text:
\section*{Introduction}
Let $q$ be a power of a prime number $p$.
Let $\ell \neq p$ be a prime number.
Let $\psi$ be a non-trivial character of
$\mathbb{F}_{q,+}=\{x \in \mathbb{F}_{q^2} \mid x^q+x=0\}$
over $\overline{\mathbb{Q}}_{\ell}$.
Let $n$ be a positive integer.
A Weil representation of
a unitary group $U_n(q)$ associated to $\psi$ is constructed in \cite{GerWeil}, which we denote by
$\rho_{U_n(q),\psi}$.
Let $m$ be a positive odd integer.

In this paper, we investigate
the behavior of $\rho_{U_n(q),\psi}$ via Shintani
lifting from $\mathbb{F}_q$ to $\mathbb{F}_{q^m}$.
Let $\Gamma$ be a cyclic group of order $m$ with
generator $\sigma$.
Let $F \colon U_n(q^m) \to U_n(q^m);\
g \mapsto ({}^t g^{\tau})^{-1}$, where
$\tau((a_{i,j}))=(a_{i,j}^q)$.
Let $\sigma$ act on $U_n(q^m)$ as $F$.
We consider the semidirect group
$U_n(q^m) \rtimes \Gamma$.
In \cite{GyoLift}, Gyoja defines a norm map
from
the set of $U_n(q^m)$-conjugacy classes in $U_n(q^m) \rtimes \sigma$ to
the set of conjugacy classes in
$U_n(q)$, which is denoted by $\mathrm{N}$.
We set $\psi_m=\psi \circ \Tr_{\mathbb{F}_{q^{2m}}/\mathbb{F}_{q^2}} \colon \mathbb{F}_{q^m,+} \to
\overline{\mathbb{Q}}_{\ell}^{\times}$.
We construct a representation $\widetilde{\rho}_{U_n(q^m),\psi_m}$ of $U_n(q^m) \rtimes \Gamma$, which is an extension of
$\rho_{U_n(q^m),\psi_m}$, and show the equality
\begin{equation*}\label{int}
\tr \widetilde{\rho}_{U_n(q^m),\psi_m}(g,\sigma)
=\tr \rho_{U_n(q),\psi}(\mathrm{N}(g,\sigma))
\end{equation*}
for any $g \in U_n(q^m)$.
This is a version of \cite[Theorem in
\S1]{HeWaWeil} in unitary cases.
As mentioned in \cite[\S1]{HeWaWeil},
the arguments in \cite{HeWaWeil} work also in unitary cases.
In this paper, we use an inductive argument similar
to that in \cite{HeWaWeil},
but replace some ingredients in the proof with geometric inputs:
In \cite{HeWaWeil}, an extended Weil representation
is constructed based on the
Schr\"{o}dinger model of a Weil representation.
On the other hand,
we construct $\widetilde{\rho}_{U_n(q^m),\psi_m}$
in a geometric way and study the representation by geometric tools.

In \cite[Theorem 1.5]{ITGeomHW}, it is known that
the representation ${\rho}_{U_n(q^m),\psi_m}$ is realized in the $\psi_m$-isotypic part of the middle cohomology of
the affine smooth variety over $\mathbb{F}_{q^2}$
defined by
\begin{equation*}\label{ft}
 z^{q^m}+z=\sum_{i=1}^n x_i^{q^m+1}
\end{equation*}
in $\mathbb{A}_{\mathbb{F}_{q^2}}^{n+1}$.
In order to extend this representation to
$U_n(q^m) \rtimes \Gamma$,
we use the Frobenius action over $\mathbb{F}_{q^2}$.

Similarly to \cite[Theorem 4.1]{HeWaWeil}, we actually show
a stronger equality in Theorem \ref{main}.
Namely, we show a similar equality for Heisenberg--Weil representations
and more general norm maps.

We briefly explain the content of each section.
In \S \ref{sec:norm}, we collect several known
facts on Gyoja's norm map.
In \S \ref{Pre}, we recall unitary Heisenberg--Weil representations.
In \S \ref{GC}, we construct an extension
$\widetilde{\rho}_{\mathit{HU}_n(q^m),\psi_m}$ of
a Heisenberg--Weil representation
geometrically and show Theorem \ref{main} when $n=1$
in Proposition \ref{prop:trzFr}.
In \S \ref{od}, we study
the behavior of $\widetilde{\rho}_{\mathit{HU}_n(q^m),\psi_m}$ restricted to
several subgroups.
In \S \ref{Sup}, we study
the support of the trace of $\widetilde{\rho}_{\mathit{HU}_n(q^m),\psi_m}$.
In \S \ref{sec:Main}, we state our main theorem.
Under the knowledges in \S \ref{od}--\S \ref{Sup}, we reduce
Theorem \ref{main} to a special case in Lemma \ref{inv} in \S \ref{Red}.
By using the character formula of a tensor induction
in \cite{GlIsTen},
we show Theorem \ref{main} in the special case
in \S \ref{Prred}.

\section{Norm map}\label{sec:norm}
We follow \cite[\S3]{HeWaWeil} and \cite[\S3]{GyoLift}.
Let $\mathbf{G}$ be a connected algebraic group
 over $\mathbb{F}_q$ with Frobenius
endomorphism $F$. Let $m$ be a positive integer.
We put $G=\mathbf{G}^{F^m}$.
Let $\Gamma$ be a cyclic group of order $m$ with a generator
$\sigma$.
Let $\sigma$ act on $G$ as
$F$.
We consider the semidirect group $G \rtimes \Gamma$.
Let $1 \leq i \leq m$ be an integer.
We set $d=(m,i)$. We choose an integer $t$ such that
$ti \equiv d \pmod m$.  We set $\mu=m/d$.
For $g \in G$, we can choose
$\alpha=\alpha(g) \in \mathbf{G}$ such that
\[
\alpha^{-1} F^d(\alpha)=(g,\sigma^i)^t (1,\sigma^{-it})
\]
by Lang's theorem.
We put
\[
\mathrm{N}_{i,t}(g,\sigma^i)=\alpha g F^i(g) \cdots F^{i(\mu-1)}(g)
\alpha^{-1}.
\]
The element $\mathrm{N}_{i,t}(g,\sigma^i)$
does belong to $G^{F^i}$. Its conjugacy class
does not depend on the choice of $\alpha$.
By \cite[Lemma 3.2(1)]{GyoLift}, $\mathrm{N}_{i,t}$
induces a bijection from the set of $G$-conjugacy classes in $G \rtimes \sigma^i$ to the set of
conjugacy classes in $G^{F^i}$.
The norm map is originally defined in \cite{ShiTworem}
in the case where $i=t=1$
and generalized in \cite{KawIrruni}.

Let $C$ be an algebraically closed field of characteristic $0$.
Let $\mathcal{C}(G \rtimes \sigma^i)$ denote the
vector space of $C$-valued functions on
$G \rtimes \sigma^i$ which are invariant under conjugation by $G$.
For a finite group $H$, let $\mathcal{C}(H)$ denote the
set of $C$-valued class functions on $H$.
Then we have the bijection
\[
\mathcal{N}_{i,t} \colon \mathcal{C}(G^{F^i}) \to \mathcal{C}(G \rtimes \sigma^i);\
\chi \mapsto \chi \circ \mathrm{N}_{i,t}.
\]
This induces the bijection
\[
 \mathcal{C}(G^{F^i})^F \stackrel{\sim}{\longrightarrow} \mathcal{C}(G \rtimes \sigma^i)^{\sigma}
\]
by \cite[Lemma 3.2(2)]{GyoLift},
where $\sigma$ acts on $G \rtimes \sigma^i$ by the conjugation.
Note that $G^{F^i}=\mathbf{G}^{F^d}$.
\begin{lem}\label{gy}
Let $\mathbf{H}$ be a connected algebraic subgroup of
$\mathbf{G}$, and $\mathbf{G}_1,\mathbf{G}_2$
two connected algebraic groups over $\mathbb{F}_q$.
\begin{enumerate}
\item\label{enu:HInd}
Let $\widetilde{\chi} \in \mathcal{C}(H \rtimes \Gamma)$,
and take $\chi \in \mathcal{C}(H^{F^i})^F$
such that $\widetilde{\chi}|_{H \rtimes \sigma^i}=\mathcal{N}_{i,t}(\chi)$.
Then we have
$(\mathrm{Ind}_{H \rtimes \Gamma}^{G \rtimes \Gamma}\widetilde{\chi})|_{G \rtimes \sigma^i}
=\mathcal{N}_{i,t}(\mathrm{Ind}_{H^{F^i}}^{G^{F^i} }\chi)$.
\item
Let
$\widetilde{\chi} \in \mathcal{C}(G \rtimes \Gamma)$,
and take $\chi \in \mathcal{C}(G^{F^i})^F$
such that $\widetilde{\chi}|_{G \rtimes \sigma^i}
=\mathcal{N}_{i,t}(\chi)$.
Then we have
$(\widetilde{\chi}|_{H \rtimes \Gamma})|_{H \rtimes
\sigma^i}=\mathcal{N}_{i,t}(\chi|_{H^{F^i}})$.
\item
Assume that $\mathbf{H}$ is normal.
We identify $G/H$ with $(\mathbf{G}/\mathbf{H})^{F^m}$.
Let
\[
p' \colon G \rtimes \Gamma \to (G/H) \rtimes \Gamma,
\quad
p \colon G^{F^i} \to (G/H)^{F^i}
\]
be the canonical projections.
Let $\widetilde{\chi} \in \mathcal{C}((G/H) \rtimes \Gamma)$,
and take $\chi \in \mathcal{C}((G/H)^{F^i})^F$
such that
$\widetilde{\chi}|_{(G/H) \rtimes \sigma^i}
=\mathcal{N}_{i,t}(\chi)$.
Then we have
$(\widetilde{\chi} \circ p')|_{G \rtimes \sigma^i}=\mathcal{N}_{i,t}(\chi \circ p)$.
\item\label{enu:chiprod}
Let
$\widetilde{\chi}_{\alpha} \in \mathcal{C}(G_{\alpha} \rtimes \Gamma)$
for $\alpha=1,2$, and take
$\chi_{\alpha} \in \mathcal{C}(G_{\alpha}^{F^i})^F$
such that
${\widetilde{\chi}_{\alpha}}|_{G_{\alpha} \rtimes \sigma^i}
=\mathcal{N}_{i,t}(\chi_{\alpha})$.
Then we have
$(\widetilde{\chi}_1 \times \widetilde{\chi}_2)|_{
(G_1 \times G_2) \rtimes \sigma^i}
=\mathcal{N}_{i,t}(\chi_1 \times \chi_2)$.
\end{enumerate}
\end{lem}
\begin{proof}
This is proved in \cite[Lemma 3.6]{GyoLift} if $C=\mathbb{C}$.
The same proof works also for a general $C$.
\end{proof}

\section{Heisenberg--Weil representation}\label{Pre}
Let $q$ be a power of a prime number $p$.
Let $V$ be a vector space over $\mathbb{F}_{q^2}$
with a hermitian form $h$.
In this paper, a hermitian form is supposed to be sesquilinear on the first coordinate.
For the hermitian space $(V,h)$,
we define $\mathbf{H}_{(V,h)}$ over $\mathbb{F}_q$ as
\[
 \mathbf{H}_{(V,h)} (R) = \{ (v,a) \in
 (V \otimes_{\mathbb{F}_q} R ) \times (\mathbb{F}_{q^2} \otimes_{\mathbb{F}_q} R)
 \mid (\Fr_q \otimes \id_R) (a) +a =h_R (v,v) \}
\]
for an $\mathbb{F}_q$-algebra $R$,
where
$h_R$ is the $R$-linear extension of $h$,
with the group operation
\[
 (v,a)(v',a')=(v+v', a+a'+h_R(v,v')).
\]
We put
$H(V,h) =\mathbf{H}_{(V,h)} (\mathbb{F}_q)$
and call it
the unitary Heisenberg group associated to $(V,h)$.

Assume that $h$ is nondegenerate in the sequel.
Let $\mathbf{U}_{(V,h)}$ be the unitary algebraic group over $\mathbb{F}_q$
defined as
\[
 \mathbf{U}_{(V,h)} (R) = \{ g \in \Aut_{\mathbb{F}_{q^2} \otimes_{\mathbb{F}_q} R}
 (V \otimes_{\mathbb{F}_q} R)
 \mid h_R (gv_1,gv_2) =h_R (v_1,v_2) \
 \textrm{for $v_1, v_2 \in V \otimes_{\mathbb{F}_q} R$} \}
\]
for an $\mathbb{F}_q$-algebra $R$,
with an action on $\mathbf{H}_{(V,h)}$ defined by
$(v,a) \mapsto (gv,a)$ for
$(v,a) \in \mathbf{H}_{(V,h)}(R)$ and
$g \in \mathbf{U}_{(V,h)}(R)$.
We put $\mathbf{HU}_{(V,h)}=\mathbf{H}_{(V,h)} \rtimes \mathbf{U}_{(V,h)}$.
We put $U(V,h) =\mathbf{U}_{(V,h)}(\mathbb{F}_q)$
and $\mathit{HU}(V,h)=\mathbf{HU}_{(V,h)}(\mathbb{F}_q)$.
We simply write
$\mathbf{H}_V$,
$\mathbf{U}_V$ and
$\mathbf{HU}_V$
for
$\mathbf{H}_{(V,h)}$,
$\mathbf{U}_{(V,h)}$ and
$\mathbf{HU}_{(V,h)}$
if it is clear which hermitian form is involved.

We take a prime number $\ell \neq p$.
For an abelian group $A$, let
$A^{\vee}=\mathrm{Hom}_{\mathbb{Z}}
 (A,\overline{\mathbb{Q}}_{\ell}^{\times})$.
For a representation $W$ of $A$ over $\overline{\mathbb{Q}}_{\ell}$
and $\chi \in A^{\vee}$,
let $W[\chi]$ denote the $\chi$-isotypic part of $W$.

Let $\mathbb{F}_{q,+}=\{x \in \mathbb{F}_{q^2} \mid x^q+x=0\}$.
Let $\psi \in \mathbb{F}_{q,+}^{\vee} \setminus \{1\}$.
Let $\rho_{H(V,h),\psi}$ denote the unique
irreducible representation of $H(V,h)$
whose restriction to the center contains $\psi$.
We put $n=\dim_{\mathbb{F}_{q^2}} V$.
The following is shown in \cite{GerWeil}
and stated in \cite[lemma 1.2]{ITGeomHW}.

\begin{lem}\label{lem:hw}
There exists a unique representation
$\rho_{\mathit{HU}(V,h),\psi}$ of
$\mathit{HU}(V,h)$ characterized by
\begin{itemize}
\item
an isomorphism
$\rho_{\mathit{HU}(V,h),\psi}|_{H(V,h)} \simeq
\rho_{H(V,h),\psi}$, and
\item
the equality
$\tr \rho_{\mathit{HU}(V,h),\psi}(g)=(-1)^n
 (-q)^{N(g)}$ for $g \in U(V,h)$, where
$N(g)=\dim_{\mathbb{F}_{q^2}} \ker (g-\id_V )$.
\end{itemize}
\end{lem}
We call $\rho_{\mathit{HU}(V,h),\psi}$
the Heisenberg--Weil representation of $\mathit{HU}(V,h)$
associated to $\psi$.
The restriction of $\rho_{\mathit{HU}(V,h),\psi}$
to the subgroup $U(V,h)$ is called
the Weil representation of $U(V,h)$
associated to $\psi$.

Let
\[
 h_n \colon
 \mathbb{F}_{q^2}^n \times \mathbb{F}_{q^2}^n \to \mathbb{F}_{q^2};\
 ((x_1,\ldots,x_n),(x'_1,\ldots,x'_n)) \mapsto
 \sum_{i=1}^n x_i^q x'_i.
\]
Any nondegenerate hermitian space $(V,h)$ of dimension $n$ is isometric to $(\mathbb{F}_{q^2}^n,h_n)$.
We simply write $H_n(q)$, $U_n(q)$ and
$\mathit{HU}_n(q)$ for
$H(\mathbb{F}_{q^2}^n,h_n)$, $U(\mathbb{F}_{q^2}^n,h_n)$ and
$\mathit{HU}(\mathbb{F}_{q^2}^n,h_n)$, respectively.

\section{Geometric construction}\label{GC}
We give a
geometric construction of an extended Heisenberg--Weil representation.
Let $m$ be an odd positive integer.
We consider the affine smooth variety over $\mathbb{F}_{q^2}$
defined by
\begin{equation*}\label{val}
z^{q^m}+z=\sum_{i=1}^n x_i^{q^m+1}
\end{equation*}
in $\mathbb{A}_{\mathbb{F}_{q^2}}^{n+1}$, which is denoted by $X_{m,n}$.
The group $\mathit{HU}_n(q^m)$ acts on $X_{m,n,\mathbb{F}_{q^{2m}}}$ by
\begin{align*}
 & ((x_i),z) \mapsto (g(x_i),z) \quad
 \textrm{for $g \in U_n(q^m)$}, \\
 & ((x_i),z) \mapsto
 ((x_i)+v,z+a+h_n(v,(x_i))) \quad
 \textrm{for $(v,a) \in H_n(q^m)$}.
\end{align*}
Let $\mathrm{Fr}_q \in \mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)$ be the geometric Frobenius automorphism.
Let $\nu \colon
\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_{q^2}) \to
\overline{\mathbb{Q}}_{\ell}^{\times}$ be the character
which sends $\mathrm{Fr}_{q^2}$ to $-q^{-1}$.
Let $\psi \in \mathbb{F}_{q,+}^{\vee} \setminus \{1\}$.
The restriction of the trace map
$\Tr_{\mathbb{F}_{q^{2m}}/\mathbb{F}_{q^2}}$ to
$\mathbb{F}_{q^m,+}$ induces
$\Tr_{\mathbb{F}_{q^{2m}}/\mathbb{F}_{q^2}} \colon
\mathbb{F}_{q^m,+} \to \mathbb{F}_{q,+}$.
We set
$\psi_m =\psi \circ \Tr_{\mathbb{F}_{q^{2m}}/\mathbb{F}_{q^2}} \in \mathbb{F}_{q^m,+}^{\vee}$.

Let $F \colon \mathit{HU}_n(q^m) \to \mathit{HU}_n(q^m)$ be the
Frobenius automorphism with respect to the rationality of
$\mathbf{HU}_{(\mathbb{F}_{q^2}^n,h_n)}$.
Let $\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_{q^2})$
act on $\mathit{HU}_n(q^m)$ by letting
$\mathrm{Fr}_{q^2}$ act as $F^2$.
We regard
\[
H_{\rm c}^n(X_{m,n,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi_m] \otimes \nu^n
\] as an
$\mathit{HU}_n(q^m) \rtimes \mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_{q^2})$-representation, which we denote by $\widetilde{\rho}_{\mathit{HU}_n(q^m),\psi_m}$.

\begin{lem}[{\cite[Theorem 1.5]{ITGeomHW}}]\label{lem:HWg}
We have
$H_{\rm c}^i(X_{m,n,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})
 [\psi_m]=0$ for $i \neq n$.
The restriction of $\widetilde{\rho}_{\mathit{HU}_n(q^m),\psi_m}$
to $\mathit{HU}_n(q^m)$
is isomorphic to $\rho_{\mathit{HU}_n(q^m),\psi_m}$.
\end{lem}

We obtain the rationality of $\rho_{\mathit{HU}_n(q^m),\psi_m}$ in the following sense.
\begin{lem}\label{rat}
We have an isomorphism
\[
\rho_{\mathit{HU}_n(q^m),\psi_m} \circ F
\simeq \rho_{\mathit{HU}_n(q^m),\psi_m}
\] as
$\mathit{HU}_n(q^m)$-representations.
\end{lem}
\begin{proof}
The $q^{m+1}$-st power geometric Frobenius morphism on
$X_{m,n,\overline{\mathbb{F}}_q}$ induces
an isomorphism on the \'{e}tale cohomology, which gives
the claimed isomorphism.
\end{proof}

\begin{lem}\label{lem:FrS}
\begin{enumerate}
\item\label{enu:X11}
The geometric Frobenius
$\mathrm{Fr}_{q^2}$ acts on
$H_{\rm c}^1(X_{1,1,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi]$ as scalar
multiplication by $-q$.
\item\label{enu:Fr2m}
We have
$\widetilde{\rho}_{\mathit{HU}_n(q^m),\psi_m}(\mathrm{Fr}_{q^{2m}})=1$.
\end{enumerate}
\end{lem}
\begin{proof}
By Lemma \ref{lem:HWg}, we have
$H_{\rm c}^i(X_{1,1,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi]=0$ for
$i \neq 1$.
Hence we have
 \[
 \Tr(\mathrm{Fr}_{q^2};H_{\rm c}^1
 (X_{1,1,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi])
 =-q^{-1}\sum_{\eta \in \mathbb{F}_{q,+}} \psi(\eta)
 \Tr((-\eta)\mathrm{Fr}_{q^2};H_{\rm c}^{\ast}
 (X_{1,1,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})).
 \]
By the Lefschetz fixed formula, we have
\begin{align*}
 \Tr((-\eta)\mathrm{Fr}_{q^2};H_{\rm c}^{\ast}
 (X_{1,1,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})) &=
 \sharp \{ (x,z) \in X_{1,1}(\overline{\mathbb{F}}_q) \mid
 x^{q^2}=x,\ z^{q^2} -\eta =z \} \\
 &=
 \begin{cases}
 q^3 & \textrm{if $\eta=0$}, \\
 0 & \textrm{otherwise}.
 \end{cases}
\end{align*}
The action of $\mathrm{Fr}_{q^2}$ commutes with
$H_1(q)$. The $H_1(q)$-representation $H_{\rm c}^1(X_{1,1,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi]$
 is irreducible by Lemma \ref{lem:HWg}. By Schur's lemma
 and $\dim H_{\rm c}^1(X_{1,1,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi]=q$, the first claim follows.

We show the second claim.
By the first assertion and the K\"{u}nneth formula,
$\mathrm{Fr}_{q^{2m}}$ acts on
$H_{\rm c}^n(X_{m,n,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi]$ as scalar multiplication by $(-1)^nq^{mn}$.
We have
$\nu^n(\mathrm{Fr}_{q^{2m}})=(-1)^{mn} q^{-mn}$.
Since $m$ is odd, the claim follows.
\end{proof}

We set $\Gamma=\mathrm{Gal}(\mathbb{F}_{q^{2m}}/\mathbb{F}_{q^2})$.
By Lemma \ref{lem:FrS} \ref{enu:Fr2m}, we can regard $\widetilde{\rho}_{\mathit{HU}_n(q^m),\psi_m}$
as an $\mathit{HU}_n(q^m) \rtimes \Gamma$-representation.
This is an extension of
$\rho_{\mathit{HU}_n(q^m),\psi_m}$ to $\mathit{HU}_n(q^m) \rtimes \Gamma$ by Lemma \ref{lem:HWg}.
The restriction of $\widetilde{\rho}_{\mathit{HU}_n(q^m),\psi_m}$
to the subgroup $U_n(q^m) \rtimes \Gamma$
is denoted by $\widetilde{\rho}_{U_n(q^m),\psi_m}$ in Introduction.
We put $\sigma=\mathrm{Fr}_{q^2}^{\frac{m+1}{2}}$.
Then $\sigma$ is the generator of $\Gamma$
acting on $\mathit{HU}_n(q^m)$ as $F$.

\begin{defn}\label{def:Vh}
Let $(V,h)$ be a nondegenerate
hermitian space of dimension $n$ over $\mathbb{F}_{q^2}$.
We take an isometry
$(V,h) \simeq (\mathbb{F}_{q^2}^n,h_n)$.
This induces $\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}) \rtimes \Gamma
\simeq \mathit{HU}_n(q^m) \rtimes \Gamma$.
Via this isomorphism and $\widetilde{\rho}_{\mathit{HU}_n(q^m),\psi_m}$, we define
a representation of $\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}) \rtimes \Gamma$, which
is denoted by $\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}$.
\end{defn}

\begin{rem}
The isomorphism class of
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}$
is independent of the choice of the isometry
$(V,h) \simeq (\mathbb{F}_{q^2}^n,h_n)$,
since the difference is a conjugation by an element of
$\mathbf{U}_V(\mathbb{F}_q)$.
\end{rem}

\begin{prop}\label{prop:trzFr}
Let $1 \leq j \leq m$. We set $(j,m)=d$.
Let $\zeta \in \mu_{q^m+1}$.
\begin{enumerate}
\item\label{enu:zFrj}
We have
\[
\tr\widetilde{\rho}_{\mathit{HU}_1(q^m),\psi_m}(\zeta, \mathrm{Fr}_{q^2}^j)
=\begin{cases}
q^d & \textrm{if $\zeta \in \mu_{\frac{q^m+1}{q^d+1}}$}, \\
-1 & \textrm{otherwise}.
\end{cases}
\]
\item\label{enu:trmd}
We have
\[
\tr\widetilde{\rho}_{\mathit{HU}_1(q^m),\psi_m}(\zeta, \mathrm{Fr}_{q^2}^j)
=\tr\widetilde{\rho}_{\mathit{HU}_1(q^d),\psi_d}(\zeta^{\frac{q^m+1}{q^d+1}}).
\]
\item\label{enu:trFrj}
We have $\tr \widetilde{\rho}_{\mathit{HU}_n(q^m),\psi_m}(\mathrm{Fr}_{q^2}^j)=q^{nd}$.
\end{enumerate}
\end{prop}
\begin{proof}
The third claim follows from the K\"{u}nneth formula
and the first one.
The second claim follows from
the first one and Lemma \ref{lem:hw}.

We show the first claim.
We consider the curve $X_m$ defined by
$z^q+z=x^{q^m+1}$ over $\mathbb{F}_{q^2}$.
We have the finite \'{e}tale morphism
$X_{m,1} \to X_m;\
(x,z) \mapsto (x,\sum_{i=0}^{m-1} (-1)^i z^{q^i})$.
The pull-back of this induces
\begin{equation}\label{eq:mm1}
H_{\rm c}^1(X_{m,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi] \xrightarrow{\sim}
H_{\rm c}^1(X_{m,1,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi_m].
\end{equation}
In the sequel,
we identify $U_1(q^i)$ with $\mu_{q^i+1}$ for any
positive integer $i$.
We have an isomorphism
\[
H_{\rm c}^1(X_{m,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi] \simeq
\bigoplus_{\chi \in \mu_{q^m+1}^{\vee} \setminus \{1\}}
\chi
\]
as $\mu_{q^m+1}$-representation by \eqref{eq:mm1}
and \cite[Lemma 1.3(1)]{ITGeomHW}.
Let $\chi \in \mu_{q^m+1}^{\vee}$. We have
\[
\chi^{q^{2j}}=\chi \iff \chi|_{\mu_{\frac{q^m+1}{q^d+1}}}
=1,
\]
because of $(q^m+1,q^{2j}-1)=q^d+1$.
We regard $\mu_{q^d+1}^{\vee}$ as a subset of
$\mu_{q^m+1}^{\vee}$ by the dual of
$\mu_{q^m+1} \to \mu_{q^d+1}; x \mapsto x^{\frac{q^m+1}{q^d+1}}$.
We have the finite morphism
$X_m \mapsto X_d;\ (x,z) \mapsto (x^{\frac{q^m+1}{q^d+1}},z)$.
Then the image of its pull-back map
\[
H_{\rm c}^1(X_{d,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi] \to
H_{\rm c}^1(X_{m,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi]
\]
equals $\bigoplus_{\chi \in \mu_{q^d+1}^{\vee} \setminus \{1\}} \chi$. This subspace
is stable under $\mathrm{Fr}_{q^2}^j$.
We write as $j=dk$.
Then $\Tr(\zeta\mathrm{Fr}_{q^2}^j;H_{\rm c}^1(X_{m,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi])$ equals
\begin{align*}
&\Tr(\zeta^{\frac{q^m+1}{q^d+1}}\mathrm{Fr}_{q^{2d}}^k;H_{\rm c}^1(X_{d,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi])
=(-q^d)^k\Tr(\zeta^{\frac{q^m+1}{q^d+1}};H_{\rm c}^1(X_{d,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi]) \\
&=(-q^d)^k \sum_{\chi \in \mu_{q^d+1}^{\vee} \setminus \{1\}} \chi(\zeta^{\frac{q^m+1}{q^d+1}})=\begin{cases}
(-1)^k q^{d(k+1)} & \textrm{if $\zeta \in \mu_{\frac{q^m+1}{q^d+1}}$}, \\
(-1)^{k+1} q^{dk} & \textrm{otherwise},
\end{cases}
\end{align*}
where the first equality follows from
Lemma \ref{lem:FrS} \ref{enu:X11} since
\[
 H_{\rm c}^1(X_{d,\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\psi]
 \xrightarrow{\sim}
 H_{\rm c}^1(X_{d,1,\overline{\mathbb{F}}_q},
 \overline{\mathbb{Q}}_{\ell})[\psi_d]
\]
similarly to \eqref{eq:mm1}.
Hence the claim follows.
\end{proof}

\section{Compatibility}\label{od}
Let $(V,h)$ be a nondegenerate hermitian space over $\mathbb{F}_{q^2}$
of dimension $n$.
\subsection{Orthogonal decomposition}
Let $(V,h)=(V_1,h_1) \oplus (V_2,h_2)$ be a decomposition of $V$
into the orthogonal direct sum of two hermitian spaces.
Then we have the natural homomorphism
\[
 \mathbf{HU}_{V_1} \times \mathbf{HU}_{V_2}
 \to \mathbf{HU}_{V}.
\]
This morphism and the projections induce
\begin{equation}\label{4-1}
 \xymatrix{
 \bigl( \mathbf{HU}_{V_1}(\mathbb{F}_{q^m}) \times
 \mathbf{HU}_{V_2} (\mathbb{F}_{q^m}) \bigr) \rtimes \Gamma
 \ar[r]^-{\delta} \ar[d]^-{i}
 & \mathbf{HU}_{V} (\mathbb{F}_{q^m}) \rtimes \Gamma \\
 \bigl( \mathbf{HU}_{V_1} (\mathbb{F}_{q^m}) \rtimes \Gamma \bigr) \times
 \bigl( \mathbf{HU}_{V_2} (\mathbb{F}_{q^m}) \rtimes \Gamma \bigr). &
 }
\end{equation}

\begin{prop}\label{prop:res}
The inflation of $\widetilde{\rho}_{\mathbf{HU}_V (\mathbb{F}_{q^m}),\psi_m}$ by $\delta$
is isomorphic to the restriction of
$\widetilde{\rho}_{\mathbf{HU}_{V_1} (\mathbb{F}_{q^m}),\psi_m} \boxtimes \widetilde{\rho}_{\mathbf{HU}_{V_2} (\mathbb{F}_{q^m}),\psi_m}$
by $i$.
\end{prop}
\begin{proof}
The claim follows from Definition \ref{def:Vh} and
the K\"{u}nneth formula.
\end{proof}

\subsection{Parabolic subgroup}
Let $W$ be an isotropic subspace of
$V$ and $W^{\perp}$ be its orthogonal.
Let $(W_0,h_0)$ be the hermitian space
$W^{\perp}/W$ with induced
hermitian form by $h$.
Let $\mathbf{P}_{W}$ be the stabilizer of $W$
in $\mathbf{U}_{V}$.
Then $\mathbf{P}_{W}$ naturally acts on
$\mathbf{H}_{W^{\perp}}$.
We have the natural homomorphism
$\mathbf{H}_{W^{\perp}} \rtimes \mathbf{P}_{W}
 \to \mathbf{HU}_{W_0}$.
This induces the homomorphism
\[
 (\mathbf{H}_{W^{\perp}} \rtimes \mathbf{P}_{W})(\mathbb{F}_{q^m}) \rtimes \Gamma
 \to \mathbf{HU}_{W_0} (\mathbb{F}_{q^m}) \rtimes \Gamma.
\]

\begin{prop}\label{prop:par}
The restriction of $\widetilde{\rho}_{\mathbf{HU}_V (\mathbb{F}_{q^m}),\psi_m}$
to $(\mathbf{H}_{V} \rtimes \mathbf{P}_{W})(\mathbb{F}_{q^m}) \rtimes \Gamma$
is isomorphic to
\[
 \mathrm{Ind}_{(\mathbf{H}_{W^{\perp}} \rtimes \mathbf{P}_{W})(\mathbb{F}_{q^m}) \rtimes \Gamma}^{(\mathbf{H}_{V} \rtimes \mathbf{P}_{W})(\mathbb{F}_{q^m}) \rtimes \Gamma} \widetilde{\rho}_{\mathbf{HU}_{W_0} (\mathbb{F}_{q^m}),\psi_m}.
\]
\end{prop}
\begin{proof}
We have $\tr\widetilde{\rho}_{\mathbf{HU}_V (\mathbb{F}_{q^m}),\psi_m}(\sigma)=q^n$
by Proposition \ref{prop:trzFr} \ref{enu:trFrj}.
Hence it suffices to check that the
trace of the induced representation at $\sigma$ is $q^n$ by \cite[Theorem 3.3(b)]{GerWeil}.
We see that the trace equals
$\lvert \mathbf{H}_V(\mathbb{F}_q)/\mathbf{H}_{W^{\perp}}(\mathbb{F}_q)
 \rvert q^{n_0}=q^n$,
where $n_0=\dim W_0$,
by the character formula of an induced representation
and Proposition \ref{prop:trzFr} \ref{enu:trFrj}
(\textit{cf.\ } \cite[the proof of Proposition 6.2]{HeWaWeil}).
Hence we obtain the claim.
\end{proof}

\section{Support of trace of extended Weil representation}\label{Sup}
We have the isomorphism
\[
 \iota \colon \mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}) \rtimes \Gamma \xrightarrow{\sim}
 \mathbf{HU}_{(V,-h)}(\mathbb{F}_{q^m}) \rtimes \Gamma;\
 (v,a,g) \mapsto (v,-a,g) .
\]
\begin{lem}\label{iso}
The $\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}) \rtimes \Gamma$-representation
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m^{-1}}$ is isomorphic to the inflation of
$\widetilde{\rho}_{\mathbf{HU}_{(V,-h)}(\mathbb{F}_{q^m}),\psi_m}$ by $\iota$.
\end{lem}
\begin{proof}
This follows from
Lemma \ref{lem:hw} and Proposition \ref{prop:trzFr} \ref{enu:trFrj}.
\end{proof}

Let $(2V,\pm h)=(V,h) \oplus (V,-h)$ be the orthogonal
direct sum.
By \eqref{4-1}, we have the group
homomorphism
\[
 \delta' \colon (\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}) \times \mathbf{HU}_{(V,-h)}(\mathbb{F}_{q^m})) \rtimes
\Gamma \to \mathbf{HU}_{(2V,\pm h)}(\mathbb{F}_{q^m}) \rtimes \Gamma.
\]
Let
\[
\Delta \colon \mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m})\rtimes \Gamma \to
 \mathbf{HU}_{(2V,\pm h)}(\mathbb{F}_{q^m}) \rtimes \Gamma;\ (g,\sigma) \mapsto
 \delta'(g,\iota(g,\sigma)).
 \]
\begin{lem}\label{lem:infd}
The inflation of $\widetilde{\rho}_{\mathbf{HU}_{(2V,\pm h)}(\mathbb{F}_{q^m}),\psi_m}$ by $\Delta$
is isomorphic to the restriction of
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m} \otimes \widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m^{-1}}$.
\end{lem}
\begin{proof}
The inflation of
$\widetilde{\rho}_{\mathbf{HU}_{(2V,\pm h)}(\mathbb{F}_{q^m}),\psi_m}$
by $\delta'$
is isomorphic to the restriction of the representation
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m} \boxtimes
 \widetilde{\rho}_{\mathbf{HU}_{(V,-h)}(\mathbb{F}_{q^m}),\psi_m}$ to
$(\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}) \times
 \mathbf{HU}_{(V,-h)}(\mathbb{F}_{q^m})) \rtimes \Gamma$.
Hence the claim follows from Lemma \ref{iso}.
\end{proof}

Let $\mathbf{Z}_{(V,h)}$ be the center of
$\mathbf{H}_{(V,h)}$.
We put
$\mathbf{ZU}_{(V,h)} = \mathbf{Z}_{(V,h)} \times \mathbf{U}_{(V,h)}$.

\begin{lem}\label{lem:indtri}
The tensor product
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m} \otimes \widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m^{-1}}$ is
isomorphic to the induction of the trivial character
of $\mathbf{ZU}_{(V,h)}(\mathbb{F}_{q^m}) \rtimes \Gamma$.
\end{lem}
\begin{proof}
Take $\zeta \in \mathbb{F}_{q^2} \setminus \{ 1 \}$ such that
$\Nr_{\mathbb{F}_{q^2}/\mathbb{F}_q} (\zeta)=1$.
Let $W$ be the Lagrangian subspace
$\{(v,\zeta v) \in V \oplus V \}$ of $2V$,
which is isotropic via $\pm h$.
Note that $W^{\perp}=W$.
By Proposition \ref{prop:par},
we have an isomorphism
\begin{equation}\label{eq:Ind2VW}
 \widetilde{\rho}_{\mathbf{HU}_{(2V,\pm h)}(\mathbb{F}_{q^m}),\psi_m}|_{(\mathbf{H}_{(2V,\pm h)} \rtimes \mathbf{P}_W )(\mathbb{F}_{q^m}) \rtimes \Gamma}
 \simeq
 \mathrm{Ind}_{(\mathbf{H}_{(W,\pm h)} \rtimes \mathbf{P}_W)(\mathbb{F}_{q^m}) \rtimes \Gamma}^{(\mathbf{H}_{(2V,\pm h)} \rtimes \mathbf{P}_W)(\mathbb{F}_{q^m}) \rtimes \Gamma} \widetilde{\psi}_m,
\end{equation}
where
\[
 \widetilde{\psi}_m \colon (\mathbf{H}_{(W,\pm h)} \rtimes \mathbf{P}_W)(\mathbb{F}_{q^m}) \rtimes \Gamma \to \overline{\mathbb{Q}}_{\ell}^{\times};\
 \left(a,(v,\zeta v),g,\sigma^i \right) \mapsto \psi_m(a).
\]
The image of $\Delta$ is contained in
$(\mathbf{H}_{(2V,\pm h)} \rtimes \mathbf{P}_W)(\mathbb{F}_{q^m}) \rtimes \Gamma$.
The map $\Delta$ induces a bijection
\begin{align*}
 (\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}) \rtimes \Gamma)/
 (\mathbf{ZU}_{(V,h)}&(\mathbb{F}_{q^m}) \rtimes \Gamma) \xrightarrow{\sim} \\
 & ((\mathbf{H}_{(2V,\pm h)} \rtimes \mathbf{P}_W)(\mathbb{F}_{q^m}) \rtimes \Gamma)/((\mathbf{H}_{(W,\pm h)} \rtimes \mathbf{P}_W)(\mathbb{F}_{q^m}) \rtimes \Gamma).
\end{align*}
Inflating \eqref{eq:Ind2VW}
under $\Delta$,
we ontain the claim by
Lemma \ref{lem:infd}.
\end{proof}

\begin{rem}
The proof of Lemma \ref{lem:indtri}
is slightly different from that of \cite[Lemma 7.3]{HeWaWeil}
in the sense that
we use $\{(v,\zeta v) \in V \oplus V\}$
instead of
$\{(v, v) \in V \oplus V \}$.
This choice makes the proof clearer.
\end{rem}

\begin{cor}\label{cor:out}
The trace of the representation
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}$
is zero outside
the conjugates of $\mathbf{ZU}_{(V,h)}(\mathbb{F}_{q^m}) \rtimes \Gamma$.
\end{cor}
\begin{proof}
We take an isomorphism $\overline{\mathbb{Q}}_{\ell}
\simeq \mathbb{C}$. We consider
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}$ and $\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m^{-1}}$ as representations over $\mathbb{C}$.
First, we show that the representation
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m^{-1}}$
is isomorphic to the complex conjugate of
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}$.
Clearly $\psi_m^{-1}$ equals the complex conjugate of $\psi_m$. Hence
$\rho_{\mathbf{H}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m^{-1}}$
is isomorphic to the complex
conjugate of $\rho_{\mathbf{H}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}$.
Therefore,
we obtain the claim by Lemma \ref{lem:hw} and
Proposition \ref{prop:trzFr} \ref{enu:trFrj},
since $\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m^{-1}}$ is irreducible as an $\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m})$-representation.

The required claim follows from this and Lemma \ref{lem:indtri}.
\end{proof}

\section{Main theorem}\label{sec:Main}
Let the notation be as in \S \ref{sec:norm}.
\begin{thm}\label{main}
Let $1 \leq i \leq m$ be an integer.
We set $d=(m,i)$. We take an integer
$t$ such that $ti \equiv d \pmod m$.
Then there is a unique extension
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}$ of
$\rho_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}$ to
$\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}) \rtimes \Gamma$ such that
\begin{equation}\label{char}
\tr \widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}(g,\sigma^i)
=\tr \widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^d}),\psi_d}(\mathrm{N}_{i,t}(g,\sigma^i))
\end{equation}
for any $g \in \mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m})$.
\end{thm}
Since the restriction
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}|_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m})}$ is irreducible,
the uniqueness in Theorem \ref{main} follows immediately from
\eqref{char}. Hence our task in the following
is to show \eqref{char}.

\section{Reduction steps}\label{Red}
To show Theorem \ref{main},
we imitate the arguments in \cite[\S8]{HeWaWeil}.
We recall a known fact.

\begin{lem}\label{lem:ss}
Let $g_0 \in U(V,h)$.
Assume that $g_0$ stabilizes no non-trivial isotropic
subspace. Then $g_0$ is semisimple.
\end{lem}
\begin{proof}
The claim is stated in \cite[proof of Theorem 4.9.2]{GerWeil}.
We recall a proof here (\textit{cf.\ } \cite[\S 8]{HeWaWeil}).
Let $g_0 =su$ be the Jordan decomposition in $U(V,h)$.
If $u \neq \id_V$, then
$\Image (u -\id_V ) \cap \Ker (u-\id_V )$
is a non-trivial isotropic subspace of $V$ stable under $g_0$.
Hence the claim follows.
\end{proof}

We fix $i$ and $t$.
Changing the base field from $\mathbb{F}_q$ to $\mathbb{F}_{q^d}$,
we may assume that $d=(m,i)=1$ (\textit{cf.\ } \cite[Remark 3.1 (ii)]{HeWaWeil}).
We set $g_0=\mathrm{N}_{i,t}(g,\sigma^i)$.
If $g_0$ does not belong to $\mathbf{ZU}_{(V,h)}(\mathbb{F}_q)$,
the both sides of \eqref{char} are $0$ by Corollary \ref{cor:out}.
Assume $g \in \mathbf{Z}_{(V,h)}(\mathbb{F}_{q^m})$.
The left hand side of \eqref{char} equals
$\psi_m(g) q^n$ by Proposition \ref{prop:trzFr} \ref{enu:trFrj}.
On the other hand,
the right hand side of \eqref{char} is
$\psi (g_0 ) \dim \widetilde{\rho}_{\mathit{HU}(V,h),\psi}
 =\psi_m(g) q^n$,
since $g_0=\Tr_{\mathbb{F}_{q^{2m}}/\mathbb{F}_{q^2}}(g)$.
Hence we have \eqref{char} on
$\mathbf{Z}_{(V,h)}(\mathbb{F}_{q^m}) \rtimes \Gamma$.
By applying Lemma \ref{gy} \ref{enu:chiprod}
to the product
$\mathbf{ZU}_{(V,h)}=\mathbf{Z}_{(V,h)} \times \mathbf{U}_{(V,h)}$,
we may assume that $g_0 \in U(V,h)$.

We show Theorem \ref{main} in the case where
$g_0 \in U(V,h)$ by induction on $n$.
The claim for $n=1$ is shown in Proposition \ref{prop:trzFr} \ref{enu:trmd}.
Assume $n>1$.
If $g_0$ stabilizes a non-trivial
orthogonal decomposition of $(V,h)$,
the claim follows from the induction hypothesis
by Lemma \ref{gy} \ref{enu:chiprod},
and Proposition \ref{prop:res}.
If $g_0$ stabilizes a non-trivial isotropic subspace of $V$,
the claim follows from the induction hypothesis
by Lemma \ref{gy} \ref{enu:HInd}
and Proposition \ref{prop:par}.

Now, we may assume that \textit{$g_0$ stabilizes no non-trivial isotropic
subspace and stabilizes no non-trivial orthogonal
decomposition}. By Lemma \ref{lem:ss}, $g_0$ is semisimple.
We write $s$ for $g_0$.

We set $A=\mathrm{End}_{\mathbb{F}_{q^2}}(V)$.
Let $\dagger_h \colon A \to A;\ f \mapsto f^{\dagger_h}$ be the adjoint
involution associated to $h$. Namely, we have
$h(f(x),y)=h(x,f^{\dagger_h}(y))$ for any $x,y \in V$.
Then we have $s^{\dagger_h}=s^{-1}$ by definition.
Hence the involution stabilizes $\mathbb{F}_{q^2}[s]$.
\begin{lem}\label{red}
The subalgebra $\mathbb{F}_{q^2}[s] \subset \mathrm{End}_{\mathbb{F}_{q^2}}(V)$ is a field.
\end{lem}
\begin{proof}
We write as $\mathbb{F}_{q^2}[s] \simeq \prod_{\alpha \in I} F_{\alpha}$, where $F_{\alpha}$ is a field.
Let $e_{\alpha}$ be the idempotent in $\mathbb{F}_{q^2}[s]$ associated to
$F_{\alpha}$.
We have a direct sum
$V=\bigoplus_{\alpha \in I} V_{\alpha}$,
where $V_{\alpha}=\{v \in V \mid e_{\alpha} v=v\}$.
The subspaces $V_{\alpha}$ are $s$-stable.
The involution $\dagger_h$ gives a permutation
$\alpha \mapsto \overline{\alpha}$ on $I$,
with an isomorphism $F_{\alpha} \simeq F_{\overline{\alpha}}$.

Assume that $I$ has at least two elements.
We take $\alpha \in I$.
If $\alpha=\overline{\alpha}$, then $V$ is the orthogonal sum of
$V_{\alpha}$ and $\bigoplus_{\beta \neq \alpha} V_{\beta}$ as hermitian spaces. Actually, we have
\[
h(x_{\alpha},x_{\beta})=h(e_{\alpha}x_{\alpha},x_{\beta})=h(x_{\alpha},e_{\overline{\alpha}} x_{\beta})=h(x_{\alpha},e_{\alpha} x_{\beta})=0
\]
for any $\alpha \neq \beta$, $x_{\alpha} \in V_{\alpha}$ and $x_{\beta} \in V_{\beta}$. Hence $V$ has a non-trivial $s$-stable orthogonal decomposition. This
contradicts to the assumption.

Assume $\alpha \neq \overline{\alpha}$.
Then $V_{\alpha}$ is a non-trivial $s$-stable isotropic
subspace of $V$.
Actually, we have
\[
h(x_{\alpha},x'_{\alpha})=h(e_{\alpha}x_{\alpha},x'_{\alpha})=h(x_{\alpha},e_{\overline{\alpha}} x'_{\alpha})=0
\]
for any $x_{\alpha},x'_{\alpha} \in V_{\alpha}$.
Again this contradicts to the assumption.
Hence $I$ consists of one element. The claim follows.
\end{proof}

We put $E=\mathbb{F}_{q^2}[s] \subset A$ and $E_+ =E^{\dagger_h}$.
We regard $V$ as an $E$-vector space.

\begin{lem}\label{inv}
\begin{enumerate}
\item
The extension
$E/E_+$ is a quadratic extension and
$[E_+:\mathbb{F}_q]$ is odd.
\item
There exists a nondegenerate hermitian form $\widetilde{h} \colon
V \times V \to E$ such that $h=\Tr_{E/\mathbb{F}_{q^2}} \circ \widetilde{h}$.
\item
We have $\dim_E V=1$.
\end{enumerate}
\end{lem}
\begin{proof}
The first claim follows from that
$\dagger_h$ on $\mathbb{F}_{q^2}$ is the $q$-th power map.

Since $\Tr_{E/\mathbb{F}_{q^2}} \colon E \times E \to \mathbb{F}_{q^2}$
is nondegenerate,
we can define a nondegenerate hermitian form
$\widetilde{h} \colon V \times V \to E$
by the condition that
$\Tr_{E/\mathbb{F}_{q^2}}(a \widetilde{h}(v,v')) =h(v,a v')$
for $a \in E, v,v' \in V$.
Hence we obtain the second claim.

The element $s \in E^{\times}$ acts on $V$ as
scalar multiplication.
Since $s$ stabilizes no non-trivial orthogonal decomposition of $(V,\widetilde{h})$ as hermitian spaces over $E$ by the assumption,
we obtain the third claim.
\end{proof}

\section{Proof in the reduced case}\label{Prred}
We will show Theorem \ref{main} in the situation
of Lemma \ref{inv}.
Note that $n$ is odd and $E=\mathbb{F}_{q^{2n}}$ by Lemma \ref{inv}.
By the natural homomorphism
$U(V,\widetilde{h}) \hookrightarrow U(V,h)$, we obtain
\[
 r \colon \mathit{HU}(V,\widetilde{h}) \to \mathit{HU}(V,h);\
 (v,a,g) \mapsto (v,\Tr_{E/\mathbb{F}_{q^2}}(a),g) .
\]
We put $\psi_E =\psi \circ \Tr_{E/\mathbb{F}_{q^2}} \in
\mathbb{F}_{q^n,+}^{\vee}$.
\begin{lem}\label{e1}
\begin{enumerate}
\item\label{enu:inf}
The inflation of $\widetilde{\rho}_{\mathit{HU}(V,h),\psi}$
by $r$ is isomorphic to
$\widetilde{\rho}_{\mathit{HU}(V,\widetilde{h}),\psi_E}$.
\item\label{enu:trs}
Let $s \in U(V,\widetilde{h})$.
Then we have
\[
\tr \widetilde{\rho}_{\mathit{HU}(V,h),\psi}(s)
=\tr \widetilde{\rho}_{\mathit{HU}(V,\widetilde{h}),\psi_E}(s)
=
\begin{cases}
q^n & \textrm{if $s=1$}, \\
-1 & \textrm{otherwise}.
\end{cases}
\]
\end{enumerate}
\end{lem}
\begin{proof}
Let $r^\ast \widetilde{\rho}_{\mathit{HU}(V,h),\psi}$
denote the inflation of $\widetilde{\rho}_{\mathit{HU}(V,h),\psi}$
by $r$.
The restriction of $r^\ast \widetilde{\rho}_{\mathit{HU}(V,h),\psi}$ to the center of $H(V,\widetilde{h})$ is a multiple of $\psi_E$ with multiplicity $q^n$. Hence we have $(r^\ast \widetilde{\rho}_{\mathit{HU}(V,h),\psi})|_{H(V,\widetilde{h})} \simeq \rho_{H(V,\widetilde{h}),\psi_E}$.
Therefore the second claim implies the first one.
The second claim follows from Lemma \ref{lem:hw}
(\textit{cf.\ } \cite[Corollary 3.5]{GerWeil}).
\end{proof}
Let $e=(m,n)$.
We put $\Gamma_e=\mathrm{Gal}(\mathbb{F}_{q^e}/\mathbb{F}_q)$.
For $\alpha \in \Gamma_e$,
we put
$E_{\alpha}=E \otimes_{\mathbb{F}_{q^e},\alpha} \mathbb{F}_{q^m}$
and
$V_{\alpha}=V \otimes_{\mathbb{F}_{q^e},\alpha} \mathbb{F}_{q^m}$.
Note that $E_{\alpha}$ is isomorphic to $\mathbb{F}_{q^{2mn/e}}$.
We have
\[
 E \otimes_{\mathbb{F}_q} \mathbb{F}_{q^m} \xrightarrow{\sim}
 \prod_{\alpha \in \Gamma_e}
 E_{\alpha} ;\
 x \otimes a \mapsto (x \otimes a)_{\alpha}, \
 V \otimes_{\mathbb{F}_q} \mathbb{F}_{q^m} \xrightarrow{\sim}
 \prod_{\alpha \in \Gamma_e}
 V_{\alpha} ;\
 v \otimes a \mapsto (v \otimes a)_{\alpha}.
\]
The base change of $\widetilde{h}$ for
$\mathbb{F}_{q^m}/\mathbb{F}_q$
induces a hermitian form
$\widetilde{h}_{\alpha} \colon V_{\alpha} \times V_{\alpha} \to E_{\alpha}$.
We put
$h_{\alpha}=\Tr_{E_{\alpha}/\mathbb{F}_{q^{2m}}} \circ \widetilde{h}_{\alpha}$,
which is a hermitian form on $V_{\alpha}$ over $\mathbb{F}_{q^{2m}}$.

Let $(V_m,h_m)$ be the base change of $(V,h)$
for $\mathbb{F}_{q^m}/\mathbb{F}_q$.
Then $(V_m,h_m)$ is isomorphic to the
orthogonal sum  $\bigoplus_{\alpha} (V_{\alpha},h_{\alpha})$.
Hence we have the homomorphisms
\begin{equation}\label{ff}
 \prod_{\alpha \in \Gamma_e}
 \mathit{HU}(V_{\alpha},\widetilde{h}_{\alpha})
 \hookrightarrow \prod_{\alpha \in \Gamma_e}
 \mathit{HU}(V_{\alpha},h_{\alpha}) \hookrightarrow
 \mathit{HU}(V_m,h_m) =
 \mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}).
\end{equation}

We put $\alpha_0 =\id_{\mathbb{F}_{q^e}} \in \Gamma_e$.
We omit the index set $\Gamma_e$ in the notation.
We consider the projection
$\mathrm{pr}_{\alpha_0} \colon \prod_{\alpha}
 \mathit{HU}(V_{\alpha},\widetilde{h}_{\alpha}) \to \mathit{HU}(V_{\alpha_0},\widetilde{h}_{\alpha_0})$.
We consider the homomorphism
\[
\left(\prod_{\alpha}\mathit{HU}(V_{\alpha},\widetilde{h}_{\alpha})\right) \rtimes \langle \sigma^e\rangle
\xrightarrow{\mathrm{pr}_{\alpha_0} \times \id}
\mathit{HU}(V_{\alpha_0},\widetilde{h}_{\alpha_0}) \rtimes \langle \sigma^e\rangle,
\]
where the left hand side is regarded as a subgroup of
$\left(\prod_{\alpha}\mathit{HU}(V_{\alpha},\widetilde{h}_{\alpha})\right) \rtimes \Gamma$.
We put $\psi_{E_{\alpha}}=\psi \circ \Tr_{E_{\alpha}/\mathbb{F}_{q^2}} \in \mathbb{F}_{q^{mn/e},+}^{\vee}$.
Let $R$ denote the tensor induction to
$\left(\prod_{\alpha}\mathit{HU}(V_{\alpha},\widetilde{h}_{\alpha})\right) \rtimes \Gamma$
of the inflation of
 $\widetilde{\rho}_{\mathit{HU}(V_{\alpha_0},\widetilde{h}_{\alpha_0}),\psi_{E_{\alpha_0}}}$ by this homomorphism.
 We take $s'_0 \in U(V_{\alpha_0},\widetilde{h}_{\alpha_0})$ with norm $s \in U(V,\widetilde{h})$.
We set $s'=(s'_0,1,\ldots,1) \in \prod_{\alpha}U(V_{\alpha},\widetilde{h}_{\alpha})$.
Recall that we assume $(m,i)=1$.

\begin{lem}\label{pp1}
We have
\[
\tr R(s',\sigma^i)=
\begin{cases}
q^n & \textrm{if $s=1$}, \\
-1 & \textrm{otherwise}.
\end{cases}
\]
\end{lem}
\begin{proof}
The claim follows from \cite[\S2]{GlIsTen}
and Proposition \ref{prop:trzFr} \ref{enu:zFrj} (\textit{cf.\ } \cite[Definitions 10 and 11]{KnoFrob}).
\end{proof}
\begin{cor}\label{cor:infpro}
\begin{enumerate}
\item
The $(\prod_{\alpha}\mathit{HU}(V_{\alpha},\widetilde{h}_{\alpha}))\rtimes \Gamma$-representation $R$ is isomorphic to the inflation of
$\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}$ by the natural
homomorphism
$(\prod_{\alpha}\mathit{HU}(V_{\alpha},\widetilde{h}_{\alpha}))\rtimes \Gamma \to \mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}) \rtimes \Gamma$.
\item\label{enu:trs'sig}
We have
\[
\tr \widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}), \psi_m}(s',\sigma^i)
=\begin{cases}
q^n & \textrm{if $s=1$}, \\
-1 & \textrm{otherwise}.
\end{cases}
\]
\end{enumerate}
\end{cor}
\begin{proof}
By \cite[\S 2]{GlIsTen}, the restriction of $R$
to the subgroup $\prod_{\alpha} \mathit{HU}(V_{\alpha},\widetilde{h}_{\alpha})$ is isomorphic to
$\boxtimes_{\alpha} \widetilde{\rho}_{\mathit{HU}(V_{\alpha},\widetilde{h}_{\alpha}),\psi_{E_{\alpha}}}$.
The inflation of $\widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}$
by \eqref{ff} is isomorphic to
$\boxtimes_{\alpha} \rho_{\mathit{HU}(V_{\alpha},\widetilde{h}_{\alpha}),\psi_{E_{\alpha}}}$
by Proposition \ref{prop:res} and Lemma \ref{e1} \ref{enu:inf}.
By Proposition \ref{prop:trzFr} \ref{enu:trFrj}, we have
$\tr \widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}),\psi_m}(\sigma)
=q^n$.
Since $\sigma$ is a generator
of $\Gamma$,
the first claim follows from Lemma \ref{pp1} (\textit{cf.\ } \cite[Remark 10.2]{HeWaWeil}).
The second claim follows from the first one and
Lemma \ref{pp1}.
\end{proof}

Note that
$(g,\sigma^i)$ is $\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m})$-conjugate to
$(s',\sigma^i)$
for any $g \in \mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m})$ satisfying
$s=\mathrm{N}_{i,t}(g,\sigma^i)$.
Hence,
it suffices to show
\[
 \tr \widetilde{\rho}_{\mathbf{HU}_{(V,h)}(\mathbb{F}_{q^m}), \psi_m}(s',\sigma^i)
 =\tr \rho_{\mathit{HU}(V,h),\psi}(s) .
\]
This follows from Lemma \ref{e1} \ref{enu:trs} and Corollary \ref{cor:infpro} \ref{enu:trs'sig}.

redpajama_set_name: RedPajamaArXiv


text:
Rally to Restore Sanity and/or Fear var en demonstration, der fandt sted den 30. oktober 2010 på The National Mall i Washington, D.C. ført an af Jon Stewart og "en i karakter" Stephen Colbert. Det er en sammensmeltning af to tidligere separate events: Stewarts Rally to Restore Sanity og Colberts eget satiriske modstykke, The March to Keep Fear Alive.

'Rally to restore sanity' er blevet kaldt "en demonstration for de folk, der har haft for travlt til at deltage i demonstrationer". Hovedformålet var at give et sted for deltagerne, hvor de kunne blive hørt, i stedet for dem der normalt bliver hørt, dem Stewart beskriver som de mere højlydte og ekstreme 15–20 procent af amerikanerne der "kontrollerer debatten" i amerikansk politik, som for eksempel Tea Party-bevægelsen og 9/11 Truth-bevægelsen. Nyhedsrapporter har kaldt demonstrationen et satirisk svar på Glenn Becks Restoring Honor-rallyet og "Reclaim the Dream" mod-rallyet.

Noter

Se også
 The Daily Show
 The Colbert Report
 Comedy Central

Demonstrationer
Begivenheder i 2010
Washington D.C.

redpajama_set_name: RedPajamaWikipedia


How can we balance increasing demands for food, fibre and fuel with the need to conserve biodiversity under a changing climate? A paper published last month by UNEP-WCMC and others suggests that a spatially explicit exploration of trade-offs can support better land use planning and reduce biodiversity losses from projected agricultural expansion in East Africa. It also highlights the role of protected areas in helping to reduce biodiversity losses at the national scale.
As population grows, the demand for food and other agricultural products increases and the expansion of agriculture into non cultivated areas seems inevitable. Yet, these may harbour important biodiversity and provide essential ecosystem services, including those necessary to support agriculture. Understanding the trade-offs between increasing agricultural production and conserving biodiversity and ecosystem services, underpinned by spatially explicit information on the effect of different potential trajectories of land use change, can help support sustainable land use planning and policy development.
A new paper led by UNEP-WCMC: Exploring future agricultural development and biodiversity in Uganda, Rwanda and Burundi: a spatially explicit scenario based assessment, assesses the potential impacts of four plausible future socio-economic and climate scenarios for agricultural production on forests and biodiversity in Uganda, Rwanda and Burundi, under different conservation policies. The paper is the result of work conducted as part of a UNEP-WCMC project entitled Commodities and Biodiversity, funded by the John D. and Catherine T. MacArthur Foundation.
For this study, UNEP-WCMC teamed-up with some of the world's leading agricultural research and modelling centres. The socio-economic scenarios for the East Africa region used in this study were developed through the CGIAR Research Program on Climate Change, Agriculture and Food Security (CCAFS). The International Food Policy Research Institute (IFPRI) then quantified the scenarios using the IMPACT agricultural economic model, whose outputs were used by the Centre for Environmental Systems Research (CESR, University of Kassel) to simulate land use and land cover change for 2005 and 2050 using the LandSHIFT model. Finally, UNEP-WCMC assessed potential impacts of these projected changes on biodiversity using a biodiversity metric based on the distribution of suitable habitat for species in the region and weighted by their global range size (i.e. how endemic a species is).
The study found that different future socio-economic scenarios show relatively consistent projections of areas of high agricultural development leading to similar spatial patterns of habitat and biodiversity loss. Yet, the results also show that protected areas help reduce forest loss at the national scale under all scenarios in Rwanda and Burundi where there are few forests outside of these areas, but not in Uganda where forest still exists outside of protected areas. Without protected areas, biodiversity losses are higher under all scenarios, while extending protected areas to include other important biodiversity sites can help reduce biodiversity losses in all three countries.
The question of how to balance agriculture development and biodiversity conservation was the subject of discussions between governments, scientists and businesses at the 8th Trondheim Conference on Biodiversity last week, which UNEP-WCMC helped to facilitate. The importance of scenarios and models to improve understanding of biodiversity-agriculture inter-linkages was well recognised at the event. UNEP-WCMC is working to ensure that these tools and knowledge support more sustainable agricultural development strategies, from global to sub-national scales.


text:
Q: Prevent calls to iPhone launching FaceTime on MacBook in Mojave? It used to be (still is on my High Sierra MacBook Pro) all I needed to do was untick "Calls from iPhone" in macOS FaceTime app prefs:

But now (on my Mojave MacBook Air) this seems to do nothing. All incoming phone calls ring on my iPhone and launch and ring on FaceTime on MacBook. Even when my iPhone is in airplane mode.
I'd rather not (because I'd still like to be able to place outgoing calls—and it's not necessary on my older machine), but I also tried disabling "Cellular / Calls on Other Devices / Allow Calls on [Mojave MBA]" (and disabling the entire "Allow Calls on Other Devices", not just the MBA), yet this didn't stop the calls either.
And I've tried rebooting both MacBook and iPhone. And logging out/in of iCloud.
Apple tech support had me reset PRAM, but this did nothing. Anyone else having this trouble?

A: Try Settings -> Phone -> Calls on Other Devices -> turn it off

redpajama_set_name: RedPajamaStackExchange


Love love love seeing that sweet smile back on her face! Praying she can get rid of these wire "toys" and get back home to her real toys ASAP!
Wow, she looks great! We had this same experience (med and wire-free at 24 hours or so) and it is such a blessing.


text:
AU-UN Joint Special Representative/Joint Chief Mediator's message on the occasion of Eid Al-Adha and the Haj
By United Nations - African Union Mission in Darfur (
EL FASHER (DARFUR), Sudan, October 1, 2014/African Press Organization (APO)/ -- On behalf of the African Union-United Nations Mission in Darfur, I wish the people and the Government of Sudan a peaceful and joyous Eid Al-Adha and Haj Mabroor to the lucky ones.
As we commemorate this occasion, we should all remember the message of peace and sacrifice that this occasion symbolises and draw lessons from the story of Ibrahim and his son (Peace Be Upon Them).
The road to achieving peace is fraught with hurdles, hardships, tests and sacrifice. It is only when we have the will and the resolve that we are able to achieve this condition, to which we all aspire; and which holds so many benefits for all.
The ritual of Haj in its essence is a quest for peace and redemption. Redemption from worldly desires and a renewal of our commitment to Almighty to be fair and compassionate in our relationships with each other. It is also a unique lesson in the value of unity as you see millions congregating in the holy sites to know each other, trade and develop peaceful relationships.
While celebrating this year's Eid, fresh hopes are looming in the horizon. The prospects of an inclusive national dialogue and the recent discussions with armed movements in Addis Ababa breathe fresh air in the quest for peace in Darfur, and indeed all of Sudan. The progress of Darfuri internal dialogue is yet another positive step in the direction of peace stability and development of Darfur. We hope that all these efforts shall converge to achieve strides in the search for sustainable peace, democracy and development for this historic country and its good people.
On this day, we also remember our courageous peacekeepers, the men and women who sacrificed their lives for the people of Darfur.
May the goodwill of this most holy occasion inspire us all to intensify our search for an all-inclusive, just and durable peace in Darfur and, indeed, all of Sudan.
Eid Mubarak to all and Haj Mabroor to those who are participating this year!
Mohamed Ibn Chambas
UNAMID Joint Special Representative / Joint Chief Mediator for Darfur

redpajama_set_name: RedPajamaCommonCrawl


Since you have a penchant for asking questions through a list, here are a few that I would like to get answers for. Before you dismiss it or trash my letter, let me also remind you that I am one of the many 'common man' who had voted and campaigned for you in Delhi Assembly elections. Your actions of late have left a lot to be desired. I don't know where you seek your feedback from these days but as a well wisher let me tell you that your current coterie that gives you a positive feedback about your methods may prove to be your greatest undoing.
Watching the overstretched love saga unfold in three hours confirms one thing - Rockstar is literally Imtiaz Ali's "labour of love" that everyone will not enjoy. Creative film-makers who want to make money spinners walk a tight rope. On one side is their creative thirst that they have to satify and on the other there are market compulsions that need to be addressed. After watching Imtiaz Ali's Rockstar, I believe the product became a victim of walking on this tight rope. The marketing hype gave a wrong impression of what the film was about.
For a nation that savors trash such as Ready and Bodyguard, Ra.One shouldn't be an untouchable. I saw the 3-D version of it and before I get into the details, let me play the Devil's advocate and argue that Ra.One isn't as bad as some people are making it out to be.
If you watch it as any other children's film, you will not trash it as much as you do it now. In fact you will appreciate SRK for spending so much on a children's film. The VFX is better than any of the Bollywood films. The action is not juvenile, though watching the romance king SRK in a superhero's garb is! Ra.One is undoubtedly a film that takes the use of visual effects in Bollywood cinema to another level. The songs are good (a must for Hindi films), SRK and Kareena have acted their parts well and Arjun Rampal impresses in his "cameo" as the main antagonist Ra.One. Several of these pluses helped Ra.One sell its satellite rights for 35 crore and distribution rights in excess of 77 crore. The holiday weekend ensured that Ra.One garnered enough to stay afloat solidly at the BO. On what basis, then, should we trash SRK's dream project?
Mr. Bharat was an enterprising young man who decided to one day open a multi-cuisine restaurant called "Little India" in his home town. The restaurant opened to a rapturous welcome from the local people. He hired staff to care of the guests coming in. 10 waiters to take care of the 30 tables, a team of three chefs, a manager and two security guards. Business started to grow and there was a steady stream of guests daily to the restaurant. Within a short span of time, the restaurant became famous in the whole city.
Pathetic may be a small word to define the despicable act of covering up for the failure of State and Central machineries in the recent Mumbai blasts. But who cares! Mumbai's resilience and ability to fight back may be spoken of but do the people really have an option besides picking up from where life and its uncertainties left them on the dark evening of 13th July? India's restraint is being talked about in international media but truly speaking it is not an option that the State is exercising but a compulsion.
An honest man doesn't become an able leader by default. One of the biggest living examples of that has to be our Prime Minister. Manmohan Singh seems a bit too frail and powerless for a guy with a 20 year long political career (during which he never faced a public election) with stints as the Finance Minister and the Prime Minister of a country. He recently was on air to lay the onus of inept governance on coalition politics. He said that compulsions arising out of coalition politics had tied him down badly and were impeding his fight against corruption. All this came a CWG, Adarsh, IPL-Kochi fiasco, 2-G and Devas later!
My first memories of Laloo take me back to watching him ride a bicycle to the Bihar Vidhan Sabha during the Iraq-Kuwait War. I remember Dad saying that this may just be the recipe that Bihar wanted. A poor man's son who started his politics with JP could re-write Bihar's destiny. No one could gauge that it could be a gimmick until seven years later, when he rode a cycle rickshaw to the court. The image of a simple young man with a purpose was replaced by that of a shrewd, plump looking middle aged man who had perfected the art of playing to the gallery.


text:
Q: UNION of non-nullable columns is nullable Running MS SQL server 2014 express. Suddenly one of the columns in a view became nullable, despite being a union of two non-nullable columns. Please help me understand what's going on here:

and the UNION:

What am I missing?
Edit with more information: when I resave IncomingTransactions view, its column now becomes null, but it shouldn't be! Here's the definition of the Quantity column: (CASE PIN.StatusId WHEN 6 THEN PIN.QuantityReceived WHEN 7 THEN 0 ELSE PIN.QuantityRevised END) AS Quantity. Each of the quantity fields are non-null, and the case statement is exhaustive. The rest of the query is a simple join on the StatusId field, which is a non-null FK, so I'm still lost here.
Edit 2: based on YB's suggestions below, I created a minimal test case that reproduces this behaviour:
Create Table ybTest1 (Q1 decimal (7,2) not null, X int not null);
GO
Create View ybTestNSB As
Select (CASE X WHEN 0 THEN Q1 ELSE CAST(0 as decimal(7,2)) END) AS Q From ybTest1
GO

The Q column in the ybTest1 view is null, even though the case statement is exhaustive. Even if I wrap the 0 in the ELSE branch with CAST(0 as decimal(7,2)), as YB suggested, it's still null. Either CASE doesn't have the semantics I thought, or this is a bug.

A: Just about every column that is computed as a result of an expression is regarded as nullable in SQL Server. The workaround is to use ISNULL as you are doing. This is mentioned in the computed columns section here

The Database Engine automatically determines the nullability of
  computed columns based on the expressions used. The result of most
  expressions is considered nullable even if only nonnullable columns
  are present ... An expression that is nullable can be turned into a
  nonnullable one by specifying ISNULL(check_expression, constant),
  where the constant is a nonnull value substituted for any null result.

but it applies wherever a column is derived as a result of a calculation, including in a view definition.
There is little or no logic to analyse whether nulls are actually possible (sometimes more difficult than it seems as various deprecated set options can induce null rather than overflow errors so even 1 + X could produce a null in your example) and it errs on the side of caution. I don't see any way your case expression can output null in reality but in my experience pretty much any computed column will be treated as nullable except ones wrapped in isnull.
So in your test case you could replace
Create View ybTestNSB As
Select (CASE X WHEN 0 THEN Q1 ELSE CAST(0 as decimal(7,2)) END) AS Q From ybTest1

With
Create View ybTestNSB As
Select ISNULL(CASE X WHEN 0 THEN Q1 END, 0) AS Q From ybTest1

To avoid having to put an annoying entirely redundant expression there.


text:
Respiração aeróbia cloacal ocorre nas tartarugas, onde o processo de substituição de gás carbônico do sangue por oxigênio (Hematose, processo de difusão) ocorre na cloaca.

Fisiologia


text:
Q: Error when installing php54-xml in Centos 6.4 I'm trying to install php54-xml in Centos 6.4 using yum.
I did yum install php54-xml in Console and I get below error:
Error: Package: 1:php-pear-1.9.4-4.el6.noarch (@base)
           Requires: php-cli >= 5.1.0-1
           Removing: php54-cli-5.4.22-1.ius.el6.x86_64 (@php-ius)
               php-cli = 5.4.22-1.ius.el6
           Obsoleted By: php55-cli-5.5.10-1.el6.x86_64 (CentALT)
               Not found
           Available: php-cli-5.3.3-26.el6.x86_64 (base)
               php-cli = 5.3.3-26.el6
           Available: php-cli-5.3.3-27.el6_5.x86_64 (updates)
               php-cli = 5.3.3-27.el6_5
           Available: php-cli-5.3.27-3.el6.x86_64 (CentALT)
               php-cli = 5.3.27-3.el6
           Available: php53u-cli-5.3.27-1.ius.el6.i686 (php-ius)
               php-cli = 5.3.27-1.ius.el6
           Available: php54-cli-5.4.21-1.ius.el6.i686 (php-ius)
               php-cli = 5.4.21-1.ius.el6
           Available: php54-cli-5.4.21-2.ius.el6.i686 (php-ius)
               php-cli = 5.4.21-2.ius.el6
           Available: php54w-cli-5.4.24-1.w6.x86_64 (webtatic)
               php-cli = 5.4.24-1.w6
           Available: php54w-cli-5.4.25-1.w6.x86_64 (webtatic)
               php-cli = 5.4.25-1.w6
           Available: php55w-cli-5.5.8-1.w6.x86_64 (webtatic)
               php-cli = 5.5.8-1.w6
           Available: php55w-cli-5.5.9-1.w6.x86_64 (webtatic)
               php-cli = 5.5.9-1.w6
 You could try using --skip-broken to work around the problem
 You could try running: rpm -Va --nofiles --nodigest

I tried installing php-cli to resolve the issue then I get below error:
Transaction Check Error:
  file /etc/php.ini from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/lib64/php-zts/modules/curl.so from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/lib64/php/modules/curl.so from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/lib64/php-zts/modules/fileinfo.so from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/lib64/php/modules/fileinfo.so from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/lib64/php-zts/modules/json.so from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/lib64/php/modules/json.so from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/lib64/php-zts/modules/phar.so from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/lib64/php/modules/phar.so from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/lib64/php-zts/modules/zip.so from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/lib64/php/modules/zip.so from install of php-common-5.3.27-3.el6.x86_64 conflicts with file from package php54-common-5.4.22-1.ius.el6.x86_64
  file /usr/bin/phar.phar from install of php-cli-5.3.27-3.el6.x86_64 conflicts with file from package php54-cli-5.4.22-1.ius.el6.x86_64
  file /usr/bin/php from install of php-cli-5.3.27-3.el6.x86_64 conflicts with file from package php54-cli-5.4.22-1.ius.el6.x86_64
  file /usr/bin/php-cgi from install of php-cli-5.3.27-3.el6.x86_64 conflicts with file from package php54-cli-5.4.22-1.ius.el6.x86_64
  file /usr/bin/phpize from install of php-cli-5.3.27-3.el6.x86_64 conflicts with file from package php54-cli-5.4.22-1.ius.el6.x86_64
  file /usr/share/man/man1/php.1.gz from install of php-cli-5.3.27-3.el6.x86_64 conflicts with file from package php54-cli-5.4.22-1.ius.el6.x86_64
  file /usr/share/man/man1/phpize.1.gz from install of php-cli-5.3.27-3.el6.x86_64 conflicts with file from package php54-cli-5.4.22-1.ius.el6.x86_64

It seems though there is a conflict between php5.3.x which is the previous version from what I can see with php5.4.x
Is there a way to install php54-xml so I can use dom_import_simplexml function as I'm getting undefined error when I run a PHP script in console?

A: You have bunch of packages from bunch of repos (php-ius, webtatic) besides normal CentOS repos. Packages from these repos might have different way of integrating its packages into the system. I would suggest to stick to one source - e.g. Red Hat supported Software Collections repository - packages installed from there can live one beside the other, i.e. you can have multiple PHP versions installed in parallel:
https://www.softwarecollections.org/en/scls/?search=php
To cleanup your system before installation of packages from Software Collections I would recommend to remove all 3rd party packages. You can display packages you want to remove by:
yum list extras --disablerepo=\* --enablerepo=base --enablerepo=updates
yum remove <all_the_listed_packages>

First command basically lists all packages you have installed from repository other than CentOS's base or updates repo.


From the Russian Aristocrat Collection, this evocative and unique double gun pendant features and oxidised silver and a gold plated gun.
Inspired by the Byronic hero Pechorin from Mikhail Lermontov's 'A Hero of Our Time' Tomasz Donocik's Russian Aristocrat Collection draws upon his opulent old-world aesthetic and deftly re-contextualizes it for the contemporary accessories wearer.
Part of a uniquely captivating collection of jewellery from Tomasz Donocik, which aims to challenge the preconceptions of self-adornment. Through the sensitive fusion of materials his bold, masculine and often androgynous creations inject jewellery design with new meaning and form.


We provide various types of delivery such as FCL/LCL cargo and RO/RO (Roll-on/Roll-off), LO/LO (Lift-on/Lift-off), long-length cargo and break bulk cargo.
Complete transport solution that can include FTL/LTL, multimodal, intermodal ways of transporting general goods, FMCG and oversized or project cargo.
Tailor made Air freight solutions which would meet each of our customer's custom needs.
JSC "Baltijos dominanta" is a fast growing company providing logistics services and solutions for cargo transportation from all over the world to Kazakhstan. Our goal is to establish a trustworthy service which would lead our Clients/Partners to reaching their operational efficiency and cutting costs, by providing them with the most optimized logistics services throughout our well established global network of partners and agents, then implemented with the efforts of our well experienced team members.
Tailor made Air freight solutions which would meet each of our customer's custom needs.
Complex logistics solutions which are provided under the exact request of our customers and combined of more than one type of service provided.
Warehousing solutions including customs bonded warehouses for import/export cargo.


text:
War on Women: Forced Sex in the Name of Religion (Part 1/2)
Iman Bibars
@_ibibars
October 21, 2013 01:42 EDT
Sex jihad should be taken seriously and demands a thorough investigation. This is the first of a two part series.
Jihad in Islam is the call to fight and go to war and be ready to die for the victory of Islam against the enemy, who is defined as an infidel. The call for Jihad is seen, by some religious scholars, as a higher spiritual value. To die for your religion is seen as an honor and is defined as martyrdom whereby the martyr is guaranteed heaven and all its privileges, including not being seen as dead but as alive in the kingdom of God.
The concept of dying and sacrificing your life for God is more common in most of the major religions than one might think. However, the problem lies in who is defined as an infidel, by whom, and the reasons behind it. If two Muslim groups go to war against one another, each defining the other as an infidel, we face a dilemma: how can both be right?
The dead from both camps could be considered martyrs. I raise this issue to show that l believe the concept of Jihad and the definition of the infidel has been abused by warlords across history in all religions, where innocent followers have been brainwashed and manipulated.
Recently, following the Arab Uprisings, Jihad has taken another serious and shocking deviation, allegedly started by a Saudi sheikh who urged Sunni Muslim women in Iraq to enter Jihad in support of those at war against Shi'ites. Jihad has spread like wildfire since the civil war erupted in Syria; but the idea of Jihad for women has been defined as gravely and fundamentally different from that of men.
When 16 Tunisian women were shipped to Syria, they discovered it was for the purpose of sex slavery. Reports state that they were forced to have intercourse with the members of the Free Syrian Army, the main Syrian opposition group known to the West as "the rebels."
Jihad al-Nikah: Sex Jihad
According to the ruling, "Jihad al-Nikah" – literally Jihad of intercourse – opposition fighters to the Assad regime, who are either single men or married but without access to their wives, can create oral contracts (marriages) with virgins, divorced or widowed women for a short period, for the sole purpose of sexual intercourse.
This period might not exceed an hour, in some cases, and is immediately followed by divorce so that other fighters can have the opportunity of intercourse with the same woman. As such, any one of those 16 Tunisian women could have had intercourse up to 10 times per day. The objective – according to the fatwa – is to encourage the fighters and provide them with their religious and natural right to have intercourse, improving their morale and making them better of the battlefield.
This Jihad al-Nikah does not follow Islamic shari'a, as it denies women the customary and obligatory three-month wait between marriages lest she is pregnant; denies her the right of choice; denies her the public announcement which gives oral marriages legitimacy within society; and, of course, denies her any legal rights as the marriage is not registered.
It also contradicts the real objective of marriage in Islam and all the Abrahamic faiths, which is of procreation — meaning the creation of a family.
Whether this fatwa was really issued from a reliable or official source or not, it quickly spread on social media outlets; reports came back from Syria and the two main Muslim Brotherhood sit-ins in Rabaa Adewaya and Nahda Square in Egypt that it was being welcomed and promoted.
One Facebook page with the name "Veiled sisters against secular and unveiled women" stated: "A woman can't fight like a man and her contribution is thus limited to Jihad al-Nikah."
Therefore, the directive is that women must help and perform Jihad by serving the needs of male fighters. They added: "This agreement follows the rules of shari'a" and that no more details could be added on the page.
The Muslim Brotherhood and Jihad al-Nikah
A security source in the Egyptian police revealed that two young women, who were Muslim Brotherhood supporters and who were among those demonstrating in the Rabaa sit-in, had filed reports at the Madinat Nasr police station that they had both been raped under the pretense of Jihad al-Nikah — stating that they had been forced into having sex and had not given their consent. Both were unmarried, aged 26 and 29, and their police report numbers are 1735 and 1736 for the year 2013.
Several weeks later, a new Facebook group called "Ikhwan (MB) without violence" reported that there had been 76 cases of forced sex in the name of Jihad al-Nikah in Rabaa and Nahda in Egypt. This was reported by Hanaa Mohammed, the head of the women's unit at the new emerging (defecting) group "MB without violence," in her live interview on Al Arabiya with Mohamed Warawi in his program, "Egyptian Incident."
In another interview on the Egyptian channel Sada al Balad, Mohammed confirmed the same story. She stated that Mohamed Badie, the Murshid (spiritual leader) of the Muslim Brotherhood, had issued a fatwa to the pro-Muslim Brotherhood sit-ins to allow Jihad al-Nikah. Badie allegedly stated that they were in a period of Jihad, which thus allowed them to establish Nikah contracts during the sit-ins.
Mohammed stated that this Jihad was not only linked to young, Syrian female refugees in Egypt being used as sex slaves, but that it also applied to female members of the Muslim Brotherhood. Mohammed finally confirmed that these contracts were not registered — they specified a duration and time for each contract — and that Jihad intercourse took place in the tents of the sit-ins.
Stories were also reported that several apartments around the sit-ins were rented by the hour, to allow for this Jihad. Women were repeatedly used by several men and stories indicate that it was usually without their consent, especially with younger Syrian refugees.
*[Read the final part on November 2.]
Image: Copyright © Shutterstock. All Rights Reserved
For more than 10 years, Fair Observer has been free, fair and independent. No billionaire owns us, no advertisers control us. We are a reader-supported nonprofit. Unlike many other publications, we keep our content free for readers regardless of where they live or whether they can afford to pay. We have no paywalls and no ads.
In the post-truth era of fake news, echo chambers and filter bubbles, we publish a plurality of perspectives from around the world. Anyone can publish with us, but everyone goes through a rigorous editorial process. So, you get fact-checked, well-reasoned content instead of noise.
We publish 2,500+ voices from 90+ countries. We also conduct education and training programs on subjects ranging from digital media and journalism to writing and critical thinking. This doesn't come cheap. Servers, editors, trainers and web developers cost money. Please consider supporting us on a regular basis as a recurring donor or a sustaining member.
Fair Observer Recommends
Russian President Vladimir Putin, Syrian President Bashar al-Assad and the Russia-Ukraine War
Arab Digest, August 6, 2022
Libyan General Gets Closer to Assad
Gulf State Analytics, June 6, 2020
The NY Times Shames "Peacenik" Tulsi Gabbard
Peter Isackson, August 23, 2019


We formed part of the renowned Grant Associates team working on the masterplan for Kuala Lumpur's (KL) new international financial district known as the Tun Razak Exchange (TRX). The vision for this 30-hectare site was for a highly functioning, innovative, and dynamic urban district that would establish TRX as a financial centre of choice. The intention was also for TRX to become a globally significant civic park and public realm, with integrated environmental design solutions reinforcing sustainability and ecological profile.
These aspirations were all in keeping with evidence that is increasingly showing that those Asian cities that are transforming themselves into clean, green and sustainable places to live and work, are subsequently attracting much greater foreign investment, skilled labour and tourism. New urban development in KL that aspires to be 'World Class' must differentiate itself from unsustainable development models and pursue genuine ecologically-based advantages over competitors.
•	Adhere to environmental policy objectives for KL, Malaysia and the ASEAN region.
•	Make the link between provision of high quality biodiverse green infrastructure and the Centre's economic success.
•	Set a benchmark for KL's emerging integrated sustainable water management strategy.
•	Celebrate Islam's respect for the environment.
•	Promote biophilic design - the instinctive bond between human beings and other living systems.
Green infrastructure proposals included the incorporation of elements of Malaysia's traditional Kampung gardens such as various traditional food and medicinal plants, potentially to be incorporated on tower block roofs and facades (sky-kampungs); inclusion of locally representative habitat analogues with existing lowland dipterocarp urban forest reserves providing references (Bukit Nanas Forest Reserve, Bukit Sungai Putih Forest Reserve and Bukit Sungai Besi Forest Reserve); and informing the architectural design to 'visually borrow' from the surrounding karst topography.
The scheme had become labelled as 'Asia's first Islamic financial centre'. It therefore seemed appropriate to consider whether the development's greenspace, and from an ecological perspective biodiverse components of this, could be inspired in part by Islamic culture. The Qur'an is replete with references to the precious resources of air, water, and land, and proscribes wastefulness.
The prophet Mohammed (may peace be upon him) is said to have encouraged the planting of trees, banned destroying vegetation even during war, loved animals and displayed great kindness to them, and encouraged other Muslims to do likewise. The Qur'an also describes the believing men and women as those who 'walk on the Earth in humility' (25:63). Some scholars have interpreted this verse, and others like it, to mean that Muslims are to protect nature's many bounties.
•	Water. Islamic gardens have always traditionally had water at their heart such as fountains, pools, and flowing watercourses, which in turn produces greenery and shade.
•	Boundary features. Enclosed rectangle courtyards/gardens are a feature of Islamic landscape design and create opportunities for greening facades.
•	Symmetrical lay-outs. Eight-pointed stars, octagons and rectangles express the ideals of harmony and order in Islam, and can be created using beautiful geometric shapes for pools, paths, pavilions or flowerbeds. There is scope here for reference to the ecological architecture of many species on the Malay peninsula to celebrate such symmetry.
It was, however, also considered important not to uniquely focus on Islamic design references, recognising that many Malaysians believe that park and garden designs should reflect a tropical theme, and specifically showcase Malaysian identity, i.e. be a harmonious fusion of Malay, Chinese, and Indian culture, referencing artistic history, legends, fables, and customs.


In a medium skillet, cook the pancetta and oil over medium-low heat, stirring occasionally, until the pancetta is browned and the fat has rendered, about 10 minutes. Remove from heat.
In a large bowl, whisk together the eggs, Pecorino, Parmesan, and pepper. Bring a pot of salted water to boil; cook pasta until al dente, drain, and immediately toss with the egg mixture. Add the pancetta along with its rendered fat, toss well, and serve immediately with extra pepper, if desired.
If the heat of the cooked pasta does not quite set the eggs, you can gently toss it again over very low heat in the pan you cooked the pancetta in. But stir continuously and make sure the heat is as low as it can go so that you don't wind up with scrambled eggs.
I'm making this tonight. Is it really 2 tablespoons of black pepper?
Made.this!! Wow! My best pasta dish yet.
I know this is heresy, but I grew up eating my mother's carbonara and my favorite way to eat it was the next morning, heated up for breakfast, when the eggs in the leftovers did indeed scramble. No matter, it was delicious.
Thank you all for catching this, Sara double-checked her notes and it should be 1 cup of each type of cheese—the recipe has been updated!
According to the USDA, 1 cup of Paresano-Reggiano is 4 oz by weight, and according to a recipe I found from Martha Stewart and according to Gourmetsleuth.com, 1 cup of shredded Pecorino Romano weighs 2.8 oz, so this recipe calls for about 7 oz of cheese rather than a pound. This is still quite excessive compared to other pasta carbonara recipes; I read several and most called for 2 ounces of cheese, with one requiring 3 ounces and another 4 ounces, but this is still much more reasonable than if you assume that 8 ounces of cheese by volume equals 8 ounces of cheese in weight. Perhaps Sara Jenkins is a die-hard cheese lover, with which I could certainly commiserate.
I had the same concern about the amount of cheese. Maybe the recipe meant to read "2 cups Pecorino Reggiano -OR- 2 cups Parmigiano-Reggiano"?
I think the cheese amount is quite off (4 cups cheese = 1 pound to a pound of spaghetti).


OCD | young, wild, and tidy.
This past year, I have been opening my heart and practicing more vulnerability in the hopes that living more exposed, is living more freely. So, with that vulnerability, I want to share something I've struggled with for many years.


text:
A Man on the Edge az Iron Maiden brit heavy metal együttes 1995-ös The X Factor albumának első kislemezen kiadott dala. Ez volt az Iron Maiden első hivatalos kiadványa a zenekarhoz 1994-ben csatlakozott Blaze Bayley énekessel. A kislemez a brit slágerlistán a 10. helyet szerezte meg.

Története
A Man on the Edge dal volt hivatott felvezetni az Iron Maiden tizedik stúdióalbumát, amelyet új énekessel, a Bruce Dickinson helyére érkezett Blaze Bayley-vel rögzítettek. Bayley és Janick Gers gitáros közös szerzeményének dalszövegét a Michael Douglas főszereplésével készült Összeomlás című film ihlette. A dal refrénje maga a film eredeti angol címe: Falling Down. A promóciós videóklipben javarészt a zenekar látható, ahogy előadják a dalt; egyedül a refrénhez készítettek animációt, amelyben öltönyös, aktatáskás emberek zuhannak ki toronyházakból.

A Man on the Edge kislemez különböző formátumokban jelent meg 1995. szeptember 25-én az EMI kiadásában. Mindegyik kiadványon szerepel a The Edge of Darkness című dal, ami végül a nagylemezre is felkerült. A The Edge of Darkness dalszövegét szintén egy mozifilm, a vietnámi háború idején játszódó, Francis Ford Coppola rendezte, 1979-es Apokalipszis most ihlette. A 12"-es maxi-single hanglemezen a csak itt hallható I Live My Way a harmadik szám. A CD formátumban kiadott kislemezekre szintén olyan dalok kerültek fel, amelyek a nagylemezen nem szerepelnek: Dave Murray és Steve Harris közös szerzeménye, a Justice of the Peace, valamint egy újabb Bayley-Gers dal Judgement Day címmel. A maxi-single mellett egy dupla CD-s boxset változat is megjelent, ahol a dalok után az új énekessel készült kétrészes interjú hallható.

A Man on the Edge kislemez három hétig szerepelt a brit slágerlistán. Legmagasabb pozíciója a 10. hely volt, ami a Bayley-érában megjelent kislemezek közül a legjobb. A 2000-ben megjelent The Wicker Man kislemez B-oldalán a zenekarba akkor visszatérő Bruce Dickinson előadásában hallható a Man on the Edge dal egy 1999-es koncertfelvételről.

Számlista
12" Maxi-Single
 Man on the Edge (Blaze Bayley, Janick Gers) – 4:19
 The Edge of Darkness (Steve Harris, Bayley, Gers) – 6:40
 I Live My Way (Harris, Bayley, Gers) – 3:48

CD Maxi-Single
 Man on the Edge (Bayley, Gers) – 4:19
 The Edge of Darkness (Harris, Bayley, Gers) – 6:40
 Justice of the Peace (Dave Murray, Harris) – 3:38
 Judgement Day (Bayley, Gers) – 4:09

Box Set – CD 1
 Man on the Edge (Bayley, Gers) – 4:19
 The Edge of Darkness (Harris, Bayley, Gers) – 6:40
 Judgement Day (Bayley, Gers) – 4:09
 Blaze Bayley interview, Part I – 5:41

Box Set – CD 2
 Man on the Edge (Bayley, Gers) – 4:19
 The Edge of Darkness (Harris, Bayley, Gers) – 6:40
 Justice of the Peace (Murray, Harris) – 3:38
 Blaze Bayley interview, Part II – 5:55

Közreműködők
 Blaze Bayley – ének
 Dave Murray – gitár
 Janick Gers – gitár
 Steve Harris – basszusgitár
 Nicko McBrain – dobok

Források

Iron Maiden-kislemezek
1995 kislemezei


S -14° 16' 0" and E 50° 10' 0"
Sambava in Sava Region with it's 31,522 residents is a town located in Madagascar about 365 mi (or 588 km) north-east of Antananarivo, the country's capital place.
Time in Sambava is now 02:35 AM (Friday). The local timezone is named Indian / Antananarivo with an UTC offset of 3 hours. We know of 8 airports nearby Sambava. The closest airport in Madagascar is Sambava Airport in a distance of 1 mi (or 2 km), South-East. Besides the airports, there are other travel options available (check left side).
There is one Unesco world heritage site nearby. It's Rainforests of the Atsinanana in a distance of 59 mi (or 95 km), North-West.
Being here already, you might want to pay a visit to some of the following locations: Ambilobe, Antsiranana, Antsohihy, Andilamena and Fenoarivo Atsinanana. To further explore this place, just scroll down and browse the available info. Let's start with some photos from the area.
TRAVERSEZ AVEC MOI LA VILLE DE SAMBAVA A MADAGASCAR !!!!!
Sava is a region of Madagascar. Its capital is Sambava. Until 2009 Sava belonged to Antsiranana Province. The region is situated at the northern part of the east coast of Madagascar. It is bordered by Diana to the north, Sofia to the west, and Analanjirofo to the south. The population was estimated to be 805,300 in 2004 and the total area is 25,518 km . The region contains wild areas such as Marojejy National Park.
Located at -14.27, 50.17 (Lat. / Lng.), about 0 miles away.
Sambava Airport is an airport in Sambava, Sava Region, Madagascar.
Located at -14.2786, 50.1747 (Lat. / Lng.), about 1 miles away.
Sambava is a city and commune (commune urbaine; Malagasy: kaominina) at the east coast of northern Madagascar. It is the capital of Sambava District and Sava Region. The population of the commune was estimated to be approximately 40,000 in 2001 commune census. Sambava is served by a local airport. In addition to primary schooling the town offers secondary education at both junior and senior levels. The town provides access to hospital services to its citizens.
Located at -14.2667, 50.1667 (Lat. / Lng.), about 0 miles away.
Nosiarina is a town and commune in northern Madagascar. It belongs to the district of Sambava, which is a part of Sava Region. The population of the commune was estimated to be approximately 12,000 in 2001 commune census. Primary and junior level secondary education are available in town. The majority 99% of the population of the commune are farmers. The most important crops are rice and vanilla, while other important agricultural products are banana and coffee.
Located at -14.2167, 50.05 (Lat. / Lng.), about 9 miles away.
Morafeno is a town and commune in northern Madagascar. It belongs to the district of Sambava, which is a part of Sava Region. The population of the commune was estimated to be approximately 9,000 in 2001 commune census. Only primary schooling is available in town. The majority 99.9% of the population the commune are farmers. The most important crop is vanilla, while other important products are coffee, cloves and rice. Services provide employment for 0.1% of the population.
Located at -14.4, 50.0333 (Lat. / Lng.), about 13 miles away.
Farahalana is a town and commune in northern Madagascar. It belongs to the district of Sambava, which is a part of Sava Region. The population of the commune was estimated to be approximately 23,000 in 2001 commune census. Primary and junior level secondary education are available in town. The majority 96% of the population the commune are farmers. The most important crop is vanilla, while other important products are coffee, cloves and rice.
Located at -14.4333, 50.1667 (Lat. / Lng.), about 12 miles away.
Bemanevika is a town and commune in northern Madagascar. It belongs to the district of Sambava, which is a part of Sava Region. The population of the commune was estimated to be approximately 13,000 in 2001 commune census. Primary and junior level secondary education are available in town. The majority 95% of the population in the commune are farmers. The most important crop is vanilla, while other important products are coffee, coconut and rice.
Located at -14.1167, 50.1 (Lat. / Lng.), about 11 miles away.
Anjangoveratra is a town and commune in northern Madagascar. It belongs to the district of Sambava, which is a part of Sava Region. The population of the commune was estimated to be approximately 16,000 in 2001 commune census. Primary and junior level secondary education are available in town. The majority 98.5% of the population in the commune are farmers, while an additional 1% receives their livelihood from raising livestock.
Located at -14.1333, 50.0833 (Lat. / Lng.), about 11 miles away.
Ambohimalaza is a town and commune in northern Madagascar. It belongs to the district of Sambava, which is a part of Sava Region. The population of the commune was estimated to be approximately 8,000 in 2001 commune census. Only primary schooling is available in town. The majority 99.99% of the population in the commune are farmers. The most important crop is vanilla, while other important products are coffee, cloves and rice. Services provide employment for 0.01% of the population.
Located at -14.3167, 50.05 (Lat. / Lng.), about 9 miles away.
These are some bigger and more relevant cities in the wider vivinity of Sambava.


Starting January 1st, 2010 Up The Ra! will open membership to all. Any political orientation will be allowed. There will be zero tolerance for trolling and blatant attacks on anyone.
Republican Socialism - The ONLY Working Class Alternative!
Our flag is the Starry Plough. Our Class is the Working Class. Our enemy is Foreign Imperialism and Native Capitalism. We are the IRP's. Here we are and here we will remain until Ireland becomes a socialist society in the image of Connolly, Mellows and Costello.
has many applied to join this republican forum so far?
Speaking of which, I'd suggest looking through the membership list and noticing how many have joined since January last year -- and how many of those have ever posted anything at all. Quite enlightening.
Provos are welcome to join and debate on here. There never offically was a "no provo" policy. When elle became a mod, she drove away those whom she disagreed with, and banned many for no reason. This forum went downhill fast because of her, and also due to myself neglecting this forum. By opening up the forum hopefully old friends will come back and that some new ones join.
I think peace lovers would know (assuming they know about members of this site) that they wouldn't be very welcomed by certain people. But hey, they aren't even interested in talking with the groups carrying on the fight, they made that quite clear when they denied to talk with the Real IRA in march, so it's their fault.
Last edited by TheBoondockSaint9mm on Sat Jan 09, 2010 12:37 am, edited 1 time in total.
"Oglaigh na hEireann will continue to strike at the British occupation forces when, wherever and whenever we decide."
To all new forum members....''Welcome to the Jungle''!....I'm the quiet one.
BISCUITS wrote: To all new forum members....''Welcome to the Jungle''!....I'm the quiet one.
Aallowing Sinn Féin members/supporters to register with the Up The Ra! Forum will encourage a lot more interesting debates - maybe we'll even be able to "re-convert" some of their members/supporters into being Republicans again.
Last edited by Costello on Fri Jan 15, 2010 1:00 am, edited 1 time in total.
"I owe my allegiance to the working class."
!! FREE ALL REPUBLICAN PRISONERS NOW !!


Cynthia D. Brittain's practice focuses on domestic and international estate planning for high-net worth individuals. As part of her private client services practice, she assists multi-generational families with business succession, family governance, and tax planning.
Prior to joining the firm, Ms. Brittain was the Senior Wealth Planning Strategist and Director of Planning for Abbot Downing, Wells Fargo's boutique family- office division. Ms. Brittain has worked in London and Hong Kong as a private client services attorney. Ms. Brittain has lectured nationwide on estate planning, international planning, and asset protection.
leaders such as Chanel, Estee Lauder, Dior, L'Oreal, and Procter & Gamble but also numerous small and mid-sized start-ups and turn-arounds. After beginning his career in marketing at Colgate-Palmolive in New York, he relocated to Europe where he spent 10 years in the fragrance industry. Upon his return to the US he founded Fragrance Solutions, a market research firm serving the prestige fine fragrance market, which he successfully sold to the NPD Group. He then founded jasmin.com a venture backed e-commerce fragrance business which partnered with Ulta and P&G. Jasmin.com was successfully sold to Asford.com. Dale's corporate experience includes serving as CEO in the US of two French luxury companies – Fauchon and Christofle and of a venture backed wireless business.
Seasoned fundraiser, Rebecca Summers began her career in development in 1993 while teaching high school in New Hampshire. What started as a means to improve education and programming within her small school of 300 students, quickly developed into a passion to improve the quality of education across the state. Soon Rebecca was fundraising for multiple school districts and educational consortiums, writing federal and foundation grants to support after school programs, innovative practices, professional development, alternative education, school reform, and more.
In 2007 she moved to Boston where she raised millions for the Massachusetts Society for the Prevention of Cruelty to Children. She later moved to Bridge Over Troubled Waters, Boston's premier organization dedicated to transforming the lives of runaway, homeless, and high-risk youth. At Bridge, Rebecca broke fundraising records, leading a campaign to build the city's first youth shelter and bringing in the largest gifts in agency history. Rebecca and Bridge were also awarded the prestigious Bank of America Neighborhood Builder's Award and the Citizen's Bank Champions in Action Award. In 2012 Rebecca returned to Santa Barbara, where she grew up, and is currently a Development Officer for the Santa Barbara Museum of Natural History.
At present she is in the midst of a $30 million campaign to transform the 100 year old institution, repairing and refreshing facilities, improving accessibility, and creating new indoor and outdoor exhibitions. She holds a Bachelor's Degree in Natural Science from Bennington College and a Master's Degree in Counseling Psychology from Goddard College.
Prior to launching JZPR (a full service public relations firm specializing in lifestyle, hospitality, entertainment and nonprofit industries), Jennifer Zacharias spent five years as Director of Public Relations for a Santa Barbara public relations and marketing firm. Before that, she spent five years at CNN's Larry King Live show, as a key member of the show's production team. There, she helped produce nightly shows and assisted the management of a team that evaluated, sought, and booked up-to-the-minute guests for cable television's highest-rated news program. Before CNN, Jennifer lived in Nice, France, where she worked as a translator and press attaché for Canal+, France's premier national cable network, reporting from the Cannes Film Festival, and working film industry. Jennifer is very involved in the local nonprofit community.
She is on the Board of Directors for Habitat for Humanity, and the co-chair for their current capital campaign, "Building Homes, Building Hope", to build 12 affordable homes in Santa Barbara. She is a member of the American Heart Association's Circle of Red, and sits on their "Go Red For Women Luncheon" committee. Jennifer was also involved in the Women's Fund of Santa Barbara, having served on their Steering Committee as their Communications Chair for 3 years. Jennifer is a past president of the Junior League of Santa Barbara, where she's been actively involved for nine years, sat on the Board of Directors for three years, and is currently an advisor to their campaign to build a new Children's Library in Santa Barbara.
Whitney Webb is the Director of Operations, as well as a head Trainer and co-director of Camp Start-Up. Before coming to IMI, Whitney was a Kiva Fellow in Kigali, Rwanda where she worked to strengthen partnerships and increase the reach of microfinance, through employee trainings and industry research. Whitney graduated from the University of Florida, and, upon graduation, accepted a position with Porter Orlin hedge fund, in NYC, where she performed investment research for the managing director. During her three years there she also volunteered at The International Center, teaching English and mentoring members who were new to the US. She left New York to travel and volunteer with various NGOs within southeast Asia and South America in the areas of education and business sustainability. Whitney is a board member and volunteer business mentor at Youth Interactive, a non-profit teen center in Santa Barbara working to promote leadership and entrepreneurial skills.
Ernest Smith spent 35 years in the Financial services industry prior to retiring in 2006. He was President of Fidelity Information Services and Co-COO of Fidelity National Financial ( both Fortune 500 Companies) after retiring he was a founding partner of Real Estate Digital and Managing Partner for Smith-Cavin Partners. He has an MBA from Oxford Brooke's University ( formerly Oxford Polytechnic). He was a former Board member of the Dreams Come True Foundation in Jacksonville, Florida and former Board member of the Oakland School of Performing Arts ( a charter school founded by Jerry Brown, governor of California. He has served as a business mentor at Youth Interactive for one year and recently joined the Board.
Kyle Ashby focuses on strategy, branding and marketing for startups, lifestyle brands, technology companies and organizations as the director of the advisory firm Kaldera. He is also the founder of StartupSB, an organization connecting Santa Barbara entrepreneurs and startups to resources, knowledge and capital, and Travel in Central America, a company providing unique experiences in Central America. Kyle was on the founding team of BrandLive and Mobile Corporation and has worked in executive, marketing and project management roles at multinational companies and technology startups. He has also taught classes in marketing, innovation, and entrepreneurship at UC Santa Barbara and Antioch University. Kyle is an active startup advisor and on the leadership team of the Startup Champions Network. Kyle has an MBA from Babson College, degrees in Film Studies and Geography from UC Santa Barbara and is a certified Project Manager and Scrum Master.
Media4Good (M4G) was started as a new model of charity and was nominated by Ernst & Young as social entrepreneur of the year! Social entrepreneur Natalie Gensac founded the production based organization in 2005 to use the power of media to reach out to a new generation of intelligent givers—people who need to know their contributions will truly make a sustainable difference in the global struggle against poverty and illiteracy. To that end she founded the Youth Interactive program in 2008 initially working with communities oversees until in 2012 she decided to concentrate on creating an impact at home and Youth Interactive Santa Barbara opened its doors in the fall of that year.
Her core focus is to support the two groups essential to social and economic advancement in our world today: women and children.