\section{Finite type \texorpdfstring{$B$}{B}}\label{fin:section}

Write \(B_n\) as the group of signed permutations of \(1,\ldots,n\).
Use the simple reflection \(s_0\) that changes the sign of the first
position and the simple reflections \(s_i\) that interchange positions
\(i\) and \(i+1\), for \(1\leq i<n\). The weights are \(b=L(s_0)\)
and \(a=L(s_i)\) for \(i>0\). The trivial group is denoted by \(B_0\).

\begin{theorem}\label{thm:finite-b}
For every \(n\) and every pair of positive integers \(a,b\), the weighted
Coxeter system \(B_n\) satisfies \textup{P1--P15}.
\end{theorem}

For a finite Hecke algebra use its trace \(\tau(N_w)=\delta_{1,w}\).
The irreducible representations are taken over the fraction field of
\(\mathbb Z[v,v^{-1}]\), extended to a splitting field if necessary.
The Schur element \(s_E(v)\) of an irreducible representation \(E\) is
normalized by
\[
 \tau(h)=\sum_E s_E(v)^{-1}\operatorname{tr}_E(h).
\]
Here \(\operatorname{tr}_E\) is the ordinary matrix trace. Write
\(A(E)=\frac12\deg_v s_E(v)\). The automorphism \(\dagger\) is given by
\(N_s\mapsto -N_s^{-1}\) for simple \(s\). On generic labels it is the
sign twist, written \(E\otimes\operatorname{sgn}\). Write \(a_{\mathrm{Sch}}(E)=-\frac12\min\deg_v s_E(v)\)
for the ordinary Schur valuation, where \(\min\deg_v\) means the
least exponent with nonzero coefficient. The
semilinear automorphism \(v\mapsto v^{-1}\), \(N_s\mapsto -N_s\)
preserves the trace and acts on the labels by the sign twist.
Consequently the ordinary Schur valuation of
\(E\otimes\operatorname{sgn}\) is \(A(E)\).

We use three assertions in the induction on \(n\):
\begin{enumerate}
 \item \textup{P1--P15} for all positive integral \(a,b\).
 \item if \(r<b/a<r+1\) with \(r\geq0\) integral, the left and right cells
 are the recording and insertion fibres of rank-\(r\) domino insertion,
 respectively, with the irreducible characters given by their
 two-quotient labels. The two-sided cells are the common-shape sets.
 Left cells with the same character have signed based isomorphic
 cell modules, with the insertion labels preserved.
 \item in the same open ranges, the original left cell module of
 character \(E\), completed at any positive logarithmic lift of a
 root of unity, is the image under the sign-twisted KZ functor of the
 standard Cherednik module labelled by \(E\otimes\operatorname{sgn}\).
\end{enumerate}
Here a signed based isomorphism sends each given basis vector to a
signed basis vector. The notions in the last assertion are specified
in Section~\ref{fin:kzsection}. The first assertion for finite type
\(A\), and its irreducible cell theory, are classical
\cite[Chapter~15]{Lusztig}. We prove its local KZ comparison as well.

\input{finite-cells}
\input{finite-kz}
\input{finite-propagation}

\subsection{Induction on the rank}

\begin{proof}[Proof of Theorem~\ref{thm:finite-b}]
In rank zero the algebra is \(\mathcal A\). In rank one the
cell lattices and the KZ standards are the two one-dimensional
modules, with labels exchanged by \(\dagger\). In rank two the
finite dihedral calculation gives \textup{P1--P15}
\cite[Proposition~5.1]{GeckFinite}. For the off-wall rank-two cell
classification, in addition to the singleton cells
\(\{1\},\{w_0\},\{s\},\{w_0s\}\), where \(s\) is the lighter of
the two simple reflections and \(w_0\) is the longest element, the left
cells are the two sets
\(\{t,st\}\) and \(\{ts,sts\}\), where \(t\) is the heavier reflection.
This follows from the generator multiplication formula in the
dihedral calculation. Each two-element set affords the unique
two-dimensional representation. Rank-\(r\) domino insertion on the
eight signed permutations gives these recording fibres: if
\(b<a\) their two-element sets are
\(\{(-2,1),(-2,-1)\}\) and \(\{(2,1),(2,-1)\}\).
If \(a<b\) they are \(\{(-1,2),(-2,1)\}\) and
\(\{(2,-1),(1,-2)\}\).
The singleton characters are determined by their left descent
sets, since \(N_s\) acts by \(v^{L(s)}\) for a descent \(s\) and
by \(-v^{-L(s)}\) otherwise. Inversion gives the right cells.
The based isomorphisms and the local
recognition in rank two follow from the proper rank-one sources as in
Lemmas~\ref{fin:based} and~\ref{fin:recognition}.
For higher ranks, assume all three assertions in smaller ranks.
Lemma~\ref{fin:domino} first identifies the off-wall actual cells
and Lemma~\ref{fin:based} gives their based isomorphisms. The third
assertion follows from Lemma~\ref{fin:recognition}.
Lemma~\ref{fin:e1} gives the spectral threshold ideals and
Lemma~\ref{fin:e2} separates the two-sided cells in each exact layer.
Proposition~\ref{fin:close} now gives \textup{P1--P15} off the walls.
By P4, P9 and P10, a two-sided path between elements of one two-sided
cell stays at a common value of \(a\), and each edge preserves
the shape. Conversely the recording and insertion fibres of a fixed
shape connect all its pairs of tableaux. This proves the two-sided
claim in the second induction assertion.

For an integral ratio \(b/a\), use the off-wall result just proved in
the same rank in Lemma~\ref{fin:wall}. This supplies the spectral
threshold ideals on the wall. Lemma~\ref{fin:e2} and
Proposition~\ref{fin:close} complete \textup{P1--P15} there, using
the wall statements in proper ranks. This proves the induction
for all positive weights.
\end{proof}

Every proper standard parabolic of either affine group in the paper
is a product of finite groups of type \(B\) and equal-parameter
finite Weyl groups. The latter include the finite factors of type
\(D\) at the affine \(B\) fork. The following observation gives the
finite results for all those products.

\begin{lemma}\label{fin:products}
A finite direct product of weighted Coxeter systems satisfying
\textup{P1--P15} also satisfies \textup{P1--P15}.
\end{lemma}

\begin{proof}
The canonical basis is the tensor product basis, by bar invariance
and triangularity. Its structure coefficients are products and its
\(a\)- and \(\Delta\)-values are sums of the factor values.
The \(a\)-formula follows by independently choosing a
maximizing product in each factor. P1 therefore identifies
distinguished tuples with tuples of distinguished elements, and the
signs and leading coefficients are products. Left, right and
two-sided preorders are product preorders, since a generator acts in
one tensor factor. This proves P1--P8 and P12--P14 componentwise.
For P9--P11 use P4 to see that equal sums along a preorder relation
give equality of every factor value. On an exact
\(a\)-layer a projected generator changes at most one factor
and preserves its value. The layer thus splits by the tuple of factor
values. On each summand the independent left and right actions
commute by the factor P15 when they act in the same position, and by
tensor factorization otherwise.
\end{proof}

\begin{lemma}\label{lem:finite-spectral}
Let \(W_I\) be a finite standard parabolic subgroup of either affine
group. For every irreducible \(E\) choose a finite balanced model
\(\rho_E\) for the convention at infinity, and put
\[
 b_E(w)=\lim_{v\to+\infty}v^{-A(E)}\rho_E(N_w)
 \qquad(w\in W_I).
\]
The limit is the same with \(C_w\) in place of \(N_w\), and
\(b_E(w)\ne0\) implies \(a_I(w)=A(E)\). For each \(d\) the
span of the \(C_w\) with \(a_I(w)\geq d\), over the fraction
field, is the sum of the generic simple matrix blocks with
\(A(E)\geq d\). On the exact \(d\)-layer the matrices \(b_E(w)\)
realize the finite asymptotic algebra, with its unit acting as the
identity on each block with \(A(E)=d\).
If \(W_J\subseteq W_I\) and \(F\) is a constituent of the restriction
of \(E\) to the smaller Hecke algebra, then \(A(F)\leq A(E)\).
\end{lemma}

\begin{proof}
Use the finite balanced models of
\cite[Proposition~2.3]{GeckLeading}, with the semilinear sign change
described above. The lower coefficients of \(C_w\) have strictly
negative degree, so the two matrix limits agree. From
\cite[Lemma~3.2 and Remark~4.2]{GeckFinite}, P1--P15 imply that the
constituents of a left cell have spectral degree equal to its
\(a_I\)-value, and that \(b_E(w)\ne0\) implies
\(a_I(w)=A(E)\).
P4 makes each threshold span a two-sided ideal. Its left-cell
filtration, and that of its quotient, therefore identify its generic
blocks. The identification of the leading matrix algebra with the
\(\gamma\)-product is \cite[Proposition~3.6]{GeckFinite}.
The unit is sent to the identity by the block realizations of
\cite[Propositions~3.4--3.6]{GeckLeading}.

For the restriction statement let \(m_{F,E'}\) be the multiplicity of
\(F\) in the restriction of \(E'\). The standard traces of the
two parabolic algebras satisfy \(\tau_I|_{H_J}=\tau_J\), so
\[
 s_F(v)^{-1}=\sum_{E'} m_{F,E'}s_{E'}(v)^{-1}.
\]
At a large positive real \(v\), the central primitive idempotents
are self-adjoint for the positive standard trace form. Indeed the
adjoint of left multiplication by such an idempotent is left
multiplication by its conjugate under \(*\), and positivity forbids
two different central idempotents from being adjoints.
The trace identity at each idempotent gives \(s_{E'}(v)>0\).
Thus the terms of highest degree on the right have positive
leading coefficients, and
\(A(F)=\min_{m_{F,E'}>0}A(E')\).
\end{proof}