$\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}=\Gamma^{\sharp}(R_{\nu}^{\;{\mathfrak{m}}}\times\\{0\\})\;.$

For every measurable subset ${\cal E}$ of
$\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$, with the map
$\pi_{{\mathfrak{m}},{\mathfrak{n}}}:G^{\prime\prime}\rightarrow\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$
introduced in Equation (44), we define

${\widetilde{{\cal E}}}=\pi_{{\mathfrak{m}},{\mathfrak{n}}}^{\;-1}({\cal
E})\subset G^{\prime\prime}\;,$ (52)

which is a measurable subset of $G^{\prime\prime}$ invariant by the
translations on the right by $G^{\prime\prime}(R_{\nu})$.

For every measurable subset $\Phi$ of
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$, with
$\operatorname{orb}_{\mathfrak{m}}:{\cal
M}_{{\mathfrak{n}},{\mathfrak{m}}}({\cal
O}_{\nu})\rightarrow\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$
the isometric map defined in Subsection 3.1, we define

${\widetilde{\Phi}}=\operatorname{orb}_{\mathfrak{m}}^{\,-1}(\Phi)\subset{\cal
M}_{{\mathfrak{n}},{\mathfrak{m}}}({\cal O}_{\nu})\;,$ (53)

which is a measurable subset of ${\cal
M}_{{\mathfrak{n}},{\mathfrak{m}}}({\cal O}_{\nu})$.

For every $r\in{\mathbb{Z}}$ and for all measurable subsets $\Phi$ of
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$ and ${\cal E}$ of
$\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$, using the notation
of Equation (50) with $\Psi={\widetilde{\Phi}}$ and ${\cal
F}={\widetilde{{\cal E}}}$, we finally define

$\Omega=U^{-}({\cal
O}_{\nu})\;G^{\prime\prime}_{\lozenge}\;Z\;U^{+}_{\Box}\subset
G^{\sharp}\quad\text{and}\quad\Omega_{\Phi,{\cal
E},r}=U^{-}_{{\widetilde{\Phi}}}\;G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\;Z_{r}\;U^{+}_{\Box}\subset\Omega\;.$ (54)

###### Lemma 4.1

With
$c^{\prime}_{1}=(q-1)\,q^{({\mathfrak{g}}-1)\,{\mathfrak{m}}\,{\mathfrak{n}}}\;q_{\nu}^{\;2\,{\mathfrak{m}}\,{\mathfrak{n}}+2}\;\frac{\prod_{i=1}^{{\mathfrak{m}}}(q_{\nu}^{\;i}-1)^{2}\prod_{i=1}^{{\mathfrak{n}}}(q_{\nu}^{\;i}-1)^{2}}{(q_{\nu}-1)^{2}\;\prod_{i=1}^{\mathbf{d}}(q_{\nu}^{\;i}-1)^{2}}$,
we have

$\mu_{G}(\Omega_{\Phi,{\cal
E},r})=c^{\prime}_{1}\;q_{\nu}^{\;\mathbf{d}\operatorname{lcm}({\mathfrak{m}},{\mathfrak{n}})\,r}\;\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}(\Phi)\;\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}({\cal
E})\;.$

Proof. By Proposition 2.5, we have

$\mu_{G}(\Omega_{\Phi,{\cal
E},r})=c_{1}\;\mu_{U^{-}}(U^{-}_{{\widetilde{\Phi}}})\;\mu_{G^{\prime\prime}}(G^{\prime\prime}_{{\widetilde{{\cal
E}}}})\;\mu_{U^{+}}(U^{+}_{\Box})\;\int_{z\in
Z_{r}}|\chi_{\mathfrak{m}}(z)|^{\mathbf{d}\,{\mathfrak{m}}}\;d\mu_{Z}(z)\;.$
(55)

By Equations (18), (53) and (33), and since
$\Phi\subset\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$, we have

$\mu_{U^{-}}(U^{-}_{{\widetilde{\Phi}}})=\operatorname{Haar}_{{\mathfrak{n}},{\mathfrak{m}}}({\widetilde{\Phi}})=\operatorname{Haar}_{{\mathfrak{n}},{\mathfrak{m}}}(\operatorname{orb}_{\mathfrak{m}}^{\;-1}(\Phi))=c_{1}\;\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}(\Phi)\;.$
(56)

By Equations (18) and (9), we have

$\mu_{U^{+}}(U^{+}_{\Box})=\operatorname{Haar}_{{\mathfrak{m}},{\mathfrak{n}}}(\Box)=\operatorname{Covol}(R_{\nu}^{\;{\mathfrak{m}}\,{\mathfrak{n}}})=q^{({\mathfrak{g}}-1)\,{\mathfrak{m}}\,{\mathfrak{n}}}\;.$
(57)

Note that $z=\big{(}\begin{smallmatrix}\lambda I_{\mathfrak{m}}&0\\\ 0&\mu
I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}$ belongs to $Z_{r}$ if and only if
$\chi_{\mathfrak{m}}(z)=\lambda=\pi_{\nu}^{\;-\frac{\operatorname{lcm}\\{{\mathfrak{m}},{\mathfrak{n}}\\}}{{\mathfrak{m}}}\;r}\;$.
Since the Haar measure $\mu_{Z}$ is normalized so that $\mu_{Z}(Z({\cal
O}_{\nu}))=1$, we have

$\int_{z\in
Z_{r}}|\chi_{\mathfrak{m}}(z)|^{\mathbf{d}\,{\mathfrak{m}}}d\mu_{Z}(z)=|\pi_{\nu}^{\;-\frac{\operatorname{lcm}\\{{\mathfrak{m}},{\mathfrak{n}}\\}}{{\mathfrak{m}}}\;r}|^{\mathbf{d}\,{\mathfrak{m}}}=q_{\nu}^{\;\mathbf{d}\operatorname{lcm}({\mathfrak{m}},{\mathfrak{n}})\,r}\;.$
(58)

Note that $\operatorname{GL}_{\mathfrak{n}}^{1}(A)$ for $A=R_{\nu},{\cal
O}_{\nu},K_{\nu}$ is stable by the Cartan involution
$\underline{g}\mapsto\widecheck{\underline{g}}=\;^{t}\underline{g}^{-1}$, and
that this map preserves the Haar measure
$\mu_{\operatorname{GL}_{\mathfrak{n}}^{1}(K_{\nu})}$ (defined in Subsection
3.2) of the selfadjoint unimodular group
$\operatorname{GL}_{\mathfrak{n}}^{1}(K_{\nu})$. For every $\lambda\in{\cal
O}_{\nu}^{\;\times}$, let

$s^{\prime}_{\mathfrak{m}}(\lambda)=\big{(}\big{(}\begin{smallmatrix}\lambda&0\\\
0&I_{{\mathfrak{m}}-1}\end{smallmatrix}\big{)},I_{{\mathfrak{n}}}\big{)}\in\operatorname{GL}_{\mathfrak{m}}({\cal
O}_{\nu})\times\operatorname{GL}_{\mathfrak{n}}({\cal O}_{\nu})\;.$

Let
$\iota:G^{\prime\prime}\rightarrow\operatorname{GL}^{1}_{\mathfrak{m}}(K_{\nu})\times\operatorname{GL}^{1}_{\mathfrak{n}}(K_{\nu})$
be the group morphism defined by $\big{(}\begin{smallmatrix}\overline{g}&0\\\
0&\underline{g}\end{smallmatrix}\big{)}\mapsto(\,\overline{g},\widecheck{\underline{g}}\,)$.
Any element of
$\operatorname{GL}^{1}_{\mathfrak{m}}(K_{\nu})\times\operatorname{GL}^{1}_{\mathfrak{n}}(K_{\nu})$
may be written as
$s^{\prime}_{\mathfrak{m}}(\lambda)\,\iota(g^{\prime\prime})$ for unique
elements $\lambda\in{\cal O}_{\nu}^{\;\times}$ and $g^{\prime\prime}\in
G^{\prime\prime}$. Hence we have a split exact sequence

$1\longrightarrow
G^{\prime\prime}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\operatorname{GL}^{1}_{\mathfrak{m}}(K_{\nu})\times\operatorname{GL}^{1}_{\mathfrak{n}}(K_{\nu})\longrightarrow{\cal
O}_{\nu}^{\;\times}\longrightarrow 1\;,$

which induces two split exact sequences

$1\longrightarrow G^{\prime\prime}({\cal
O}_{\nu})\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\operatorname{GL}_{\mathfrak{m}}({\cal
O}_{\nu})\times\operatorname{GL}_{\mathfrak{n}}({\cal
O}_{\nu})\longrightarrow{\cal O}_{\nu}^{\;\times}\longrightarrow 1\;,$
$1\longrightarrow
G^{\prime\prime}(R_{\nu})\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\operatorname{GL}_{\mathfrak{m}}(R_{\nu})\times\operatorname{GL}_{\mathfrak{n}}(R_{\nu})\longrightarrow
R_{\nu}^{\;\times}\longrightarrow 1\;.$

By the normalisation of the measures (see Equations (36), (16) and (34)), the
Haar measure
$\mu_{\operatorname{GL}_{\mathfrak{m}}^{1}(K_{\nu})}\otimes\mu_{\operatorname{GL}_{\mathfrak{n}}^{1}(K_{\nu})}$
of
$\operatorname{GL}^{1}_{\mathfrak{m}}(K_{\nu})\times\operatorname{GL}^{1}_{\mathfrak{n}}(K_{\nu})$
satisfies

$d(\mu_{\operatorname{GL}_{\mathfrak{m}}^{1}(K_{\nu})}\otimes\mu_{\operatorname{GL}_{\mathfrak{n}}^{1}(K_{\nu})})(s^{\prime}_{\mathfrak{m}}(\lambda)\iota(g^{\prime\prime}))=(1-q_{\nu}^{\;-1})\;d\mu_{G^{\prime\prime}}(g^{\prime\prime})\;d\mu_{{\cal
O}_{\nu}^{\;\times}}(\lambda)\;.$ (59)

We endow from now on the product space
$(\operatorname{GL}^{1}_{\mathfrak{m}}(K_{\nu})/\operatorname{GL}_{\mathfrak{m}}(R_{\nu}))\times(\operatorname{GL}^{1}_{\mathfrak{n}}(K_{\nu})/\operatorname{GL}_{\mathfrak{n}}(R_{\nu}))$
with the product measure
$\mu_{\operatorname{Lat}^{1}_{\mathfrak{m}}}\otimes\mu_{\operatorname{Lat}^{1}_{\mathfrak{n}}}$
(see Subsection 3.2 for the definition of the measures
$\mu_{\operatorname{Lat}^{1}_{k}}$ for
$k\in{\mathbb{N}}{\smallsetminus}\\{0\\}$) and the quotient space
$G^{\prime\prime}/G^{\prime\prime}(R_{\nu})$ with the measure
$\mu_{G^{\prime\prime}/G^{\prime\prime}(R_{\nu})}$ so that the two canonical
projections (that are covering maps)

$\operatorname{GL}^{1}_{\mathfrak{m}}(K_{\nu})\times\operatorname{GL}^{1}_{\mathfrak{n}}(K_{\nu})\rightarrow(\operatorname{GL}^{1}_{\mathfrak{m}}(K_{\nu})/\operatorname{GL}_{\mathfrak{m}}(R_{\nu}))\times(\operatorname{GL}^{1}_{\mathfrak{n}}(K_{\nu})/\operatorname{GL}_{\mathfrak{n}}(R_{\nu}))$

and $G^{\prime\prime}\rightarrow G^{\prime\prime}/G^{\prime\prime}(R_{\nu})$
locally preserve the measures. Using Equation (43) for the first equality
below, and Equation (59) for the second one, since we have
${\operatorname{Card}}\;R_{\nu}^{\;\times}=q-1$ by Equation (2) and again by
Equation (34), we hence have

$\|\,\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}\,\|=\|\,\mu_{\operatorname{Lat}^{1}_{\mathfrak{m}}}\otimes\mu_{\operatorname{Lat}^{1}_{\mathfrak{n}}}\,\|=\frac{(1-q_{\nu}^{\;-1})^{2}}{q-1}\;\|\,\mu_{G^{\prime\prime}/G^{\prime\prime}(R_{\nu})}\,\|\;.$

Since the measures
$\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}$ and
$\mu_{G^{\prime\prime}/G^{\prime\prime}(R_{\nu})}$ on
$\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}=G^{\prime\prime}/G^{\prime\prime}(R_{\nu})$
are both $G^{\prime\prime}$-invariant (see Equation (42) for the first one),
we thus have
$\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}=\frac{(q_{\nu}-1)^{2}}{q_{\nu}^{\;2}(q-1)}\;\mu_{G^{\prime\prime}/G^{\prime\prime}(R_{\nu})}$.
In particular, we have

$\mu_{G^{\prime\prime}}(G^{\prime\prime}_{{\widetilde{{\cal
E}}}})=\mu_{G^{\prime\prime}}(G^{\prime\prime}_{\lozenge}\cap{\widetilde{{\cal
E}}})=\mu_{G^{\prime\prime}/G^{\prime\prime}(R_{\nu})}({\cal
E})=\frac{(q-1)\,q_{\nu}^{\;2}}{(q_{\nu}-1)^{2}}\;\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}({\cal
E})\;.$ (60)

Lemma 4.1 follows from Equation (55) by plugging in the computations of
Equations (56), (60), (57) and (58), by defining the constant
$c^{\prime}_{1}=c_{1}^{\;2}\;\frac{(q-1)\,q_{\nu}^{\;2}}{(q_{\nu}-1)^{2}}\;q^{({\mathfrak{g}}-1)\,{\mathfrak{m}}\,{\mathfrak{n}}}$
and by expliciting $c_{1}$ using Proposition 2.5. $\Box$

The following result gives a precise 1-to-1 correspondence between partial
lattices in $\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}=\Gamma^{\sharp}(R_{\nu}^{\;{\mathfrak{m}}}\times\\{0\\})$
and appropriate matrices in the discrete group
$\Gamma=\operatorname{SL}_{\mathbf{d}}(R_{\nu})$.

###### Proposition 4.2

The map $g\mapsto\Lambda_{g}=g(R_{\nu}^{\;{\mathfrak{m}}}\times\\{0\\})$ from
$\Gamma\cap\Omega$ to $\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$ is a bijection such that for every
nonzero ideal $I$ of $R_{\nu}$, for every $r\in{\mathbb{Z}}$ and for all
measurable subsets $\Phi$ of
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$ and ${\cal E}$ of
$\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$, the following two
assertions are equivalent

  1. (1)

the integral matrix $g\in\Gamma\cap\Omega$ lies in $\Omega_{\Phi,{\cal
E},r}\cap\Gamma_{I}$,

  2. (2)

the primitive ${\mathfrak{m}}$-lattice $\Lambda_{g}\in\operatorname{{\cal
P}\\!{\cal L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$ satisfies that
$\Lambda_{g}\in\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}(I)$ (as defined in the beginning of Subsection
2.4), $V_{\Lambda_{g}}\in\Phi$ (as defined in Subsection 2.2),
$\llbracket\Lambda_{g}\rrbracket\in{\cal E}$ (as defined in Equation (49)) and
$\overline{\operatorname{Covol}}(\Lambda_{g})=\overline{\operatorname{Covol}}((\Lambda_{g})^{\perp})=q_{\nu}^{\;\operatorname{lcm}\\{{\mathfrak{m}},{\mathfrak{n}}\\}r}$.

Proof. Since $\Omega\subset G^{\sharp}$, if $g\in\Gamma\cap\Omega$, then
$g\in\Gamma^{\sharp}$, thus
$\Lambda_{g}=g(R_{\nu}^{\;{\mathfrak{m}}}\times\\{0\\})\in\operatorname{{\cal
P}\\!{\cal L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$. Hence the above map is
well defined.

For all $g=\big{(}\begin{smallmatrix}\alpha&\gamma\\\
\beta&\delta\end{smallmatrix}\big{)}\in G^{\sharp}$ and
$p^{+}=\big{(}\begin{smallmatrix}\alpha^{\prime}&\gamma^{\prime}\\\
0&\delta^{\prime}\end{smallmatrix}\big{)}\in
P^{+}=G^{\prime\prime}ZU^{+}\subset G^{\sharp}$, since the first column of
$g\,p^{+}$ is $\big{(}\begin{smallmatrix}\alpha\alpha^{\prime}\\\
\beta\alpha^{\prime}\end{smallmatrix}\big{)}$, since
$(\beta\alpha^{\prime})(\alpha\alpha^{\prime})^{-1}=\beta\alpha^{-1}$ and
since
$\nu(\det(\alpha\alpha^{\prime}))=\nu(\det\alpha)+\nu(\det\alpha^{\prime})$,
we have $g\,p^{+}\in G^{\sharp}$. In particular, the action by right
translations on $\Gamma=\operatorname{SL}_{\mathbf{d}}(R_{\nu})$ of its
subgroup $P^{+}(R_{\nu})$ defined in Equation (11), which satisfies
$P^{+}(R_{\nu})=G^{\prime\prime}(R_{\nu})\,U^{+}(R_{\nu})$, preserves
$\Gamma^{\sharp}=\Gamma\cap G^{\sharp}$. By the identification given just
after the statement of Lemma 2.1, we thus have

$\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}=\Gamma^{\sharp}/P^{+}(R_{\nu})\;.$
(61)

Let us prove that

$G^{\sharp}=\coprod_{\gamma\in P^{+}(R_{\nu})}\Omega\,\gamma\;.$ (62)

Since $P^{+}(R_{\nu})\subset\Gamma$, this will imply that
$\Gamma^{\sharp}=\Gamma\cap G^{\sharp}=\coprod_{\gamma\in
P^{+}(R_{\nu})}(\Gamma\cap\Omega)\gamma$. By Equation (61), this will imply
that the map $g\mapsto\Lambda_{g}=g(R_{\nu}^{\;{\mathfrak{m}}}\times\\{0\\})$
from $\Gamma\cap\Omega$ to $\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$ is a bijection.

In order to prove Equation (62), let us fix $g\in G^{\sharp}$. By Proposition
2.5, there exist unique elements $u^{-}\in U^{-}({\cal O}_{\nu})$,
$g^{\prime\prime}=\big{(}\begin{smallmatrix}\overline{g}&0\\\
0&\underline{g}\end{smallmatrix}\big{)}\in G^{\prime\prime}$, $z\in Z$ and
$u^{+}\in U^{+}$ such that $g=u^{-}\,g^{\prime\prime}\,z\,u^{+}$. By the
definition of the fundamental domain $G^{\prime\prime}_{\lozenge}$, there
exist a unique $f^{\prime\prime}=\big{(}\begin{smallmatrix}\overline{f}&0\\\
0&\underline{f}\end{smallmatrix}\big{)}\in G^{\prime\prime}_{\lozenge}$ and a
unique $\gamma^{\prime\prime}=\big{(}\begin{smallmatrix}\overline{\gamma}&0\\\
0&\underline{\gamma}\end{smallmatrix}\big{)}\in G^{\prime\prime}(R_{\nu})$
such that

$\overline{f}\;\overline{\gamma}=\overline{g},\quad\underline{f}\;\underline{\gamma}=\underline{g}\quad\text{and}\quad
g^{\prime\prime}=f^{\prime\prime}\,\gamma^{\prime\prime}\;.$ (63)

Since $Z$ centralizes $G^{\prime\prime}$, we have
$g=u^{-}\,f^{\prime\prime}\,z\,\gamma^{\prime\prime}\,u^{+}$. Since
$G^{\prime\prime}$ normalizes $U^{+}$, and by the definition of $\Box$, there
exist unique elements $u^{+}_{0}\in U^{+}_{\Box}$ and $\gamma_{0}\in
U^{+}(R_{\nu})$ such that
$\gamma^{\prime\prime}\,u^{+}(\gamma^{\prime\prime})^{-1}=u^{+}_{0}\,\gamma_{0}$.
Defining $\gamma=\gamma_{0}\gamma^{\prime\prime}\in P^{+}(R_{\nu})$, we have

$g=u^{-}\,f^{\prime\prime}\,z\,(\gamma^{\prime\prime}\,u^{+}(\gamma^{\prime\prime})^{-1})\,\gamma^{\prime\prime}=u^{-}\,f^{\prime\prime}\,z\,u^{+}_{0}\gamma$
(64)

and $u^{-}\,f^{\prime\prime}\,z\,u^{+}_{0}\in\Omega$ by the definition of
$\Omega$ in Equation (54). Since the writing $h=u^{+}_{0}\gamma$ of an element
$h\in P^{+}(R_{\nu})=U^{+}(R_{\nu})\,G^{\prime\prime}(R_{\nu})$ with
$u^{+}_{0}\in U^{+}(R_{\nu})$ and $\gamma\in G^{\prime\prime}(R_{\nu})$ is
unique, this proves Equation (62).

Let us now assume that $g\in\Gamma\cap\Omega$. By the uniqueness of the
writing in Equation (64), we may uniquely write
$g=u^{-}\,f^{\prime\prime}\,z\,u^{+}_{0}$ with $u^{-}\in U^{-}({\cal
O}_{\nu})$, $f^{\prime\prime}\in G^{\prime\prime}_{\lozenge}$, $z\in Z$ and
$u^{+}_{0}\in U^{+}_{\Box}$. By the definition of $\Omega_{\Phi,{\cal E},r}$
in Equation (54), we have $g\in\Omega_{\Phi,{\cal E},r}$ if and only if
$u^{-}\in U^{-}_{{\widetilde{\Phi}}}$, $f^{\prime\prime}\in
G^{\prime\prime}_{{\widetilde{{\cal E}}}}$ and $z\in Z_{r}$.

We have $u^{-}\in U^{-}_{{\widetilde{\Phi}}}$ if and only if there exists
$\beta\in{\widetilde{\Phi}}$ with
$u^{-}=\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&\;0\\\
\beta&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}$, hence if and only if there
exists
$\beta\in{\widetilde{\Phi}}={\operatorname{orb}_{\mathfrak{m}}}^{-1}(\Phi)$
with
$\operatorname{orb}_{\mathfrak{m}}(\beta)=u^{-}(K_{\nu}^{\;{\mathfrak{m}}}\times\\{0\\})=V_{\Lambda_{u^{-}}}$
by the definition of $\operatorname{orb}_{\mathfrak{m}}$ in Subsection 3.1,
therefore if and only if $V_{\Lambda_{g}}\in\Phi$ by Proposition 2.8 (i).

By Equation (52) and by the definition of
$\pi_{{\mathfrak{m}},{\mathfrak{n}}}$ in Equation (44), we have

$G^{\prime\prime}_{{\widetilde{{\cal
E}}}}=G^{\prime\prime}_{\lozenge}\cap\pi_{{\mathfrak{m}},{\mathfrak{n}}}^{\;-1}({\cal
E})=\big{\\{}\big{(}\begin{smallmatrix}\overline{g}&0\\\
0&\underline{g}\end{smallmatrix}\big{)}\in
G^{\prime\prime}_{\lozenge}:(\;\overline{g}\,R_{\nu}^{\;{\mathfrak{m}}},\,\widecheck{\underline{g}}\,R_{\nu}^{\;{\mathfrak{n}}}\,)\in{\cal
E}\big{\\}}\;.$

Hence by Equation (49) and since $f^{\prime\prime}\in
G^{\prime\prime}_{\lozenge}$, we have $f^{\prime\prime}\in
G^{\prime\prime}_{{\widetilde{{\cal E}}}}$ if and only if
$\llbracket\Lambda_{g}\rrbracket\in{\cal E}$.

We have $z\in Z_{r}$ if and only if
$z=\begin{pmatrix}\pi_{\nu}^{\;-\frac{\operatorname{lcm}\\{{\mathfrak{m}},{\mathfrak{n}}\\}}{{\mathfrak{m}}}\,r}I_{\mathfrak{m}}&0\\\
0&\pi_{\nu}^{\;\frac{\operatorname{lcm}\\{{\mathfrak{m}},{\mathfrak{n}}\\}}{{\mathfrak{n}}}\,r}I_{\mathfrak{n}}\end{pmatrix}$,
hence if and only if
$\overline{\operatorname{Covol}}(\Lambda_{g})=q_{\nu}^{\;\operatorname{lcm}\\{{\mathfrak{m}},{\mathfrak{n}}\\}\,r}$
by Proposition 2.8 (ii) and by Equation (28). Note that we have
$\overline{\operatorname{Covol}}((\Lambda_{g})^{\perp})=\overline{\operatorname{Covol}}(\Lambda_{g})$
by Proposition 2.8 (ii)⟂.

The fact that $g\in\Gamma_{I}$ if and only if
$\Lambda_{g}\in\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}(I)$ has been shown in Lemma 2.3 (1). This
concludes the proof of Proposition 4.2. $\Box$

###  4.2 Counting in well-rounded families

A crucial tool of this paper is a counting result of lattice points by
Gorodnik and Nevo [GN]. In this subsection, after the necessary definitions,
we recall from [HP1] an adaptation of the Gorodnik-Nevo result, and we proceed
to the construction of the well-rounded family of subsets to which we will
apply it.

Let ${\bf G^{\prime}}$ be an absolutely connected and simply connected semi-
simple algebraic group over $K_{\nu}$, which is almost $K_{\nu}$-simple. Let
$G^{\prime}={\bf G^{\prime}}(K_{\nu})$ be the locally compact group of
$K_{\nu}$-points of ${\bf G^{\prime}}$. Let $\Gamma^{\prime}$ be a
nonuniform222This implies that ${\bf G^{\prime}}$ is isotropic over $K_{\nu}$,
as part of the assumptions of [GN]. lattice in $G^{\prime}$, and let
$\mu_{G^{\prime}}$ be any (left) Haar measure of $G^{\prime}$. Note that
$G^{\prime}=G$ and $\Gamma^{\prime}=\Gamma_{I}$ (defined in Subsection 2.4)
satisfy these assumptions for every nonzero ideal $I$ of $R_{\nu}$.

Let $\rho>0$. Let $({\cal V}^{\prime}_{\epsilon})_{\epsilon>0}$ be a
fundamental system of neighborhoods of the identity in $G^{\prime}$, which

$\bullet$ is symmetric (that is, $x\in{\cal V}^{\prime}_{\epsilon}$ if and
only if $x^{-1}\in{\cal V}^{\prime}_{\epsilon}$),

$\bullet$ is nondecreasing with $\epsilon$ (that is, ${\cal
V}^{\prime}_{\epsilon}\subset{\cal V}^{\prime}_{\epsilon^{\prime}}$ if
$\epsilon\leq\epsilon^{\prime}$), and

$\bullet$ has upper local dimension $\rho$, that is, there exist
$m_{1},\epsilon_{1}>0$ such that $\mu_{G^{\prime}}({\cal
V}^{\prime}_{\epsilon})\geq m_{1}\,\epsilon^{\rho}$ for every
$\epsilon\in\;]0,\epsilon_{1}[\,$.

Let $C\geq 0$. Let $({\cal Z}_{n})_{n\in{\mathbb{N}}}$ be a family of
measurable subsets of $G^{\prime}$. We define

$({\cal Z}_{n})^{+\epsilon}={\cal V}^{\prime}_{\epsilon}{\cal Z}_{n}{\cal
V}^{\prime}_{\epsilon}=\bigcup_{g,h\in{\cal V}^{\prime}_{\epsilon}}g{\cal
Z}_{n}h\;\;\;{\rm and}\;\;\;({\cal Z}_{n})^{-\epsilon}=\bigcap_{g,h\in{\cal
V}^{\prime}_{\epsilon}}g{\cal Z}_{n}h\;.$

The family $({\cal Z}_{n})_{n\in{\mathbb{N}}}$ is $C$-Lipschitz well-rounded
with respect to $({\cal V}^{\prime}_{\epsilon})_{\epsilon>0}$ if there exists
$\epsilon_{0}>0$ and $n_{0}\in{\mathbb{N}}$ such that for all
$\epsilon\in\;]0,\epsilon_{0}[$ and $n\geq n_{0}$, we have

$\mu_{G^{\prime}}(({\cal
Z}_{n})^{+\epsilon})\leq(1+C\,\epsilon)\;\mu_{G^{\prime}}(({\cal
Z}_{n})^{-\epsilon})\;.$

We refer to [HP1, Theo. 4.1] for a proof of the following adaptation of
results of Gorodnik-Nevo [GN].

###### Theorem 4.3

For every $\rho>0$, there exists
$\tau(\Gamma^{\prime})\in\;]0,\frac{1}{2(1+\rho)}]$ such that for every
symmetric nondecreasing fundamental system $({\cal
V}^{\prime}_{\epsilon})_{\epsilon>0}$ of neighborhoods of the identity in
$G^{\prime}$ with upper local dimension $\rho$, for every $C\geq 0$, for every
family $({\cal Z}_{n})_{n\in{\mathbb{N}}}$ of measurable subsets of
$G^{\prime}$ that is $C$-Lipschitz well-rounded with respect to $({\cal
V}^{\prime}_{\epsilon})_{\epsilon>0}$, and for every $\delta>0$, we have that,
as $n\rightarrow+\infty$,

$\Big{|}{\operatorname{Card}}({\cal
Z}_{n}\cap\Gamma^{\prime})-\frac{1}{\|\mu_{G^{\prime}/\Gamma^{\prime}}\|}\,\mu_{G^{\prime}}({\cal
Z}_{n})\Big{|}=\operatorname{O}\big{(}\mu_{G^{\prime}}({\cal
Z}_{n})^{1-\tau(\Gamma^{\prime})+\delta}\big{)}\;,$

where the function $\operatorname{O}(\cdot)$ depends only on
$G^{\prime},\Gamma^{\prime},\delta,C,({\cal
V}^{\prime}_{\epsilon})_{\epsilon>0},\rho$. $\Box$

We will use, as a fundamental system of neighborhoods of the identity element
in $G$, a family of compact-open subgroups of $G({\cal O}_{\nu})$ given by the
kernels of the morphisms of reduction modulo $\pi_{\nu}^{\;N}{\cal O}_{\nu}$
for appropriate $N\in{\mathbb{N}}$. For every $\epsilon>0$, let
$N_{\epsilon}=\big{\lfloor}-\log_{q_{\nu}}\epsilon\,\big{\rfloor}$ so that
$N_{\epsilon}\geq 1$ if and only if $\epsilon\leq\frac{1}{q_{\nu}}$. Let
${\cal V}_{\epsilon}=G({\cal O}_{\nu})$ if $\epsilon>\frac{1}{q_{\nu}}$ and
otherwise let

$\displaystyle{\cal V}_{\epsilon}$ $\displaystyle=\ker(G({\cal
O}_{\nu})\rightarrow\operatorname{SL}_{\mathbf{d}}({\cal
O}_{\nu}/\pi_{\nu}^{\;N_{\epsilon}}{\cal O}_{\nu}))$
$\displaystyle=\big{\\{}I_{\mathbf{d}}+\pi_{\nu}^{\;N_{\epsilon}}X:X\in{\cal
M}_{\mathbf{d}}({\cal O}_{\nu})\big{\\}}\cap G\;.$ (65)

The family $({\cal V}_{\epsilon})_{\epsilon>0}$ is indeed nondecreasing and we
have $\bigcap_{\epsilon>0}{\cal V}_{\epsilon}=\\{\operatorname{id}\\}$. Note
that for all $\epsilon_{1},\dots,\epsilon_{k}>0$, we have

$\displaystyle\min\\{N_{\epsilon_{1}},\cdots,N_{\epsilon_{k}}\\}$
$\displaystyle\geq\min\\{-\log_{q_{\nu}}\epsilon_{1},\cdots,-\log_{q_{\nu}}\epsilon_{k}\\}-1$
$\displaystyle\geq-\log_{q_{\nu}}(\epsilon_{1}+\cdots+\epsilon_{k})-1\geq
N_{q_{\nu}(\epsilon_{1}+\cdots+\epsilon_{k})}\,,$

hence

${\cal V}_{\epsilon_{1}}{\cal V}_{\epsilon_{2}}\cdots{\cal
V}_{\epsilon_{k}}\;\;\subset\;\;{\cal
V}_{q_{\nu}(\epsilon_{1}+\cdots+\epsilon_{k})}\,.$ (66)

For every subgroup $H$ of $G$, let ${\cal V}_{\epsilon}^{H}={\cal
V}_{\epsilon}\cap H$. The index of ${\cal V}_{\epsilon}$ in $G({\cal
O}_{\nu})$ is given by Lemma 2.6 with $N=N_{\epsilon}$.

We denote the operator norm of a linear operator $\ell$ of the normed
$K_{\nu}$-algebra ${\cal M}_{\mathbf{d}}(K_{\nu})$ (for the supremum norm
defined before Lemma 3.1) by

$\|\ell\|=\max\Big{\\{}\frac{\|\ell(X)\|}{\|X\|}:X\in{\cal
M}_{\mathbf{d}}(K_{\nu})-\\{0\\}\Big{\\}}\in
q_{\nu}^{\;{\mathbb{Z}}}\cup\\{0\\}\,,$

so that $\ell({\cal M}_{\mathbf{d}}({\cal O}_{\nu}))\subset{\cal
M}_{\mathbf{d}}(\pi_{\nu}^{\;-\log_{q_{\nu}}\|\ell\|}{\cal O}_{\nu})$ if
$\ell$ is invertible. For every $g\in G$, recall that $\operatorname{Ad}g$ is
the linear automorphism $x\mapsto gxg^{-1}$ of ${\cal
M}_{\mathbf{d}}(K_{\nu})$. Also recall that $P^{-}=U^{-}G^{\prime\prime}Z$.

###### Lemma 4.4

For all $\epsilon\in\;]0,\frac{1}{q_{\nu}}]$ and $g\in G$, we have

$g\,{\cal V}_{\epsilon}\,g^{-1}\;\subset\;{\cal
V}_{\|\operatorname{Ad}g\,\|\;\epsilon}\;,\;\;\;{\cal V}_{\epsilon}={\cal
V}^{P^{-}}_{\epsilon}\;{\cal V}^{U^{+}}_{\epsilon}\;\;\;{\rm and}\;\;\;{\cal
V}^{P^{-}}_{\epsilon}={\cal V}^{U^{-}}_{\epsilon}\;{\cal
V}^{G^{\prime\prime}}_{\epsilon}\;.$

Furthermore, the number $\rho=\mathbf{d}^{2}-1$ is an upper local dimension of
the family $({\cal V}_{\epsilon})_{\epsilon>0}$.

Proof. The first claim follows from the fact that

$\displaystyle g\,{\cal V}_{\epsilon}\,g^{-1}$
$\displaystyle=\big{(}I_{\mathbf{d}}+\pi_{\nu}^{\;N_{\epsilon}}g{\cal
M}_{\mathbf{d}}({\cal O}_{\nu})g^{-1}\big{)}\cap G$
$\displaystyle\subset\big{(}I_{\mathbf{d}}+\pi_{\nu}^{\;N_{\epsilon}-\log_{q_{\nu}}\|\operatorname{Ad}g\,\|}{\cal
M}_{\mathbf{d}}({\cal O}_{\nu})\big{)}\cap G={\cal
V}_{\epsilon\,\|\operatorname{Ad}g\,\|}\;.$

Note that ${\cal V}_{\epsilon}$ is contained in ${\cal U}_{G}$ (defined in
Equation (15)). Indeed, if $g=\big{(}\begin{smallmatrix}\alpha&\gamma\\\
\beta&\delta\end{smallmatrix}\big{)}\in{\cal V}_{\epsilon}$ then $\alpha\in
I_{\mathfrak{m}}+\pi_{\nu}^{N_{\epsilon}}{\cal M}_{\mathfrak{m}}({\cal
O}_{\nu})$, hence $\det(\alpha)\in 1+\pi_{\nu}^{N_{\epsilon}}{\cal O}_{\nu}$,
so that
$\nu(\det\alpha)=0\in\operatorname{lcm}\\{{\mathfrak{m}},{\mathfrak{n}}\\}{\mathbb{Z}}$
since $N_{\epsilon}\geq 1$ and therefore $g\in{\cal U}_{G}$.

By Proposition 2.5, we may hence uniquely write any
$g=\big{(}\begin{smallmatrix}\alpha&\gamma\\\
\beta&\delta\end{smallmatrix}\big{)}\in{\cal V}_{\epsilon}$ as
$g=u^{-}\,g^{\prime\prime}\,z\,u^{+}$ with $u^{-}\in U^{-}$,
$g^{\prime\prime}\in G^{\prime\prime}$, $z\in Z$ and $u^{+}\in U^{+}$. By
Equation (22) and since $\nu(\det\alpha)=0$, we have
$\alpha\in\operatorname{GL}_{\mathfrak{m}}({\cal O}_{\nu})$, $u^{\pm}\in
U^{\pm}({\cal O}_{\nu})$, $g^{\prime\prime}\in G^{\prime\prime}({\cal
O}_{\nu})$ and $z=I_{\mathbf{d}}$. Furthermore, since $g\in{\cal
V}_{\epsilon}$, we have
$\alpha=I_{\mathfrak{m}}\mod\pi_{\nu}^{\;N_{\epsilon}}$,
$\gamma=0\mod\pi_{\nu}^{\;N_{\epsilon}}$,
$\beta=0\mod\pi_{\nu}^{\;N_{\epsilon}}$ and
$\delta=I_{\mathfrak{n}}\mod\pi_{\nu}^{\;N_{\epsilon}}$. Therefore again by
Equation (22), we have $u^{\pm}\in{\cal V}_{\epsilon}^{U^{\pm}}$ and
$g^{\prime\prime}\in{\cal V}_{\epsilon}^{G^{\prime\prime}}$. This proves the
second and third claims.

In order to prove the last claim, let us apply Lemma 2.6 with
$N=N_{\epsilon}=\big{\lfloor}-\log_{q_{\nu}}\epsilon\,\big{\rfloor}$, so that
$N_{\epsilon}-1\leq-\log_{q_{\nu}}\epsilon$. Since $\mu_{G}(G({\cal
O}_{\nu}))=1$, we hence have

$\displaystyle\mu_{G}({\cal V}_{\epsilon})$
$\displaystyle=\frac{\mu_{G}(G({\cal O}_{\nu}))}{[G({\cal O}_{\nu}):{\cal
V}_{\epsilon}]}=\frac{q_{\nu}-1}{q_{\nu}^{\,(N_{\epsilon}-1)(\mathbf{d}^{2}-1)}}\prod_{i=0}^{\mathbf{d}-1}(q_{\nu}^{\;\mathbf{d}}-q_{\nu}^{\;i})^{-1}\geq\epsilon^{\mathbf{d}^{2}-1}(q_{\nu}-1)\prod_{i=0}^{\mathbf{d}-1}(q_{\nu}^{\;\mathbf{d}}-q_{\nu}^{\;i})^{-1}\;.$

This proves the result. $\Box$

We will need the following effective version of the refined LU decomposition
by blocks given in Proposition 2.5. We denote by $c:h\mapsto c_{h}$ the
continuous function from $G$ to $[0,+\infty[$ defined by
$c_{h}=\|\operatorname{Ad}h\,\|$ for every $h\in G$.

###### Lemma 4.5

For all $\epsilon\in\;]0,\frac{1}{q_{\nu}}]$, $u^{-}\in U^{-}$,
$g^{\prime\prime}\in G^{\prime\prime}$, $z\in Z$ and $u^{+}\in U^{+}$, if
$|\chi_{\mathfrak{m}}(z)|\geq 1$ and $g=u^{-}\,g^{\prime\prime}\,z\,u^{+}$,
then

${\cal V}_{\epsilon}\;g\;{\cal V}_{\epsilon}\;\subset\;u^{-}\;{\cal
V}_{c_{g^{\prime\prime}}q_{\nu}(c_{u^{-}g^{\prime\prime}}+c_{u^{+}})\,\epsilon}^{U^{-}}\;g^{\prime\prime}\;{\cal
V}_{q_{\nu}(c_{u^{-}g^{\prime\prime}}+c_{u^{+}})\,\epsilon}^{G^{\prime\prime}}\;z\;{\cal
V}_{q_{\nu}(c_{u^{-}g^{\prime\prime}}+2c_{u^{+}})\,\epsilon}^{U^{+}}\,u^{+}\;.$

Proof. In order to simplify the notation, let $p=u^{-}g^{\prime\prime}$ and
$u=u^{+}$, so that $g=pzu$. If $z=\big{(}\begin{smallmatrix}\lambda
I_{{\mathfrak{m}}}&0\\\ 0&\mu I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}$, as
seen in Equation (19) for the first equality, we have

$z\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&0\\\
\beta&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}z^{-1}=\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&0\\\
\mu\,\lambda^{-1}\beta\;&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}\quad\text{and}\quad
z^{-1}\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&\gamma\\\
0&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}z=\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&\;\lambda^{-1}\mu\,\gamma\\\
0&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}.$

Since $|\lambda|=|\chi_{\mathfrak{m}}(z)|\geq 1$ and $\det z=1$ so that
$|\mu|=|\lambda|^{-\frac{{\mathfrak{m}}}{{\mathfrak{n}}}}\leq 1$, for every
$\epsilon^{\prime}>0$, we have

$z\;{\cal V}_{\epsilon^{\prime}}^{U^{-}}z^{-1}\subset{\cal
V}_{\epsilon^{\prime}}^{U^{-}}\quad\text{and}\quad z^{-1}\,{\cal
V}_{\epsilon^{\prime}}^{U^{+}}z\subset{\cal V}_{\epsilon^{\prime}}^{U^{+}}\;.$
(67)

Using for the following sequence of equalities and inclusions respectively

$\bullet$ the first claim of Lemma 4.4 for the first inclusion,

$\bullet$ the second and third claims of Lemma 4.4 for the second equality,

$\bullet$ the claim on the left in Equation (67) and the fact that $Z$
centralises $G^{\prime\prime}$ for the second inclusion,

$\bullet$ the fact that ${\cal V}_{c_{u}\epsilon}$ is a normal subgroup of
$G({\cal O}_{\nu})$ that contains ${\cal V}_{c_{u}\epsilon}^{U^{-}}\,{\cal
V}_{c_{u}\epsilon}^{G^{\prime\prime}}$ for the third inclusion,

$\bullet$ twice the claim on the right in Equation (67) for the fourth
inclusion,

$\bullet$ twice Equation (66) with $k=2$ and $k=3$, defining the constants
$c^{\prime\prime}_{1}=q_{\nu}(c_{p}+c_{u})$ and
$c^{\prime\prime}_{2}=q_{\nu}(c_{p}+2c_{u})$ for the fifth inclusion,

$\bullet$ again the third claim of Lemma 4.4 for the sixth equality,

$\bullet$ the fact that $G^{\prime\prime}$ normalizes $U^{-}$ and again the
first claim of Lemma 4.4 for the last inclusion,

we have

$\displaystyle{\cal V}_{\epsilon}\,g\,{\cal V}_{\epsilon}$
$\displaystyle=p\;p^{-1}{\cal V}_{\epsilon}\,p\;z\;u\,{\cal
V}_{\epsilon}\,u^{-1}\;u\;\subset\;p\,{\cal V}_{c_{p}\epsilon}\,z\,{\cal
V}_{c_{u}\epsilon}\,u=p\,{\cal V}_{c_{p}\epsilon}^{P^{-}}\,{\cal
V}_{c_{p}\epsilon}^{U^{+}}\,z\;{\cal V}_{c_{u}\epsilon}^{U^{-}}\,{\cal
V}_{c_{u}\epsilon}^{G^{\prime\prime}}\,{\cal V}_{c_{u}\epsilon}^{U^{+}}\,u$
$\displaystyle=p\,{\cal V}_{c_{p}\epsilon}^{P^{-}}\,{\cal
V}_{c_{p}\epsilon}^{U^{+}}\,z\;{\cal
V}_{c_{u}\epsilon}^{U^{-}}\,z^{-1}\,z\,{\cal
V}_{c_{u}\epsilon}^{G^{\prime\prime}}\,{\cal
V}_{c_{u}\epsilon}^{U^{+}}\,u\subset p\,{\cal
V}_{c_{p}\epsilon}^{P^{-}}\,{\cal V}_{c_{p}\epsilon}^{U^{+}}\,{\cal
V}_{c_{u}\epsilon}^{U^{-}}\,{\cal
V}_{c_{u}\epsilon}^{G^{\prime\prime}}\,z\;{\cal V}_{c_{u}\epsilon}^{U^{+}}\,u$
$\displaystyle\subset p\,{\cal V}_{c_{p}\epsilon}^{P^{-}}\,{\cal
V}_{c_{u}\epsilon}\,{\cal V}_{c_{p}\epsilon}^{U^{+}}\,\,z\;{\cal
V}_{c_{u}\epsilon}^{U^{+}}\,u=p\,{\cal V}_{c_{p}\epsilon}^{P^{-}}\,{\cal
V}_{c_{u}\epsilon}^{P^{-}}{\cal V}_{c_{u}\epsilon}^{U^{+}}\,{\cal
V}_{c_{p}\epsilon}^{U^{+}}\,\,z\;{\cal V}_{c_{u}\epsilon}^{U^{+}}\,u$
$\displaystyle=p\,{\cal V}_{c_{p}\epsilon}^{P^{-}}\,{\cal
V}_{c_{u}\epsilon}^{P^{-}}\,z\,z^{-1}\,{\cal
V}_{c_{u}\epsilon}^{U^{+}}\,z\,z^{-1}\,{\cal
V}_{c_{p}\epsilon}^{U^{+}}\,\,z\;{\cal V}_{c_{u}\epsilon}^{U^{+}}\,u$
$\displaystyle\subset p\,{\cal V}_{c_{p}\epsilon}^{P^{-}}\,{\cal
V}_{c_{u}\epsilon}^{P^{-}}\,z\,{\cal V}_{c_{u}\epsilon}^{U^{+}}\,{\cal
V}_{c_{p}\epsilon}^{U^{+}}\,{\cal V}_{c_{u}\epsilon}^{U^{+}}\,u\subset
p\,{\cal V}_{c^{\prime\prime}_{1}\epsilon}^{P^{-}}\,z\,{\cal
V}_{c^{\prime\prime}_{2}\epsilon}^{U^{+}}\,u$
$\displaystyle=u^{-}\,g^{\prime\prime}\,{\cal
V}_{c^{\prime\prime}_{1}\epsilon}^{U^{-}}\,{\cal
V}_{c^{\prime\prime}_{1}\epsilon}^{G^{\prime\prime}}\,z\,{\cal
V}_{c^{\prime\prime}_{2}\epsilon}^{U^{+}}\,u=u^{-}\,g^{\prime\prime}\,{\cal
V}_{c^{\prime\prime}_{1}\epsilon}^{U^{-}}\,{g^{\prime\prime}}^{-1}\,g^{\prime\prime}\,{\cal
V}_{c^{\prime\prime}_{1}\epsilon}^{G^{\prime\prime}}\,z\,{\cal
V}_{c^{\prime\prime}_{2}\epsilon}^{U^{+}}\,u$ $\displaystyle\subset
u^{-}\,{\cal
V}_{c_{g^{\prime\prime}}c^{\prime\prime}_{1}\epsilon}^{U^{-}}\;g^{\prime\prime}\,{\cal
V}_{c^{\prime\prime}_{1}\epsilon}^{G^{\prime\prime}}\,z\,{\cal
V}_{c^{\prime\prime}_{2}\epsilon}^{U^{+}}\,u\;,$

as wanted. $\Box$

Let $\Phi$ be a closed ball of radius less than $1$ in the metric space
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}$, contained in
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$. By Lemma 3.1 and
with the notation of Equation (53), the set
${\widetilde{\Phi}}=\operatorname{orb}_{\mathfrak{m}}^{\;-1}(\Phi)$ is a
closed ball of same radius in ${\cal M}_{{\mathfrak{n}},{\mathfrak{m}}}({\cal
O}_{\nu})$. Let ${\cal E}$ be a closed ball in
$\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$, small enough so that
there exists a closed ball in the clopen fundameental domain
$G^{\prime\prime}_{\lozenge}$ which maps isometrically to ${\cal E}$ by the
locally isometric map
$\pi_{{\mathfrak{m}},{\mathfrak{n}}}:G^{\prime\prime}\rightarrow\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$
defined in Equation (44). Let ${\widetilde{{\cal
E}}}=\pi_{{\mathfrak{m}},{\mathfrak{n}}}^{-1}({\cal E})$ and let
$r\in{\mathbb{N}}$. Using the notation $\Omega_{\Phi,{\cal E},r}$ defined in
Equation (54), the family of Lipschitz well-rounded subsets with respect to
$({\cal V}_{\epsilon})_{\epsilon>0}$ that we will use in order to apply
Theorem 4.3 is given by the following result.

###### Proposition 4.6

With $\Phi$ and ${\cal E}$ as above, the family $\big{(}\Omega_{\Phi,{\cal
E},r}\big{)}_{r\in{\mathbb{N}}}$ is $0$-Lipschitz well-rounded with respect to
$({\cal V}_{\epsilon})_{\epsilon>0}$.

Proof. Recall that $\Omega_{\Phi,{\cal
E},r}=U^{-}_{{\widetilde{\Phi}}}\;G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\;Z_{r}\;U^{+}_{\Box}$ with the notation at the beginning of Subsection
4.1. We will actually prove (as allowed by the ultrametric situation) the
stronger statement that given $\Phi$ and ${\cal E}$ as above, if $\epsilon$ is
small enough, then for every $r\in{\mathbb{N}}$, we have

$\big{(}U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}\big{)}^{-\epsilon}=U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}=\big{(}U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}\big{)}^{+\epsilon}\;.$

Let

$c=\max\big{\\{}q_{\nu}\max\\{c_{g^{\prime\prime}}(c_{u^{-}g^{\prime\prime}}+c_{u^{+}}),c_{u^{-}g^{\prime\prime}}+2\,c_{u^{+}}\\}:u^{-}\in
U^{-}_{{\widetilde{\Phi}}},\;g^{\prime\prime}\in
G^{\prime\prime}_{{\widetilde{{\cal E}}}},\;u^{+}\in U^{+}_{\Box}\big{\\}}\,,$
(68)

which is finite since $U^{-}_{{\widetilde{\Phi}}}$,
$G^{\prime\prime}_{{\widetilde{{\cal E}}}}$ and $U^{+}_{\Box}$ are compact
subsets of $G$. Since ${\widetilde{\Phi}}$ is a ball of radius less than $1$
in ${\cal M}_{{\mathfrak{n}},{\mathfrak{m}}}(K_{\nu})$, let $v_{0}\in{\cal
M}_{{\mathfrak{n}},{\mathfrak{m}}}(K_{\nu})$ and
$k\in{\mathbb{N}}\\!\smallsetminus\\!\\{0\\}$ be such that

${\widetilde{\Phi}}=v_{0}+\pi_{\nu}^{\;k}{\cal
M}_{{\mathfrak{n}},{\mathfrak{m}}}({\cal O}_{\nu})\;.$

Let $r_{\cal E}$ be the radius of the ball ${\cal E}$ (satisfying the
assumptions of Proposition 4.6). By Equation (46) and since the map
$\underline{g}\mapsto\widecheck{\underline{g}}$ is an isometry of
$\operatorname{GL}_{\mathfrak{n}}(K_{\nu})$, there exists
$\big{(}\begin{smallmatrix}{\overline{g}}_{0}&0\\\
0&\underline{g}_{0}\end{smallmatrix}\big{)}\in G^{\prime\prime}$ such that

$G^{\prime\prime}_{{\widetilde{{\cal
E}}}}=\big{\\{}\big{(}\begin{smallmatrix}{\overline{g}}&0\\\
0&\underline{g}\end{smallmatrix}\big{)}\in
G^{\prime\prime}:\max\\{d({\overline{g}},{\overline{g}}_{0}),\;d(\underline{g},\underline{g}_{0})\\}\leq
r_{\cal E}\big{\\}}\;.$ (69)

Let us now consider $\epsilon_{0}=\frac{1}{c}\,q_{\nu}^{-\max\\{k,\,r_{\cal
E}+\log_{q_{\nu}}(\frac{1}{r_{\cal
E}}\|{\overline{g}}_{0}\|\,\|{{\overline{g}}_{0}}^{-1}\|),\,r_{\cal
E}+\log_{q_{\nu}}(\frac{1}{r_{\cal
E}}\|\underline{g}_{0}\|\,\|{\underline{g}_{0}}^{-1}\|)\\}-2}>0$, so that for
every $\epsilon\in\;]0,\epsilon_{0}[\,$, we have

$N_{c\epsilon}>1+\max\big{\\{}k,r_{\cal
E}+\log_{q_{\nu}}\big{(}\frac{1}{r_{\cal
E}}\|{\overline{g}}_{0}\|\,\|{{\overline{g}}_{0}}^{-1}\|\big{)},r_{\cal
E}+\log_{q_{\nu}}\big{(}\frac{1}{r_{\cal
E}}\|\underline{g}_{0}\|\,\|{\underline{g}_{0}}^{-1}\|\big{)}\big{\\}}\geq
1\;.$ (70)

Let us prove that for every $\epsilon\in\;]0,\epsilon_{0}[\,$, we have

$U_{{\widetilde{\Phi}}}^{-}\;{\cal
V}_{c\epsilon}^{U^{-}}=U_{{\widetilde{\Phi}}}^{-},\quad
G^{\prime\prime}_{{\widetilde{{\cal E}}}}\;{\cal
V}_{c\epsilon}^{G^{\prime\prime}}=G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\quad\text{and}\quad{\cal
V}_{c\epsilon}^{U^{+}}\,U_{\Box}^{+}=U_{\Box}^{+}\;.$ (71)

For all $u\in U_{{\widetilde{\Phi}}}^{-}$ and $u^{\prime}\in{\cal
V}_{c\epsilon}^{U^{-}}$, let $\beta,\beta^{\prime}\in{\cal
M}_{{\mathfrak{n}},{\mathfrak{m}}}({\cal O}_{\nu})$ be such that
$u=\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&0\\\
v_{0}+\pi_{\nu}^{\;k}\beta\;&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}$ and
$u^{\prime}=\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&\;0\\\
\pi_{\nu}^{\;N_{c\epsilon}}\beta^{\prime}\;&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}$.
Then since $N_{c\epsilon}>k$, we have
$uu^{\prime}=\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&0\\\
v_{0}+\pi_{\nu}^{\;k}\beta+\pi_{\nu}^{\;N_{c\epsilon}}\beta^{\prime}\;&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}\in
U_{{\widetilde{\Phi}}}^{-}$. Therefore we have
$U_{{\widetilde{\Phi}}}^{-}{\cal V}_{c\epsilon}^{U^{-}}\subset
U_{{\widetilde{\Phi}}}^{-}$ and the opposite inclusion is clear. This proves
the equality on the left-hand side of Formula (71).

The proof of the equality on the right-hand side is similar. For all $u\in
U_{\Box}^{+}$ and $u^{\prime}\in{\cal V}_{c\epsilon}^{U^{+}}$, let
$\gamma\in\Box$ and $\gamma^{\prime}\in{\cal
M}_{{\mathfrak{m}},{\mathfrak{n}}}({\cal O}_{\nu})$ be such that
$u=\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&\;\gamma\\\
0&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}$ and
$u^{\prime}=\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&\;\pi_{\nu}^{\;N_{c\epsilon}}\gamma^{\prime}\\\
0&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}$. Then since $N_{c\epsilon}>1$
and since $\Box+\pi_{\nu}{\cal M}_{{\mathfrak{m}},{\mathfrak{n}}}({\cal
O}_{\nu})=\Box$ by the construction of the fundamental domain ${\mathcal{D}}$
at the beginning of Subsection 4.1, we have
$u^{\prime}u=\big{(}\begin{smallmatrix}I_{{\mathfrak{m}}}&\;\gamma+\pi_{\nu}^{\;N_{c\epsilon}}\gamma^{\prime}\\\
0&I_{{\mathfrak{n}}}\end{smallmatrix}\big{)}\in U_{\Box}^{+}$. Therefore we
have ${\cal V}_{c\epsilon}^{U^{+}}U_{\Box}^{+}\subset U_{\Box}^{+}$ and the
opposite inclusion is clear.

Let $g=\Big{(}\begin{smallmatrix}\overline{g}&0\\\
0&\underline{g}\end{smallmatrix}\Big{)}\in G^{\prime\prime}_{{\widetilde{{\cal
E}}}}$. For every
$g^{\prime}=\Big{(}\begin{smallmatrix}\overline{g^{\prime}}&0\\\
0&\underline{g^{\prime}}\end{smallmatrix}\Big{)}\in{\cal
V}_{c\epsilon}^{G^{\prime\prime}}$, there exist $\alpha\in{\cal
M}_{{\mathfrak{m}}}({\cal O}_{\nu})$ and $\delta\in{\cal
M}_{{\mathfrak{n}}}({\cal O}_{\nu})$ such that

$\quad\overline{g^{\prime}}=I_{{\mathfrak{m}}}+\pi_{\nu}^{\;N_{c\epsilon}}\alpha\quad\text{and}\quad\underline{g^{\prime}}=I_{{\mathfrak{n}}}+\pi_{\nu}^{\;N_{c\epsilon}}\delta\;.$

We have
$gg^{\prime}=\Big{(}\begin{smallmatrix}\overline{g}\,\overline{g^{\prime}}&0\\\
0&\underline{g}\,\underline{g^{\prime}}\end{smallmatrix}\Big{)}\in
G^{\prime\prime}$. Let us prove that
$d(\overline{g}\,\overline{g^{\prime}},\,\overline{g}_{0})\leq r_{\cal E}$. A
similar proof gives that
$d(\underline{g}\,\underline{g^{\prime}},\,\underline{g}_{0})\leq r_{\cal E}$,
thus proving that $G^{\prime\prime}_{{\widetilde{{\cal E}}}}\;{\cal
V}_{c\epsilon}^{G^{\prime\prime}}\subset G^{\prime\prime}_{{\widetilde{{\cal
E}}}}$ by Equation (69). The opposite inclusion being clear, this proves the
middle equality of Formula (71).

By the submultiplicativity of the supremum norm, since $\alpha\in{\cal
M}_{\mathfrak{m}}({\cal O}_{\nu})$ so that $\|\alpha\|\leq 1$, by Lemma 3.3
since $\overline{g}\in B(\overline{g}_{0},r_{\cal E})$ and by Equation (70),
we have

$\|\pi_{\nu}^{\;N_{c\epsilon}}\,\overline{g}\,\alpha\,{\overline{g}_{0}}^{-1}\|\leq
q_{\nu}^{\;-N_{c\epsilon}}\|\overline{g}\|\,\|\alpha\|\,\|{\overline{g}_{0}}^{-1}\|\leq
q_{\nu}^{\;-N_{c\epsilon}+r_{\cal
E}}\|\overline{g}_{0}\|\,\|{\overline{g}_{0}}^{-1}\|\leq\frac{r_{\cal
E}}{q_{\nu}}\;.$

We also have, by the ultrametric triangle inequality,

$\displaystyle\|\,\overline{g}\,\overline{g^{\prime}}\,{\overline{g}_{0}}^{-1}-I_{\mathfrak{m}}\|$
$\displaystyle=\|\,\overline{g}(I_{{\mathfrak{m}}}+\pi_{\nu}^{\;N_{c\epsilon}}\alpha)\,{\overline{g}_{0}}^{-1}-I_{\mathfrak{m}}\|=\|\,\overline{g}\,{\overline{g}_{0}}^{-1}-I_{\mathfrak{m}}+\pi_{\nu}^{\;N_{c\epsilon}}\,\overline{g}\,\alpha\,{\overline{g}_{0}}^{-1}\|$
$\displaystyle\leq\max\\{\|\,\overline{g}\,{\overline{g}_{0}}^{-1}-I_{\mathfrak{m}}\|,\;\|\pi_{\nu}^{\;N_{c\epsilon}}\,\overline{g}\,\alpha\,{\overline{g}_{0}}^{-1}\|\\}\;.$

Thus since $\overline{g}\in B(\overline{g}_{0},r_{\cal E})$ and
$\operatorname{ln}(1+t)\leq t$ for every $t\geq 0$, we have

$\log_{q_{\nu}}(1+\|\,\overline{g}\,\overline{g^{\prime}}\,{\overline{g}_{0}}^{-1}-I_{\mathfrak{m}}\|)\leq\max\\{r_{\cal
E},\log_{q_{\nu}}(1+\frac{r_{\cal E}}{q_{\nu}})\\}\leq r_{\cal E}\;.$

Since $N_{c\epsilon}\geq 1$, the standard formula for the inverse of
$I_{\mathfrak{m}}+X$ when $\|X\|<1$ gives that there exists
$\alpha^{\prime}\in{\cal M}_{\mathfrak{n}}({\cal O}_{\nu})$ such that
$\overline{g^{\prime}}^{\;-1}=I_{{\mathfrak{m}}}+\pi_{\nu}^{\;N_{c\epsilon}}\alpha^{\prime}$.
A proof similar to the one above thus gives that
$\log_{q_{\nu}}(1+\|\,{\overline{g}_{0}}(\,\overline{g}\,\overline{g^{\prime}}\,)^{-1}-I_{\mathfrak{m}}\|)\leq
r_{\cal E}$, which proves as wanted that
$d(\overline{g}\,\overline{g^{\prime}},\,\overline{g}_{0})\leq r_{\cal E}$.
This concludes the proof of Formula (71).

Now, for every $r\in{\mathbb{N}}$, we have by Lemma 4.5 and by Equations (68)
and (71) that

$\displaystyle\big{(}U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}\big{)}^{+\epsilon}={\cal
V}_{\epsilon}\,U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}\,{\cal V}_{\epsilon}$ $\displaystyle\subset\;\;$
$\displaystyle U^{-}_{{\widetilde{\Phi}}}\,{\cal
V}^{U^{-}}_{c\epsilon}\,G^{\prime\prime}_{{\widetilde{{\cal E}}}}\,{\cal
V}^{G^{\prime\prime}}_{c\epsilon}\,Z_{r}\,{\cal
V}^{U^{+}}_{c\epsilon}\,U^{+}_{\Box}=U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}\;.$

Since the converse inclusion is immediate, we have
$\big{(}U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}\big{)}^{+\epsilon}=U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}$.

Since ${\cal V}_{\epsilon}$, being a subgroup, is stable by $g\mapsto g^{-1}$,
this implies that
$g\;U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}\;h$ contains
$U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}$ for all $g,h\in{\cal V}_{\epsilon}$, so that
$\big{(}U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}\big{)}^{-\epsilon}\supset
U^{-}_{{\widetilde{\Phi}}}\,G^{\prime\prime}_{{\widetilde{{\cal
E}}}}\,Z_{r}\,U^{+}_{\Box}$. Since the converse inclusion is immediate, this
concludes the proof of Proposition 4.6. $\Box$

###  4.3 The main statement and its proof

Error terms in equidistribution results usually require smoothness properties
on test functions. The appropriate smoothness regularity of functions defined
on ultrametric spaces as $\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}$ and
$\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$ is the locally
constant one. The locally constant regularity on such homogeneous spaces of
totally discontinuous groups could be defined (as for instance in [AtGP],
[KPS, §4.3]) by using the familly of small compact-open subgroups $({\cal
V}_{\epsilon})_{\epsilon\,\in\,]0,1]}$ of $G$ defined in Subsection 4.2, and
by defining an $\epsilon$-locally constant map on
$\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$ to be a map which is
constant on every orbit of ${\cal V}_{\epsilon}\cap G^{\prime\prime}$ on
$\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$. But it turns out to
be more convenient in this paper to use a general purely metric definition.
For every ultrametric space $E$ and $\epsilon\in\;]0,1]$, a bounded map
$f:E\rightarrow{\mathbb{R}}$ is $\epsilon$-locally constant if it is constant
on every closed ball of radius $\epsilon$ in $E$. With
$\|f\|_{\infty}=\sup_{x\in E}|f(x)|$ the supremum norm of $f$, the
$\epsilon$-locally constant norm of $f$ is
$\|f\|_{\epsilon}=\frac{\|f\|_{\infty}}{\epsilon}$.

The key result of this paper is the following one. Let
$\ell=\operatorname{lcm}({\mathfrak{m}},{\mathfrak{n}})$. For every nonzero
ideal $I$ of $R_{\nu}$, let $\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}(I)=\Gamma_{I}^{\sharp}(R_{\nu}^{\;{\mathfrak{m}}}\times\\{0\\})$
(see Subsection 2.4 for the definition of $\Gamma_{I}$ and Equation (51) for
the one of $\Gamma_{I}^{\sharp}$) and

$\displaystyle c_{I}=$ (72)
$\displaystyle\frac{q^{\;({\mathfrak{g}}-1)(\mathbf{d}^{2}-\mathbf{d}+1-{\mathfrak{m}}\,{\mathfrak{n}})}\,(q_{\nu}^{\;\mathbf{d}}-1)^{2}\,\prod_{i=2}^{\mathbf{d}}((q_{\nu}^{\;i-1}-1)\;\zeta_{K}(i))\,N(I)^{{\mathfrak{m}}\,{\mathfrak{n}}}\prod_{{\mathfrak{p}}\,|I}\prod_{i=1}^{\mathfrak{m}}\frac{N({\mathfrak{p}})^{i}-N({\mathfrak{p}})^{-{\mathfrak{n}}}}{N({\mathfrak{p}})^{i}-1}}{(q-1)\,q_{\nu}^{\;2\,{\mathfrak{m}}\,{\mathfrak{n}}+2}\,\prod_{i=1}^{{\mathfrak{m}}}(q_{\nu}^{\;i}-1)^{2}\prod_{i=2}^{{\mathfrak{n}}}(q_{\nu}^{\;i}-1)^{2}}\;.$

###### Theorem 4.7

For every nonzero ideal $I$ of $R_{\nu}$, for the weak-star convergence of
Borel measures on the locally compact space
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}\times\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$,
we have

$\lim_{i\rightarrow+\infty}\;\frac{c_{I}}{q_{\nu}^{\;\ell\,\mathbf{d}\,i}}\sum_{\Lambda\in\operatorname{{\cal
P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}(I)\;:\;\overline{\operatorname{Covol}}\;\Lambda\,=\,q_{\nu}^{\;\ell\,i}}\;\Delta_{V_{\Lambda}}\otimes\Delta_{\llbracket\Lambda\rrbracket}={\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}}_{\mid\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}}\otimes\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}\;.$
(73)

Furthermore, there exists $\tau\in\;]0,\frac{1}{2\,\mathbf{d}^{2}}]$ such that
for all $\delta\in\;]0,\tau[$ and $\epsilon\in\;]0,1]$, there is an additive
error term of the form
$\operatorname{O}_{\nu,\delta,I}\big{(}q_{\nu}^{\;\ell\,\mathbf{d}\,i(-\tau+\delta)}\,\|f\|_{\epsilon}\,\|g\|_{\epsilon}\big{)}$
in the above equidistribution claim when evaluated on pairs $(f,g)$ for all
compactly supported $\epsilon$-locally constant maps
$f:\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}\rightarrow{\mathbb{R}}$
and
$g:\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}\rightarrow{\mathbb{R}}\;$:
as $i\rightarrow+\infty$, we have

$\displaystyle\frac{c_{I}}{q_{\nu}^{\;\ell\,\mathbf{d}\,i}}\sum_{\Lambda\in\operatorname{{\cal
P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}(I)\;:\;\overline{\operatorname{Covol}}\;\Lambda\,=\,q_{\nu}^{\;\ell\,i}}f(V_{\Lambda})\;g(\llbracket\Lambda\rrbracket)$
$\displaystyle=\;$
$\displaystyle\Big{(}\int_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}}f\,d\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}\Big{)}\Big{(}\int_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}g\,d\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}\Big{)}+\operatorname{O}_{\nu,\delta,I}\big{(}q_{\nu}^{\;\ell\,\mathbf{d}\,i(-\tau+\delta)}\,\|f\|_{\epsilon}\,\|g\|_{\epsilon}\big{)}\;.$

Proof. Let $\Lambda\in\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$. Then there exists
$g\in\Gamma^{\sharp}$ (thus $g\in{\cal U}_{G}$) such that we have
$\Lambda=\Lambda_{g}=g(R_{\nu}^{\;{\mathfrak{m}}}\times\\{0\\})$, so that if
$g=u^{-}\,g^{\prime\prime}\,z\,u^{+}$ is the decomposition given by
Proposition 2.5, then $u^{-}\in U^{-}({\cal O}_{\nu})$. Hence by Proposition
2.8 (i), we have

$V_{\Lambda}=V_{\Lambda_{g}}=V_{\Lambda_{u^{-}}}\in\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}=U^{-}({\cal
O}_{\nu})V_{R_{\nu}^{\;{\mathfrak{m}}}\times\\{0\\}}\;.$

Furthermore, we have
$\llbracket\Lambda\rrbracket\in\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$
by Equation (49), so that the statement of Theorem 4.7 is well defined.

Let $I$ be a nonzero ideal of $R_{\nu}$. Let
$\tau=\tau(\Gamma_{I})\in\;]0,\frac{1}{2\,\mathbf{d}^{2}}]$ be as in Theorem
4.3 applied with $G^{\prime}=G$, with $\Gamma^{\prime}=\Gamma_{I}$ and with
the family $({\cal V}^{\prime}_{\epsilon})_{\epsilon>0}=({\cal
V}_{\epsilon})_{\epsilon>0}$ given by Equation (65), which has an upper local
dimension $\rho=\mathbf{d}^{2}-1$ according to the final claim of Lemma 4.4.
Let $\delta\in\;]0,\tau[$.

Let $\Phi$ be a closed ball in $\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}$
of radius
$r_{\Phi}\in\;]0,c_{1}^{\;\frac{1}{{\mathfrak{m}}\,{\mathfrak{n}}}}]$, where
$c_{1}\leq 1$ is given by Proposition 2.5. Besides, we assume that $\Phi$ is
contained in $\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$. By
Lemma 3.1 and with the notation of Equation (53), it follows that
${\widetilde{\Phi}}=\operatorname{orb}_{\mathfrak{m}}^{\;-1}(\Phi)$ is a
closed ball of radius $r_{\Phi}$ in ${\cal
M}_{{\mathfrak{n}},{\mathfrak{m}}}({\cal O}_{\nu})$. Let $\chi_{\Phi}$ be the
characteristic function of $\Phi$, which is $r_{\Phi}$-locally constant with
$\|\chi_{\Phi}\|_{r_{\Phi}}=\frac{1}{r_{\Phi}}\geq 1$. By Equation (56), we
have

$\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}(\Phi)=\frac{1}{c_{1}}\,\operatorname{Haar}_{{\mathfrak{n}},{\mathfrak{m}}}({\widetilde{\Phi}})=\frac{1}{c_{1}}\,r_{\Phi}^{\;{\mathfrak{m}}\,{\mathfrak{n}}}\leq
1\;,$

so that, since
$\tau\leq\frac{1}{2\,\mathbf{d}^{2}}\leq\frac{1}{{\mathfrak{m}}\,{\mathfrak{n}}}$,
we have

$\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}(\Phi)^{-\tau+\delta}\leq\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}(\Phi)^{-\tau}=\operatorname{O}(\|\,\chi_{\Phi}\|_{r_{\Phi}})\;.$
(74)

Let ${\cal E}$ be a closed ball in
$\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$ of radius $r_{\cal
E}\in\;]0,1]$ small enough so that
$\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}({\cal E})\leq 1$
and there exists a closed ball ${\widetilde{{\cal E}}}_{0}$ in
$G^{\prime\prime}_{\lozenge}$ mapping isometrically to ${\cal E}$ by
$\pi_{{\mathfrak{m}},{\mathfrak{n}}}:G^{\prime\prime}\rightarrow\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}$.
Let ${\widetilde{{\cal E}}}=\pi_{{\mathfrak{m}},{\mathfrak{n}}}^{\;-1}({\cal
E})=\bigsqcup_{\gamma\in G^{\prime\prime}(R_{\nu})}{\widetilde{{\cal
E}}}_{0}\,\gamma$. Let $\chi_{\cal E}$ be the characteristic function of
${\cal E}$, which is $r_{\cal E}$-locally constant with $\|\chi_{\cal
E}\|_{r_{\cal E}}=\frac{1}{r_{\cal E}}\geq 1$. By Equation (60) and by the
Alhfors regularity of the homogeneous measure $\mu_{G^{\prime\prime}}$ of the
group $G^{\prime\prime}$ with dimension $\dim
G^{\prime\prime}\leq{\mathfrak{m}}^{2}+{\mathfrak{n}}^{2}\leq 2\mathbf{d}^{2}$
for the distance $d$ defined in Section 3.3 (see in particular Equation (45)),
there exists a constant $c>0$ such that

$\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}({\cal
E})=\frac{(q_{\nu}-1)^{2}}{(q-1)\,q_{\nu}^{\;2}}\;\mu_{G^{\prime\prime}}(G^{\prime\prime}_{{\widetilde{{\cal
E}}}})=\frac{(q_{\nu}-1)^{2}}{(q-1)\,q_{\nu}^{\;2}}\;\mu_{G^{\prime\prime}}({\widetilde{{\cal
E}}}_{0})\geq c\,r_{\cal E}^{\;\dim G^{\prime\prime}}\geq c\,r_{\cal
E}^{\;2\mathbf{d}^{2}}\;,$

so that, since $\tau\leq\frac{1}{2\,\mathbf{d}^{2}}$ and
$\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}({\cal E})\leq
1$, we have

$\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}({\cal
E})^{-\tau+\delta}=\operatorname{O}(\|\,\chi_{\cal E}\|_{r_{\cal E}})\;.$ (75)

For every $r\in{\mathbb{N}}$, let us define

$\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}(I,\Phi,{\cal
E},r)=\\{\Lambda\in\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}(I):V_{\Lambda}\in\Phi,\;\llbracket\Lambda\rrbracket\in{\cal
E},\;\overline{\operatorname{Covol}}\;\Lambda\,=\,q_{\nu}^{\;\ell\,r}\\}\;$

Using respectively

$\bullet$ Proposition 4.2 for the first equality,

$\bullet$ Theorem 4.3 applied to the family $({\cal Z}_{r}=\Omega_{\Phi,{\cal
E},r})_{r\in{\mathbb{N}}}$ (where the set $\Omega_{\Phi,{\cal E},r}$ is
defined in Equation (54)), which is $0$-Lipschitz well-rounded with respect to
$({\cal V}_{\epsilon})_{\epsilon>0}$ by Proposition 4.6 for the second
equality,

$\bullet$ Lemma 4.1 for the third equality,

$\bullet$ Equations (74) and (75) for the last equality (and the fact that
$\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}$ and
$\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}$ are finite
measures),

we have

$\displaystyle{\operatorname{Card}}\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}(I,\Phi,{\cal
E},r)={\operatorname{Card}}(\Omega_{\Phi,{\cal E},r}\cap\Gamma_{I})$
$\displaystyle=\;$
$\displaystyle\frac{1}{\|\mu_{G/\Gamma_{I}}\|}\,\mu_{G}(\Omega_{\Phi,{\cal
E},r})\big{(}1+\operatorname{O}\big{(}\mu_{G}(\Omega_{\Phi,{\cal
E},r})^{-\tau+\delta}\big{)}\big{)}$ $\displaystyle=\;$
$\displaystyle\frac{c^{\prime}_{1}\;q_{\nu}^{\;\mathbf{d}\,\ell\,r}}{\|\mu_{G/\Gamma_{I}}\|}\,\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}(\Phi)\,\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}({\cal
E})\big{(}1+\operatorname{O}\big{(}q_{\nu}^{\;\mathbf{d}\,\ell\,r(-\tau+\delta)}\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}(\Phi)^{-\tau+\delta}\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}({\cal
E})^{-\tau+\delta}\big{)}\big{)}$ $\displaystyle=\;$
$\displaystyle\frac{c^{\prime}_{1}\;q_{\nu}^{\;\mathbf{d}\,\ell\,r}}{\|\mu_{G/\Gamma_{I}}\|}\,\big{(}\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}(\Phi)\,\mu_{\operatorname{Lat}^{1}_{{\mathfrak{m}},{\mathfrak{n}}}}({\cal
E})+\operatorname{O}\big{(}q_{\nu}^{\;\mathbf{d}\,\ell\,r(-\tau+\delta)}\;\|\,\chi_{\Phi}\|_{r_{\Phi}}\;\|\,\chi_{\cal
E}\|_{r_{\cal E}}\,\big{)}\big{)}\;.$ (76)

Let
$c_{I}=\frac{\|\mu_{G/\Gamma_{I}}\|}{c^{\prime}_{1}}=\frac{[\Gamma:\Gamma_{I}]\,\|\mu_{G/\Gamma}\|}{c^{\prime}_{1}}$.
With the value of $[\Gamma:\Gamma_{I}]$ given by Lemma 2.3 (2), the value of
$\|\mu_{G/\Gamma}\|$ given by Equation (40) with $k=\mathbf{d}$ and the value
of $c^{\prime}_{1}$ given in Lemma 4.1, we have

$\displaystyle
c_{I}=\frac{N(I)^{{\mathfrak{m}}\,{\mathfrak{n}}}\prod_{{\mathfrak{p}}\,|I}\prod_{i=1}^{\mathfrak{m}}\frac{N({\mathfrak{p}})^{i}-N({\mathfrak{p}})^{-{\mathfrak{n}}}}{N({\mathfrak{p}})^{i}-1}\,q^{\;({\mathfrak{g}}-1)(\mathbf{d}^{2}-\mathbf{d}+1)}\prod_{i=1}^{\mathbf{d}-1}\frac{\zeta_{K}(1+i)}{q_{\nu}^{\;i}-1}}{(q-1)\,q^{({\mathfrak{g}}-1)\,{\mathfrak{m}}\,{\mathfrak{n}}}\;q_{\nu}^{\;2\,{\mathfrak{m}}\,{\mathfrak{n}}+2}\;\frac{\prod_{i=1}^{{\mathfrak{m}}}(q_{\nu}^{\;i}-1)^{2}\prod_{i=1}^{{\mathfrak{n}}}(q_{\nu}^{\;i}-1)^{2}}{(q_{\nu}-1)^{2}\;\prod_{i=1}^{\mathbf{d}}(q_{\nu}^{\;i}-1)^{2}}}\;,$

as wanted in Equation (72). Note that every compactly supported
$\epsilon$-locally constant map on an ultrametric space is a finite linear
combination of characteristic functions of balls of radius $\epsilon$. By a
finite bilinearity argument, Theorem 4.7 follows from Equation (76). $\Box$

###### Corollary 4.8

For every nonzero ideal $I$ of $R_{\nu}$, for the weak-star convergence of
Borel measures on the locally compact space
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}\times\operatorname{Sh}^{1}_{{\mathfrak{m}}}\times\operatorname{Sh}^{1}_{{\mathfrak{n}}}$,
we have

$\displaystyle\lim_{i\rightarrow+\infty}\;\frac{c_{I}\,(1-q_{\nu}^{\;-1})^{-2}}{q_{\nu}^{\;\ell\,\mathbf{d}\,i}}$
$\displaystyle\sum_{\Lambda\in\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}(I)\;:\;\overline{\operatorname{Covol}}\;\Lambda\,=\,q_{\nu}^{\;\ell\,i}}\;\Delta_{V_{\Lambda}}\otimes\Delta_{\operatorname{sh}(\Lambda)}\otimes\Delta_{\operatorname{sh}(\Lambda^{\perp})}$
$\displaystyle=\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}\otimes\mu_{\operatorname{Sh}^{1}_{\mathfrak{m}}}\otimes\mu_{\operatorname{Sh}^{1}_{\mathfrak{n}}}\;.$
(77)

Furthermore, there exists $\tau\in\;]0,\frac{1}{2\,\mathbf{d}^{2}}]$ such that
for all $\delta\in\;]0,\tau[$ and $\epsilon\in\;]0,1]$, there is an additive
error term of the form
$\operatorname{O}_{\nu,\delta,I}\big{(}q_{\nu}^{\;\ell\,\mathbf{d}\,i(-\tau+\delta)}\,\|f\|_{\epsilon}\,\|f_{1}\|_{\infty}\|f_{2}\|_{\infty}\big{)}$
in the above equidistribution claim when evaluated on $(f,f_{1},f_{2})$ for
every compactly supported $\epsilon$-locally constant map
$f:\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}\rightarrow{\mathbb{R}}$ and
for all finitely supported maps
$f_{1}:\operatorname{Sh}^{1}_{{\mathfrak{m}}}\rightarrow{\mathbb{R}}$ and
$f_{2}:\operatorname{Sh}^{1}_{{\mathfrak{n}}}\rightarrow{\mathbb{R}}\;$: as
$i\rightarrow+\infty$, we have

$\displaystyle\frac{c_{I}\,(1-q_{\nu}^{\;-1})^{-2}}{q_{\nu}^{\;\ell\,\mathbf{d}\,i}}\sum_{\Lambda\in\operatorname{{\cal
P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}(I)\;:\;\overline{\operatorname{Covol}}\;\Lambda\,=\,q_{\nu}^{\;\ell\,i}}f(V_{\Lambda})\;f_{1}(\operatorname{sh}(\Lambda))\;f_{2}(\operatorname{sh}(\Lambda^{\perp}))$
$\displaystyle=\;$
$\displaystyle\Big{(}\int_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}f\,d\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}\Big{)}\Big{(}\int_{\operatorname{Sh}^{1}_{{\mathfrak{m}}}}f_{1}\,d\mu_{\operatorname{Sh}^{1}_{{\mathfrak{m}}}}\Big{)}\Big{(}\int_{\operatorname{Sh}^{1}_{{\mathfrak{n}}}}f_{2}\,d\mu_{\operatorname{Sh}^{1}_{{\mathfrak{n}}}}\Big{)}$
$\displaystyle+\operatorname{O}_{\nu,\delta,I}\big{(}q_{\nu}^{\;\ell\,\mathbf{d}\,i(-\tau+\delta)}\,\|f\|_{\epsilon}\,\|f_{1}\|_{\infty}\|f_{2}\|_{\infty}\big{)}\;.$

Theorem 1.2 in the Introduction follows from the first claim of this corollary
by taking $I=R_{\nu}$.

Proof. Step 1.  We first prove the result with
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$ instead of
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}$ and $\operatorname{{\cal
P}\\!{\cal L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}(I)$ instead of
$\operatorname{{\cal P}\\!{\cal L}}_{{\mathfrak{m}},\mathbf{d}}(I)$.

Since the map $\varphi_{{\mathfrak{m}},{\mathfrak{n}}}$ (defined in Equation
(48)) is proper by Lemma 3.4, the pushforward map
$(\varphi_{{\mathfrak{m}},{\mathfrak{n}}})_{*}$ of Borel measures by
$\varphi_{{\mathfrak{m}},{\mathfrak{n}}}$ is linear and weak-star continuous.
Hence applying the map
$(\operatorname{id}\times\varphi_{{\mathfrak{m}},{\mathfrak{n}}})_{*}$ to
Equation (73), using Lemma 3.4 (2) on the left hand side of Equation (73), and
Lemma 3.4 (1) on the right hand side of Equation (73), we have

$\displaystyle\lim_{i\rightarrow+\infty}\;$
$\displaystyle\frac{c_{I}}{q_{\nu}^{\;\ell\,\mathbf{d}\,i}}\sum_{\Lambda\in\operatorname{{\cal
P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}(I)\;:\;\overline{\operatorname{Covol}}\;\Lambda\,=\,q_{\nu}^{\;\ell\,i}}\;\Delta_{V_{\Lambda}}\otimes\Delta_{\operatorname{sh}(\Lambda)}\otimes\Delta_{\operatorname{sh}(\Lambda^{\perp})}$
$\displaystyle=(1-q_{\nu}^{\;-1})^{2}\,{\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}}_{\mid\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}}\otimes\mu_{\operatorname{Sh}^{1}_{\mathfrak{m}}}\otimes\mu_{\operatorname{Sh}^{1}_{\mathfrak{n}}}\;.$

It follows from Lemma 3.4 (3) and from the error term in Theorem 4.7 applied
with the compactly supported $\rho_{0}$-locally constant function
$g=(f_{1}\times f_{2})\circ\varphi_{{\mathfrak{m}},{\mathfrak{n}}}$ that we
have an additive error term of the form
$\operatorname{O}_{\nu,\delta,I}\big{(}q_{\nu}^{\;\ell\,\mathbf{d}\,i(-\tau+\delta)}\,\|f\|_{\epsilon}\,\|f_{1}\|_{\infty}\|f_{2}\|_{\infty}\big{)}$
in this equidistribution claim when evaluated on $(f,f_{1},f_{2})$ for every
compactly supported $\epsilon$-locally constant map
$f:\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}\rightarrow{\mathbb{R}}$
and for all finitely supported maps
$f_{1}:\operatorname{Sh}^{1}_{{\mathfrak{m}}}\rightarrow{\mathbb{R}}$ and
$f_{2}:\operatorname{Sh}^{1}_{{\mathfrak{n}}}\rightarrow{\mathbb{R}}\;$.

Step 2.  We now explain how to deduce Corollary 4.8 from the equidistribution
of
$(V_{\Lambda},\operatorname{sh}(\Lambda),\operatorname{sh}(\Lambda^{\perp}))$
in
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}\times\operatorname{Sh}^{1}_{\mathfrak{m}}\times\operatorname{Sh}^{1}_{\mathfrak{n}}$
when $\Lambda$ varies in $\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}(I)$ with the appropriate covolume.
The key technical lemma is the following one.

Let us denote by $W_{\mathbf{d}}$ the Weyl subgroup of
$\operatorname{GL}_{\mathbf{d}}(K_{\nu})$ consisting in the permutation
matrices of the canonical basis of $K_{\nu}^{\;\mathbf{d}}$. Note that
$W_{\mathbf{d}}$ is contained in
$\operatorname{GL}_{\mathbf{d}}(R_{\nu})\cap\operatorname{GL}_{\mathbf{d}}({\cal
O}_{\nu})$.

###### Lemma 4.9

For every $g\in\operatorname{GL}_{\mathbf{d}}(K_{\nu})$, there exists
$\sigma\in W_{\mathbf{d}}$ such that if $\sigma
g=\big{(}\begin{smallmatrix}\alpha&\gamma\\\
\beta&\delta\end{smallmatrix}\big{)}$ with $\alpha\in{\cal
M}_{\mathfrak{m}}(K_{\nu})$, then
$\alpha\in\operatorname{GL}_{\mathfrak{m}}(K_{\nu})$ and
$\beta\,\alpha^{-1}\in{\cal M}_{{\mathfrak{n}},{\mathfrak{m}}}({\cal
O}_{\nu})$.

Proof. For every $g\in{\cal M}_{\mathbf{d}}(K_{\nu})$, we denote by
$g_{{\mathfrak{m}}\mid}$ the submatrix of $g$ consisting of its first
${\mathfrak{m}}$ columns. For all $\alpha\in{\cal M}_{\mathfrak{m}}(K_{\nu})$
and $j,k\in\llbracket 1,{\mathfrak{m}}\rrbracket$, we denote by
$\alpha_{{\widehat{j}},{\widehat{k}}}$ the submatrix of $\alpha$ where the
$j$-th row and $k$-th column have been removed. Recall that the $(j,k)$
coefficient of the comatrix $\operatorname{Comm}(\alpha)$ of $\alpha$ is
$\operatorname{Comm}(\alpha)_{j,k}=(-1)^{j+k}\det\alpha_{{\widehat{j}},{\widehat{k}}}$.

Note that the statement of Lemma 4.9 is invariant by multiplication of the
left of $g$ by an element of $W_{\mathbf{d}}$. Since multiplying on the left
$g$ by an element of $W_{\mathbf{d}}$ amounts to permuting the rows of $g$, up
to such a multiplication, we may assume that the absolute value of the upper-
left ${\mathfrak{m}}\times{\mathfrak{m}}$ minor of $g_{{\mathfrak{m}}\mid}$
(hence of $g$) is maximal over the absolute values of all
${\mathfrak{m}}\times{\mathfrak{m}}$ minors of $g_{{\mathfrak{m}}\mid}$. Let
us then prove that if $g=\big{(}\begin{smallmatrix}\alpha&\gamma\\\
\beta&\delta\end{smallmatrix}\big{)}$ with $\alpha\in{\cal
M}_{\mathfrak{m}}(K_{\nu})$, then
$\alpha\in\operatorname{GL}_{\mathfrak{m}}(K_{\nu})$ and
$\beta\,\alpha^{-1}\in{\cal M}_{{\mathfrak{n}},{\mathfrak{m}}}({\cal
O}_{\nu})$, which proves Lemma 4.9 by taking $\sigma=\operatorname{id}$.

Since $g$ is invertible, the rank of its submatrix $g_{{\mathfrak{m}}\mid}$ is
${\mathfrak{m}}$. Hence $g_{{\mathfrak{m}}\mid}$ has at least one nonzero
${\mathfrak{m}}\times{\mathfrak{m}}$ minor, so that $|\det\alpha|\neq 0$ by
the above maximality property.

For all $i\in\llbracket 1,{\mathfrak{n}}\rrbracket$ and $j\in\llbracket
1,{\mathfrak{m}}\rrbracket$, let us prove that the $(i,j)$-coefficient
$(\beta\alpha^{-1})_{i,j}$ of the matrix $\beta\alpha^{-1}\in{\cal
M}_{{\mathfrak{n}},{\mathfrak{m}}}(K_{\nu})$ has absolute value at most $1$,
which proves Lemma 4.9. We denote by $A(i,j)\in{\cal
M}_{{\mathfrak{m}}}(K_{\nu})$ the matrix $\alpha$ where its $j$-th row has
been replaced by the $(i+{\mathfrak{m}})$-th row of $g_{{\mathfrak{m}}\mid}$.
By the above maximality property and since $\det A(i,j)$ is an
${\mathfrak{m}}\times{\mathfrak{m}}$ minor of $g_{{\mathfrak{m}}\mid}$, we
have

$|\det A(i,j)|\leq|\det\alpha|\;.$

Since the $i$-th row of $\beta$ is the $(i+{\mathfrak{m}})$-th row of
$g_{{\mathfrak{m}}\mid}$, since
$\alpha^{-1}=\frac{1}{\det\alpha}\;^{t}\\!\operatorname{Comm}(\alpha)$, and by
the Laplace expansion formula for the determinant of $A(i,j)$ with respect to
its $j$-th row, we have

$\displaystyle(\beta\alpha^{-1})_{ij}$
$\displaystyle=\sum_{k=1}^{\mathfrak{m}}\beta_{i,k}(\alpha^{-1})_{k,j}=\frac{1}{\det\alpha}\sum_{k=1}^{\mathfrak{m}}g_{i+{\mathfrak{m}},k}\operatorname{Comm}(\alpha)_{j,k}$
$\displaystyle=\frac{1}{\det\alpha}\sum_{k=1}^{\mathfrak{m}}(-1)^{j+k}g_{i+{\mathfrak{m}},k}\det\alpha_{{\widehat{j}},{\widehat{k}}}=\frac{\det
A(i,j)}{\det\alpha}\;.$

Therefore $|(\beta\alpha^{-1})_{ij}|\leq 1$, as wanted. $\Box$

The linear action of an element $\sigma$ of the Weyl group $W_{\mathbf{d}}$ on
an element $\Lambda\in{\cal P}{\cal L}_{{\mathfrak{m}},\mathbf{d}}$ satisfies
the following properties.

$\bullet$ Since $\sigma\in\operatorname{GL}_{\mathbf{d}}(R_{\nu})$, the
${\mathfrak{m}}$-lattice $\sigma\Lambda$ is primitive, and by Equation (8), we
have

$\overline{\operatorname{Covol}}(\sigma\Lambda)=\overline{\operatorname{Covol}}(\Lambda)\;.$

$\bullet$ We have $V_{\sigma\Lambda}=\sigma V_{\Lambda}$ by the left hand side
of Equation (7).

$\bullet$ Since $\sigma\in\operatorname{GL}_{\mathbf{d}}({\cal O}_{\nu})$, and
by the construction of the shape map $\operatorname{sh}$ in and above Equation
(27), we have $\operatorname{sh}(\sigma\Lambda)=\operatorname{sh}(\Lambda)$.

$\bullet$ Let $R_{\nu}^{\;\mathbf{d},*}$ be the standard full
$R_{\nu}$-lattice of the dual space of $K_{\nu}^{\;\mathbf{d}}$, which is
invariant under the dual action of $\sigma$ since
$\widecheck{\sigma}=\;^{t}\sigma^{\,-1}\in\operatorname{GL}_{\mathbf{d}}(R_{\nu})$.
As seen in Equation (12), we have
$(\sigma\Lambda)^{\perp}=\widecheck{\sigma}(\Lambda^{\perp})$. Hence
$\operatorname{sh}((\sigma\Lambda)^{\perp})=\operatorname{sh}(\Lambda^{\perp})$
since $\widecheck{\sigma}\in\operatorname{GL}_{\mathbf{d}}({\cal O}_{\nu})$.

$\bullet$ By the $\operatorname{GL}_{\mathbf{d}}({\cal O}_{\nu})$-invariance
of the probability measure
$\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}$ (see Subsection 3.1),
and since $\sigma\in\operatorname{GL}_{\mathbf{d}}({\cal O}_{\nu})$, we have
$\sigma_{*}\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}=\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}$.

$\bullet$ Since $\sigma\in\operatorname{GL}_{\mathbf{d}}({\cal O}_{\nu})$, the
left action of $\sigma$ on $\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}$ is
isometric for the distance $d$ on
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}$ constructed in Subsection 3.1.

By Lemma 4.9, for every $\Lambda\in{\cal P}{\cal
L}_{{\mathfrak{m}},\mathbf{d}}{\smallsetminus}{\cal P}{\cal
L}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$ with
$\overline{\operatorname{Covol}}(\Lambda)\in q_{\nu}^{\;\ell\,{\mathbb{Z}}}$,
there exists $\sigma\in W_{\mathbf{d}}$ such that $\sigma\Lambda\in{\cal
P}{\cal L}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$ and $V_{\sigma\Lambda}=\sigma
V_{\Lambda}\in\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$.
Furthermore, $\sigma$ maps a small ball centered at $V_{\Lambda}$ contained in
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}{\smallsetminus}\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$
to a ball contained in
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}$ centered at
$V_{\sigma\Lambda}$ of the same radius, by the last point above. Hence the
equidistribution with error term as $i\rightarrow+\infty$ of
$(V_{\Lambda},\operatorname{sh}(\Lambda),\operatorname{sh}(\Lambda^{\perp}))$
in
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}\times\operatorname{Sh}^{1}_{\mathfrak{m}}\times\operatorname{Sh}^{1}_{\mathfrak{n}}$
when $\Lambda$ varies in $\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}(I)$ with
$\overline{\operatorname{Covol}}(\Lambda)=q_{\nu}^{\;\ell\,i}$ follows from
the equidistribution with error term as $i\rightarrow+\infty$ of
$(V_{\Lambda},\operatorname{sh}(\Lambda),\operatorname{sh}(\Lambda^{\perp}))$
in
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}\times\operatorname{Sh}^{1}_{\mathfrak{m}}\times\operatorname{Sh}^{1}_{\mathfrak{n}}$
when $\Lambda$ varies in $\operatorname{{\cal P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}^{\sharp}(I)$ with
$\overline{\operatorname{Covol}}(\Lambda)=q_{\nu}^{\;\ell\,i}$. $\Box$

We conclude this paper with a proof of Corollary 1.1 in the introduction.

Proof that Theorem 1.2 implies Corollary 1.1.  For
$k={\mathfrak{m}},{\mathfrak{n}}$, with
${\widetilde{\Gamma}}_{k}=\operatorname{GL}_{k}(R_{\nu})$, let us consider the
map $\iota:D\mapsto D^{-1}$ defined in the Introduction from the discrete set
$\operatorname{Sh}^{1}_{k}=\operatorname{GL}_{k}({\cal
O}_{\nu})\,\backslash\operatorname{GL}_{k}^{1}(K_{\nu})\,/\,{\widetilde{\Gamma}}_{k}$
to the discrete set ${\widetilde{\Gamma}}_{k}\backslash V_{0}{\cal
I}_{\nu,k}={\widetilde{\Gamma}}_{k}\,\backslash\operatorname{GL}_{k}^{1}(K_{\nu})\,/\,\operatorname{GL}_{k}({\cal
O}_{\nu})$. By Equation (36) , this map satisfies

$\iota_{*}\mu_{\operatorname{Sh}^{1}_{k}}=\mu_{\operatorname{GL}_{k}^{1}(K_{\nu})}(\operatorname{GL}_{k}({\cal
O}_{\nu}))\,\mu_{{\widetilde{\Gamma}}_{k}\backslash V_{0}{\cal
I}_{\nu,k}}=(1-q_{\nu}^{\;-1})\,\mu_{{\widetilde{\Gamma}}_{k}\backslash
V_{0}{\cal I}_{\nu,k}}\;.$

Let
$\varphi^{\prime}:\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}\times\operatorname{Sh}^{1}_{{\mathfrak{m}}}\times\operatorname{Sh}^{1}_{{\mathfrak{n}}}\rightarrow{\widetilde{\Gamma}}_{\mathfrak{m}}\backslash
V_{0}{\cal
I}_{\nu,{\mathfrak{m}}}\times{\widetilde{\Gamma}}_{\mathfrak{n}}\backslash
V_{0}{\cal I}_{\nu,{\mathfrak{n}}}$ be the continuous map defined by
$(x,y,z)\mapsto(y^{-1},z^{-1})$. Since
$\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}$ is compact, this map
$\varphi^{\prime}$ is proper, and the pushforward map $\varphi^{\prime}_{*}$
of Borel measures by $\varphi^{\prime}$ is linear and weak-star continuous.
With $\ell=\operatorname{lcm}({\mathfrak{m}},{\mathfrak{n}})$, the image by
$\varphi^{\prime}_{*}$ of the left hand side of Equation (1) is hence

$\lim_{i\rightarrow+\infty}\;\frac{c^{\prime\prime}}{q_{\nu}^{\;\ell\,\mathbf{d}\,i}}\sum_{\Lambda\in\operatorname{{\cal
P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}\;:\;\overline{\operatorname{Covol}}(\Lambda)=q_{\nu}^{\;\ell\,i}}\Delta_{\operatorname{sh}(\Lambda)^{-1}}\otimes\Delta_{\operatorname{sh}(\Lambda^{\perp})^{-1}}\;.$

Since $\|\mu_{\operatorname{Gr}_{{\mathfrak{m}},\mathbf{d}}}\|=1$ by Equation
(32), the image by $\varphi^{\prime}_{*}$ of the right hand side of Equation
(1) is
$(1-q_{\nu}^{\;-1})^{2}\,\mu_{{\widetilde{\Gamma}}_{\mathfrak{m}}\backslash
V_{0}{\cal
I}_{\nu,{\mathfrak{m}}}}\otimes\mu_{{\widetilde{\Gamma}}_{\mathfrak{n}}\backslash
V_{0}{\cal I}_{\nu,{\mathfrak{n}}}}$. Hence with
$c^{\prime}=\frac{c^{\prime\prime}}{(1-q_{\nu}^{\;-1})^{2}}$ as given in
Corollary 1.1, Theorem 1.2 does imply Corollary 1.1. $\Box$

Remark. Proceeding as in the proof of Corollary 4.8, for every nonzero ideal
$I$ of $R_{\nu}$, we have the following version with error term and
congruences of Corollary 1.1 : There exists
$\tau\in\;]0,\frac{1}{2\,\mathbf{d}^{2}}]$ such that for all finitely
supported maps $f_{1}:{\widetilde{\Gamma}}_{\mathfrak{m}}\backslash V_{0}{\cal
I}_{\nu,{\mathfrak{m}}}\rightarrow{\mathbb{R}}$ and
$f_{2}:{\widetilde{\Gamma}}_{\mathfrak{n}}\backslash V_{0}{\cal
I}_{\nu,{\mathfrak{n}}}\rightarrow{\mathbb{R}}\;$ and for every
$\delta\in\;]0,\tau[$, we have

$\displaystyle\frac{c^{\prime}}{q_{\nu}^{\;\ell\,\mathbf{d}\,i}}\sum_{\Lambda\in\operatorname{{\cal
P}\\!{\cal
L}}_{{\mathfrak{m}},\mathbf{d}}(I)\;:\;\overline{\operatorname{Covol}}\;\Lambda\,=\,q_{\nu}^{\;\ell\,i}}f_{1}(\operatorname{sh}(\Lambda)^{-1})\;f_{2}(\operatorname{sh}(\Lambda^{\perp})^{-1})$
$\displaystyle=\;$ $\displaystyle\Big{(}\int
f_{1}\,d\mu_{{\widetilde{\Gamma}}_{\mathfrak{m}}\backslash V_{0}{\cal
I}_{\nu,{\mathfrak{m}}}}\Big{)}\Big{(}\int
f_{2}\,d\mu_{{\widetilde{\Gamma}}_{\mathfrak{n}}\backslash V_{0}{\cal
I}_{\nu,{\mathfrak{n}}}}\Big{)}+\operatorname{O}_{\nu,\delta,I}\big{(}q_{\nu}^{\;\ell\,\mathbf{d}\,i(-\tau+\delta)}\,\|f_{1}\|_{\infty}\|f_{2}\|_{\infty}\big{)}\;.$
(78)

##  Appendix A Dual and factor partial lattices

Let $V$ be a $K_{\nu}$-vector space with finite dimension $D\geq 2$ endowed
with an ultrametric norm $\|\;\|$. Let $k\in\llbracket 1,D-1\rrbracket$ and
let $W$ be a $k$-dimensional $K_{\nu}$-vector subspace of $V$, endowed with
the restriction norm. We endow the quotient $(D-k)$-dimensional
$K_{\nu}$-vector space $V/W$ with the quotient norm. We denote by
$\pi:V\rightarrow V/W$ the canonical projection. Then the Haar measures
$\mu_{V}$, $\mu_{W}$ and $\mu_{V/W}$ on respectively $V$, $W$, $V/W$,
normalized by these choices of norms as explained in Subsection 2.2, satisfy
the following Weil’s normalization process (see [Wei3, §9]). For all
${\overline{x}}\in V/W$ and $x\in V$ such that $\pi(x)={\overline{x}}$, let
$\mu_{\pi^{-1}({\overline{x}})}$ be the measure on the $K_{\nu}$-affine
subspace $\pi^{-1}({\overline{x}})=x+W$ such that the translation
$\tau_{x}:y\mapsto x+y$, which is a homeomorphism from $W$ to
$\pi^{-1}({\overline{x}})$, satisfies
$(\tau_{x})_{*}\mu_{W}=\mu_{\pi^{-1}({\overline{x}})}$ (this does not depend
on the choice of $x$ in $\pi^{-1}({\overline{x}})$). Then Weil’s normalisation
is asking that we have the following disintegration property of the measure
$\mu_{V}$ over the measure $\mu_{V/W}$ by $\pi$:

$d\mu_{V}=\int_{{\overline{x}}\in
V/W}d\mu_{\pi^{-1}({\overline{x}})}\;d\mu_{V/W}({\overline{x}})\;.$ (79)

This formula implies that the normalizations of $\mu_{V}$ and $\mu_{W}$
uniquely determine the normalization of $\mu_{V/W}$. In order to check that
this normalization coincides with the one coming from the quotient norm on
$V/W$, we apply the above formula on $B_{V}(0,1)$, noting that
$\pi(B_{V}(0,1))=B_{V/W}(0,1)$ and that, by the ultrametric property, for
every $x\in B_{V}(0,1)$, we have $-x+B_{V}(0,1)\cap(x+W)=B_{W}(0,1)$.

Let $\Lambda$ be a $k$-lattice in $V$. Its dual lattice is the
$R_{\nu}$-submodule of the dual $K_{\nu}$-vector space $V_{\Lambda}^{*}$
defined by

$\Lambda^{*}=\\{\ell\in V_{\Lambda}^{*}:\forall\;x\in\Lambda,\;\;\ell(x)\in
R_{\nu}\\}\;.$

###### Lemma A.1

The dual lattice $\Lambda^{*}$ is a full $R_{\nu}$-lattice in
$V_{\Lambda}^{*}$, we have $(\Lambda^{*})^{*}=\Lambda$ and

$\operatorname{Covol}(\Lambda^{*})\operatorname{Covol}(\Lambda)=q^{2k({\mathfrak{g}}-1)}\;.$

Proof. Let $(b_{1},\dots,b_{k})$ be an $R_{\nu}$-basis of $\Lambda$. Then
$(b_{1},\dots,b_{k})$ is a $K_{\nu}$-basis in $V_{\Lambda}$, and we denote by
$(b_{1}^{*},\dots,b_{k}^{*})$ its dual basis in $V_{\Lambda}^{*}$. A linear
form $\ell=\sum_{i=1}^{k}\lambda_{i}b_{1}^{*}\in V_{\Lambda}^{*}$ takes
integral values on all elements of $\Lambda$ if and only if it takes integral
values on $b_{1},\dots,b_{k}$, that is, if and only if its coordinates
$\lambda_{1},\dots,\lambda_{k}$ are integral. Thus
$\Lambda^{*}=\oplus_{i=1}^{k}R_{\nu}b_{i}^{*}$, which is a $k$-lattice in the
$k$-dimensional vector space $V_{\Lambda}^{*}$. Since the dual basis in
$(V_{\Lambda}^{*})^{*}=V_{\Lambda}$ of the $K_{\nu}$-basis
$(b_{1}^{*},\dots,b_{k}^{*})$ of $V_{\Lambda}^{*}$ is the $K_{\nu}$-basis
$(b_{1},\dots,b_{k})$ of $V_{\Lambda}$, we have $(\Lambda^{*})^{*}=\Lambda$.

The duality pairing $V_{\Lambda}^{*}\times V_{\Lambda}\rightarrow K_{\nu}$
defined by $(\ell,x)\mapsto\ell(x)$, which sends $\Lambda^{*}\times\Lambda$ to
$R_{\nu}$ induces a $K_{\nu}$-linear isomorphism $V_{\Lambda}^{*}\times
V_{\Lambda}\rightarrow{K_{\nu}}^{2k}$ that sends the full $R_{\nu}$-lattice
$\Lambda^{*}\times\Lambda$ to ${R_{\nu}}^{2k}$ and the supremum product norm
of the norm on $V_{\Lambda}$ and of its dual norm on $V_{\Lambda}^{*}$ to the
standard supremum norm on ${K_{\nu}}^{2k}$. Hence, using Equation (9) for the
last equality, we have

$\operatorname{Covol}(\Lambda^{*})\operatorname{Covol}(\Lambda)=\operatorname{Covol}(\Lambda^{*}\times\Lambda)=\operatorname{Covol}({R_{\nu}}^{2k})=q^{2k({\mathfrak{g}}-1)}\;.\quad\Box$

Assume that $V$ is also endowed with an integral structure $V_{R_{\nu}}$. Let
$\Lambda$ be a primitive $k$-lattice in the integral $K_{\nu}$-space $V$. The
factor lattice of $\Lambda$ is the $R_{\nu}$-submodule $\Lambda^{\pi}$ of the
quotient $K_{\nu}$-vector space $V/V_{\Lambda}$ which is the image of
$V_{R_{\nu}}$ by the canonical projection $\pi:V\rightarrow V/V_{\Lambda}$.

###### Lemma A.2

The factor lattice $\Lambda^{\pi}$ is a full $R_{\nu}$-lattice in
$V/V_{\Lambda}$. The canonical $K_{\nu}$-linear isomorphism
$V/V_{\Lambda}\rightarrow(V_{\Lambda}^{\perp})^{*}$ maps $\Lambda^{\pi}$ to
$(\Lambda^{\perp})^{*}$. We have

$\operatorname{Covol}(\Lambda^{\pi})=\operatorname{Covol}((\Lambda^{\perp})^{*})\quad\text{and}\quad\operatorname{Covol}(\Lambda^{\pi})\operatorname{Covol}(\Lambda)=\operatorname{Covol}(V_{R_{\nu}})$

Proof. Since the $k$-lattice $\Lambda$ is primitive, there exists an
$R_{\nu}$-basis $(b_{1},\dots,b_{D})$ of $V_{R_{\nu}}$ such that
$(b_{1},\dots,b_{k})$ is an $R_{\nu}$-basis of $\Lambda$, hence a
$K_{\nu}$-basis of $V_{\Lambda}$. Then $(\pi(b_{k}+1),\dots,\pi(b_{D}))$ is a
$K_{\nu}$-basis of $V/V_{\Lambda}$, and an $R_{\nu}$-basis of $\Lambda^{\pi}$
by definition. Hence $\Lambda^{\pi}$ is a $(D-k)$-lattice in the
$(D-k)$-dimensional vector space $V/V_{\Lambda}$.

Identifying a $K_{\nu}$-vector space $W$ with its bidual $(W^{*})^{*}$ by the
map $x\mapsto(\ell\mapsto\ell(x))$ as usual, the map
${\widetilde{\Theta^{\prime\prime}}}:V\rightarrow(V_{\Lambda}^{\perp})^{*}$
defined by $x\mapsto x_{\mid V_{\Lambda}^{\perp}}$ induces a $K_{\nu}$-linear
isomorphism
$\Theta^{\prime\prime}:V/V_{\Lambda}\rightarrow(V_{\Lambda}^{\perp})^{*}$.
With $(b_{1},\dots,b_{D})$ as above, we have seen in the proof of Proposition
2.2 that $(b_{k+1}^{*},\dots,b_{D}^{*})$ is an $R_{\nu}$-basis of
$\Lambda^{\perp}$, hence a $K_{\nu}$-basis of
$V_{\Lambda^{\perp}}=V_{\Lambda}^{\perp}$. As seen in the proof of Lemma A.1,
the dual $K_{\nu}$-basis $({b_{k+1}^{*}}^{*},\dots,{b_{D}^{*}}^{*})$ of
$(b_{k+1}^{*},\dots,b_{D}^{*})$ is an $R_{\nu}$-basis of
$(\Lambda^{\perp})^{*}$. But $({b_{k+1}^{*}}^{*},\dots,{b_{D}^{*}}^{*})$ is
exactly
$\big{(}\Theta^{\prime\prime}(\pi(b_{k+1})),\dots,\Theta^{\prime\prime}(\pi(b_{k+1}))\big{)}$.
Hence $\Theta^{\prime\prime}(\Lambda^{\pi})=(\Lambda^{\perp})^{*}$.

When $V/V_{\Lambda}$ is endowed with the quotient norm, and
$(V_{\Lambda}^{\perp})^{*}$ with the dual norm of the restriction to
$V_{\Lambda}^{\perp}$ of the dual norm on $V^{*}$, the above map
$\Theta^{\prime\prime}$ is an isometry. Hence
$\operatorname{Covol}(\Lambda^{\pi})=\operatorname{Covol}((\Lambda^{\perp})^{*})$.

Let $F$ be a clopen strict fundamental domain for the action of $R_{\nu}$ on
$K_{\nu}$. The formula
$\operatorname{Covol}(V_{R_{\nu}})=\operatorname{Covol}(\Lambda)\operatorname{Covol}(\Lambda^{\pi})$
follows by integrating Equation (79) with $W=V_{\Lambda}$ on the strict
fundamental domain $F\,b_{1}+\dots+F\,b_{D}$ of $V_{R_{\nu}}$. $\Box$

Proof of Equation (13).  Under its assumptions, we may assume that
$V={K_{\nu}}^{D}$ and $V_{R_{\nu}}={R_{\nu}}^{D}$ and that the norm of $V$ is
the standard supremum norm of ${K_{\nu}}^{D}$. We then have
$\operatorname{Covol}(V_{R_{\nu}})=q^{D({\mathfrak{g}}-1)}$ by Equation (9).
Hence respectively by Lemma A.1 (recalling that $\Lambda^{\perp}$ is a
$(D-k)$-lattice), by the first equality in Lemma A.2, and by the second
equality in Lemma A.2, we have

$\displaystyle\operatorname{Covol}(\Lambda^{\perp})$
$\displaystyle=q^{2(D-k)({\mathfrak{g}}-1)}\operatorname{Covol}((\Lambda^{\perp})^{*})^{-1}=q^{2(D-k)({\mathfrak{g}}-1)}\operatorname{Covol}(\Lambda^{\pi})^{-1}$
$\displaystyle=q^{2(D-k)({\mathfrak{g}}-1)}\operatorname{Covol}(\Lambda)q^{-D({\mathfrak{g}}-1)}=\operatorname{Covol}(\Lambda)\;q^{(D-2k)({\mathfrak{g}}-1)}\;,$

as wanted. $\Box$

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Department of Mathematics, Rämistrasse 101
---
ETH Zurich, 8092 Zurich, Switzerland
e-mail<EMAIL_ADDRESS>

Laboratoire de mathématique d’Orsay, UMR 8628 CNRS
---
Université Paris-Saclay, 91405 ORSAY Cedex, FRANCE
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