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\title{The normalization constant of \\ a certain invariant measure on $GL_n(D_\bA)$}
\author{Yoshihide Nakamura and Takao Watanabe
\footnote{The second author was partly supported by Grant-in-Aid for Scientific Research, Ministry of Education, Culture, Science and Technology, Japan. }}

\begin{document}
\maketitle
\footnotetext{2000 Mathematics Subject Classification. Primary 11R52; Secondary 11H50}
\begin{abstract}
The ratio of the Tamagawa measure and a certain invariant measure on the group $GL_n(D_\bA)$ is computed, where $D_\bA$ is the ad\`ele of a division algebra $D$ over a global field.  An explicit formula of the ratio is described in terms of the special values of the zeta function of $D$. This formula yields ({\romannumeral 1}) an explicit lower bound of the Hermite--Rankin constant $\gamma_{n,m}(D)$ of $D$ and ({\romannumeral 2}) an explicit asymptotic behavior of the distribution of rational points on Brauer--Severi variety.
\end{abstract}


\subsection*{Introduction}
Let $G$ be a connected reductive algebraic group defined over a global field $k$ and $G(\bA)$ the ad\`ele group of $G$. Since $G(\bA)$ is a locally compact unimodular group, it has a non-trivial invariant measure. The invariant measure $\omega^G_\bA$ on $G(\bA)$ induced from the invariant gauge form $\omega^G$ on $G$ defined over $k$ is called the Tamagawa measure, which is a canonical invariant measure on $G(\bA)$ in a sense.
There is another useful invariant measure on $G(\bA)$ defined as follows: We fix a parabolic subgroup $R$ of $G$ defined over $k$ and a maximal compact subgroup $K$ of $G(\bA)$ which possesses an Iwasawa decomposition $G(\bA) = R(\bA)K$. Let $\omega^R_\bA$ denote the Tamagawa measure of $R(\bA)$ and $\omega_K$ the invariant measure on $K$ normalized so that $\omega_K(K) = 1$. Then the product $\omega^R_\bA \cdot \omega_K$ defines an invariant measure, say $\omega_{(G(\bA), R(\bA))}$, on $G(\bA)$. Since an invariant measure is unique up to constant, there is the positive constant $C_{G,R,K}$ such that $\omega^G_\bA = C_{G,R,K} \cdot \omega_{(G(\bA), R(\bA))}$. We call $C_{G,R,K}$ the normalization constant of $\omega_{(G(\bA), R(\bA))}$.

In general, the constant $C_{G, R, K}$ has a description by an Euler product such as
$$ C_{G, R, K} = \prod_{v} \epsilon_v J_v \,,$$
where $v$ runs over all places of $k$ and $\epsilon_v$ are elementary constants determined by $G$ and $R$.  Every $J_v$ is an integral of the form
$$ J_v = \int_{U_R^-(k_v)} \eta_v(u_v) d\omega_v^{U_R^-}(u_v) \,, $$
where $U_R^-$ denotes the unipotent radical of the opposite $k$-parabolic subgroup of $R$ and $\eta_v$ the function on $G(k_v)$ induced by the modular character of $R(k_v)$. In \S 2.1, we will show this formula in detail. 
In principle, the constant $C_{G, R, K}$ can be explicitly computed by using this formula and the reduction of $J_v$ to the cases of semisimple rank one groups due to Gindikin--Karpelevi\v{c} formula (see \S 2.2 and \S 2.3). Indeed, an explicit formula of $C_{G, R, K}$ is known in the case where $G$ is a $k$-quasisplit group ([L]), an orthogonal group ([Ik]) and a unitary group ([Ic]). However, except for the case that $G$ is a $k$-quasisplit group, its actual computation is not easy.

In this paper, we give an explicit formula of $C_{G, R, K}$ in the case that $G$ is an inner $k$-from of general linear groups, \textit{i.e.}, $G$ is the algebraic group determined by $G(k) = M_n(D)^\times = GL_n(D)$, where $D$ is a division $k$-algebra. We fix a minimal $k$-parabolic subgroup $P$ of $G$ and a certain maximal compact subgroup $K$ of $G(\bA)$ such that $G(\bA) = P(\bA)K$. Since $C_{G, R, K} = C_{G, P, K}/C_{M_R, M_R \cap P, M_R(\bA) \cap K}$ holds for any standard $k$-parabolic subgroup $R$ of $G$ with a Levi subgroup $M_R$, it is sufficient to compute $C_{G, P, K}$. Then the integral $J_v$ occurring in the Euler product of $C_{G, P, K}$ is decomposed into a product of integrals over a division $k_v$-algebra $D(v)$ which is equivalent to $D \otimes_k k_v$ in the Brauer group of $k_v$. By computing the integrals over $D(v)$, we obtain the value of $J_v$, and as a consequence, the explicit formula of $C_{G, P, K}$ is described in terms of special values of the zeta function $Z_D(s)$ of $D$ (see \S 3.6).

Our motivation of computing $C_{G, R, K}$ is the following. In [Wa], the second author introduced the fundamental Hermite constant $\gamma(G,Q;k)$ of the pair of a connected reductive $k$-group $G$ and a maximal $k$-parabolic subgroup $Q$ of $G$. Then the constant $C_{G,Q,K}$ appeared in the Minkowski--Hlawka type lower bound of $\gamma(G,Q;k)$. Thus an explicit formula of $C_{G,R,K}$ yields an explicit lower bound of $\gamma(G,Q,k)$. In the case of $G(k) = GL_n(D)$, we will take up this application in \S 4.2. Moreover, we will apply the formula of $C_{G,R,K}$ to give an explicit asymptotic behavior of the distribution of rational points on Brauer--Severi variety in \S 4.3.





\vskip 8mm
\subsection*{Notations}
Let $k$ be a global field, \textit{i.e.}, an algebraic number field of finite degree over $\bQ$ or an algebraic function field of one variable over a finite field. In the latter case, we identify the constant field of $k$ with the finite field $\bF_q$ with $q$ elements. Let $\fV$ be the set of all places of $k$.  We write $\fV_\infty$, $\fV_\bR$, $\fV_\bC$ and $\fV_f$ for the sets of all infinite places, all real places, all imaginary places and all finite places of $k$, respectively. For $v \in \fV$, $k_v$ denotes the completion of $k$ at $v$. If $v \in \fV_f$, $\fo_v$ denotes the maximal compact subring of $k_v$ and $q_v$ the cardinality of the residual field of $k_v$. We fix, once and for all, a Haar measure $\mu_v$ on $k_v$ normalized so that $\mu_v(\fo_v) = 1$ if $v \in \fV_f$, $\mu_v([0,1]) = 1$ if $v \in \fV_\bR$ and $\mu_v(\{a \in k_v \,:\, a \ol a \le 1\}) = 2\pi$ if $v \in \fV_\bC$. Then the absolute value $|\cdot|_v$ on $k_v$ is defined as $|a|_v = \mu_v(aC)/\mu_v(C)$, where $C$ is an arbitrary compact subset of $k_v$ with nonzero measure. Let $\bA$ be the ad\`ele ring of $k$, $|\cdot |_\bA = \prod_{v \in \fV} |\cdot|_v$ the idele norm on the idele group $\bA^\times$ and $\mu_\bA = \prod_{v \in \fV} \mu_v$ an invariant measure on $\bA$. The measure $\mu_\bA$ is characterized by
\begin{equation*}
\mu_\bA(\bA/k) = \left \{
\begin{array}{cl}
|D_k|^{1/2} & (\mbox{if $k$ is an algebraic number field of discriminant $D_k$}). \\
q^{g(k)-1} & (\mbox{if $k$ is a function field of genus $g(k)$}).
\end{array} \right.
\end{equation*}
The zeta function $\zeta_k(s)$ of $k$ is defined to be
$$ \zeta_k(s) = \prod_{v \in \fV_f} (1 - q_v^{-s})^{-1} \,. $$
The residue of $\zeta_k(s)$ at $s = 1$ is denoted by $\rho_k$.

Let $k_1$ be an arbitrary field. If $\fA_1$ is a central simple $k_1$-algebra, then $\Nr_{\fA_1/k_1}$ and $\tau_{\fA_1/k_1}$ stand for the reduced norm and the reduced trace of $\fA_1$, respectively. The unit group of $\fA_1$ is denoted by $\fA_1^\times$.



\vskip 8mm
\subsection*{1 Normalization constant of an invariant measure}

\subsubsection*{1.1 Tamagawa measure}
Let $G$ be a connected affine algebraic group defined over $k$. For any $k$-algebra $A$, $G(A)$ stands for the set of $A$-rational points of $G$. Let $\bX^*(G)$ and $\bX^*_k(G)$ be the free $\bZ$-modules consisting of all rational characters and all $k$-rational characters of $G$, respectively. The absolute Galois group $\Gal(\ol k/k)$ acts on $\bX^*(G)$. The representation of $\Gal(\ol k/k)$ in the space $\bX^*(G) \otimes_\bZ \bQ$ is denoted by $\sigma_G$ and the corresponding Artin $L$-function is denoted by $L(s, \sigma_G) = \prod_{v \in \fV_f} L_v(s, \sigma_G)$. We set $\sigma_k(G) = \lim_{s \to 1} (s - 1)^{n}L(s, \sigma_G)$, where $n = \rank\, \bX^*_k(G)$. Let $\omega^G$ be a nonzero right invariant gauge form on $G$ defined over $k$. From $\omega^G$ and the fixed Haar measure $\mu_v$ on $k_v$, one can construct a right invariant Haar measure $\omega^G_v$ on $G(k_v)$. Then, the Tamagawa measure on $G(\bA)$ is well defined by
\begin{equation*}
\omega^G_\bA = \mu_\bA(\bA/k)^{-\dim G}\omega^G_\infty \omega^G_f \,,
\end{equation*}
where
\begin{equation*}
\omega^G_\infty = \prod_{v \in \fV_\infty} \omega^G_v \;\; \text{and} \;\; \omega^G_f = \sigma_k(G)^{-1}\prod_{v \in \fV_f} L_v(1, \sigma_G)\omega^G_v \,.
\end{equation*}

For each $g \in G(\bA)$, we define the homomorphism $\vartheta_G(g) \,:\, \bX^*_k(G) \longrightarrow \bR_+$ by $\vartheta_G(g)(\chi) = |\chi(g)|_\bA $ for $\chi \in \bX^*_k(G)$. Then $\vartheta_G$ is a homomorphism from $G(\bA)$ into $\Hom_\bZ(\bX^*_k(G), \bR_+)$. We write $G(\bA)^1$ for the kernel of $\vartheta_G$. The Tamagawa measure $\omega_{G(\bA)^1}$ on $G(\bA)^1$ is defined as follows:

\begin{itemize}
\item The case of $\ch(k) = 0$. If a $\bZ$-basis $\chi_1, \cdots, \chi_{n}$  of $\bX^*_k(G)$ is fixed, then $\Hom_\bZ(\bX^*_k(G), \bR_+)$ is identified with $(\bR_+)^n$ and $\vartheta_G$ gives rise to an isomorphism from $G(\bA)^1 \backslash G(\bA)$ onto $(\bR_+)^{n}$. Put the Lebesgue measure $dt$ on $\bR$ and the invariant measure $dt/t$ on $\bR_+$. Then $\omega_{G(\bA)^1}$ is the measure on $G(\bA)^1$ such that the quotient measure $\omega_{G(\bA)^1} \backslash \omega^G_\bA$ is  the pullback of the measure $\prod_{i = 1}^{n} dt_i/t_i$ on $(\bR_+)^{n}$ by $\vartheta_G$. The measure $\omega_{G(\bA)^1}$ is independent of the choice of the $\bZ$-basis $\chi_1, \cdots, \chi_{n}$.

\item The case of $\ch(k) > 0$. The value group of the idele norm $|\cdot |_\bA$ is the cyclic group $q^\bZ$ generated by $q$. Thus the image $\Im \vartheta_G$ of $\vartheta_G$ is contained in $\Hom_\bZ(\bX^*_k(G), q^\bZ)$ and $G(\bA)^1$ is an open normal subgroup of $G(\bA)$. Since the index of $\Im \vartheta_G$ in $\Hom_\bZ(\bX^*_k(G), q^\bZ)$ is finite ([O, I, Proposition 5.6]), $$ d^*_G = (\log q)^{\rank \bX^*_k(G)} [\Hom_\bZ(\bX^*_k(G), q^\bZ) \,:\, \Im \vartheta_G]  $$
is well defined. The measure $\omega_{G(\bA)^1}$ is defined to be the restriction of the measure $(d^*_G)^{-1}\omega^G_\bA$ to $G(\bA)^1$.
\end{itemize}

In both cases, we put the counting measure $\omega_{G(k)}$ on $G(k)$. The volume of $G(k) \backslash G(\bA)^1$ with respect to the measure $\omega_{G(k)} \backslash \omega_{G(\bA)^1}$ is called the Tamagawa number of $G$ and denoted by $\tau(G)$.



\vskip 5mm
\subsubsection*{1.2 Another Haar measure on $G(\bA)$ and its normalization constant}
In the following, let $G$ be a connected reductive group defined over $k$. We fix a maximal $k$-split torus $S$ in $G$ and a minimal $k$-parabolic subgroup $P$ of $G$ which contains $S$. The centralizer of $S$ in $G$ gives a Levi subgroup $M_P$ of $P$. Thus $P$ has a Levi decomposition: $P = M_PU_P$, where $U_P$ denotes the unipotent radical of $P$. Let $R$ be a $k$-parabolic subgroup of $G$ such that $P \subset R$. Such $R$ is called a standard $k$-parabolic subgroup. There exists a unique  Levi subgroup $M_R$ of $R$ such that $M_P \subset M_R$. The unipotent radical of $R$ is denoted by $U_R$. We fix a maximal compact subgroup $K$ of $G(\bA)$ satisfying the following property; For every standard $k$-parabolic subgroup $R$ of $G$, $K \cap M_R(\bA)$ is a maximal compact subgroup of $M_R(\bA)$,  and furthermore  $M_R(\bA)$ possesses an Iwasawa decomposition $(M_R(\bA) \cap U_P(\bA)) M_P(\bA) (K \cap M_R(\bA))$.

If a standard $k$-parabolic subgroup $R$ of $G$ is given, then one can define another Haar measure $\omega_{(G(\bA), R(\bA))}$ of $G(\bA)$ as follows. Let $\omega^{M_R}_\bA$ and $\omega^{U_R}_\bA$ be the Tamagawa measures of $M_R(\bA)$ and $U_R(\bA)$, respectively. The modular character $\delta_{R}^{-1}$ of $R(\bA)$ is a function on $M_R(\bA)$ which satisfies the integration formula
\begin{equation*}
\int_{U_R(\bA)} f(mum^{-1}) d\omega^{U_R}_{\bA}(u) = \delta_{R}(m)^{-1} \int_{U_R(\bA)} f(u) d\omega^{U_R}_{\bA}(u).
\end{equation*}
Let $\omega_K$ be the Haar measure on $K$ normalized so that the total volume equals one. Then the mapping
\begin{equation*}
 f \mapsto \int_{U_R(\bA) \times M_R(\bA) \times K} f(umh) \delta_{R}(m)^{-1} d\omega^{U_R}_\bA(u)d\omega^{M_R}_\bA(m)d\omega_K(h)\,, \quad (f \in C_0(G(\bA)))
\end{equation*}
defines an invariant measure on $G(\bA)$ and is denoted by $\omega_{(G(\bA), R(\bA))}$.

Since a non-trivial invariant measure on $G(\bA)$ is unique up to constant, there exists a positive constant $C_{G,R,K}$ such that
\begin{equation*}
\omega^{G}_\bA = C_{G,R,K} \cdot \omega_{(G(\bA), R(\bA))}\,.
\end{equation*}
We call $C_{G,R,K}$ the normalization constant of $\omega_{(G(\bA), R(\bA))}$. For simplicity, we often write $C_{G, R}$ for $C_{G, R, K}$.  It is easy to show the following compatibility of three constants $C_{G, R, K}$, $C_{G, P, K}$ and $C_{M_R, M_R \cap P, M_R(\bA) \cap K}$:
\begin{equation*}
 C_{G, R, K} = \frac {C_{G, P, K}} {C_{M_R, M_R \cap P, M_R(\bA) \cap K}}.
\end{equation*}





\vskip 8mm
\subsection*{2 A formula of $C_{G,R}$}

\subsubsection*{2.1 An expression of $C_{G, R}$ by a product of integrals}
Let $G$, $R$ and $K$ be the same as in \S 1.2. We consider the right $G$-homogeneous space $\fX_R = U_R \backslash G$. Since $U_R$ is a split unipotent subgroup, one has $\fX_R(\bA) = U_R(\bA) \backslash G(\bA)$. Since both $U_R$ and $G$ are unimodular, $\omega^{U_R} \backslash \omega^G$ gives a unique (up to constant) $G$-invariant gauge form on $\fX_R$ defined over $k$. The $G(\bA)$-invariant measure on $\fX_R(\bA)$ defined from $\omega^{U_R} \backslash \omega^G$ is equal to
\begin{equation}
\omega^{U_R}_\bA \backslash \omega^G_\bA = C_{G, R} \delta_R^{-1} \omega^{M_R}_\bA \omega_K \,.
\end{equation}

We take the opposite parabolic subgroup $R^{-}$ of $R$. We denote by $U_{R}^{-}$ the unipotent radical of $R^{-}$, \textit{i.e.}, $U_{R}^{-} = U_{R^-}$. Then one has the Levi decomposition $R^{-} = U_{R}^{-}M_R$ and $R \cap R^{-} = M_R$. By [B-T, Proposition 4.10 d)], the product morphism $U_R \times R^{-} \longrightarrow G$ is injective and gives an isomorphism of variety from $U_R \times R^{-}$ onto a Zariski open set in $G$. Thus $R^{-}$ is regarded as a Zariski open subset of $\fX_R$. Since $(\omega^{U_R} \backslash \omega^G)|_{R^-}$ yields a right invariant gauge form on $R^-$ defined over $k$, there exists a constant $\lambda \in k^\times$ such that
\begin{equation}
(\omega^{U_R} \backslash \omega^G)|_{R^-} = \lambda \omega^{U_R^-} \omega^{M_R} \,.
\end{equation}
For each $v \in \fV$, define the function $\eta_v \,:\, G(k_v) \longrightarrow \bR_+$ by $\eta_v(umh) = \delta_R(m)$ for $u \in U_R(k_v), m \in M_R(k_v)$ and $h \in K_v$. We take a right $K$-invariant $\Phi \in C_0(\fX_R(\bA))$ of the form $\Phi = \prod_{v \in \fV} \Phi_v$, $\Phi_v \in C_0(\fX_R(k_v))$.
On the one hand, by (1), we have
\begin{eqnarray}
\int_{\fX_R(\bA)} \Phi(x) d(\omega^{U_R}_\bA \backslash \omega^G_{\bA})(x)
& = & C_{G,R} \int_{M_R(\bA) \times K} \Phi(mh) \delta_R(m)^{-1} d\omega^{M_R}_{\bA}(m)d\omega_K(h) \nonumber \\
& = & C_{G,R} \int_{M_R(\bA)} \Phi(m) \delta_R(m)^{-1} d\omega^{M_R}_{\bA}(m).
\end{eqnarray}
On the other hand, by (2),
\begin{eqnarray*}
&& \int_{\fX_R(\bA)} \Phi(x) d(\omega^{U_R}_\bA \backslash \omega^G_{\bA})(x) \\
& = & \frac {\mu_\bA(\bA/k)^{\dim U_R - \dim G}} {\sigma_k(G)} \prod_{v \in \fV_\infty} \int_{\fX_R(k_v)} \Phi_v(x_v)d(\omega^{U_R}_v \backslash \omega^G_v)(x_v) \\
& & \qquad \times \prod_{v \in \fV_f} L_v(1, \sigma_G) \int_{\fX_R(k_v)} \Phi_v(x_v)d(\omega^{U_R}_v \backslash \omega^G_v)(x) \\
& = & \frac {\mu_\bA(\bA/k)^{\dim U_R - \dim G}} {\sigma_k(G)} \prod_{v \in \fV_\infty} \int_{M_R(k_v) \times U_R^-(k_v)} \Phi_v(m_vu_v) \delta_R(m_v)^{-1} |\lambda|_v d\omega^{M_R}_v(m_v)d\omega^{U_R^-}_v(u_v) \\
& & \qquad \times \prod_{v \in \fV_f} L_v(1, \sigma_G)\int_{M_R(k_v) \times U_R^-(k_v)} \Phi_v(m_vu_v) \delta_R(m_v)^{-1} |\lambda|_v d\omega^{M_R}_v(m_v)d\omega^{U_R^-}_v(u_v). \\
& = & \frac {\mu_\bA(\bA/k)^{\dim U_R - \dim G}} {\sigma_k(G)} \prod_{v \in \fV_\infty} \int_{M_R(k_v) \times U_R^-(k_v)} \Phi_v(m_vu_v) \delta_R(m_v)^{-1} d\omega^{M_R}_v(m_v)d\omega^{U_R^-}_v(u_v) \\
& & \qquad \times \prod_{v \in \fV_f} L_v(1, \sigma_G)\int_{M_R(k_v) \times U_R^-(k_v)} \Phi_v(m_vu_v) \delta_R(m_v)^{-1} d\omega^{M_R}_v(m_v)d\omega^{U_R^-}_v(u_v)
\end{eqnarray*}
since $|\lambda|_\bA = 1$. We decompose $u_v \in U_R^-(k_v)$ into $u_v^\prime m_v^\prime h_v^\prime$, $u_v^\prime \in U_R(k_v), m_v^\prime \in M_R(k_v)$ and $h_v^\prime \in K_v$. Then one has
\begin{eqnarray*}
& & \int_{M_R(k_v) \times U_R^-(k_v)} \Phi_v(m_vu_v) \delta_R(m_v)^{-1} d\omega^{M_R}_v(m_v)d\omega^{U_R^-}_v(u_v) \\
& & = \int_{M_R(k_v) \times U_R^-(k_v)} \Phi_v((m_vu_v^\prime m_v^{-1})(m_v m_v^\prime) h_v^\prime) \delta_R(m_v)^{-1} d\omega^{M_R}_v(m_v)d\omega^{U_R^-}_v(u_v) \\
& & = \int_{M_R(k_v) \times U_R^-(k_v)} \Phi_v(m_v m_v^\prime) \delta_R(m_v)^{-1} d\omega^{M_R}_v(m_v)d\omega^{U_R^-}_v(u_v) \\
& & = \int_{M_R(k_v) \times U_R^-(k_v)} \Phi_v(m_v) \delta_R(m_v(m_v^\prime)^{-1})^{-1} d\omega^{M_R}_v(m_v)d\omega^{U_R^-}_v(u_v) \\
& & = \left (\int_{M_R(k_v)} \Phi_v(m_v)\delta_R(m_v)^{-1} d\omega^{M_R}_v(m_v) \right ) \left ( \int_{U_R^-(k_v)} \eta_v(u_v) d\omega^{U_R^-}_v(u_v) \right ).
\end{eqnarray*}
By the definition of Tamagawa measures,
\begin{eqnarray*}
\int_{M_R(\bA)} \Phi(m)\delta_R(m)^{-1} d\omega^{M_R}_\bA(m) & = & \frac {\mu_\bA(\bA/k)^{-\dim M_R}} {\sigma_k(M_R)} \prod_{v \in \fV_\infty} \int_{M_R(k_v)} \Phi_v(m_v)\delta_R(m_v)^{-1} d\omega^{M_R}_v(m_v) \\
& & \qquad \times \prod_{v \in \fV_f} L_v(1, \sigma_{M_R}) \int_{M_R(k_v)} \Phi_v(m_v)\delta_R(m_v)^{-1} d\omega^{M_R}_v(m_v).
\end{eqnarray*}
Therefore,
\begin{eqnarray}
\int_{\fX_R(\bA)} \Phi(x) d(\omega^{U_R}_\bA \backslash \omega^G_{\bA})(x)
& = & \frac {\mu_\bA(\bA/k)^{\dim R - \dim G} \sigma_k(M_R)} {\sigma_k(G)} \int_{M_R(\bA)} \Phi(m)\delta_R(m)^{-1} d\omega^{M_R}_{\bA}(m) \nonumber \\
& & \qquad \times \prod_{v \in \fV_\infty} J_v \prod_{v \in \fV_f} \frac {L_v(1, \sigma_G)} {L_v(1, \sigma_{M_R})} J_v \,,
\end{eqnarray}
where
\begin{equation*}
J_v = \int_{U_R^-(k_v)} \eta_v(u_v) d\omega_v^{U_R^-}(u_v) \,.
\end{equation*}
From (3), (4) and $\dim R - \dim G = -\dim U_R$, we obtain the following.

\medskip
\begin{theorem} Notations being as above, we have
\begin{equation*}
 C_{G,R} = \frac {\mu_\bA(\bA/k)^{-\dim U_R} \sigma_k(M_R)} {\sigma_k(G)}\prod_{v \in \fV_\infty} J_v \prod_{v \in \fV_f} \frac {L_v(1, \sigma_G)} {L_v(1, \sigma_{M_R})} J_v \,.
\end{equation*}
\end{theorem}



\vskip 5mm
\subsubsection*{2.2 Reduction of $J_v$ to the case of minimal $k_v$-parabolic subgroups}
We explain how to compute the local integral $J_v$. Let $P^{(v)}$ be a minimal parabolic subgroup of $G$ defined over $k_v$ such that $P^{(v)}(k_v) \subset R(k_v)$. Then $P^{(v)}$ has a Levi subgroup $M^{(v)}$ such that $M^{(v)}(k_v) \subset M_R(k_v)$. Let $U^{(v)}$ be the unipotent radical of $P^{(v)}$ and $U^{(v)-}$ be the unipotent radical of the opposite parabolic subgroup of $P^{(v)}$. We set $P_{M_R}^{(v)} = P^{(v)} \cap M_R$, $U_{M_R}^{(v)} = U^{(v)} \cap M_R$ and $U_{M_R}^{(v)-} = U^{(v)-} \cap M_R$. Then $P_{M_R}^{(v)}$ is a minimal parabolic subgroup of $M_R$ defined over $k_v$ with the unipotent radical $U_{M_R}^{(v)}$ and a Levi subgroup $M^{(v)}$. The unipotent group $U_R(k_v)$ is a normal subgroup of $U^{(v)}(k_v)$, and $U^{(v)}(k_v)$ has a semidirect product decomposition $U_R(k_v)U_{M_R}^{(v)}(k_v)$. Let $\delta_{P^{(v)}}^{-1} \,:\, M^{(v)}(k_v) \longrightarrow \bR_+$ and  $\delta_{P_{M_R}^{(v)}}^{-1} \,:\, M^{(v)}(k_v) \longrightarrow \bR_+$ be the modular characters of $P^{(v)}(k_v)$ and $P_{M_R}^{(v)}(k_v)$, respectively. One has a relation
\begin{equation}
 \delta_R^{-1}\Big|_{M^{(v)}(k_v)} = \delta_{P^{(v)}}^{-1} \cdot \delta_{P_{M_R}^{(v)}} \;.
\end{equation}
Define the function $\eta^G_{P^{(v)}} \,:\, G(k_v) \longrightarrow \bR_+$ by $\eta^G_{P^{(v)}}(umh) = \delta_{P^{(v)}}(m)$ for $u \in U^{(v)}(k_v)$, $m \in M^{(v)}(k_v)$ and $h \in K_v$. In a similar fashion, the function $\eta^{M_R}_{P_{M_R}^{(v)}} \,:\, M_R(k_v) \longrightarrow \bR_+$ is defined by $\eta^{M_R}_{P_{M_R}^{(v)}}(umh) = \delta_{P_{M_R}^{(v)}}(m)$ for $u \in U_{M_R}^{(v)}(k_v)$, $m \in M^{(v)}(k_v)$ and $h \in K_v \cap M_R(k_v)$. We set
\begin{equation*}
 J^G_v = \int_{U^{(v)-}(k_v)} \eta^G_{P^{(v)}}(u) d\omega_{U^{(v)-}(k_v)}(u), \qquad J^{M_R}_v = \int_{U_{M_R}^{(v)-}(k_v)} \eta^{M_R}_{P_{M_R}^{(v)}}(u) d\omega_{U_{M_R}^{(v)-}(k_v)}(u) \,.
\end{equation*}
Here, we fix invariant measures $\omega_{U^{(v)-}(k_v)}$ on $U^{(v)-}(k_v)$ and $\omega_{U_{M_R}^{(v)-}(k_v)}$ on $U_{M_R}^{(v)-}(k_v)$ such that
$$ \omega_{U^{(v)-}(k_v)} = \omega_v^{U_R^-} \cdot \omega_{U_{M_R}^{(v)-}(k_v)} \,.
$$
Let us compute $J^G_v$ following the decomposition $U^{(v)-} = U_R^{-}(k_v)U_{M_R}^{(v)-}(k_v)$:
\begin{equation*}
 J^G_v = \int_{U^{(v)-}(k_v)} \eta^G_{P^{(v)}}(u) d\omega_{U^{(v)-}(k_v)}(u) = \int_{U_{M_R}^{(v)-}(k_v)} d\omega_{U_{M_R}^{(v)-}(k_v)}(u_2) \int_{U_R^{-}(k_v)} \eta^G_{P^{(v)}}(u_1u_2) d\omega_v^{U_R^{-}}(u_1) \,.
\end{equation*}
Let $u_i = \alpha_i \beta_i \gamma_i$, $\alpha_i \in U^{(v)}(k_v), \beta_i \in M^{(v)}(k_v)$ and $\gamma_i \in K_v$ for $i = 1,2$. Since
$$ \eta^G_{P^{(v)}}(u_1u_2) = \eta^G_{P^{(v)}}(\alpha_2\beta_2 (\alpha_2\beta_2)^{-1}u_1 (\alpha_2\beta_2)) = \delta_{P^{(v)}}(\beta_2) \eta^G_{P^{(v)}}((\alpha_2\beta_2)^{-1}u_1 (\alpha_2\beta_2)), $$
one has
\begin{eqnarray*}
J^G_v & = & \int_{U_{M_R}^{(v)-}(k_v)} d\omega_{U_{M_R}^{(v)-}(k_v)}(u_2) \int_{U_R^{-}(k_v)} \delta_{P^{(v)}}(\beta_2) \delta_{R^-}(\beta_2)\eta^G_{P^{(v)}}(u_1) d\omega_v^{U_R^{-}}(u_1) \\
& = & \int_{U_{M_R}^{(v)-}(k_v)} \delta_{P^{(v)}}(\beta_2) \delta_{R^-}(\beta_2) d\omega_{U_{M_R}^{(v)-}(k_v)}(u_2) \int_{U_R^{-}(k_v)} \delta_{P^{(v)}}(\beta_1) d\omega_v^{U_R^{-}}(u_1) \,.
\end{eqnarray*}
By $\delta_{R^-}(\beta_2) = \delta_R(\beta_2)^{-1}$, $\delta_{R}(\beta_1) = \delta_{P^{(v)}}(\beta_1)$ and (5), we obtain
\begin{eqnarray*}
 J^G_v & = & \int_{U_{M_R}^{(v)-}(k_v)}
\delta_{P_{M_R}^{(v)}}(\beta_2)d\omega_{U_{M_R}^{(v)-}(k_v)}(u_2) \cdot
\int_{U_R^{-}(k_v)} \delta_{R}(\beta_1) d\omega_v^{U_R^{-}}(u_1) \\
& = & J^{M_R}_v \cdot J_v \,.
\end{eqnarray*}
Therefore, one has
\begin{equation}
 J_v = \frac {J^G_v} {J^{M_R}_v} \,.
\end{equation}



\vskip 5mm
\subsubsection*{2.4 Gindikin-Karpelevi\v{c} formula of $J_v^G$}
We set
\begin{equation*}
 J^G_v(s) = \int_{U^{(v)-}(k_v)} \eta^G_{P^{(v)}}(u)^{s + 1/2} d\omega_{U^{(v)-}(k_v)}(u),
\end{equation*}
where $s$ is a complex number with $\Re(s) > 0$. We recall the Gindikin--Karpelevi\v{c} formula of $J^G_v(s)$ (cf. [K, Chap.VII, \S 5, Corollary 7.5]). Let $S^{(v)}$ be a maximal $k_v$-split torus of $M^{(v)}$, $\Sigma_v(G)$ the relative root system of $G$ with respect to $S^{(v)}$ and $\Sigma_v^+(G)$ the set of positive roots of $\Sigma_v(G)$ corresponding to the minimal $k_v$-parabolic subgroup $P^{(v)}$. We set
\begin{equation*}
 \fa_v = X^*_{k_v}(S^{(v)}/Z_G^{(v)}) \otimes_\bZ \bR \,,
\end{equation*}
where $Z_G^{(v)}$ denotes the maximal central $k_v$-split torus of $G$. Note that the real vector space $\fa_v$ is identified with $X^*_{k_v}(M^{(v)}/Z_G^{(v)}) \otimes_\bZ \bR$ since $M^{(v)}/S^{(v)}$ is anisotropic over $k_v$. The set of simple roots of $\Sigma_v^+(G)$ gives a basis of $\fa_v$, and hence $\Sigma_v(G)$ is regarded as a subset of $\fa_v$. Thus, for each $\beta \in \Sigma_v(G)$, the function $\xi^G_\beta \,:\, G(\bA) \longrightarrow \bR_+$ is well defined by $\xi^G_\beta(umh) = |\beta(m)|_v$ for $u \in U^{(v)}(k_v)$, $m \in M^{(v)}(k_v)$ and $h \in K_v$.
We fix an admissible inner product $(\cdot, \cdot)$ on $\fa_v$ and define the coroot $\beta^\vee$ of $\beta \in \Sigma_v(G)$ by
\begin{equation*}
 \beta^\vee = \frac {2} {(\beta, \beta)} \beta \,.
\end{equation*}
For $\beta \in \Sigma_v^+(G)$, the connected component $(\Ker \beta)^0$ of the kernel of $\beta$ is a subtorus of $S^{(v)}$. We denote by $G_{(\beta)}$ the centralizer of $(\Ker \beta)^0$ in $G$. Then $G_{(\beta)}$ is a reductive $k_v$-subgroup of $G$ with semisimple $k_v$-rank one. We set $P_{(\beta)} = G_{(\beta)} \cap P^{(v)}$, $M_{(\beta)} = G_{(\beta)} \cap M^{(v)}$, $U_{(\beta)} = G_{(\beta)} \cap U^{(v)}$, $U_{(\beta)}^- = G_{(\beta)} \cap U^{(v)-}$ and $K_{(\beta)} = G_{(\beta)}(k_v) \cap K_v$. We assume that $G_{(\beta)}(k_v) = P_{(\beta)}(k_v)K_{(\beta)}$ holds for all $\beta \in \Sigma_v^+(G)$. Then we define the function $\eta_\beta \,:\, G_{(\beta)}(k_v) \longrightarrow \bR_+$ by $\eta_\beta(umh) = \delta_{P_{(\beta)}}(m)$ for $u \in U_{(\beta)}(k_v)$, $m \in M_{(\beta)}(k_v)$ and $h \in K_{(\beta)}$, where $\delta_{P_{(\beta)}}^{-1} \,:\, M_{(\beta)}(k_v) \longrightarrow \bR_+$ denote the modular character of $P_{(\beta)}(k_v)$. Moreover, we write $\rho^G_v$ for the half-sum of positive roots and $\xi_{\rho^G_v} \,:\, G(\bA) \longrightarrow \bR_+$ for the function corresponding to $\rho^G_v$, \textit{i.e.},
\begin{equation*}
 \rho^G_v = \frac {1} {2} \sum_{\beta \in \Sigma^+_v} (\dim U_{(\beta)}) \beta \,, \qquad \xi_{\rho^G_v} = \prod_{\beta \in \Sigma^+_v} (\xi^G_\beta)^{\dim U_{(\beta)}/2} \,.
\end{equation*}
There is a relation $\xi_{\rho^G_v}^2 = \eta^G_{P^{(v)}}$.
With these notations, the Gindikin--Karpelevi\v{c} formula of $J^G_v(s)$ is stated as follows:
\begin{equation}
 J^G_v(s) = \prod_{{\scriptsize\begin{array}{c} \beta \in \Sigma_v^+(G) \\ \beta/2 \not\in \Sigma_v^+(G) \end{array}}} \int_{U_{(\beta)}^-(k_v)} \xi^G_\beta(u)^{(\rho^G_v, \beta^\vee)s}\eta_\beta(u)^{1/2} d\omega_{U_{(\beta)}^-(k_v)}(u) \,.
\end{equation}
Here, we fix a family of invariant measures $\omega_{U_{(\beta)}^-(k_v)}$, $\beta \in \Sigma_v^+(G)$ such that
$$ \omega_{U^{(v)-}(k_v)} = \prod_{{\scriptsize\begin{array}{c} \beta \in \Sigma_v^+ \\ \beta/2 \not\in \Sigma_v^+ \end{array}}} \omega_{U_{(\beta)}^-(k_v)} $$
holds. In principle, $C_{G,R}$ can be computed by Theorem 1 and formulas (6), (7).





\vskip 8mm
\subsection*{3 An explicit formula of $C_{G, P}$ in the case of $G(k) = GL_n(D)$}

\subsubsection*{3.1 Central simple algebras}
Let $D$ be a central division $k$-algebra of degree $d^2$. Let $D_v = D \otimes_k k_v$ for $v \in \fV$ and $D_\bA = D \otimes_k \bA$.
Since $D_v$ is a central simple $k_v$-algebra, it is isomorphic with an algebra $M_{d/d_v}(D(v))$, where $D(v)$ is a division $k_v$-algebra of degree $d_v^2$. The set $\fV$ is divided into two subsets $\fV_1 = \{ v \in \fV \,:\, d_v = 1 \}$ and $\fV_2 = \{ v \in \fV \,:\, d_v > 1 \}$. We write $\fV_{\bR,1}$, $\fV_{\bR,2}$, $\fV_{f,1}$ and $\fV_{f,2}$ for $\fV_{\bR} \cap \fV_1$, $\fV_{\bR} \cap \fV_2$, $\fV_f \cap \fV_1$ and $\fV_f \cap \fV_2$, respectively. We fix a maximal order $\fO_D$ of $D$. For $v \in \fV_f$, the completion of $\fO_D$ in $D_v$ is denoted by $\fO_{D_v}$, which is a maximal order of $D_v$. Since any maximal order of $D_v$ is conjugate to $\fO_{D_v}$, there is an isomorphism from $D_v$ onto $M_{d/d_v}(D(v))$ such that the image of $\fO_{D_v}$ equals $M_{d/d_v}(\fO_{D(v)})$, where $\fO_{D(v)}$ denotes a unique maximal order of $D(v)$.

For every $v \in \fV_f$, we denote by $\fd_v$ the different of $\fO_{D_v}/\fo_v$, \textit{i.e.},
$$
 \fd_v^{-1} = \{ a \in D_v \,:\, \tau_{D_v/k_v}(a\fO_{D_v}) \subset \fo_v \}.
$$
Then the different $\fd_{\fO_D}$ of $\fO_D$ is given by $\prod_{v \in \fV_f} \fd_v$. The absolute norm $\Nfd_{D/k}$ of $\fd_{\fO_D}$ is defined to be
$$ \Nfd_{D/k} = \prod_{v \in \fV_f} |\fO_{D_v}/\fd_v| \,, $$
which is independent of the choice of the maximal order $\fO_D$ (cf. [R, Theorems (25.3) and (25.7)])

Now we consider the central simple $k$-algebra $\fA = M_n(D)$ and its maximal order $\fO_{\fA} = M_n(\fO_D)$. We identify $\fA_v = \fA \otimes_k k_v$ with $M_n(D_v)$ for $v \in \fV$ and $\fA_\bA = \fA \otimes_k \bA$ with $M_n(D_\bA)$. For $v \in \fV_f$, set $\fO_{\fA_v} = M_n(\fO_{D_v})$, which is a maximal order of $\fA_v$. Hereafter, $G$ denotes an affine algebraic $k$-group defined by $G(k) = \fA^\times = GL_n(D)$. The ad\`ele group $G(\bA)$ of $G$ is the unit group of $\fA_\bA$. If $v \in \fV_\infty$, we define an involution $a \mapsto a^*$ of $\fA_v$ as follows. We fix an algebra isomorphism $\fA_v \cong M_{nd/d_v}(D(v))$. Then, for $a = (a_{ij}) \in \fA_v$ ($a_{ij} \in D(v)$), the involution $a^*$ is defined to be $a^* = (\ol a_{ij})^{t}$, where the superscript $t$ means the transpose of a matrix and $a_{ij} \mapsto \ol a_{ij}$ denotes the canonical involution of the division algebra $D(v)$, \textit{i.e.}, it is the identity map, the complex conjugate or the quaternion conjugate according as $v \in \fV_{\bR,1}$, $v \in \fV_{\bC}$ or $v \in \fV_{\bR,2}$. By using this involution, we define the subgroup $K_v$ of $G(k_v) = \fA_v^\times$ by $K_v = \{ a \in \fA_v^\times \,:\, a^{-1} = a^* \}$. If $v \in \fV_f$, set $K_v = \fO_{\fA_v}^\times$. Then $K = \prod_{v \in \fV} K_v$ gives a maximal compact subgroup of $G(\bA)$. Let $P$ be the minimal $k$-parabolic subgroup of $G$ which consists of upper triangular matrices in $G$. We will compute the constant $C_{G, P} = C_{G,P,K}$.



\vskip 5mm
\subsubsection*{3.2 Self-dual measures}

It is convenient to use a self-dual measure on $D_\bA$ in order to compute $C_{G,P}$. We recall its construction. We fix a non-trivial character $\psi \,:\, \bA/k \longrightarrow \bC^1$ as follows. If $\ch(k) > 0$, we arbitrarily choose a non-trivial $\psi$. If $\ch(k) = 0$, we define the character $\psi_0$ on the ad\`ele group $\bA_\bQ$ of $\bQ$ by
$$ \psi_0(x) = e^{-2\pi\sqrt{-1} x_\infty} \prod_{p \,:\, \textrm{prime}} e^{2\pi\sqrt{-1}(x_p \; \textrm{mod}\, \bZ_p)} $$
for $x = (x_\infty, x_2, x_3, \cdots) \in \bA_\bQ$, and then set $\psi = \psi_0 \circ \Tr_{k/\bQ}$.
For every $v \in \fV$, $\psi$ induces a character $\psi_v \,:\, k_v \longrightarrow \bC^1$. Let $\fC$ be an arbitrary central simple $k$-algebra and $\fC_v = \fC \otimes_k k_v$ for $v \in \fV$ and $\fC_\bA = \fC \otimes_k \bA$. An invariant measure $\nu_{\fC_v}$ on the locally compact additive group $\fC_v$ is called the self-dual measure with respect to $\psi_v$ if
$$ \Phi(x) = \int_{\fC_v} \left \{ \int_{\fC_v} \Phi(z) \psi_v(\tau_{\fC_v/k_v}(yz)) d\nu_{\fC_v}(z) \right \} \psi_v(-\tau_{\fC_v/k_v}(xy)) d\nu_{\fC_v}(y) $$
holds for any Schwartz--Bruhat function $\Phi$ on $\fC_v$. The product measure $\nu_{\fC_\bA} = \prod_{v \in \fV} \nu_{\fC_v}$ on $\fC_\bA$ satisfies
$$ \Phi(x) = \int_{\fC_\bA} \left \{ \int_{\fC_\bA} \Phi(z) \psi_v(\tau_{\fC/k}(yz)) d\nu_{\fC_\bA}(z) \right \} \psi_v(-\tau_{\fC/k}(xy)) d\nu_{\fC_\bA}(y) $$
for any Schwartz--Bruhat function $\Phi$ on $\fC_\bA$. The invariant measure $\nu_{\fC_\bA}$ is called the self-dual measure of $\fC_\bA$ with respect to $\psi$.

For $v \in \fV$, let $\nu_{D(v)}$ be the self-dual measure on $D(v)$ with respect to $\psi_v$. It is known by [T, Propositions 5, 6, 7 and 8] that the product measure $\nu_{D(v)}^{d^2/d_v^2}$ coincides with the self-dual measure on $M_{d/d_v}(D(v))$ with respect to $\psi_v$. Hence one can identify $\nu_{D_v}$ with $\nu_{D(v)}^{d^2/d_v^2}$. Note that this identification is independent of the choice of the algebra isomorphism $D_v \cong M_{d/d_v}(D(v))$ because of Skolem--Noether theorem. Therefore, we have
$$ \nu_{D_\bA} = \prod_{v \in \fV} \nu_{D_v} = \prod_{v \in \fV} \nu_{D(v)}^{d^2/d_v^2} \,. $$
As was shown in the proof of [T, Theorem 2], $\nu_{D_\bA}$ is the Tamagawa measure of $D_\bA$, namely $\nu_{D_\bA}(D_\bA/D) = 1$.

We define another invariant measure $\mu_{D_\bA}$ on $D_\bA$. If $v \in \fV_1$, \textit{i.e.}, $D(v) = k_v$, then we put $\mu_{D(v)} = \mu_v$, where $\mu_v$ is the measure on $k_v$ introduced in Notations. For $v \in \fV_2$, $\mu_{D(v)}$ is defined to be the invariant measure on $D(v)$ normalized so that $\mu_{D(v)}(\fO_{D(v)}) = 1$ if $v \in \fV_{f,2}$ and $\mu_{D(v)}(\{ x \in D(v) \,:\, \Nr_{D(v)/k_v}(x) \le 1 \}) = 4\pi^2$ if $v \in \fV_{\bR,2}$. For every $v \in \fV$, we set $\mu_{D_v} = \mu_{D(v)}^{d^2/d_v^2}$, which gives an invariant measure on $D_v \cong M_{d/d_v}(D(v))$. By Skolem--Noether Theorem, $\mu_{D_v}$ is independent of the choice of the algebra isomorphism $D_v \cong M_{d/d_v}(D(v))$. In particular, one has $\mu_{D_v}(\fO_{D_v}) = 1$ for $v \in \fV_f$.  The product measure $\mu_{D_\bA} = \prod_{v \in \fV}\mu_{D_v}$ is an invariant measure on $D_\bA$.
For every $v \in \fV$, there is the positive constant $\kappa_v$ such that $\mu_{D(v)} = \kappa_v \nu_{D(v)}$. One has $\mu_{D_v} = \kappa_v^{d^2/d_v^2}\nu_{D_v}$.

\medskip
\begin{lemma}
$\mu_{D_{\bA}} (D_\bA/D) = \prod_{v \in \fV} \kappa_v^{d^2/d_v^2}
= \mu_{\bA} (\bA/k)^{d^2} \Nfd_{D/k}^{1/2}$.
\end{lemma}
\textit{Proof}.
We define the Schwartz--Bruhat function $\Phi_\bA = \prod_{v \in \fV} \Phi_v$ on $D_\bA$ as follows: If $v \in \fV_f$, let $\Phi_v$ be the characteristic function of $\fO_{D_v}$. If $v \in \fV_\infty$, we set $\Phi_v (x) = e^{- [k_v : \bR]d_v\pi \Tr(x^*x)}$, where $\Tr(x^*x)$ denotes the trace of the Hermitian matrix $x^*x$. One hand, we have
$$ \int_{D_{\bA}} \Phi_{\bA} (x) d\mu_{D_{\bA}}(x)
 = 1.
$$
On the other hand, by [T, \S II, Propositions 1 and 2],
$$  \int_{D_\bA} \Phi_{\bA} (x) d\nu_{D_{\bA}}(x)
 = \mu_\bA(\bA/k)^{-d^2}\Nfd_{D/k}^{-1/2} \,,
$$
which proves the lemma.



\vskip 5mm
\subsubsection*{3.3 A formula of $C_{G, P}$}

Let $M_P$ be the Levi subgroup of $P$ consisting of diagonal matrices in $G$ and $S$ be the maximal $k$-split torus of $M_P$, \textit{i.e.},
\begin{eqnarray*}
M_P(k) & = & \left \{ \diag(a_1, \cdots, a_n) = \left ( \begin{array}{ccc} a_1 &  & 0 \\ & \ddots & \\ 0 & & a_n \end{array} \right ) \;:\; a_1, \cdots, a_n \in D^\times \right \} \\
S(k) & = & \{ \diag(a_1, \cdots, a_n) \;:\; a_1, \cdots, a_n \in k^\times \}.
\end{eqnarray*}
Let $\Sigma(G)$ be the relative root system of $G$ with respect to $S$ and $\Sigma^+(G)$ be the set of positive roots of $\Sigma(G)$ corresponding to $P$. For each $\alpha \in \Sigma(G)$, $U_{\alpha}$ denotes the root subgroup of $G$. We fix an isomorphism $U_{\alpha}(k) \cong D$ and define the invariant measures $\nu_{U_{\alpha}(k_v)}$ on $U_{\alpha}(k_v)$ for $v \in \fV$ and $\nu_{U_{\alpha}(\bA)}$ on $U_{\alpha}(\bA)$ as
$$ \nu_{U_{\alpha}(k_v)} = \nu_{D_v}, \qquad \nu_{U_{\alpha}(\bA)} = \prod_{v \in \fV} \nu_{U_{\alpha}(k_v)} = \nu_{D_\bA}.$$
We set
$$ \nu_{U_P^-(k_v)} = \prod_{\alpha \in \Sigma^+(G)} \nu_{U_{-\alpha}(k_v)}, \qquad \nu_{U_P^-(\bA)} = \prod_{\alpha \in \Sigma^+(G)} \nu_{U_{-\alpha}(\bA)} = \prod_{v \in \fV} \nu_{U_P^-(k_v)}. $$
Since $\nu_{D_\bA}$ is the Tamagawa measure on $D_\bA$, $\nu_{U_P^-(\bA)}$ coincides with the Tamagawa measure on the unipotent group $U_P^-(\bA)$, \textit{i.e.}, $\omega^{U_P^-}_\bA = \nu_{U_P^-(\bA)}$.

For $v \in \fV$, we define the local integral $I_v$ by
$$ I_v = \int_{U_P^-(k_v)} \eta_v(u_v) d\nu_{U_P^-(k_v)}(u_v) \,, $$
where the function $\eta_v \,:\, G(k_v) \longrightarrow \bR_+$ is defined by
$$ \eta_v(u \cdot \diag(a_1, \cdots, a_n) \cdot h) = \prod_{i=1}^n |\Nr_{D_v/k_v}(a_i)|_v^{d(n-2i+1)}$$
for $u \in U_P(k_v)$, $a_1, \cdots, a_n \in D_v^\times$ and $h \in K_v$. Since
$$ \frac{\sigma_k(M_P)}{\sigma_k(G)} = \rho_k^{n-1}, \qquad \frac{L_v(1, \sigma_G)}{L_v(1, \sigma_{M_P})} = (1 - q_v^{-1})^{n-1} $$
and
$$ \omega^{U_P^-}_\bA = \mu_\bA(\bA/k)^{-\dim U_P}\prod_{v \in \fV} \omega_v^{U_P^-} = \prod_{v \in \fV} \nu_{U_P^-(k_v)}, $$
Theorem 1 leads us to
\begin{equation}
C_{G, P} = \rho_k^{n-1} \prod_{v \in \fV_\infty} I_v \prod_{v \in \fV_f} (1 - q_v^{-1})^{n-1} I_v \,.
\end{equation}



\vskip 5mm
\subsubsection*{3.4 Reduction of $I_v$ to the case of $GL_2(D(v))$}

We fix a place $v \in \fV$.
Let $S^{(v)}$ be the maximal $k_v$-split torus in $M_P$ and $P^{(v)}$ be a minimal $k_v$-parabolic subgroup of $G$ such that $S^{(v)} \subset P^{(v)} \subset P$.
The unipotent radical of $P^{(v)}$ is denoted by $U^{(v)}$.
The centralizer $M^{(v)}$ of $S^{(v)}$ in $G$ is a Levi subgroup of $P^{(v)}$.
As in \S 2.3, we set $P_{M_P}^{(v)} = P^{(v)} \cap M_P$, $U_{M_P}^{(v)} = U^{(v)} \cap M_P$ and $U_{M_P}^{(v)-} = U^{(v)-} \cap M_P$.
Let $\Sigma_v(G)$ be the relative root system of $G$ with respect to $S^{(v)}$ and $\Sigma_v^+(G)$ be the set of positive roots of $\Sigma_v(G)$ corresponding to $P^{(v)}$.
For every $\beta \in \Sigma_v(G)$, $U_{(\beta)}$ stands for the root subgroup of $G$.
We fix an isomorphism $U_{(\beta)}(k_v) \cong D(v)$ and define the invariant measures $\nu_{U_{(\beta)}(k_v)}$ on $U_{(\beta)}(k_v)$, $\nu_{U^{(v)-}(k_v)}$ on $U^{(v)-}(k_v)$ and $\nu_{U_{M_P}^{(v)-}(k_v)}$ on $U_{M_P}^{(v)-}(k_v)$ as
$$ \nu_{U_{(\beta)}(k_v)} = \nu_{D(v)}, \qquad \nu_{U^{(v)-}(k_v)} = \prod_{\beta \in \Sigma_v^+(G)} \nu_{U_{(-\beta)}(k_v)}, \qquad  \nu_{U_{M_P}^{(v)-}(k_v)} = \prod_{\stackrel{\scriptstyle \beta \in \Sigma_v^+(G)}{\beta|_S = 0}} \nu_{U_{(-\beta)}(k_v)} \,. $$
For a $k$-root $\alpha \in \Sigma(G)$, one has
$$ U_\alpha(k_v) = \prod_{\stackrel{\scriptstyle \beta \in \Sigma_v(G)}{\beta|_S = \alpha}} U_{(\beta)}(k_v) \,. $$
From $\nu_{D_v} = \nu_{D(v)}^{d^2/d_v^2}$, it follows
$$ \nu_{U_\alpha(k_v)} = \prod _{\stackrel{\scriptstyle \beta \in \Sigma_v(G)}{\beta|_S = \alpha}} \nu_{U_{(\beta)}(k_v)}\,. $$
This implies the relation $\nu_{U^{(v)-}(k_v)} = \nu_{U_P^-(k_v)} \cdot \nu_{U_{M_P}^{(v)-}(k_v)}$. Therefore, if we set
\begin{eqnarray*}
I_v^G(s) & = & \int_{U^{(v)-}(k_v)} \eta^G_{P^{(v)}}(u)^{s + 1/2} d\nu_{U^{(v)-}(k_v)}(u) \,, \\
I_v^{M_P}(s) & = & \int_{U_{M_P}^{(v)-}(k_v)} \eta^{M_P}_{P^{(v)}_{M_P}}(u)^{s + 1/2} d\nu_{U_{M_P}^{(v)-}(k_v)}(u)
\end{eqnarray*}
for $\Re(s) > 0$ with the notations in \S 2.3, then $I_v \cdot I_v^{M_P}(1/2) = I_v^G(1/2)$ holds similarly as (6).

Let $K_v^{GL_2}$ be a maximal compact subgroup of $GL_2(D(v))$ defined by the same way as $K_v$. We define the function $\eta^{GL_2}_v \,:\, GL_2(D(v)) \longrightarrow \bR_+$ as follows:
$$ \eta^{GL_2}_v \left (\left ( \begin{array}{cc} 1 & b \\ 0 & 1 \end{array} \right ) \left ( \begin{array}{cc} a_1 & 0 \\ 0 & a_2 \end{array} \right ) h \right ) = |\Nr_{D(v)/k_v}(a_1)|_v^{d_v}|\Nr_{D(v)/k_v}(a_2)|_v^{-d_v} $$
for $b \in D(v)$, $a_1, a_2 \in D(v)^\times$ and $h \in K_v^{GL_2}$. We set
$$ I^{GL_2}_v(s) = \int_{D(v)} \eta^{GL_2}_v \left (\left ( \begin{array}{cc} 1 & 0 \\ b & 0 \end{array} \right ) \right )^{s + 1/2} d\nu_{D(v)}(b)  $$
for $\Re(s) > 0$. Then, by the Gindikin--Karpelevi\v{c} formula,
\begin{eqnarray*}
I_v^G(s) & = & \prod_{\beta \in \Sigma_v^+(G)} \int_{U_{(-\beta)}(k_v)} \xi^G_{\beta}(u)^{(\rho^G_v, \beta^\vee)s} \eta_\beta(u)^{1/2} d\nu_{U_{(-\beta)}(k_v)}(u) \\
& = & \prod_{\beta \in \Sigma_v^+(G)} I^{GL_2}_v((\rho_v^G, \beta^\vee)s/d_v^2)\\
& = & \prod_{1 \le i < j \le nd/d_v} I^{GL_2}_v((j - i)s),
\end{eqnarray*}
and, in a similar fashion,
\begin{equation*}
I_v^{M_P}(s) = \left (\prod_{1 \le i < j \le d/d_v} I^{GL_2}_v((j-i)s) \right )^n \,.
\end{equation*}
Therefore,
\begin{equation}
I_v = \left (\prod_{1 \le i < j \le d/d_v} I^{GL_2}_v((j - i)/2) \right )^{-n} \prod_{1 \le i < j \le nd/d_v} I^{GL_2}_v((j - i)/2) \,.
\end{equation}



\vskip 5mm
\subsubsection*{3.5 Computations of $I_v^{GL_2}(s)$}

An Iwasawa decomposition of the unipotent matrix $\left (\begin{array}{cc} 1 & 0 \\ x & 1 \end{array} \right ) \in GL_2(D(v))$ is given as follows:
\begin{itemize}
\item If $v \in \mathfrak{V}_f$,
$$
 \left ( \begin{array}{cc}
  1 & 0 \\
  x & 1
 \end{array} \right )
 =
 \left \{ \begin{array}{ll}
  \left ( \begin{array}{cc}
   1 & 0 \\
   x & 1
  \end{array} \right )
  & (x \in \fO_{D(v)}).
  \\
  \left ( \begin{array}{cc}
   1 & x^{-1} \\
   0 & 1
  \end{array} \right )
  \left ( \begin{array}{cc}
   x^{-1} & 0 \\
   0 & x
  \end{array} \right )
  \left ( \begin{array}{cc}
   0 & -1 \\
   1 & x^{-1}
  \end{array} \right )
  \hspace{3em}
  & (x \notin \fO_{D(v)}).
  \\
 \end{array} \right .
$$
\item If $v \in \fV_{\bR,1}$,
$$
 \left ( \begin{array}{cc}
  1 & 0 \\
  x & 1
 \end{array} \right )
 =
 \left ( \begin{array}{cc}
  1 & \frac{x}{1 + x^2} \\
  0 & 1
 \end{array} \right )
 \left ( \begin{array}{cc}
  \frac{1}{\sqrt{1 + x^2}} & 0 \\
  0 & \sqrt{1 + x^2}
 \end{array} \right )
 \left ( \begin{array}{cc}
  \frac{1}{\sqrt{1 + x^2}} & - \frac{x}{\sqrt{1 + x^2}} \\
  \frac{x}{\sqrt{1 + x^2}} & \frac{1}{\sqrt{1 + x^2}}
 \end{array} \right ) .
$$
\item If $v \in \fV_{\bC}$,
$$
 \left ( \begin{array}{cc}
  1 & 0 \\
  x & 1
 \end{array} \right )
 =
 \left ( \begin{array}{cc}
  1 & \frac{\bar{x}}{1 + |x|_v} \\
  0 & 1
 \end{array} \right )
 \left ( \begin{array}{cc}
  \frac{1}{\sqrt{1 + |x|_v}} & 0 \\
  0 & \sqrt{1 + |x|_v}
 \end{array} \right )
 \left ( \begin{array} {cc}
  \frac{1}{\sqrt{1 + |x|_v}} & - \frac{\bar{x}}{\sqrt{1 + |x|_v}} \\
  \frac{x}{\sqrt{1 + |x|_v}} & \frac{1}{\sqrt{1 + |x|_v}}
 \end{array} \right ).
$$
\item If $v \in \fV_{\bR,2}$,
$$
 \left ( \begin{array}{cc}
  1 & 0 \\
  x & 1
 \end{array} \right )
 =
 \left ( \begin{array}{cc}
  1 & \frac{\bar{x}}{1 + |x|^2} \\
  0 & 1
 \end{array} \right )
 \left ( \begin{array}{cc}
  \frac{1}{\sqrt{1 + |x|^2}} & 0 \\
  0 & \sqrt{1 + |x|^2}
 \end{array} \right )
 \left ( \begin{array}{cc}
  \frac{1}{\sqrt{1 + |x|^2}} & - \frac{\bar{x}}{\sqrt{1 + |x|^2}} \\
  \frac{x}{\sqrt{1 + |x|^2}} & \frac{1}{\sqrt{1 + |x|^2}}
 \end{array} \right ),
$$
where $|x| = \Nr_{D(v)/k_v}(x)^{1/2}$ for $x \in D(v)$.
\end{itemize}

\medskip
\begin{lemma}
$$  I_v^{GL_2} (s) = \kappa_v^{-1} \times
  \left \{ \begin{array}{ll}
   \displaystyle\frac{1 - q_v^{-2d_vs-d_v}}{1 - q_v^{-2d_vs}}
	& (v \in \fV_f). \\
   \pi^{1/2} \displaystyle\frac{\Gamma(s)}{\Gamma(s+1/2)}
	& (v \in \fV_{\bR,1}). \\
   \pi / s
	& (v \in \fV_{\bC}). \\
    \displaystyle\frac{\pi^2}{s(4s+1)}
	& (v \in \fV_{\bR,2}).
  \end{array} \right .
$$
\end{lemma}

\textit{Proof}.
Let $v \in \mathfrak{V}_f$ and $\pi_{D(v)}$ be a prime element of $D(v)$. Since $\kappa_v \nu_{D(v)} = \mu_{D(v)}$, one has
\begin{eqnarray*}
 \kappa_v I_v^{GL_2}(s) & = & 1 + \sum_{t = 1}^\infty
	\int_{\pi^{-t}_{D(v)}\mathfrak{O}_{D(v)}^{\times}} \left|
	N_{D(v)/k_v}(x)\right|_v^{-2d_vs-d_v} d \mu_{D(v)} (x)
	\\
 & = &  1 + \sum_{t = 1}^\infty
	q_v^{-(2s+1)td_v}
	\int_{\pi_{D(v)}^{-t}\mathfrak{O}_{D(v)}^{\times}}
	d \mu_{D(v)} (x)  \\
 & = &  1 + \sum_{t = 1}^\infty
	q_v^{-2td_vs} (1 - q_v^{-d_v})  \\
 & = & 1 + (1 - q_v^{-d_v})
	\frac{q_v^{-2d_vs}}{1 - q_v^{-2d_vs}}  \\
 & = & \frac{1 - q_v^{-2d_vs-d_v}}{1 - q_v^{-2d_vs}}.
\end{eqnarray*}
If $v \in \fV_{\bR,2}$,
\begin{eqnarray*}
 \kappa_v I_v^{GL_2}(s) & = & \int_{D(v)}
	(1 + |x|^2)^{-4s-2} d \mu_{D(v)} (x) \\
 & = & 4 \int_0^{\infty} (1+r^2)^{-4s-2} r^3 dr
	 \int_0^{2 \pi} d\theta \int_{- \pi
	/2}^{\pi /2} \cos \varphi d\varphi \int_{- \pi /2}^{\pi /2}
	( \cos \psi )^2 d\psi \\
 & = & \frac{\pi^2}{s(4s+1)}.
\end{eqnarray*}
The other cases are also easy.



\vskip 5mm
\subsubsection*{3.6 An explicit formula of $C_{G,P}$}

To describe $I_v$, we define functions $F_1(s), F_2(s), F_3(s)$ in $s \in \bC$ as
$$ F_1(s) = \pi^{-s/2}\Gamma(s/2), \qquad F_2(s) = (2\pi)^{1-s}\Gamma(s), \qquad F_3(s) = (2\pi)^{2-s}\Gamma(s) \,. $$
By the formula (9) and Lemma 2, we have the following conclusion.

\medskip
\begin{lemma}
Notations being as above, we have
$$
 I_v = \kappa_v^{-\frac {1}{2} \frac {d^2} {d_v^2} n(n - 1)} \times
  \left \{ \begin{array}{ll}
   \displaystyle\prod_{\stackrel{\scriptstyle 1 \le i \le d}{i \equiv 0 \; (d_v)}} (1 - q_v^{-i})^{-(n-1)} \prod_{\stackrel{\scriptstyle d+1 \le i \le nd}{i \equiv 0 \; (d_v)}} (1 - q_v^{-i})^{-1}
	& (v \in \fV_f) \\
   \displaystyle \prod_{1 \le i \le d} F_1(i)^{n-1} \prod_{d+1 \le i \le nd} F_1(i)^{-1}
	& (v \in \fV_{\bR,1}) \\
   \displaystyle\prod_{1 \le i \le d} F_2(i)^{n-1} \prod_{d+1 \le i \le nd} F_2(i)^{-1}
	& (v \in \fV_{\bC}) \\
   \displaystyle \prod_{\stackrel{\scriptstyle 1 \le i \le d}{i \equiv 0 \; (2)}} F_3(i)^{n-1} \prod_{\stackrel{\scriptstyle d+1 \le i \le nd}{i \equiv 0 \; (2)}}F_3(i)^{-1}
	& (v \in \fV_{\bR,2}).
  \end{array} \right .
$$

\end{lemma}
\medskip

It is convenient to introduce a zeta function of $D$ in order to formulate an explicit formula of $C_{G, P}$. We first define the constant $C_D$ as follows:
\begin{itemize}
\item If $\ch(k) = 0$,
\begin{eqnarray*}
 C_D & = & \rho_k \mu_{D_\bA}(D_\bA/D)
	 \prod_{2 \le i \le d} \zeta_k(i) F_1(i)^{r_1 + r_3}
	 F_2(i)^{r_2} \\
     & & \times \prod_{v \in \fV_{f,2}}
	 \left( \prod_{\stackrel{\scriptstyle 1 \le i \le d-1}{i \not\equiv 0 \; (d_v)}}
	 1-q_v^{-i} \right)
	 \cdot \prod_{\stackrel{\scriptstyle 1 \le i \le d-1}{i \not\equiv 0 \; (2)}} i^{r_3} \,,
\end{eqnarray*}
where $r_1$, $r_2$ and $r_3$ denote the cardinality of $\fV_{\bR,1}$, $\fV_{\bC}$ and $\fV_{\bR,2}$, respectively.
\item If $\ch(k) > 0$,
\begin{equation*}
 C_D = (\log q) \rho_k \mu_{D_\bA}(D_\bA/D)\prod_{2 \le i \le d} \zeta_k(i)
	 \cdot \prod_{v \in \fV_{f,2}} \prod_{\stackrel{\scriptstyle 1 \le i \le d-1}{i \not\equiv 0 \; (d_v)}} (1-q_v^{-i}).
\end{equation*}
\end{itemize}

Then the zeta function of $D$ is defined by
\begin{eqnarray*}
 Z_D(s) & = & C_D^{-1} \prod_{0 \le i \le d-1} \zeta_k(s-i) F_1(s-i)^{r_1+r_3} F_2(s-i)^{r_2} \\
	& & \times \prod_{v \in \fV_{f,2}}
	 \left( \prod_{\stackrel{\scriptstyle 1 \le i \le d-1}{i \not\equiv 0 \; (d_v)}}
	 (1 - q_v^{- (s-i)}) \right)
	 \cdot \prod_{\stackrel{\scriptstyle 1 \le i \le d-1}{i \not\equiv 0 (2)}} (s-i)^{r_3} \,.
\end{eqnarray*}
By [T, Propositions 7 and 8], $Z_D(s)$ has a simple pole at $s=d$ with the residue
$$ \rho_D = \left \{ \begin{array}{ll}
\mu_{D_\bA}(D_\bA/D)^{-1} & \quad (\ch(k) = 0) \\
(\log q)^{-1}\mu_{D_\bA}(D_\bA/D)^{-1} & \quad (\ch(k) > 0)
\end{array}
\right .
$$
By the formula (8) and Lemmas 1 and 3, the constant $C_{G, P}$ is expressed in terms of $Z_D(s)$.

\begin{theorem}
If $G(k) = GL_n(D)$ and $P$ a minimal $k$-parabolic subgroup of $G$, then
$$ C_{G,P} = \mu_{D_\bA}(D_{\bA}/D)^{-n(n-1)/2}\rho_D^{n-1} \prod_{2 \le i \le n} Z_D(id)^{-1}.
$$
\end{theorem}

We take positive integers $n_1, \cdots, n_t$ such that $n = n_1 + \cdots + n_t$. For such $n_1, \cdots, n_t$, $R_{(n_1, \cdots, n_t)}$ denotes the standard $k$-parabolic subgroup of $G$ whose Levi subgroup $M_{R_{(n_1, \cdots, n_t)}}(k)$ is isomorphic with $GL_{n_1}(D) \times \cdots \times GL_{n_t}(D)$.
\medskip
\begin{cor}
 Let $R = R_{(n_1, \cdots, n_t)}$ be a standard $k$-parabolic subgroup of $G$. Then we have
$$ C_{G,R} = \mu_{D_\bA}(D_{\bA}/D)^{-\frac{1}{2}
	   (n^2 - \sum_{1 \leq j \leq t} n_j^2)} \rho_D^{t-1}
	   \frac{\prod_{1 \leq j \leq t}
	   \prod_{2 \leq i \leq n_j} Z_D(id)}
	   {\prod_{2 \leq i \leq n } Z_D(id)}.
$$
\end{cor}

This is a consequence of Theorem 2 and the relation $C_{G, R} = C_{G, P}/C_{M_R, M_R \cap P}$.





\vskip 8mm
\subsection*{4 Applications}

\subsubsection*{4.1 Fundamental Hermite constants of $GL_n(D)$}

We use the same notations as in \S 3.
For $1 \le m \le n-1$, $Q_m$ denotes the standard maximal $k$-parabolic subgroup $R_{(m, n-m)}$ of $G$. We recall the fundamental Hermite constants $\gamma(G, Q_m, k)$ introduced in [Wa].

In the following, we fix $m$ and write $Q$ for $Q_m$. The Levi subgroup $M_Q$ is given by
$$ M_Q(k) = \left \{ \diag(a,b) = \left ( \begin{array}{cc} a & 0 \\ 0 & b \end{array} \right ) \, : \, a \in GL_m(D), \;\; b \in GL_{n-m}(D) \right \}. $$
Denote by $Z_G$ and $Z_Q$ the central maximal $k$-split tori of $G$ and $M_Q$, respectively, \textit{i.e.},
$$ Z_G(k) = \{ \lambda I_n \, : \, \lambda \in k^\times \} \quad \textrm{and} \quad Z_Q(k) = \{ \diag(\lambda I_m, \mu I_{n-m}) \; : \; \lambda, \mu \in k^\times \}. $$
We define the $k$-rational characters $\alpha_Q \in \bX^*_k(Z_Q)$ and $\what\alpha_Q \in \bX^*_k(M_Q)$ as follows:
$$ \alpha_Q(\diag(\lambda I_m, \mu I_{n-m})) = \lambda \mu^{-1} $$
for $\diag(\lambda I_m, \mu I_{n-m}) \in Z_Q(k)$ and
$$ \what \alpha_Q(\diag(a, b)) = \Nr_{M_m(D)/k}(a)^{(n-m)/\gcd(m,n-m)} \Nr_{M_{n-m}(D)/k}(b)^{-m/\gcd(m,n-m)} $$
for $\diag(a,b) \in M_Q(k)$. Then $\alpha_Q$ (resp. $\what\alpha_Q$) is trivial on $Z_G$ and forms a $\bZ$-basis of the module $\bX^*_k(Z_G \backslash Z_Q)$ (resp. $\bX^*_k(Z_G \backslash M_Q)$).

Define the unimodular subgroups $G(\bA)^1$, $M_Q(\bA)^1$ and $Q(\bA)^1$ as follows:
\begin{eqnarray*}
G(\bA)^1 & = & \{ g \in G(\bA) \, : \, |\Nr_{M_n(D)/k}(g)|_\bA = 1 \}, \\
M_Q(\bA)^1 & = & \{ \diag(a, b) \in M_Q(\bA) \, : \, |\Nr_{M_m(D)/k}(a)|_\bA = |\Nr_{M_{n-m}(D)/k}(b)|_\bA = 1 \}, \\
Q(\bA)^1 & = & U_Q(\bA)M_Q(\bA)^1.
\end{eqnarray*}
The height function $H_Q \,:\, G(\bA) \longrightarrow \bR_+$ is well defined by
$$ H_Q(u \cdot \diag(a,b) \cdot h) = |\what\alpha_Q(\diag(a,b))|_\bA^{-1} $$
for $u \in U_Q(\bA), \diag(a,b) \in M_Q(\bA)$ and $h \in K$, and this is left $Z_G(\bA)Q(\bA)^1$ and right $K$ invariant.
We set $X_Q = Q(k) \backslash G(k)$ and $Y_Q = Q(\bA)^1 \backslash G(\bA)^1$. Then $X_Q$ is a subset of $Y_Q$ and the natural map $Y_Q \longrightarrow (Z_G(\bA)Q(\bA)^1) \backslash G(\bA)$ is injective. Thus $H_Q$ is restricted to $Y_Q$. Then the Hermite constants $\gamma(G,Q,k)$ and $\wtilde\gamma(G, Q, k)$ are defined to be
$$ \gamma(G, Q, k) = \max_{g \in G(\bA)^1} \min_{x \in X_Q} H_Q(xg).
$$
We write $\gamma_{n,m}(D)$ for $\gamma(G, Q_m, k)$, and especially $\gamma_n(D)$ for $\gamma(G, Q_1,k)$ since it is an analogue of Hermite--Rankin's constant.



\vskip 5mm
\subsubsection*{4.2 An explicit lower bound of $\gamma_{n,m}(D)$}

Since $Q = Q_m$ is maximal, there is a positive constant $\what e_Q$ such that $\delta_Q(g)  = |\what\alpha_Q(g)|_\bA^{\what e_Q}$  holds for all $g \in M_Q(\bA)$. It was proved in [Wa] that

\begin{equation}
\left ( \frac{D_{G,Q} \cdot  E_Q} {C_{G,Q}} \cdot \frac {\tau(G)} {\tau(Q)} \right )^{1/\what{e_Q}} \le \gamma(G, Q, k) \,,
\end{equation}
where $D_{G,Q}$ and $E_Q$ are given as follows with the notations in \S 1.1:
\begin{eqnarray*}
D_{G,Q} & = & \left \{ \begin{array}{ll} [\bX^*_k(Z_G) : \bX^*_k(G)]/[\bX^*_k(Z_Q) : \bX^*_k(M_Q)] & \quad (\ch(k) = 0), \\ d_G^*/d_{M_Q}^* & \quad (\ch(k) > 0),
\end{array} \right . \\
E_Q & = & \left \{ \begin{array}{ll} \what e_Q [\bX^*_k(Z_Q/Z_G) : \bX^*_k(M_Q/Z_G)] & \quad (\ch(k) = 0). \\ (1 - q_0^{-\what e_Q}) & \quad (\ch(k) > 0).
\end{array} \right.
\end{eqnarray*}
Here, $q_0 > 1$ stands for the generator of the subgroup $|\what\alpha_Q(M_Q(\bA) \cap G(\bA)^1)|_\bA$ of the cyclic group $q^\bZ$. The inequality (10) is strict if $\ch(k) > 0$. It is easy to see
$$ \begin{array}{c}
[\bX^*_k(Z_G) : \bX^*_k(G)] = dn, \qquad [\bX^*_k(Z_Q) : \bX^*_k(M_Q)] = d^2m(n-m),
\cr
[\bX^*_k(Z_Q/Z_G) : \bX^*_k(M_Q/Z_G)] = dm(n-m)/\gcd(m,n-m), \qquad \what e_Q = d \cdot \gcd(m, n-m)
\cr
d^*_G = \log q, \qquad d^*_{M_Q} = (\log q)^2, \qquad q_0 = q^{n/\gcd(m,n-m)}.
\end{array}
$$
Therefore,
$$ D_{G,Q} \cdot E_Q = \left \{ \begin{array}{ll}
dn & \quad (\ch(k) = 0). \\
(1 - q^{-dn})/(\log q) & \quad (\ch(k) > 0).
\end{array} \right .
$$
Since $\tau(G) = \tau(Q) = 1$ is known, Cororally 1 gives the following.

\medskip
\begin{theorem}
If $\ch(k) = 0$, then
$$ \left \{dn \cdot \mu_{D_\bA}(D_\bA/D)^{m(n-m)} \cdot \rho_D^{-1} \cdot \frac {\prod_{j=n-m+1}^n Z_D(jd)} {\prod_{j=2}^m Z_D(jd)} \right \}^{\frac {1} {d \cdot \gcd(m, n-m)}} \le \gamma_{n,m}(D) \,. $$
If $\ch(k) > 0$, then
$$ \left \{ \frac {1 - q^{-dn}} {\log q} \cdot \mu_{D_\bA}(D_\bA/D)^{m(n-m)} \cdot \rho_D^{-1} \cdot \frac {\prod_{j=n-m+1}^n Z_D(jd)} {\prod_{j=2}^m Z_D(jd)} \right \}^{\frac {1} {d \cdot \gcd(m, n-m)}} < \gamma_{n,m}(D) \,. $$
\end{theorem}
\medskip

For example, if $D$ is a quaternion division algebra over $\bQ$ and $m = 1$, then
one has $\rho_\bQ = 1$, $\mu_{D_\bA}(D_\bA/D) = \Nfd_{D/\bQ}^{1/2} = \prod_{p \in \fV_{f,2}} p$ and hence
$$ \left \{ \frac {12n(2n-1)^{r_3}}{\pi^{2n + 1/2}} \zeta_\bQ(2n)\zeta_\bQ(2n-1)\Gamma(n)\Gamma(n - \frac {1}{2}) \prod_{p \in \fV_{f,2}} p^{n-1} \left ( \frac {1 - p^{-(2n-1)}}{1 - p^{-1}} \right ) \right \}^{1/2} \le \gamma_n(D) \,, $$
where $r_3 = 1$ or $0$ according as $D$ is definite or indefinite. We denote the value of the left-hand side by $[n,D]$. For a square-free integer $N > 1$, let $D_N$ be a quaternion algebra over $\bQ$ such that $\Nfd_{D_{N}/\bQ}^{1/2} = N$, e.g., $D_2 = (-1,-1)$, $D_3 = (-1, -3)$, $D_5 = (-2, -5)$, $D_6 = (-1,3)$, $D_7 = (-1,-7)$ and $D_{10} = (-2,5)$, where $(a,b)$ stands for the quaternion algebra generated by $\mathbf{i}$ and $\mathbf{j}$ with $\mathbf{i}^2 = a$, $\mathbf{j}^2 = b$ and $\mathbf{ij} = -\mathbf{ji}$. The following tables give numerical examples of $[n,D_N]$:


\begin{center}
\begin{tabular}{c|cccc}
$n$ & $[n,D_2]$ & $[n, D_3]$ & $[n, D_5]$ & $[n, D_7]$ \\
\hline
2 & 1.297258519 & 1.443456027 & 1.726586552 & 1.978704389 \\
3 & 1.515273677 & 1.995775367 & 3.042255888 & 4.115273864 \\
4 & 2.530418525 & 4.040765897 & 7.938578156 & 12.70444456 \\
5 & 5.393737367 & 10.52001705 & 26.67683122 & 50.51365650 \\
6 & 13.94246428 & 33.28151972 & 108.9521040 & 244.1035544 \\
7 & 42.33203429 & 123.7370964 & 522.9445997 & 1386.303048 \\
8 & 147.6045644 & 528.3922475 & 2882.945637 & 9042.800847 \\
9 & 581.1565361 & 2547.947350 & 17947.12248 & 66607.84112 \\
10& 2549.878172 & 13691.81879 & 124505.8889 & 546744.5241
\end{tabular}
\end{center}
By [C-W], it is known $\gamma_2(D_2) = 2$, $\gamma_2(D_3) = 3$ and $\gamma_2(D_5) = 5$.
\begin{center}
\begin{tabular}{c|cccc}
$n$ & $[n,D_6]$ & $[n, D_{10}]$ & $[n, D_{14}]$ & $[n, D_{15}]$ \\
\hline
2& 1.559110703& 1.864926623& 2.137245010& 2.075098781 \\
3& 2.484720294& 3.787578034& 5.123474644& 4.988640043 \\
4& 6.085153489& 11.95502729& 19.13213909& 19.09070223 \\
5& 19.81735311& 50.25316799& 95.15640162& 98.01444678 \\
6& 80.25844451& 262.7381944& 588.6561594& 627.1722287 \\
7& 388.2457592& 1640.825823& 4349.756821& 4796.155594 \\
8& 2182.851359& 11909.79207& 37356.88820& 42634.46615 \\
9& 13982.96635& 98492.61985& 365539.4219& 431818.2696 \\
10& 100515.7012& 914034.6441& 4013813.651& 4907997.900

\end{tabular}
\end{center}
There is no example of the exact value of $\gamma_n(D)$ for indefinite quaternion algebras.


\vskip 5mm
\subsubsection*{4.3 The asymptotic distribution of rational points on $Y_Q$}

Let $Q = Q_m$, $X_Q = Q(k) \backslash G(k)$ and $Y_Q = Q(\bA)^1 \backslash G(\bA)^1$ be the same as in \S 4.1. The projective variety $Q \backslash G$ is a $k$-form of Grassmannian and is called the Brauer--Severi variety. The set $X_Q$ is considered as the set of $k$-rational points of $Q \backslash G$. For a positive real number $T$, let us define the subset $B_T$ of $Y_Q$ by
$$ B_T = \{ y \in Y_Q \;:\; H_Q(y) \le T \}\,. $$
For $g \in G(\bA)^1$, the subset $B_Tg$ is the translation of $B_T$ by $g$. The constant $\gamma_{n,m}(D)$ measures the existence of rational points in $B_Tg$, \textit{i.e.}, we have $B_Tg \cap X_Q \ne \emptyset$ for every $g \in G(\bA)^1$ if $\gamma_{n,m}(D) \le T$. In the case that $k$ is an algebraic number field, the cardinality of $B_Tg \cap X_Q$ is increasing to proportion to the volume of $B_T$ as $T \to \infty$. More precisely, it was proved in [Wa2] that
$$ \lim_{T \to \infty} \sharp(B_Tg \cap X_Q) \cdot \frac {D_{G,Q} \cdot E_Q} {C_{G,Q}} T^{-\what e_Q} = \frac {\tau(Q)} {\tau(G)} \,. $$
Therefore, we obtain the following.

\medskip
\begin{theorem}
We assume $k$ is an algebraic number field. Then the asymptotic behavior
$$ \sharp(B_Tg \cap X_Q) \sim \frac {T^{d \cdot \gcd(m,n-m)}} {dn |D_k|^{d^2(m(n-m)+1)/2} \Nfd_{D/k}^{(m(n-m)+1)/2}} \frac {\prod_{j=2}^m Z_D(jd)} {\prod_{j=n-m+1}^n Z_D(jd)}  \quad \text{as $T \to \infty$} $$
holds for all $g \in G(\bA)^1$.
\end{theorem}
\medskip

For example, if $k = \bQ$, $m = 1$ and $D = D_N$ as defined above, then we have
$$ \sharp(B_Tg \cap X_Q) \sim \frac {T^2} {[n, D_N]^2} \quad \text{as $T \to \infty$}.$$

\vskip 8mm
{\large \bf References}

\begin{list}{}{\setlength{\labelwidth}{3em}}
\item[{[B-T]}] Borel, A and Tits, J., Groupes r\'eductifs, Publ. Math. I.H.E.S. 27 (1965) 55 - 150.

\item[{[C-W]}] Coulangeon, R. and Watanabe, T., Hermite constant and Vorono\"i theory over a quaternion skew field, preprint, 2003.

\item[{[Ic]}] Ichino, A., A regularized Siegel--Weil formula for unitary groups, Math. Z. to appear.

\item[{[Ik]}] Ikeda, T., On the residue of the Eisenstein series and the Siegel--Weil formula, Comp. Math. 103 (1996) 183 - 218.

\item[{[K]}] Knapp, A., Representation Theory of Semisimple Groups, Princeton Univ. Press, Princeton, 1986.

\item[{[L]}] Lai, K. F., Tamagawa numbers of reductive algebraic groups, Comp. Math. 41 (1980) 153 -188.

\item[{[O]}] Oesterl\'e, J., Nombres de Tamagawa et groupes unipotents en caract\'eristique $p$, Invent. Math. 78 (1984) 13 - 88.

\item[{[R]}] Reiner, I., Maximal Orders, Academic Press, London etal., 1975.

\item[{[T]}] Turner, S., Zeta-functions of central simple algebras over global fields, An. Acad. Brasil Ci\^enc. 48 (1976) 171 - 186.

\item[{[V]}] Vign\'eras, M.-F., Arithm\'etique des alg\'ebres de quaternions, Springer-Verlag, Lecture Note in Mathematics 800, Heidelberg, 1980.

\item[{[Wa]}] Watanabe, T., Fundamental Hermite constants of linear algebraic groups, J. Math. Soc. Japan 55 (2003) 1061 - 1080.

\item[{[Wa2]}] Watanabe, T,. The Hardy--Littlewood property of flag varieties, Nagoya Math. J. 170 (2003) 185 - 211.

\item[{[We]}] Weil, A., Basic Number Theory, Springer-Verlag, Berlin etal., 1974.

\end{list}


\vskip 8mm
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043 Japan

nakamura@gaia.math.wani.osaka-u.ac.jp ; watanabe@math.wani.osaka-u.ac.jp
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