\section{The zero endpoint and type \texorpdfstring{$\widetilde B_n$}{B-tilde}}
\label{ab:section}

Fix $n\geq3$. Let $W^0$ be the affine Coxeter group of type
$\widetilde C_n$, with generators $s_0,\ldots,s_n$ in path
order. Let $H^0$ be its Hecke algebra with weight zero at $s_0$,
weight $b$ at $s_n$, and weight $a$ at all other generators.
This is the specialization $c=0$ of the $\widetilde C_n$ datum.
Write $W$ and $H$ for the group and algebra of type $\widetilde B_n$.
The fork generators $t_0,t_1$ are adjacent to $t_2$, followed by
the chain $t_2,\ldots,t_n$ with an edge of order four at the last
pair. The weights are $a$ at $t_0,\ldots,t_{n-1}$ and $b$ at $t_n$.
A superscript $0$ distinguishes the standard basis and the
coefficient-of-identity trace of $H^0$.

\begin{lemma}
\label{ab:crossed}
The assignments
\[
t_0=s_0s_1s_0,\qquad t_i=s_i\quad(1\leq i\leq n).
\]
embed $W$ in $W^0$ as an index-two subgroup. Let $\gamma$ be the
diagram automorphism interchanging $t_0$ and $t_1$ and fixing
$t_2,\ldots,t_n$, acting on $H$ by permuting the normalized generators.
Setting $U=N^0_{s_0}$ gives
\[
H^0=H\rtimes\langle U\rangle,\qquad
U^2=1,\qquad UhU=\gamma(h)\quad(h\in H).
\]
Under this embedding, $N_x=N_x^0$ and $N_xU=N_{xs_0}^0$ for $x\in W$.
In particular, $\tau^0|_H=\tau$ and $\tau^0(HU)=0$.
\end{lemma}

\begin{proof}
On $\mathbb R^n$ take the $\widetilde C_n$ alcove
$1/2>x_1>\cdots>x_n>0$. Its walls give $s_0$ at $x_1=1/2$,
$s_i$ at $x_i=x_{i+1}$ for $1\leq i<n$, and $s_n$ at $x_n=0$.
The $s_0$-walls are the planes $x_i\in\mathbb Z+1/2$. Removing them gives
the $\widetilde B_n$ arrangement. The union of the two adjacent
$\widetilde C_n$ alcoves across the $s_0$ wall, with the intervening
open facet included, is the $\widetilde B_n$ alcove. Its fork
walls are $x_1=x_2$ and $x_1+x_2=1$, with reflections $s_1$ and
$s_0s_1s_0$. Its reflection
group is the kernel of the character of $W^0$ that sends $s_0$ to
$-1$ and $s_1,\ldots,s_n$ to $1$. In the affine permutation
realization this character is the parity of the sum of the
translation coordinates.

In $H^0$ the relation for $U$ is $U^2=1$. Conjugation by $U$
exchanges $U N^0_{s_1}U$ and $N^0_{s_1}$ and fixes
$N^0_{s_i}$ for $2\leq i\leq n$. The order-four braid relation for $s_0,s_1$
becomes the commutation of the fork generators. The other braid
relations for the fork follow by conjugating those for $s_1$.
Conversely these relations and the action of $U$ give the
$\widetilde C_n$ relations. The quadratic relations have the
stated weights, so the maps on generators yield the crossed product.

To check the basis, expand a reduced $W$ expression by replacing
each $t_0$ by $s_0s_1s_0$. Its total weight equals the weighted
length of its element in $W^0$ because the separating positive-weight
walls are precisely the separating walls of the larger alcoves.
The Coxeter deletion process reduces this expression by braid moves
and removal of adjacent equal generators. The braid moves preserve
weight and each removal of a positive-weight generator would lower
the total weight. Hence all removals have weight zero, for which
$U^2=1$ holds in the Hecke algebra. Appending $s_0$ leaves the
weighted length unchanged because the two resulting alcoves are
adjacent across a zero-weight wall. The same argument gives
$N_xU=N^0_{xs_0}$. The trace assertions follow from these basis
identities.
\end{proof}

The common finite Weyl group $W_0$ is the group of signed permutations
on $n$ coordinates. The translation lattice of $H$ is the index-two
sublattice of the translation lattice $\mathbb Z^n$ of $H^0$ given by
\[
Q(C_n)=\left\{x\in\mathbb Z^n:\sum_{i=1}^n x_i\text{ is even}\right\}.
\]
The Bernstein elements of $H$ are those of $H^0$ indexed by this
sublattice. For a dominant lattice element the Bernstein element is
the standard basis element of its translation, so the equality follows
from Lemma~\ref{ab:crossed}. Every lattice element is a difference of
dominant ones, and the Bernstein group law extends the equality. The center of $H$ is
therefore contained in the center of $H^0$.

\begin{proposition}
\label{ab:facet}
Every irreducible tempered representation of $H$ has a constituent
of the same spectral degree on a finite standard parabolic Hecke
subalgebra. Its spectral level is \eqref{eq:root-degree} at $c=0$,
where $R=(R_1,\ldots,R_n)$ is obtained from the moduli $v^{2R_i}$
of a lift of a central torus character to the torus of $H^0$.
\end{proposition}

\begin{proof}
The two lifts differ by the character $x\mapsto(-1)^{\sum_i x_i}$
of $\mathbb Z^n$, which simultaneously changes the signs of the
torus coordinates. Thus their real exponent vectors agree. The
term for $2e_i$ in the reduced datum is $K_b(2R_i)=2K_{b/2}(R_i)$,
the sum of the two endpoint terms at $c=0$. The other terms of
\eqref{eq:root-degree} agree.

By Lemma~\ref{ab:crossed}, induction and restriction for this
crossed product preserve temperedness. Indeed, the regular
representation of $H^0$ restricts to two copies of the regular
representation of $H$, up to the automorphism $\gamma$, and is
itself induced from the regular representation of $H$. These
identifications extend induction and restriction to the reduced
$C^*$-algebras. For each irreducible tempered representation
$\rho$ of $H$ one can therefore choose an irreducible tempered
representation $\pi$ of $H^0$ above $\rho$. Its restriction is
$\rho$ if $\rho$ is $\gamma$-invariant and is
$\rho\oplus\gamma(\rho)$ otherwise. Since the center of $H$ lies in
the center of $H^0$, all constituents in the restriction have the
same central character on $H$. The finite-facet attainment statement of
Proposition~\ref{prop:facets} at the actual numerical endpoint
$v^{2c}=1$ gives a finite-facet constituent of $\pi$ of degree
$A_R$.

For $0\le k\le n$, let $F_k$ be the finite subgroup of $W^0$
obtained by deleting $s_k$. For $k=0$ it is the finite type-$B_n$
subgroup generated by $t_1,\ldots,t_n$ and lies in $W$.
For $k=1$ it is $\langle s_0\rangle\times B_{n-1}$ and its
intersection with $W$ is the standard subgroup generated by
$t_2,\ldots,t_n$, contained in either fork facet. For $k\geq2$ it is
$B_k\times B_{n-k}$, where $B_0$ means the trivial group. The
first factor acts on the first $k$ coordinates as the signed
permutation group centered at $(1/2,\ldots,1/2)$ in that block.
Its translation parity is the parity of the number of sign
changes, so its intersection with $W$ is the type-$D_k$ group
generated by $t_0,\ldots,t_{k-1}$. Thus for $k\geq2$ the
intersection $F_k\cap W$ is the standard parabolic obtained
by deleting $t_k$. Here $D_2=A_1\times A_1$ and $D_3=A_3$.
Conjugation by $U$ fixes the intersections for $k>0$ and sends
the intersection for $k=0$ to the other standard fork facet.

The inclusion at each intersection preserves the standard trace.
For $k>0$ the finite algebra of $F_k$ is a crossed product of the
intersection algebra by $\langle U\rangle$. Let $\sigma$ be a simple
type of the larger algebra and $\eta$ a simple type in its restriction.
Over a splitting field of characteristic zero, Clifford theory gives
one simple $\sigma$ above an orbit of two inequivalent types $\eta$ and
$\gamma(\eta)$, with both restriction multiplicities one. Comparing
the coefficient of the character of $\eta$ in the restricted trace
gives $s_\sigma^{-1}=s_\eta^{-1}$. For an invariant $\eta$ there are
two extensions, each of restriction multiplicity one. Their reciprocal
Schur elements sum to $s_\eta^{-1}$. They are equal because the
quotient sign interchanges the extensions and preserves the trace.
Thus $s_\sigma=2s_\eta$ in this case. The algebras agree for $k=0$,
so in every case the finite spectral degree is preserved.

Choose $\sigma$ to be an attaining constituent of $\pi$ on
$F_k$. At least one constituent of its restriction to $F_k\cap W$
occurs in the restriction of $\rho$ or of $\gamma(\rho)$. In
the latter case conjugate this constituent and the intersection
by $\gamma$. This gives a constituent on a standard finite
parabolic of $H$ in the restriction of the prescribed $\rho$.
Its spectral degree is $A_R$ in either case. For integral $a,b$,
this degree is integral by the type $B$ symbol formula for the
original constituent on $F_k$, with endpoint weights $0,b$.
\end{proof}

The $\widetilde B_n$ application of
Proposition~\ref{prop:leading} uses the induction tori and
intertwiners for the reduced datum $R_0=C_n$, $X=Q(C_n)$.
The stabilizer commutant is described by
\cite[Theorem~5.5]{DelormeOpdam}, and \cite[Proposition~6.8]{DelormeOpdam}
gives the groupoid comparison for this datum. The projected
semisimple lattice can be seen by subtracting each type-$A$
block's mean from its coordinates and retaining the tail
coordinates. Applied to $\mathbb Z^n$ this gives the product
of the type-$A$ weight lattices and the weight lattice
$\mathbb Z^\ell$ of the type-$C_\ell$ tail, where $\ell$ is
the tail size. Lifts to $\mathbb Z^n$ of a point of this product
differ by adding an integer constant to each type-$A$ block.
If a block size is odd, one such shift changes the total parity
and every point has a lift to $Q(C_n)$. If all block sizes are
even, parity is well defined on the product lattice and the
projected lattice from $Q(C_n)$ is its index-two kernel.
When there is at least one type-$A$ block, the order-two
character of this product lattice trivial on the kernel is
nontrivial on an even-sized block. On that block it is the
nontrivial quadratic twist of the Steinberg torsion label,
which acts freely on the set of such labels. The resulting
groupoid comparison is precisely that of
\cite[Proposition~6.8]{DelormeOpdam}. For the full Levi there
are no type-$A$ blocks, the unramified torus is a point, and
the stabilizer description is the discrete-series block itself.
In every case the compact model and the generalized Bernstein
weights use the actual projected lattice and the unramified
quotient of $Q(C_n)$. The normalized
relative density on each torus has a Laurent polynomial limit
as used in Proposition~\ref{prop:leading}. In fact its root
factors, after restricting to the torus, have the form
$1-\zeta v^d z^\nu$ and their conjugates, where $z^\nu$ is a
torus character, $d$ is a real exponent and $|\zeta|=1$.
Nonstationary factors have uniform normalized limits. A stationary
denominator is a product of primitive Laurent binomials. Real
analyticity of the exact density on the compact torus cancels
each such factor because the compact points of every component
of its divisor are Zariski dense
\cite[Theorem~3.25 and Proposition~3.27]{Opdam}. After cancellation
the normalized density converges uniformly. The proof of
Proposition~\ref{prop:leading}, using the finite Bernstein
expansion and the actual stabilizer action, now gives the full
equivariant Laurent polynomial range for $H$. In applying its
central cutoff construction one chooses the integral multiples
of the dual real exponents in $Q(C_n)$. This is a full-rank
lattice, so the choice is possible. Proposition~\ref{ab:facet}
gives a same-degree finite source in every unitary stabilizer
block. Hence the ordinary ideal generation and the canonical
filtration in Proposition~\ref{prop:kl-filtration} apply to $H$.

The trace used for the signed-unit argument can be computed by
the Plancherel formula of $H^0$. To see its compatibility with
the leading product of $H$, restrict all the compact families
of $H^0$ to $H$. The spectral level of every constituent is
the level of the family by Proposition~\ref{ab:facet}.
For a compact family $\pi^0_{\alpha,v,z}$ of $H^0$ write
$\mu^0_{\alpha,v}$ for its Plancherel measure on the family
torus. Lemma~\ref{ab:crossed} gives the exact identity
\[
 \tau(h)=\sum_\alpha\int
     \operatorname{Tr}(\pi^0_{\alpha,v,z}(h))\,
     d\mu^0_{\alpha,v}(z)\qquad (h\in H).
\]
After normalizing the matrices by $v^{-A_R}$ and the measures
by $v^{2A_R}$, its two-factor limit gives an isometry on the
orthonormal basis indexed by $W$. The integral spectral projections in Proposition~\ref{prop:leading}
act on each restricted fiber by the corresponding scalar cutoff.
Indeed the central Bernstein elements act on each fiber by its central
character on $H$, and the polynomial iteration defining the projections
then selects the real exponent orbit. We now identify the product under
the isometry. For fixed
$x,y\in W$ the leading product column is the constant term of
the normalized multiplication of $N_x$ on $N_y$ from
Proposition~\ref{prop:leading}. The three-factor Plancherel
formula identifies its image with the orthogonal projection of
the pointwise product onto the closed image of the isometry.
The norm of the column converges in the standard basis Hilbert
space by the operator-norm convergence of that normalized
multiplication. The four-factor Plancherel formula identifies
the limit of the squared norms with the squared norm of the
pointwise product itself. Equality of these norms places the
product in the image. For the unit, apply \(P_+(v)\) from
Proposition~\ref{prop:leading} to \(N_1\). In the restricted families
this gives \(v^{A_R}\) times the identity matrix. Its squared norm
tends to the sum over the families of the integrals of their
dimensions against the limiting measures. Pairing with each \(N_w\)
shows that the image of its constant coefficient is the orthogonal
projection of the family identities onto the closed image of the
isometry. Since the squared norms agree, this projection is the
family identity itself.