\section{The signed units}
\label{du:section}

In the trace calculation for $\widetilde C_n$ use its own
Plancherel families. For $\widetilde B_n$ use the restricted
families of $H^0$ in Section~\ref{ab:section}. Thus in this
section the family algebras used in the trace always come from
the $\widetilde C_n$ datum, with $c>0$ or $c=0$.

\subsection{Integral characteristic polynomials}

Fix one compact induction family $\pi_{v,z}$ with Levi
discrete-series representation $\delta_v$, where $z$ varies
in its unramified torus. In the $\widetilde C_n$ datum a
finite standard Levi consists of type-$A$ blocks and a
possibly empty type-$B$ tail in the finite root system.
The semisimple Hecke algebra is a product of the extended
type-$A$ algebras on the weight lattices and the nonextended
affine-$C$ tail. By \cite[Proposition~6.5]{DelormeOpdam},
the finite torus twisting group acts freely on the type-$A$
discrete series and trivially on the tail. We choose the real
Steinberg representation on each type-$A$ factor and absorb
its torsion twists into $z$. These representatives have
rational characters in the parameter. The parameter value
$v^{2c}=1$ is positive and the same Levi description applies.

We first verify rationality for the tail. Work in the field
$\mathbb Q(v^{1/e})$ for a positive integer $e$ clearing the
residual exponents. For a discrete series of the nonextended
affine-$C$ tail the residual coordinates are signed powers
of $v^{1/e}$, by the signed residual string description of
\cite[Theorem~1.1 and Section~4.2.1]{CiubotaruOpdam}.
This description applies to the discrete series surviving
at the specified weights, including $c=0$. The joint
Bernstein weights lie in the Weyl orbit of this central
character and hence also in the field. To see directly why the residual phases are signs at a specialized ray, the pole roots of a point residual coset span the full rank \cite[Theorem~2.38 and proof of Proposition~2.56]{OpdamSolleveld}. In the standard torus coordinates \(x_i\), their equations have the forms \(x_i x_j^{\pm1}=v^k\) and \(x_i=\pm v^k\) for integer exponents \(k\). For the unitary phases, regard the two-coordinate equations as edges joining equal or inverse phases. A connected component with a one-coordinate equation has all phases equal to signs. In a component without one, every phase is the phase of one chosen coordinate or its inverse. Unless a cycle identifies that phase with its inverse, one complex parameter varies freely, contradicting full rank. Thus in this component also every phase is a sign. Galois conjugation
fixes the joint weights. The ranks determining their
generalized weight multiplicities are unchanged by
conjugation. These ranks and the simplicity of the
algebraic branches persist at all sufficiently large
regular specializations, by the finite central quotient
construction in Section~\ref{sec:spectral}. The strict
Casselman inequalities therefore hold for the specialized
conjugate, so it too is a discrete series.

Every finite parabolic of the tail has split Hecke algebra
over $\mathbb Q(v^{1/e})$. In the cyclotomic presentation,
the split semisimplicity criterion
\cite[Theorem~4.2]{ChlouverakiJacon} uses quantum integers
at $v^{2a}$ and cross factors of the form $1+v^j$,
with $j\in\mathbb Z$. These factors are nonzero, including
when an endpoint weight is zero. The finite simple
multiplicities are therefore Galois invariant. For a face $f$ of the closed
fundamental alcove, denote by $m_f(\delta,\sigma)$ the
multiplicity of the finite simple $\sigma$ on restriction to
the Hecke algebra of the stabilizer of $f$. For this
nonextended datum the finite-face formula for the
Euler--Poincar\'e pairing gives
\[
\operatorname{EP}(\delta,\delta')
 =\sum_{f}(-1)^{\dim f}
   \sum_{\sigma}m_f(\delta,\sigma)m_f(\delta',\sigma)
.
\]
This is the specialization of \cite[Equations~(57)--(58)]{OpdamSolleveld}
to the nonextended datum. The sum is over all faces, including the
alcove itself.
The group of length-zero elements is trivial for this datum.
Taking $\delta'$ to be a Galois conjugate gives the
self-pairing of $\delta$, which is one. Distinct discrete
series are orthogonal
\cite[Corollary~2.34]{OpdamSolleveld}, so the conjugate has
the same character. Thus $\delta_v$ has character over
$\mathbb Q(v^{1/e})$. Enlarging $e$ will also allow the
monomial $v^{-A_R}$.

Let $P$ denote the finite Levi simple roots and $W_P$ its
finite Weyl group. In the Bernstein lattice $X$ the
unramified quotient is $X/(X\cap\mathbb RP)$, and the
projected semisimple lattice is
$X/(X\cap(\mathbb RP^\vee)^\perp)$. For $x\in X$ write
$\nu(x)$ and $\bar x$ for its respective images. A torus
monomial $z^{\nu(x)}$ means evaluation at the character
$\nu(x)$. The twisted quotient defining parabolic
induction takes the Bernstein basis element $\theta_x N_p$,
for $p\in W_P$, to
$z^{\nu(x)}\theta_{\bar x}N_p$
\cite[Proposition~4.16]{Opdam}. Put
$\mathcal O=\mathbb Z[v^{1/e},v^{-1/e}]$ for the coefficient
ring in this argument. Over $\mathcal O$ the Bernstein
presentation decomposes the ambient algebra as a right
module over the parabolic algebra, with basis $N_d$ for
the minimal representatives $d$ of $W_0/W_P$. The Bernstein
cross-relations lie in this ring because the differences
of lattice monomials in their numerators are divisible
by the corresponding root binomial.
For any fixed standard basis element $N_w$, expand
\[
N_wN_d=\sum_{d',\,x,\,p}a_{d',d,x,p}N_{d'}\theta_xN_p,
\qquad a_{d',d,x,p}\in\mathcal O .
\]
This is a finite expansion and its parabolic coefficients
give the compact induction matrix. In particular the
$(d',d)$ entry of its coefficient at $z^\nu$ is
\[
\delta_v\!\left(
  \sum_{\substack{x,p\\\nu(x)=\nu}}
        a_{d',d,x,p}\theta_{\bar x}N_p
\right).
\]
A coefficient matrix is thus the entrywise image under
$\delta_v$ of a matrix over the integral semisimple
Levi algebra.

Impose the residual central character of $\delta_v$ on this
Levi algebra. The Bernstein lattice algebra is finite over
its Weyl invariants: each of its lattice generators and
its inverse satisfy the monic polynomial given by their
Weyl orbit. The invariant elements take values in
$\mathcal O$ at the chosen signed residual monomials.
The resulting central quotient is consequently finite
over $\mathcal O$, and so is its matrix algebra. The
determinant trick makes every element of that matrix
algebra integral over $\mathcal O$. The eigenvalues of
its image under $\delta_v$ are integral as well. The
characteristic polynomial of the image belongs to
$\mathbb Q(v^{1/e})[T]$: traces of all powers are rational
by character rationality of $\delta_v$, and the Newton
identities give the coefficients. Integral closedness
puts the polynomial in $\mathcal O[T]$.

The compact Gram form is independent of $z$
\cite[Proposition~4.19]{Opdam}. Passing to the
orthonormal compact basis of Section~\ref{sec:spectral}
conjugates every torus coefficient by the same matrix.
For a finite integral Laurent polynomial
$\mathfrak d(z)$, multiplication by $\mathfrak d$ replaces the
coefficient matrices by integral linear combinations.
After the conjugation and multiplication by $v^{-A_R}$,
their characteristic polynomials still belong to
$\mathcal O[T]$. Write
\[
c_w(z)=\lim_{v\to\infty}v^{-A_R}\pi_{v,z}(N_w)
\]
in this orthonormal compact basis. The limit exists uniformly by
Proposition~\ref{prop:leading}. The fixed finite torus
support also gives coefficientwise convergence. For a
Laurent polynomial in $z$, let $[z^\nu]$ denote the
coefficient of its character monomial $z^\nu$, with
absent coefficients equal to zero. Each characteristic
polynomial coefficient has a finite limit and is a
Laurent polynomial in $v^{1/e}$ with integer
coefficients. Its limit is its integer constant term.
We obtain
\begin{equation}
\label{du:spectra}
\det\bigl(TI-[z^\nu](c_w(z)\mathfrak d(z))\bigr)\in\mathbb Z[T].
\end{equation}
For the restricted $H^0$ families, the standard basis
identification of Lemma~\ref{ab:crossed} gives the same
statement for $w\in W$ of type $\widetilde B_n$.

\subsection{The limiting measures}

We compute the stationary factors of the relative
Plancherel density. In the $\widetilde C_n$ lattice
$X=\mathbb Z^n$, let the type-$A$ blocks have sizes
$k_1,\ldots,k_r$ and let the tail have size
$n-\sum_j k_j$. The unramified quotient records the sum
of the lattice coordinates in each block. It therefore
has primitive torus coordinates $z_1,\ldots,z_r$.
The coordinates of the real Steinberg factor on a
block of size $k_j$ are $z_jv^u$, with
\[
u\in S_{k_j},\qquad
S_k=\{a(k-1-2i):0\le i<k\}.
\]
A tail coordinate is $\varepsilon v^u$ with
$\varepsilon\in\{1,-1\}$ and a fixed real exponent $u$.

Write $x_i$ for the resulting ambient coordinates.
The exact positive Plancherel product, with the Levi
roots removed, has long-root factors
\[
 \frac{|1-x_i x_j^{\pm1}|^2}
      {|1-v^{2a}x_i x_j^{\pm1}|^2},
\]
and doubled short-root factors
\[
 \frac{|1-x_i^2|^2}
      {|1-v^{b+c}x_i|^2\,|1+v^{b-c}x_i|^2},
\]
up to the overall monomial and the tail formal-degree
factor. In the long-root display each choice of sign
denotes one factor, when the corresponding root is
outside the Levi. These are the absolute-square
expressions on the tempered torus in
\cite[Theorem~3.25 and Proposition~3.27]{Opdam}. The
Weyl element that inverts the residual real part and
fixes the unitary part gives the absolute-square
expression from the product of opposite roots.
When $b=c$ the second short denominator is
$|1+x_i|^2$. At $c=0$ the two short denominators
multiply to $|1-v^{2b}x_i^2|^2$.

For a unitary monomial $Z$ set
$D_\varepsilon(Z)=|1-\varepsilon Z|^2$ for
$\varepsilon=\pm1$, and write $D_+,D_-$ for the two signs. The normalized limit of
$|1-\varepsilon v^d Z|^2$ is one when $d\ne0$
after division by $v^{2\max(d,0)}$. For $d=0$
its factor is $D_\varepsilon(Z)$. We count these
stationary factors, with denominator factors counted
negatively.

\begin{itemize}
\item Between two blocks, the ratio-root multiplicity
is $|S_k\cap S_h|-|S_k\cap(S_h-2a)|$ for block sizes
$k,h$. If their parities differ, both intersections
are empty. For equal parity and unequal sizes both
have cardinality $\min(k,h)$, and for $k=h$ the
cardinalities are $k$ and $k-1$. The sum-root
multiplicity is the same by $S_h=-S_h$. Thus the
factor for equal-size blocks $i,j$ is
$D_+(z_i/z_j)D_+(z_i z_j)$.
\item Within one block only sum roots are outside the
type-$A$ Levi. The numerator pairs whose exponents
sum to zero number $\lfloor k/2\rfloor$. The
denominator pairs whose exponents sum to $-2a$
number $\lfloor (k-1)/2\rfloor$. The doubled short
numerator adds one factor $|1-z_j^2|^2$ when $k$ is
odd. Hence the net internal numerator is
$|1-z_j^2|^2=D_+(z_j)D_-(z_j)$. The short
denominators subtract one from the positive and
negative exponents when $b+c\in S_k$ and
$b-c\in S_k$, respectively.
\item A tail coordinate $\varepsilon v^u$ contributes
only at $z_j=\varepsilon$. Counting the sum and
ratio roots gives the signed multiplicity
\[
 2\mathbf 1_{u\in S_k}
 -\mathbf 1_{u+2a\in S_k}
 -\mathbf 1_{u-2a\in S_k},
\]
where $\mathbf 1$ denotes the indicator of the
specified condition. The two numerator tests are
$u,-u\in S_k$ and the two denominator tests are
$u-2a,-u-2a\in S_k$. The displayed form uses
$S_k=-S_k$.
\end{itemize}

Thus, for each size $k$, the short factors
$D_\varepsilon(z_j)$ have a common net exponent on
its size-$k$ blocks. This exponent is nonnegative,
since a negative exponent would give a pole of the
exact density at a generic point of $z_j=\varepsilon$,
contrary to its real analyticity on the compact
tempered torus \cite[Theorem~3.25]{Opdam}.
If the exponent is positive, that divisor lies in the
zero set of the density. A short root supported on
a coordinate of block \(j\) projects to the direction of
the primitive block character \(z_j\) on the unramified
torus. The divisor \(z_j=\varepsilon\) is therefore the
connected translate of the projected root kernel through
any of its points. It is a Plancherel mirror in
\cite[Definition~4.1]{DelormeOpdam}. The relative
$c$-function has a simple pole on a mirror
\cite[Theorem~4.3(iii)]{DelormeOpdam}. This theorem
allows zero label exponents, since the labels in
\cite[\S2]{DelormeOpdam} are arbitrary positive real
numbers. Inverting the unramified character preserves
the divisor $z_j=\varepsilon$. The density is proportional
to the reciprocal of the product of the $c$-function
with its value at that inverted character. Its zero
has order two, which is the order of $D_\varepsilon$.
The exponent is therefore one. At $b=c$ the negative
test is $0\in S_k$. At $c=0$ the tests for the two
signs use the same number $b$ but the divisors
$z_j=1$ and $z_j=-1$ are distinct. The argument on
each divisor is unchanged.

On the blocks of a fixed size, take the product
\[
 \prod_{i<j}(1-z_i^{-1}z_j)(1-z_i^{-1}z_j^{-1}),
\]
where the indices are ordered within that set of
blocks. If there are only positive short factors,
multiply by $\prod_i(1-z_i^{-1})$. If there are
only negative short factors, multiply by
$\prod_i(1+z_i^{-1})$. If both occur, multiply by
$\prod_i(1-z_i^{-2})$. With neither short factor
use the displayed product alone. These are the
Weyl denominators of the corresponding types
$B$, its simultaneous sign translate, $C$ and
$D$. In the last case with one block the product
is one, and with two blocks it is the product
for $A_1\times A_1$. The Weyl denominator
identity expresses each as the alternating sum
with exponents $w\rho-\rho$, where $\rho$ is half
the sum of the positive roots. These exponents
are integral and distinct by regularity of
$\rho$ in the corresponding root system.
The sign translation also has coefficients
$\pm1$. The sets of coordinates for different
sizes are disjoint, so the product over sizes
has coefficients $\pm1$.

Let $\mathfrak d$ be this product. The normalized
limiting measure on the family is
$\kappa|\mathfrak d|^2\,dz$, where $dz$ is probability
Haar measure and $\kappa>0$ includes the finite
normalization factors and the leading tail
formal degree. If the torus is a point,
$\mathfrak d=1$ and $dz$ is the point mass. The same
measure is used on restricting a family to $H$,
by Lemma~\ref{ab:crossed}.

\subsection{The trace inequality}

\begin{lemma}
\label{du:matrix}
Let $D$ be a complex matrix with characteristic
polynomial in $\mathbb Z[T]$. For the adjoint $D^*$
with respect to any fixed positive Hermitian form,
\[
 \operatorname{Tr}(DD^*)\geq|\operatorname{Tr}D|.
\]
In the equality case $D$ is zero or an orthogonal
projection multiplied by the sign of its nonzero trace.
\end{lemma}

\begin{proof}
In an orthonormal basis Schur triangularization bounds
the squared Hilbert--Schmidt norm below by the sum of
the squared absolute eigenvalues, with equality only
for a normal matrix. Factor the monic characteristic
polynomial into monic irreducibles over $\mathbb Q$,
counting multiplicities. For a factor other than $T$,
let its roots be $\lambda_1,\ldots,\lambda_m$ and put
$S=\sum_i|\lambda_i|^2$. The absolute value of their
product is a nonzero integer by Gauss's lemma.
The arithmetic--geometric mean inequality gives
$S\geq m$, so
\[
 \left|\sum_i\lambda_i\right|\leq\sqrt{mS}\leq S.
\]
The factor $T$ contributes zero. Summing over the
factors gives the inequality. Equality makes all
roots of each nonzero factor equal and of absolute
value one, so that factor is $T-1$ or $T+1$.
Triangle equality makes their signs agree. Normality
then gives the asserted projection.
\end{proof}

\begin{lemma}
\label{du:alternant}
Let $\mathfrak d$ be a nonzero Laurent polynomial with all
nonzero coefficients in $\{1,-1\}$, and let $M(z)$ be
a matrix Laurent polynomial on the same torus.
Suppose every matrix coefficient of $M(z)\mathfrak d(z)$
has characteristic polynomial in $\mathbb Z[T]$.
For a torus-independent positive Hermitian form,
\[
 \int\operatorname{Tr}(MM^*)|\mathfrak d|^2\,dz
 \ \geq\
 \left|\int\operatorname{Tr}(M)|\mathfrak d|^2\,dz\right|.
\]
Equality makes $M$ constant, either zero or an
orthogonal projection multiplied by the sign of the
nonzero integral on the right.
\end{lemma}

\begin{proof}
Write $D_\nu=[z^\nu](M\mathfrak d)$ and
$\varepsilon_\nu=[z^\nu]\mathfrak d$.
Parseval identifies the left side with
$\sum_\nu\operatorname{Tr}(D_\nu D_\nu^*)$ and
the integral inside the absolute value with
$\sum_\nu\varepsilon_\nu\operatorname{Tr}(D_\nu)$.
Apply Lemma~\ref{du:matrix} term by term and use
the triangle inequality. In the equality case
$D_\nu=0$ whenever $\varepsilon_\nu=0$.
For any nonzero scalar entry $f$ of $M$, the
convex hull of the support of $f\mathfrak d$ is the
Minkowski sum of the convex hulls of the supports
of $f$ and $\mathfrak d$. Indeed, leading terms for
each supporting linear functional multiply
without vanishing in the Laurent polynomial
domain. This sum lies in the convex hull of the
support of $\mathfrak d$. Evaluating every linear
functional shows that $f$ has support $\{0\}$.
Hence $M$ is constant. For any \(\nu\) with
\(\varepsilon_\nu\ne0\), the coefficient is
\(D_\nu=\varepsilon_\nu M\). The equality case of
Lemma~\ref{du:matrix} then makes \(M\) zero or a signed
orthogonal projection, with sign that of \(\operatorname{Tr} M\).
The integral of \(|\mathfrak d|^2\) is positive, so the integral
on the right has the same sign.
\end{proof}

\begin{proposition}
\label{du:units}
The integral leading algebra has a finite unit which is
a sum of signed basis elements. The corresponding
signed elements are mutually orthogonal self-adjoint
idempotents. In the original orthonormal basis both
their left and their right multiplication operators
are coordinate projections.
\end{proposition}

\begin{proof}
Use the finite integral unit $e$ from
Proposition~\ref{prop:leading} and write
$e=\sum_w m_w t_w$, where $t_w$ is its leading
basis indexed by $w\in W$. Denote the positive
trace of this algebra by $\tau_J$. Its basis
satisfies $\tau_J(t_wt_y^*)=\delta_{w,y}$.
Since the unit is self-adjoint,
$\tau_J(t_w)=\langle t_w,e\rangle=m_w$.
The Plancherel realization sends each $t_w$
to the family matrices $c_w$. In the
$\widetilde B_n$ case this follows with
the restricted matrices by Section~\ref{ab:section}.
For every family Lemma~\ref{du:alternant}
applies by \eqref{du:spectra} and the computed
density. Multiplying its inequality by $\kappa$
and summing over the families yields
\[
  1=\tau_J(t_wt_w^*)\geq|\tau_J(t_w)|=|m_w|.
\]
The integers $m_w$ are consequently $0$ or $\pm1$.
For $m_w\ne0$, all the inequalities used in the
sum are equalities. Since the masses $\kappa$
are positive, the nonzero projections in the
family matrices have the common sign $m_w$.
Thus $m_wt_w$ is a self-adjoint idempotent.
These finitely many positive projections sum to the
identity. For a vector \(\xi\) in the range of one of them,
the inner product of the sum with \(\xi\) is \(\|\xi\|^2\),
which is already the contribution of that projection.
Each other term is the squared norm of the corresponding
projection of \(\xi\) and therefore vanishes. Thus the
projections are pairwise orthogonal.

The corresponding left and right multiplication
operators are orthogonal projections with
integer matrix entries. For such a projection
$P$ the diagonal entry $P_{ii}$ is the squared
norm of its $i$th column. This diagonal entry is
an integer between zero and one, and hence all
off-diagonal entries of that column vanish.
This proves the coordinate assertion.
\end{proof}