ to $(\chi\circ N)\theta$ where $N(x)=x^{q+1}$ for
$x\in k_{2}^{*}$ in 4).

Any irreducible $R$-representation $\tau$ of $G^{\prime}$ is contained in the
restriction $\sigma|_{G^{\prime}}$ to $G^{\prime}$ of an irreducible
$R$-representation $\sigma$ of $G$. The representation $\sigma|_{G^{\prime}}$
is semi-simple of multiplicity $1$ and its irreducible components are
$G$-conjugate. The stabilizer of $\tau$ contains $ZG^{\prime}$ and
$G/ZG^{\prime}$ is isomorphic to $k^{*}/(k^{*})^{2}$. We have
$|k^{*}/(k^{*})^{2}|=1$ when $p=2$ and $|k^{*}/(k^{*})^{2}|=2$ when $p$ is
odd. Therefore $\sigma|_{G^{\prime}}$ is irreducible when $p=2$ and
$\sigma|_{G^{\prime}}$ has length $1$ or $2$ when $p$ is odd.

When $\operatorname{char}_{R}\neq 2$, the length $\lg(\sigma|_{G^{\prime}})$
of $\sigma|_{G^{\prime}}$ is the number of $R$-characters $\chi$ of $k^{*}$
such that $(\chi\circ\det)\otimes\sigma\simeq\sigma$, so

(7.1) $\lg(\sigma|_{G^{\prime}})=\begin{cases}2\ &\text{ in case 3) if
$(\chi_{1}/\chi_{2})^{2}=1$ and in case 4) if $(\theta^{q-1})^{2}=1$}\\\ 1\
&\text{ otherwise}\end{cases}.$

The restrictions $\sigma_{1}|_{G^{\prime}},\sigma_{2}|_{G^{\prime}}$ of two
irreducible representations $\sigma_{1},\sigma_{2}$ of $G$ are isomorphic if
and only $\sigma_{1},\sigma_{2}$ are twists of each other by an $R$-character
of $G$. Otherwise $\sigma|_{G^{\prime}},\sigma_{2}|_{G^{\prime}}$ are
disjoint. So, we have a classification of the (isomorphism classes of)
irreducible representations of $G^{\prime}$ when $\operatorname{char}_{R}\neq
2$.

######  Remark 7.2.

The restriction to $B$ of a cuspidal representation of $G$ is the Kirillov
representation $\kappa$ of $B$ (the irreducible $R$-representation of B
induced by any non-trivial $R$-character of $U$). The restriction of $\kappa$
to $U$ is the direct sum of all non-trivial $R$-characters of $U$. The group
$B$ acts transitively on such characters, whereas $B^{\prime}=B\cap
G^{\prime}$ acts with two orbits. It follows that the restriction of $\kappa$
to $B^{\prime}$ has two inequivalent irreductible components. Consequently a
cuspidal representation of $G$ restricts to $G^{\prime}$ with length $1$ or
$2$.

Let $\ell$ be an odd prime number different from $p$. Let us consider the
reduction modulo $\ell$ of the previous irreducibles $\sigma$ over
$\mathbb{Q}_{\ell}^{ac}$ (since $G$ is finite they are integral). For an
integral $\mathbb{Q}_{\ell}^{ac}$-character $\chi$ (with values in
$\mathbb{Z}_{\ell}^{ac}$) let $\overline{\chi}$ denote its reduction modulo
$\ell$. Reduction modulo $\ell$ is compatible with twisting by a
$\mathbb{Q}_{\ell}^{ac}$-character $\chi\circ\det$ in the sense that the
reduction of $(\chi\circ\det)\otimes\sigma$ is the twist by
$\overline{\chi}\circ\det$ of the reduction of $\sigma$.

1) The trivial $\mathbb{Q}_{\ell}^{ac}$-character of $G$ reduces to the
trivial $\mathbb{F}_{\ell}^{ac}$-character.

2) When $\ell$ does not divide $q+1$, the Steinberg
$\mathbb{Q}_{\ell}^{ac}$-representation reduces to the Steinberg
$\mathbb{F}_{\ell}^{ac}$-representation.

2’) When $\ell$ divides $q+1$, the Steinberg
$\mathbb{Q}_{\ell}^{ac}$-representation reduces to a length $2$ representation
with subrepresentation $\sigma_{0}$ and trivial quotient (for the natural
integral structure).

3) The irreducible principal series
$\operatorname{ind}_{B}^{G}(\chi_{1}\otimes\chi_{2})$ reduces to the
irreducible principal series
$\operatorname{ind}_{B}^{G}(\overline{\chi}_{1}\otimes\overline{\chi}_{2})$
when $\overline{\chi}_{1}\neq\overline{\chi}_{2}$, and to
$(\overline{\chi}_{1}\circ\det)\otimes\operatorname{ind}_{B}^{G}(1)$ (of
length $2$ when $\ell$ does not divide $q+1$, and length $3$ otherwise) when
$\overline{\chi}_{1}=\overline{\chi}_{2}$ (for the natural integral
structure).

4) The supercuspidal $\mathbb{Q}_{\ell}^{ac}$-representation $\sigma(\theta)$,
reduces to the supercuspidal $\mathbb{F}_{\ell}^{ac}$-representation
$\sigma(\overline{\theta})$ if
$\overline{\theta}\neq(\overline{\theta})^{q}=\overline{\theta^{q}}$, and
otherwise (which can happen only if $\ell$ divides $q+1$) to
$(\eta\circ\det)\otimes\sigma_{0}$ where $\eta$ is the
$\mathbb{F}_{\ell}^{ac}$-character of $\mathbb{F}_{q}^{*}$ such that
$\eta\circ N=\overline{\theta}$.

A given $\mathbb{F}_{\ell}^{ac}$-character of $k^{*}$ or $k_{2}^{*}$ has a
unique lift to a $\mathbb{Z}_{\ell}^{ac}$-character of the same order, and
from the above it is clear that any irreducible
$\mathbb{F}_{\ell}^{ac}$-representation $\sigma$ of $G$ lifts to a
$\mathbb{Q}_{\ell}^{ac}$-representation. Moreover, one can choose a lift of
$\sigma$ with the same length on restriction to $G^{\prime}$, thus proving the
theorem when $\ell$ is odd.

Let us finally assume $\operatorname{char}_{R}=2$. Then $p$ is odd and $q+1=0$
in $R$. Write $q-1=2^{s}m$ with a positive integer $s$ and an odd integer $m$.
The number of irreducible $R$-representations of $G$ (resp. $ZG^{\prime}$) is
the number of conjugacy classes in $G$ (resp. $ZG^{\prime}$) of elements of
odd order. Let $g\in G$ of odd order. Then $\det(g)\in k^{*}$ has odd order so
$\det(g)\in(k^{*})^{2}$ and $g\in ZG^{\prime}$. The $G$-conjugacy class of $g$
is equal to its $ZG^{\prime}$-conjugacy class unless the $G$-centralizer of
$g$ is entirely in $ZG^{\prime}$. In that exceptional case, the
$G$-equivalence class of $g$ is the union of two $ZG^{\prime}$-equivalence
classes. This happens only when $g=zu$ where $z\in Z$ (of odd order) and
$u\neq 1$ is unipotent. That shows that $m$ is the number of
$ZG^{\prime}$-conjugacy classes of elements of odd order minus the number of
$G$-conjugacy of such elements. Consequently $m$ is the number of irreducible
$R$-representations of $ZG^{\prime}$ minus the number of irreducible
$R$-representations of $G$.

Consider first $\sigma(\theta)$ for a $\mathbb{Q}_{2}^{ac}$-character $\theta$
of $k_{2}^{*}$ of order $2^{s+1}$. Certainly $\overline{\theta}$ is trivial so
that the reduction of $\sigma(\theta)$ modulo $2$ is $\sigma_{0}$. But
$\ell(\sigma(\theta)|_{G^{\prime}})=2$ by (7.1), from which it follows that
$\ell(\sigma_{0}|_{G^{\prime}})\geq 2$. We have seen however that
$\ell(\sigma_{0}|_{G^{\prime}})\leq 2$ (Remark 7.2), so
$\ell(\sigma_{0}|_{G^{\prime}})=2$, and each irreducible component of
$\sigma_{0}|_{G^{\prime}}$ lifts to an irreducible component of
$\sigma(\theta)|_{G^{\prime}}$. The $\mathbb{F}_{2}^{ac}$-characters $\chi$ of
$k^{*}$ have an odd order, their number is $m$, and the representations
$(\chi\circ\det)\otimes\sigma_{0}$ are not equivalent (the order of $\chi$ is
odd). We deduce:

######  Lemma 7.3.

All irreducible $\mathbb{F}_{2}^{ac}$-representations of $G$ restrict
irreducibly to $G^{\prime}$ except the twists of $\sigma_{0}$ by characters.

The reduction modulo $2$ of any supercuspidal
$\mathbb{Q}_{2}^{ac}$-representation of $G^{\prime}$ is irreducible.

We deduce the classification of irreducible $R$-representations of
$G^{\prime}$ when $\operatorname{char}_{R}=2$ and Theorem 7.1 when $\ell=2$.

######  Remark 7.4.

For use in the main text we summarize:

a) When $q+1=0$ in $R$, then $\sigma_{0}|_{SL_{2}(k)}$ is irreducible if
$\operatorname{char}_{R}\neq 2$, and has length $2$ if
$\operatorname{char}_{R}=2$.

b) In 4) let $b\in k_{2}$ be an element of order $q+1$. We have
$\theta\neq\theta^{q}\Leftrightarrow\theta(b)\neq 1$ and
$\sigma(\theta)|_{SL_{2}(k)}$ is irreducible if $\theta^{2}(b)\neq 1$, and has
length $2$ if $\theta^{2}(b)=1$.

When $\operatorname{char}_{R}=2$, or when $p=2$ hence $(2,q+1)=1$, we have
$\theta(b)\neq 1\Leftrightarrow\theta(b^{2})\neq 1$ hence
$\sigma(\theta)|_{SL_{2}(k)}$ is irreducible for all $\theta\neq\theta^{q}$.

When $\operatorname{char}_{R}\neq 2$ and $p$ is odd, there exists $\theta$
such that $\theta(b)\neq 1,\theta(b)^{2}=1$, unique modulo the twist by a
character $\chi$ such that $\chi(b)=1$. The corresponding representations
$\sigma(\theta)$ of $G$ are twists of each other by a character of $G$. Their
restrictions to $SL_{2}(k)$ are isomorphic and reducible of length $2$.

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# Object Aware Egocentric Online Action Detection

Joungbin An Yunsu Park Hyolim Kang Seon Joo Kim
Yonsei University
{joungbinan, ysp7954, hyolimkang<EMAIL_ADDRESS>
###### Abstract

Advancements in egocentric video datasets like Ego4D, EPIC-Kitchens, and Ego-
Exo4D have enriched the study of first-person human interactions, which is
crucial for applications in augmented reality and assisted living. Despite
these advancements, current Online Action Detection methods, which efficiently
detect actions in streaming videos, are predominantly designed for exocentric
views and thus fail to capitalize on the unique perspectives inherent to
egocentric videos. To address this gap, we introduce an Object-Aware Module
that integrates egocentric-specific priors into existing OAD frameworks,
enhancing first-person footage interpretation. Utilizing object-specific
details and temporal dynamics, our module improves scene understanding in
detecting actions. Validated extensively on the Epic-Kitchens 100 dataset, our
work can be seamlessly integrated into existing models with minimal overhead
and bring consistent performance enhancements, marking an important step
forward in adapting action detection systems to egocentric video analysis.

##  1 Introduction

Recent advancements in large-scale egocentric video datasets, such as EPIC-
Kitchens [6, 5], Ego4D [10], and Ego-Exo4D [11], have revolutionized the study
of human interactions and behaviors from a first-person perspective. These
datasets offer a wealth of varied examples of daily activities, fostering
developments in fields like activity recognition, social interaction analysis,
and personal assistant technologies. Such progress promises to enhance user
experiences across diverse applications.

A primary challenge in this domain is the understanding of continuous, long,
fluid video streams, central for applications such as augmented reality and
robotics. These applications require technologies that can efficiently parse
and interpret continuous video feeds, ensuring context-aware and timely
interactions. Consequently, there is an urge for advanced methodologies to
meet the real-time demands of dynamic environments in understanding first-
person view streaming videos.

Figure 1:  Overview of our methodology.

There has been ongoing work in Online Action Detection (OAD) [7], which
involves identifying actions in a streaming video without access to future
frames. However, existing works [7, 21, 22, 1] mainly focus on third-person
view exo-centric videos, as its development has progressed using third-person
view datasets such as THUMOS [12], TVSeries [7], and FineAction [13]. With the
recent developments in egocentric datasets, recent works [15, 11] have
released benchmarks for understanding key steps in streaming egocentric videos
by employing existing OAD models. While these benchmarks have provided
valuable insights, employing existing models specialized in exocentric view
fails to fully exploit the unique attributes of egocentric video that set it
apart from traditional third-person footage.

The main limitation stems from the inherent differences in perspective and
context between exocentric and egocentric videos. Egocentric videos,
characterized by their first-person viewpoint, necessitate a nuanced, object-
centered understanding of the scene. This is due to the lack of pose
information, a consequence of the camera’s positioning in egocentric setups.
For example, in a cooking scenario, an egocentric video might capture close-up
actions such as the user’s hands slicing vegetables or stirring a pot. In
contrast, an exocentric video could provide a broader view of the user’s full
body and the kitchen layout but fail to detail the fine-grained movements and
object interactions typical of the egocentric perspective. Therefore, models
designed for egocentric videos must prioritize hand-object interactions to
more accurately identify specific actions like chopping, stirring, or pouring.

Recognizing this need, we instantiate an effort to integrate egocentric-
specific priors into existing methodologies, thereby boosting its
effectiveness in interpreting the first-person footage. We specifically focus
on the fact that the perceived objects present in the scene offer a rich
contextual background for understanding fine-grained activity. For instance,
an apple placed on top of the cutting board with a knife immediately signals
the impending action of ‘Cutting Apple.’ Similarly, the interaction patterns
observed with objects commonly found in certain environments—such as utensils,
appliances, and ingredients—can provide important cues for recognizing
specific activities.

As such, we aim to enhance scene object awareness in prior models by
introducing an Object-Aware Module. This module begins by integrating an off-
the-shelf detector [9] with the existing OAD framework. It utilizes learnable
vectors to attend to the detected objects’ outputs, extracting object scene
information. These vectors are further refined by incorporating temporal cues
from the OAD model, creating enriched, object and temporally aware
representations. These can then be processed to determine the impending
action. The overview of our methodology is shown in Fig. 1.

We validate our methodology through extensive experiments on the Epic-
Kitchens-100 dataset using the most recent state-of-the-art OAD models. Due to
its lightweight and versatile design, our Object-Aware Module can be
seamlessly integrated into any existing OAD works with minimal overhead,
notably enhancing the interpretation of egocentric data. Our results
demonstrate consistent performance improvements across various models,
confirming the module’s broad applicability and effectiveness.

##  2 Related Works

Online Action Detection has seen considerable advancements since its
introduction [7]. Recent developments primarily fall into two categories:
traditional RNN variants [7, 20, 1] and transformer-based models [21, 22, 17].
Notably, RNN-based approaches have recently been revisited [1], demonstrating
state-of-the-art performance with high efficiency. This resurgence is
attributed to innovations in minimal RNN architectures and the adoption of
specialized loss functions tailored for streaming video processing.

However, the majority of advancements have predominantly focused on
traditional third-person view datasets. Our work diverges by extending OAD
systems beyond their standard third-person constraints. We introduce a plug-
and-play module that enhances object awareness, a crucial element in
interpreting first-person footage. This initiative not only advances the
application of OAD to egocentric videos but also lays the groundwork for
future research in egocentric video analysis.

##  3 Methodology

We present a method to augment existing OAD models in interpreting the first-
person footage by making them aware of the objects in the scene. The key to
this lies in extracting object information in the scene (Sec. 3.1), adequately
integrating object information and temporal cues (Sec. 3.2), and refining the
updated pieces of information to detect actions more accurately (Sec. 3.3).

###  3.1 Extracting Object Information

Recognizing that scene awareness of objects provides informative contextual
information for understanding actions, we initiate our process by extracting
object data from the scene. Drawing inspiration from notable works in the
field [14, 8], we employ Faster-RCNN [9]. We apply this detector to the last
frame of every video snippet inputted into our model. From the output of $K$
detected bounding boxes, we aggregate their confidence scores corresponding to
their category. This results in a vector $f\in\mathbb{R}^{1\times C}$ where
$C$ is the number of classes. Each value in $C$ will indicate the likeliness
at which the object is present in the scene. This object representation
effectively encodes the objects present in the scene and as we validate in the
experimental section, enriches scene comprehension.

Figure 2:  Object-Aware Module.

###  3.2 Object-Aware Module

We introduce an Object-Aware Module designed to effectively integrate object
information with temporal cues. As illustrated in Fig. 2, our module is
composed of two transformer layers. In the first layer, learnable query
vectors and object information—extracted as detailed in Sec. 3.1—interact to
update the query vectors. These vectors are then processed through the second
transformer layer, where they undergo cross-attention with temporal cues
captured by the OAD model. Subsequently, the updated queries pass through a
feed-forward layer, preparing them for action classification. The first layer
enhances the query vectors’ sensitivity to scene-specific object information,
fostering object awareness. The second layer employs cross-attention to merge
this object information with temporal cues, crucial for accurately identifying
ongoing actions.

Despite its straightforward design and implementation, the module is effective
in enhancing action identification from the first-person footage as validated
in Section 4.3. This efficacy is attributable to the query-propagating design,
which has demonstrated considerable success across various domains, including
object detection [2], instance segmentation [4], and action localization [16].
We have adapted and refined this design for our module, enabling it to serve
as an efficient information bottleneck that effectively encapsulates object
information.

###  3.3 Classifying Actions

Finally, the updated information must be aggregated effectively to enhance
action detection accuracy. The Object-Aware Module generates $N$ learnable
query vectors, capturing both object and temporal data essential for action
classification. A global max pooling operation then aggregates these vectors
to consolidate the $N$ vectors into a single vector. This consolidated vector
is subsequently classified using standard classifiers. Given that the Epic-
Kitchens dataset requires the classification of Verbs, Nouns, and Actions
separately, we utilize three distinct classifiers tailored to each category.

##  4 Experimental Results

###  4.1 Dataset

We conduct our experiments using the Epic-Kitchens-100 dataset [6], which
comprises 100 hours of egocentric video featuring 90,000 action segments.
These segments are annotated and categorized into 97 verb classes, 300 noun
classes, and 3,806 action classes. Our experiments follow the training and
validation split outlined by Furnari et al. [8].

###  4.2 Experimental Setup

OAD Models. We employ the open-source implementations of TeSTra [22], MAT
[18], and MiniROAD [1] in their default configurations. Notably, since TeSTra
and MAT implement the MixCLIP data augmentation technique, we also apply
MixCLIP in our use of MiniROAD.

Setting. We follow the standard OAD protocol: videos are processed at 24 fps
and frames are fed into a TSN model [19] pretrained on the Kinetics-400
dataset [3], using a window size and stride of 6. Regarding the module
configuration, we utilize a single block of the Object-Aware Module with 16
learnable queries and an embedding dimension of 1024 within the module. We
project the object information with a single feed-forward layer to match the
dimension of query vectors.

Evaluation Metrics. We follow the dataset setting [6] and use the mean Top-5
Recall to measure Verb, Noun, and Action accuracy.

Method | Overall
---|---
Verb | Noun | Action
TeSTra [22] | 39.7 | 45.6 | 25.1
MAT [17] | 44.5 | 48.3 | 26.3
MiniROAD [1] | 39.2 | 42.7 | 22.7
TeSTra + OA-M | 50.5 (+10.8) | 47.6 (+2.0) | 25.2 (+0.1)
MAT + OA-M | 52.9 (+8.4) | 48.7 (+0.4) | 26.4 (+0.1)
MiniROAD + OA-M | 49.2 (+10.0) | 48.5 (+5.8) | 26.7 (+4.0)

Table 1: Recent OAD models (top) vs. integration of our Object-Aware Module
(bottom).

Object Information | Verb | Noun | Action
---|---|---|---
None (baseline) | 39.2 | 42.7 | 22.7
To Input (with RGB) | 38.6 | 42.5 | 22.6
OA-Module | 49.2 | 48.5 | 26.7

Table 2: Position to input object information.

###  4.3 Results

#### Main Results

The results of applying our methodology to three recent OAD models are
presented in Table 1. We observe consistent improvements in Verb, Noun, and
Action accuracy across all models tested. These findings confirm the
versatility of our Object-Aware Module, demonstrating its capability to be
seamlessly integrated into any existing OAD framework. Importantly, as
validated by improvements in all Verb, Noun, and Action accuracies, it
enhances object awareness, which is crucial for accurately interpreting
egocentric scenes. We observe the most significant improvements in Verb
accuracy, as identifying the object of interaction provides important clues
about the impending action, such as opening, squeezing, cutting, etc. However,
gains in Action accuracy are less pronounced, which we attribute to the
complex nature of the dataset’s action labels, encompassing over 3,800
distinct classes.

#### How to utilize object information

Table 2 shows the outcomes from various methods of integrating object
information using the state-of-the-art OAD model [1]. The baseline, presented
in the first row, does not include object information. The second row
illustrates results when object information is concatenated with RGB at the
input level. In contrast, the last row highlights our approach using the OA-
Module. These results indicate that naïvely adding object information does not
yield optimal outcomes, emphasizing the critical role of the OA-Module in
effectively leveraging this data.

##  5 Limitations

As this represents the preliminary stage of an ongoing project, our
experimental results are currently somewhat limited. We acknowledge that our
incorporation of object scores for providing object information, as described
in Section 3, could be further improved. We are currently working on using
only the “active” objects that the user is actually interacting with, instead
of all the objects present in the scene. Also, instead of a naïve objectness
score, we are experimenting to incorporate the active object’s spatial
information. We plan to extend this framework to more diverse datasets,
including Ego4D and Ego-Exo4D.

##  6 Conclusion

This work aims to enhance existing Online Action Detection frameworks for more
effective application in egocentric videos by leveraging egocentric priors. We
introduce the Object-Aware Module, a flexible module that can be seamlessly
integrated into any current OAD method. This module utilizes object
information as important contextual cues to predict impending actions better.
Through this, we improve the adaptability and accuracy of OAD systems in
interpreting first-person video data, paving the way for advancements in
applications that require precise real-time analysis.

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# AGILE Observations of the Gravitational Wave Source GW170104

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55affiliationmark:  , A. Ursi33affiliationmark:  , A. Argan33affiliationmark:
, C. Pittori11affiliation<EMAIL_ADDRESS>22affiliationmark:  , I. Donnarumma1515affiliationmark:  , A.
Bulgarelli66affiliationmark:  , F. Fuschino66affiliationmark:  , C.
Labanti66affiliationmark:  , M. Marisaldi1616affiliationmark:  , Y.
Evangelista33affiliationmark:  , G. Minervini33affiliationmark:  , A.
Giuliani77affiliationmark:  , M. Cardillo33affiliationmark:  , F.
Longo99affiliationmark:  , F. Lucarelli11affiliation:
<EMAIL_ADDRESS>22affiliationmark:  , P. Munar-
Adrover1717affiliationmark:  , G. Piano33affiliationmark:  , M.
Pilia88affiliationmark:  , V. Fioretti66affiliationmark:  , N.
Parmiggiani66affiliationmark:  , A. Trois88affiliationmark:  , E. Del
Monte33affiliationmark:  , L.A. Antonelli11affiliation:
<EMAIL_ADDRESS>22affiliationmark:  , G.
Barbiellini99affiliationmark:  , P. Caraveo77affiliationmark:  , P.W.
Cattaneo1010affiliationmark:  , S. Colafrancesco1414affiliationmark:  , E.
Costa33affiliationmark:  1515affiliationmark:  , F.
D’Amico1515affiliationmark:  , M. Feroci33affiliationmark:  , A.
Ferrari1111affiliationmark:  , A. Morselli1212affiliationmark:  , L.
Pacciani33affiliationmark:  , F. Paoletti1818affiliationmark:
33affiliationmark:  , A. Pellizzoni88affiliationmark:  , P.
Picozza1212affiliationmark:  , A. Rappoldi1010affiliationmark:  , S.
Vercellone1313affiliationmark:  1ASI Space Science Data Center (SSDC), via del
Politecnico, I-00133 Roma, Italy 2INAF-OAR, via Frascati 33, I-00078 Monte
Porzio Catone (Roma), Italy 3INAF-IAPS, via del Fosso del Cavaliere 100,
I-00133 Roma, Italy 4Dipartimento di Fisica, Università di Roma “Tor Vergata”,
via della Ricerca Scientifica 1, I-00133 Roma, Italy 5Gran Sasso Science
Institute, viale Francesco Crispi 7, I-67100 L’Aquila, Italy 6INAF-IASF-
Bologna, via Gobetti 101, I-40129 Bologna, Italy 7INAF-IASF Milano, via
E.Bassini 15, I-20133 Milano, Italy 8INAF, Osservatorio Astronomico di
Cagliari, via della Scienza 5, I-09047 Selargius (CA), Italy 9Dip. di Fisica,
Università di Trieste and INFN, via Valerio 2, I-34127 Trieste, Italy 10INFN-
Pavia, via Bassi 6, I-27100 Pavia, Italy 11CIFS, c/o Physics Department,
University of Turin, via P. Giuria 1, I-10125, Torino, Italy 12INFN Roma Tor
Vergata, via della Ricerca Scientifica 1, I-00133 Roma, Italy 13INAF,
Osservatorio Astronomico di Brera, via Emilio Bianchi 46, I-23807 Merate (LC),
Italy 14University of Witwatersrand, Johannesburg, South Africa 15ASI, via del
Politecnico snc, I-00133 Roma, Italy 16Birkeland Centre for Space Science,
Department of Physics and Technology, University of Bergen, Norway 17Unitat de
Física de les Radiacions, Departament de Física, and CERES-IEEC, Universitat
Autònoma de Barcelona, E-08193 Bellaterra, Spain  18East Windsor RSD, 25A
Leshin Lane, Hightstown, NJ 08520, USA

###### Abstract

The LIGO/Virgo Collaboration (LVC) detected on 2017 January 4 a significant
gravitational-wave (GW) event (now named GW170104). We report in this Letter
the main results obtained from the analysis of hard X-ray and gamma-ray data
of the AGILE mission that repeatedly observed the GW170104 localization region
(LR). At the LVC detection time $T_{0}$ AGILE observed about 36% of the LR.
The gamma-ray imaging detector did not reveal any significant emission in the
energy range 50 MeV–30 GeV. Furthermore, no significant gamma-ray transients
were detected in the LR that was repeatedly exposed over timescales of
minutes, hours, and days. We also searched for transient emission using data
near $T_{0}$ of the omnidirectional detector MCAL operating in the energy band
0.4–100 MeV. A refined analysis of MCAL data shows the existence of a weak
event (that we call “E2”) with a signal-to-noise ratio of 4.4 $\sigma$ lasting
about 32 ms and occurring $0.46$ $\pm\,0.05\,\rm s$ before $T_{0}$. A study of
the MCAL background and of the false-alarm rate of E2 leads to the
determination of a post-trial significance between 2.4$\,\sigma$ and
2.7$\,\sigma$ for a temporal coincidence with GW170104. We note that E2 has
characteristics similar to those detected from the weak precursor of GRB
090510. The candidate event E2 is worth consideration for simultaneous
detection by other satellites. If associated with GW170104, it shows emission
in the MeV band of a short burst preceding the final coalescence by 0.46 s and
involving $\sim 10^{-7}$ of the total rest mass energy of the system.

gravitational waves, gamma rays: general.

††slugcomment: Submitted to Astrophysical Journal Letters, May 31, 2017.
Accepted, July 20, 2017.

##  1 Introduction

The LIGO/Virgo Collaboration (LVC) detected a gravitational-wave (GW) event on
2017 January 4 The LIGO Scientific Collaboration and Virgo Collaboration
(2017a, b, c). This event, originally labeled G268556, occurred at time
$T_{0}$ = 10:11:$58.599$ UTC; it is now named GW170104 (Abbott et al. 2017,
hereafter A17). The event is identified as a compact binary coalescence (CBC)
of two black holes of masses $\sim 31\,M_{\odot}$ and $\sim 19\,M_{\odot}$.
The estimated distance is $880^{+450}_{-390}\,\rm Mpc$ corresponding to a
redshift of $z\,=\,0.18^{+0.08}_{-0.07}$. GW170104 is an event of great
interest because its “false-alarm rate” (FAR) is less than 1 in 70,000 years
as determined by a refined analysis (A17). This event is the third of a set of
high-significance and confirmed GW events detected by LVC, the first one being
GW150914 and the second one being GW151226 Abbott et al. (2016a, b, c, d, e,
f).

A first sky map of GW170104 was distributed through a LVC-GCN on 2017 January
4th The LIGO Scientific Collaboration and Virgo Collaboration (2017a),
including an initial localization generated by the BAYESTAR pipeline Singer &
Price (2016). An updated sky map was distributed on Jan. 17, 2017 The LIGO
Scientific Collaboration and Virgo Collaboration (2017c), based on
LALInference Veitch et al. (2015). The probability is concentrated in two
long, thin arcs in the sky. The 50% credible region spans about 500 $\rm
deg^{2}$ and the 90% region about 2000 $\rm deg^{2}$. A final sky map was
issued on 2017 May 17 The LIGO Scientific Collaboration and Virgo
Collaboration (2017b). AGILE reacted to the initial LVC notification of
GW170104, and started a quicklook analysis within a few hours as discussed
below Tavani et al. (2017a, b). In the following, we name differently the two
arcs of the GW170104 LR differently: the northern arc is named “A”, and the
southern arc is named “B”.

In this Letter we present the main results of the analysis of AGILE data
concerning GW170104. Sect. 2 summarizes the observation capability of AGILE
for the search of GW event counterparts. Sect. 3 presents the results
concerning gamma-ray emission above 50 MeV from GW170104. Sect. 4 includes the
analysis of gamma-ray data obtained by the MCAL detector in the energy range
0.4-100 MeV with a discussion of an interesting candidate event occurring near
$T_{0}$. Sect. 5 presents a brief discussion of our results and conclusions
are reported in Sect. 6.

##  2 The AGILE satellite searching for GW event counterparts

The AGILE satellite that is orbiting the Earth in a near equatorial orbit at
an altitude of $\sim$ 500 km Tavani et al. (2009) is currently exposing 80% of
the entire sky every 7 minutes in a “spinning mode”. The instrument consists
of an imaging gamma-ray Silicon Tracker (sensitive in the energy range 30
MeV–30 GeV), Super-AGILE (SA; operating in the energy range 20–60 keV), and
the Mini-Calorimeter (MCAL; working in the range 0.35–100 MeV; Labanti et al.
2009; Marisaldi et al. 2008; Fuschino et al. 2008) with an omnidirectional
field of view (FoV) and self-triggering capability in burst-mode for various
trigger timescales. The anticoincidence (AC) system completes the instrument
(for a summary of the AGILE mission features, see Tavani et al. 2009). The
combination of Tracker, MCAL, and AC working as a gamma-ray imager constitutes
the AGILE-GRID. The instrument is capable of detecting gamma-ray transients
and GRB-like phenomena on timescales ranging from submilliseconds to tens of
hundreds of seconds. In preparation of the Advanced LIGO second Observing run
(O2) and the first Advanced Virgo one, the AGILE data acquisition system and
trigger configuration were substantially improved. The data obtained for
GW170104 greatly benefited from this improvement.

Both MCAL and SA are particularly efficient in detecting GRBs of the long and
short classes as demonstrated in recent years222GRB 080514B, Giuliani et al.
(2008), GRB 090401B, Moretti et al. (2009), GRB 090510, Giuliani et al.
(2010), GRB 100724B, Del Monte et al. (2011), an evaluation of GRID upper
limits for a sample of GRB in Longo et al. (2012), GRB 130327B, Longo et al.
(2013), GRB 130427A, Verrecchia et al. (2013), GRB 131108A, Giuliani et al.
(2013, 2014), and the first catalog of GRB detected by MCAL Galli et al.
(2013).. Because of its triggering capabilities, in particular a special
submillisecond search for transient events detected by MCAL Labanti et al.
(2009); Tavani et al. (2009), AGILE so far has further detected more than 1000
terrestrial gamma-ray flashes with durations ranging from hundreds to
thousands of microseconds Marisaldi et al. (2014); Tavani et al. (2011).

The characteristics that make AGILE in spinning mode an important instrument
for follow-up observations of wide GW source LRs are: a very large FoV of the
GRID (2.5 sr), an accessible region of 80% of the whole sky that can be
exposed every 7 minutes, 100–150 useful passes every day for any region in the
accessible sky, a gamma-ray exposure of $\sim$ 2 minutes of any field in the
accessible sky every 7 min, sensitivity of $\sim 10^{-8}\,\rm
erg\,cm^{-2}\,s^{-1}$ above 30 MeV for typical single-pass of unocculted sky
regions, a submillisecond MCAL trigger for very fast events in the range
0.4–100 MeV, and hard X-ray (20–60 keV) triggers of GRB-like events with a
localization accuracy of 2–3 arcmin in the SA FoV ($\sim 1$ sr) when operating
in imaging mode.

Satellite data are transmitted to the ground currently on average every
passage over the ASI Malindi ground station in Kenya. Scientific data are then
processed by a fast processing that was recently improved for the search of
electromagnetic (e.m.) counterparts of GW sources. The current data processing
can typically produce an alert for a transient gamma-ray source and/or GRB-
like events within 20 min–2 hrs from satellite onboard acquisition Bulgarelli
et al. (2014); Pittori (2013).

##  3 GRID observations of GW170104

###  3.1 Prompt emission

Figure 1: AGILE-GRID $E>50$ MeV sensitivity map (in $\rm erg\,cm^{-2}s^{-1}$)
in Galactic coordinates based on the gamma-ray exposure at the detection time
$T_{0}$ of GW170104. The shadowed areas show the Earth-occulted region and the
sky fraction not directly accessible by AGILE for solar panel constraints. The
magenta contours show the LALinference 90% c.l. LR of GW170104 The LIGO
Scientific Collaboration and Virgo Collaboration (2017c). The two parts of the
northern arc of the LR are marked with “A1” and “A2” while the southern arc is
marked by “B”. The AGILE instrument has significant exposure of the B-region
at $T_{0}$, covering about 35% of the total LR of GW170104.

AGILE sky scanning observation covered the GW170104 event (at time $T_{0}$ =
10:11:58.599 UTC on 2017 January 4; The LIGO Scientific Collaboration and
Virgo Collaboration 2017a) with the GRID FoV only marginally occulted by the
Earth. Fig. 1 shows the gamma-ray sensitivity map above 50 MeV at the GW170104
detection time. The Earth partially covers the lower part of the B arc
(obtained from the 90% GW170104 LR from the refined localization map; A17).
The remaining part of the B arc and a small area of the upper A arc (A2) are
not occulted and inside the AGILE exposure region. We performed a search for
(1) the prompt event involving the GRID, MCAL, AC and SA and (2) precursor and
delayed emission on multiple timescales involving the GRID. Interestingly, the
AGILE-GRID exposed a good fraction (36%) of the B arc on time interval of
about 200 s centered at $T_{0}$.

Figure 2: Gamma-ray light curve obtained by the AGILE-GRID for the short GRB
090510 repositioned at redshift $z=0.18$ of GW170104 (blue points). The GRID
upper limits obtained for the exposed fraction of the B arc of GW170104 at
different timescales are marked in red.

No significant gamma-ray emission was detected by the GRID near $T_{0}$. We
obtained $3\,\sigma$ flux upper limits (ULs) in the band 50 MeV–10 GeV,
$UL=1.8\times 10^{-6}\rm\,erg\,cm^{-2}\,s^{-1}$ and $UL=2.9\times
10^{-8}\rm\,erg\,cm^{-2}\,s^{-1}$ for 2 s and 100 s integrations,
respectively. As shown in Fig.2, we compare our gamma-ray ULs for GW170104
with the short GRB 090510 gamma-ray light curve detected by AGILE Giuliani et
al. (2010). The short GRB 090510 (at the redshift $z=0.9$) has been considered
a reference for potential electromagnetic gamma-ray emission that could be
associated to a GW event Ackermann et al. (2016); Tavani et al. (2016). In
Fig. 2 we show the GRB 090510 light curve rescaled (in flux and time
corrected) as if it were emitted at the GW170104 estimated distance
($z=0.18$).

Table 1: Analysis of individual passes over the GW170104 localization region
Expo- Bin 3 $\sigma$ UL Fractionc sure Central Rangeb No Timea Total B A -10
-950 2.9/84.0 50% 100% 20% -9 -850 4.1/105.0 72% 100% 10% -8 -750 2.9/95.0 39%
22% 53% -7 -650 11.0/59.0 34% 0% 62% -6 -550 8.5/108.0 33% 62% 10% -5 -450
2.9/74.0 50% 100% 9% -4 -350 4.1/91.0 63% 100% 36% -3 -250 2.9/103.0 36% 20%
58% -2 -150 7.9/51.0 32% 0% 58% -1 -50 11.0/89.0 30% 56% 9% 0 0 2.9/110.0 36%
64% 9% +1 +50 4.5/98.0 43% 56% 33% +2 +150 3.0/66.0 32% 0% 28% +3 +250
5.9/42.0 32% 0% 29% +4 +350 17.0/105.0 25% 19% 5% +5 +450 3.0/42.0 25% 8% 5%
+6 +550 4.8/110.0 29% 17% 12% +7 +650 3.2/73.0 30% 29% 1% +8 +750 4.8/33.0 31%
29% 2% +9 +850 34.0/108.0 15% 10% 5% +10 +950 3.2/54.0 15% 10% 5%

Notes. aafootnotetext: Centroid of the window time interval with respect to
$T_{0}$, $t-T_{0}$,
in seconds. All integrations have a duration of 100 s. bbfootnotetext:
$3\,\sigma$ flux upper limit ($10^{-8}\,\rm erg\,cm^{-2}\,s^{-1}$) obtained
for emission in
the range 50 MeV–10 GeV and for a spectrum assumed to be
similar to the delayed gamma-ray emission of the short
GRB 090510. ccfootnotetext: Fraction of the GRID exposed whole LVC LR of
GW170104
and of the B and A arcs respectively.

###  3.2 Precursor and Delayed Emission

Figure 3: Sequence of ($E>50$ MeV) maps in Galactic coordinates showing the
AGILE-GRID passes with the best sensitivity (in $\rm erg\,cm^{-2}s^{-1}$) over
the GW170104 LR obtained during the period (- 1000 s, + 1000 s) with respect
to $T_{0}$. The color maps show the gamma-ray flux 3 $\sigma$ upper limits in
the range 50 MeV–10 GeV with the most stringent values being $UL=(1-2)\times
10^{-8}\rm\,erg\,cm^{-2}\,s^{-1}$. The sequence shows 21 maps for all the one-
orbit passes of Table 1, corresponding to the 100 s interval numbers (from top
left to bottom right): from $-10$ to $+10$. The GW170104 LR is marked by the
purple contour, the LALinference 90% c.l. The LIGO Scientific Collaboration
and Virgo Collaboration (2017c).

AGILE was also optimally positioned in the GW170104 LR at interesting time
intervals preceding and following the prompt event. We show in Fig. 3 the
sequence of passes of 100 s duration over the LR within $T_{0}\pm\,1000$ s,
and we summarize in Table 1 the 3 $\sigma$ flux UL range, minimum and maximum
value within the covered LR portion. We label time intervals of the highest
GRID exposure with progressive numbers with respect to the prompt interval
$\Delta_{0}$. It is interesting to note that exposures obtained from Table 1
reached 60-70% of the LR. We also carried out a long-timescale search for
transient gamma-ray emission during the hours immediately following or
anticipating the prompt event. No significant gamma-ray emission in the
GW170104 LR was detected over timescales of hours up to two-day integrations.

Figure 4: Light curve of MCAL data which includes the GW170104 event time.
Data are displayed with a 32 ms time binning for the MCAL full energy band
(0.35–100 MeV), after refined data processing. The E1, E2 and E3 event times
are marked by vertical magenta, green and light blue lines, while the $T_{0}$
is marked by a dashed red line. The horizontal dashed grey line indicates the
4 $\sigma$ level, estimated on the whole data acquisition interval (12.6 s).
The orange horizontal line marks the average background level.

##  4 Analysis of MCAL data and identification of an interesting event

The onboard MCAL processing procedure identified an above-threshold event that
triggered full telemetry data acquisition for a 12.6 s segment starting at
time $T_{1}$ = 10:11:47.4 UTC and ending at $T_{2}$ = 10:12:00.0 UTC. This
event occurred on the 16 ms trigger timescale for which the average MCAL
occurrence rate333This rate is dominated by background events. An analysis of
the triggered MCAL data with 16 ms bin shows that the rate of data
acquisitions with peaks of signal-to-noise (S/N) above 5 $\sigma$ is
$\sim\,5\,\times 10^{-4}\,\rm Hz$. is 7.5 $\times 10^{-3}\rm Hz$.
Interestingly, this time interval includes the GW170104 detection time
$T_{0}$. Analysis in this time interval reveals a significant peak at the
onboard trigger time $T_{E1}=T_{0}-6.87\rm\,s$ (10:11:51.70 $\pm 0.05\,\rm s$
UTC, that we call “E1”). This event appears to be significant with 26 ms
binning resulting in 37 raw counts in the full energy band and a preliminary
signal-to-noise (S/N) ratio of 5.2 $\sigma$. After having carried out standard
off-line processing and event pile-up removal, the E1 S/N ratio becomes
$3.6\,\sigma$. Furthermore, two more events within 1 s from $T_{0}$ are
noticed in the raw telemetry for the full MCAL energy band. An event (that we
call “E2”) occurred at $T_{E2}=T_{0}-0.46\,\rm s$ (10:11:58.10 $\pm 0.05\,\rm
s$ UTC) and is significant with 32 ms binning. This event shows 37 raw counts
and pre-filtering S/N ratio of 4.2 $\sigma$. After filtering and pile-up
removal its S/N ratio slightly increases to 4.4 $\sigma$ (see Fig. 4). A third
feature after $T_{0}$ (that we call “E3”) is noticed in the 96 ms binned light
curve occurring on $T_{E3}=T_{0}+0.30\,\rm s$ (10:11:58.90 $\pm 0.05\,\rm s$
UTC). This feature shows 89 raw counts and a pre-filtering S/N ratio of 3.7
$\sigma$. After filtering and pile-up removal its S/N ratio is 3 $\sigma$(in
Figure 5 a 4s interval centered at E2 time of the 32 ms light curve, in the
full band and in the E $\leq$ 1.4 MeV, E $\geq$ 1.4 MeV energy bands).

Therefore, with these preliminary indications, we focused on detecting signals
in specific time bins that turn out to be the most effective ones in revealing
short GRBs as detected by the AGILE-MCAL. We performed a refined search (with
time shifts) in three binning time scales (26, 32, and 96 ms). One relevant
timescale is of the order of 100 ms as demonstrated by the range of T50
measured in MCAL data (see Fig. 4 of Galli et al. 2013). This timescale is
optimal for searching emission during the prompt phase of short GRBs (e.g.,
GRB 090510, and more recently GRB 170127C, Ursi et al. 2017). The other two
timescales (26 and 32 ms) are chosen to optimize searches of ”GRB
090510-precursor-like events” known to last between 20 and 30 ms. The latter
class of events constitutes the physical reference of our optimized onboard
trigger logic for data acquisition that AGILE implemented since November 2016
to improve its sensitivity to GW events. We applied to each binned light curve
different phase shifts: 4 equally spaced shifts for the 32 and 96 ms binned
light curves and 2 shifts for the 26 ms one, corresponding to shifts of $1/4$
and $1/2$ of the time bins, respectively. These events/ features are presented
in the 32 ms binned light curve of Fig. 4 showing filtered and pile-up removed
data. We verified that the South Atlantic Anomaly passage occurs
$\sim\,1700\,\rm s$ after $T_{0}$.

Figure 5:  MCAL light curve binned at 32 ms for the 4 s time interval centered
on the E2 event. Data displayed for the MCAL full energy band (0.35 - 100 MeV,
top panel), and for the soft ($\leq 1.4\,\rm MeV$) and hard ($\geq 1.4\,\rm
MeV$) energy bands (bottom panel). The GW170104 event time is at the origin of
the abscissa. In the top panel the horizontal dashed grey line indicates the 4
$\sigma$ level, estimated on the whole data acquisition interval (12.6 s). The
orange horizontal line marks the average background level.

In order to assess the post-trial probability of detecting these three events
close to $T_{0}$ we developed a targeted procedure to identify untriggered
peaks with the characteristics of E1, E2, and E3, within their respective
light curves (of binning equal to 26, 32, and 96 ms). This procedure was
carried out over a time interval of 2 weeks centered at $T_{0}$, for a total
livetime of 104,557 s. Given the MCAL filtered telemetry, this analysis
produces FARs for these events and specific choices of time binning. The
results444Our signal is a superposition of Poisson noise and sparse non-
Poissonian events. We conservatively ignore the subtraction of the latter
events and determine the FAR as deduced from the total signal. are:
$FAR_{E1}=1\,\times\,10^{-3}$ Hz for 26 ms binning,
$FAR_{E2}=1\,\times\,10^{-4}$ Hz for 32 ms binning, and
$FAR_{E3}=3\,\times\,10^{-3}$ Hz for 96 ms binning. Only event E2 potentially
emerges as an interesting candidate; further analysis supports this
conclusion. It is then important to obtain the FARs for the 26 and 96 ms
binned light curves corresponding to detections with the same S/N of E2. We
obtained ${1.6\,\times\,10^{-4}}$ and ${5.7\,\times\,10^{-5}}$ Hz for the 26
and 96 ms binned light curves, respectively. We take
${FAR_{E2}=1\,\times\,10^{-4}}$ Hz as representative of our search of events
similar to E2. In addition to the refined search binnings, for completeness in
the analysis of the relevant time scales in the number of trials calculation,
we also include the 16 and 64 ms unshifted binnings that refer to the onboard
trigger logic time scales initially used for our data processing. We determine
the FARs with the same S/N of E2 to be ${2.4\,\times\,10^{-4}}$ and
${7.5\,\times\,10^{-5}}$ Hz, respectively.

Having obtained the FAR over a 2-week timescale, we then estimate the post-
trial probability $P$ of these events accidentally occurring within a time
interval $\delta\,t$ from $T_{0}$ with two different methods. The first one
follows the formalism and definition laid out by Connaughton et al. (2016) for
the analysis of Fermi-GBM data of GW150914. In that work, the probability is
defined as $P\,=\,N\times\,FAR\,\times\,\delta\,t\,\times(1+\ln(\Delta
t/t_{bin}))$, where $N$ is the number of trials, $FAR$ is the false-alarm
rate, $\delta\,t$ is the time difference between the event time and $T_{0}$,
$\Delta\,t$ is the one-sided time interval over which the search is done, and
$t_{bin}$ is the time binning used in the analysis. We considered as an event
of interest only E2 for which we obtained: $P^{E2}=\,1{2}\times\,(1\times
10^{-4}\,{\rm Hz})\,\times(0.46\,{\rm s})\times(2+\ln(11.2\,{\rm
s}/0.0{1}6\,{\rm s})+\ln(1.4\,{\rm s}/0.0{1}6\,{\rm s}))\simeq
0.00{719}\,(\sim 2.4\,\sigma)$, where we included both the backward and
forward time intervals $\Delta t$ of 11.2 and 1.4 $\rm s$ respectively, and
$\delta\,t=0.46\,\rm s$. We adopted a factor N$\,=\,(4\times 2)+(2\times
1){+1+1}\,=\,12$ to take into account the number of trials implemented in the
refined analysis procedure, which includes three independent searches and
their corresponding time shifts. This number of trials is overestimated
because the results for overlapping phase shifts are not independent and for
the effect of unsubtracted non-Poissonian events in the FAR calculation.

The second method to estimate the post-trial probability of E2 is based on an
independent procedure. We considered the whole MCAL data stream of the same
two weeks used for FAR estimate with ”a-posteriori” choices: 32 ms binning,
and fixed time shift. We simulated the occurrence of an external trigger, at a
random time $t^{*}$, and determined the time difference ($\tau$) between
$t^{*}$ and the closest peak with S/N $\geq\,$4.4 $\sigma$. This procedure was
repeated 6 million times and produced a distribution of $\tau$’s. We determine
the probability of occurrence of a signal with characteristics similar to E2
($\tau\,\leq\,$0.46 s) to be 3.7 $\times\,10^{-3}$, that corresponds to 2.7
$\sigma$. We consider this value to be an alternative estimate of the post-
trial probability of E2 events.

We conclude that the E2 event turns out to have a post-trial coincidence
probability between 2.4 and 2.7$\,\sigma$.

##  5 Discussion

AGILE observed the field containing GW170104 with good coverage and
significant gamma-ray exposure of its LR. The AGILE-GRID exposed 36% of the B
arc at the detection time $T_{0}$. No transient gamma-ray source was detected
near $T_{0}$ over timescales of 2, 20, and 200 s starting at $T_{0}$. Fig. 2
shows these prompt upper limits compared to a rescaled gamma-ray light curve
of GRB 090510. Our gamma-ray upper limits are quite relevant in constraining
prompt gamma-ray emission from GW170104. Furthermore, we obtain constraints
for gamma-ray emission above 50 MeV also for possible precursor and delayed
emission as detailed in Table 1. It is important to note that the co-axial SA
detector had a partial coverage of the GW170104 LR at $T_{0}$ (within 1s),
that was observed between 0 and 30∘ off-axis. No significant detection was
obtained by SA imaging or ratemeter data in the 20–60 keV band during the time
interval $T_{0}\pm$ 100 s for a stable background. The $3\,\sigma$ UL has been
derived for a 1 s integration time and varies between $F^{SAc}\,=\,1.5\times
10^{-8}\,\rm erg\,cm^{-2}$ for an on-axis position and $F^{SAs}\,=\,6.6\times
10^{-8}\,\rm erg\,cm^{-2}$ at 30∘ off-axis position. A search in the SA
ratemeter data does not produce a significant detection with a $2\,\sigma$
fluence upper limit of $2.4\times 10^{-8}\,\rm erg\,cm^{-2}$ (1 s integration
time).

MCAL detected three short-timescale events/features, of which only one
remained significant after a refined analysis. The E2 event has a pre-trial
S/N ratio of 4.4 $\sigma$, and a post-trial probability of occurring within
0.46 s before $T_{0}$ between 2.4 and 2.7 $\sigma$. This event is the most
interesting and is worthy of additional investigation and searches for
possible coincident detections by other space instruments. We estimated the
observed energy flux in the energy range 0.4–40 MeV at two positions on the B
arc, and at one on the A1 arc. We used a mean photon index taking into account
the detector response and fitted the normalization parameter. We obtained on
the B arc $F_{Bc}\,=\,2.8_{-1.0}^{+1.9}\times 10^{-6}\,\rm
erg\,cm^{-2}\,s^{-1}$ assuming a power-law model with a fixed mean photon
index of -2 and estimating it at the a position with the best sensitivity (on-
axis), while F${}_{Bs}\,=\,2.5_{-1.0}^{+2.7}\times 10^{-6}\,\rm
erg\,cm^{-2}\,s^{-1}$ with the same photon index at a position with lower
sensitivity. For the A1 arc, we obtained F${}_{A1}\,=\,3.7_{-1.4}^{+5.1}\times
10^{-6}\,\rm erg\,cm^{-2}\,s^{-1}$ with the same photon index. We consider the
$F_{Bc}$ value the best flux estimate for the B arc, corresponding to a
fluence $F^{\prime}_{Bc}\,=\,8.9_{-3.0}^{+6.1}\times 10^{-8}\,\rm
erg\,cm^{-2}$. Moreover, we evaluate the flux at this position also assuming a
photon index of -1.6 (observed, e.g., during the prompt phase of GRB 090510),
obtaining $F_{hBc}\,=\,4.6_{-2.0}^{+5.5}\times 10^{-6}\,\rm
erg\,cm^{-2}\,s^{-1}$ and
$F^{\prime}_{hBc}\,=\,1.5_{-0.6}^{+1.8}\times\,10^{-7}\,\rm erg\,cm^{-2}$.

We obtained the fluence extrapolations to the SA energy band to be
$\sim\,2.1\,\times\,10^{-8}\,\rm erg\,cm^{-2}$ for photon index of -2 and
$\sim\,4.8\,\times\,10^{-9}\,\rm erg\,cm^{-2}$ for -1.6. We note that the SA
UL fluence at the center of its FoV, $F^{SAc}$, is slightly lower than the
former value, therefore providing a constraint on the spectrum. The latter
value of the extrapolated fluence for the index of -1.6 is not constrained by
the SA observations. We also checked the extrapolation of the E2 flux to the
GRID energy band and compared it to the 2 s UL fluence. We obtained an
extrapolated fluence at energies above 50 MeV of
$\sim\,8.9\,\times\,10^{-8}\,\rm erg\,cm^{-2}$ for a -2 photon index. Again,
we end up with a lower value with respect to the GRID 2 s UL fluence,
$5.8\,\times\,10^{-6}\,\rm erg\,cm^{-2}$.

In conclusion, for the GW170104 ’s most likely distance of 880 Mpc Abbott et
al. (2017), we obtain for the E2 event in the MCAL band an isotropic
luminosity $L_{\rm iso}=2.6\times 10^{50}\,\rm erg\,s^{-1}$, and total energy
$E_{\rm iso}=8.3\times 10^{48}\,\rm erg$.

It is interesting to note that the E2 event appears to be quite similar in its
timing and flux characteristics compared to the precursor event detected about
0.46 s before the onset of the brightest part of the short GRB 090510. This
weak precursor was simultaneously detected by the AGILE-MCAL and Fermi-GBM
Abdo et al. (2009); Giuliani et al. (2010), and was therefore confirmed by the
simultaneous detection of two space instruments. In our case, this precursor
of GRB 090510 played an important role in calibrating the new MCAL data
acquisition system that has been optimized just for GW counterpart searches.
We notice that both the triggering events E1 and E2 (which occurred within the
telemetered time window after E1) satisfy flux and timing conditions set up
following the MCAL detection of the GRB 090510 precursor555The MCAL triggering
procedure allows only one trigger within the allocated time window following a
first event. The E2 event could have independently triggered the MCAL system,
but it did not because it was too close to E1.. The E2 event appears to be
harder than the 2009 precursor.

If associated with GW170104 , the E2 event shows interesting physical
properties. Its fluence and $L_{\rm iso}$ are comparable to those of weak
short GRBs previously detected by several space instruments. A total energy
estimated in the isotropic approximation of $\simeq 10^{49}\,\rm erg$ is
$10^{-7}$ times smaller than the total black hole rest mass of
$50\,M_{\odot}\,c^{2}$. Beaming effects would make this ratio even smaller. It
is clear that, if any association between E2 and GW170104 were confirmed by
simultaneous detections of different instruments, the e.m. output following
the GW170104 gravitational coalescence would be severely constrained. MCAL
cannot determine the position of the event in the absence of simultaneous
onboard detection by the GRID or SA. However, from Fig. 1 we obtain the
definite sky region belonging to the B arc that was not occulted by the Earth
and exposed by MCAL for off-axis angles up to 45∘ at the $T_{0}$ of GW170104.
We also have large off-axis angle MCAL exposures for regions labeled A1 and A2
in Fig. 1.

##  6 Conclusions

The LVC detection of the gravitational wave GW170104 produced by a BH–BH
coalescence of a total mass of $50\,M_{\odot}$ is a remarkable event, the
third in the LVC series of confirmed detections. AGILE data are useful in
assessing the possible e.m. emission associated with GW events. AGILE already
obtained interesting upper limits for gamma-ray and hard X-ray emission from
the first GW event, GW150914 Tavani et al. (2016). In that case, the AGILE-
GRID exposure of the LVC LR occurred 100 s before and 300 s after the event
$T_{0}$.

GW170104 is much more interesting from the point of view of AGILE
observations. In this case, significant exposure of the LR was obtained
simultaneously with the detection time $T_{0}$. The gamma-ray and hard X-ray
upper limits obtained by the GRID and SA are therefore even more constraining
than for GW150914.

Even more interestingly, we identify in the MCAL data the E2 candidate event
with noticeable characteristics. We note its similarity in flux and timing
properties to the precursor of the short GRB 090510. Our estimate of the post-
trial probability of being temporally associated with GW170104
(${2.4}\,$–$\,2.7\,\sigma$). As in the case of the GBR 090510 precursor, a
confirmation by different space instruments would provide the definite proof
of its association with the GW.

AGILE continues its observations of the sky in spinning mode and is fully
operational in the search of GW source counterparts.

We thank the LIGO/Virgo Collaboration for sharing information on GW170104 and
for discussion and comments on our manuscript. In particular, we thank F.
Ricci, M. Branchesi, L. Singer, and E. Katsavounidis. We thank the anonymous
referee for his/her comments. We thank the ASI management, the technical staff
at the ASI Malindi ground station, the technical support team at the ASI Space
Science Data Center, and the Fucino AGILE Mission Operation Center. AGILE is
an ASI space mission developed with programmatic support by INAF and INFN. We
acknowledge partial support through the ASI grant no. I/028/12/2.

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