\section{Finite parabolic constituents in type \texorpdfstring{\(\widetilde C_n\)}{C-tilde}}
\label{fac:attainment}

Fix \(n\geq 2\). Use the \(\widetilde C_n\) realization \(W=\mathbb Z^n\rtimes W_0\), with \(W_0\) the group of signed permutations. The middle weight is \(a>0\), the finite endpoint weight is \(b>0\), and the affine endpoint weight is \(c\geq0\). For \(0\leq k\leq n\), the finite standard subgroup \(W_k\) obtained by deleting node \(k\), with nodes numbered \(0,\ldots,n\) from the affine endpoint, has factors of types \(B_k\) and \(B_{n-k}\) with endpoint weights \(c,b\), respectively. A factor of rank zero is trivial.

For a simple representation \(\sigma\) of a finite parabolic Hecke algebra, its Schur element \(s_\sigma\) is defined by the trace expansion
\[
 \tau(h)=\sum_\sigma s_\sigma^{-1}\operatorname{Tr}(\sigma(h)),
\]
where \(\tau\) is the coefficient-of-identity trace of that algebra and \(\operatorname{Tr}\) is the ordinary matrix trace. Its finite spectral degree is half the largest exponent of \(v\) in \(s_\sigma\). For real weights we use the same definition with real exponents in the finite Schur formula.

\begin{proposition}\label{prop:facets}
Let \(R\) be a real central exponent vector of an irreducible tempered representation of the \(\widetilde C_n\) Hecke algebra with weights \(a,b,c\) as above. Its restriction to the finite parabolic Hecke algebra of some \(W_k\) contains a simple type of finite spectral degree \(A_R\), the spectral level in \eqref{eq:root-degree}.
\end{proposition}

For the rest of this section use \(v^a\) as the base of exponents, so that the middle weight is one. Divide \(b,c\) and the coordinates of \(R\) by \(a\) and retain their notation. To recover degrees in \(v\), multiply both sides of the degree equality by \(a\).

\input{fac-combinatorics}

\subsection{Tempered types and their finite restrictions}
\label{fac:characters}

Write the central torus character as its unitary phase times its positive real part, and choose a representative \(z=(z_1,\ldots,z_n)\) for the unitary phase up to signed permutation. Coordinates with phase \(+1\) and with phase \(-1\) give two signed-permutation factors of the stabilizer \((W_0)_z\). Within each other group of phases equivalent under inversion, choose the coordinates all to have the same phase. The stabilizer on that group is the symmetric group permuting its coordinates. These factors form a direct product, and Lusztig's reduction associates a real-central graded tempered factor to each of them \cite[Corollary~1.12]{CK}\cite[Section~2.2.1]{CiubotaruOpdam}. In the normalized units the unnormalized middle label is \(q=v^{2a}\), and the normalized endpoint parameters of \cite[Remark~1.3 and Theorem~1.10]{CK} are \(q^{b/2}\) and \(q^{c/2}\). Thus the graded short parameters at phases \(+1,-1\) are \(m,d\) of \eqref{fac:centers}, with the other graded parameter equal to one. The type \(A\) factors at the other phases have zero real exponent in their free central direction, by the tempered inequalities. Multiplying each short reflection by \(-1\), leaving the polynomial algebra fixed, changes the sign of the short graded parameter and interchanges the two finite colors.

For the symmetric-group factors, real-central tempered type \(A\) induction uses Steinberg factors \cite[Theorem 4.5]{CK}. A Steinberg factor of size \(k\) has the exponents \(R_k\) of Section~\ref{fac:degree} and finite character the sign of \(\mathfrak S_k\). If several such factors share a unitary phase, their joint finite character is the induction of their sign characters. The same strings occur in the real-central tempered signed-permutation factors. Away from half-integer short parameter, those factors are the irreducible inductions of one discrete-series factor and a list of such strings, as in \cite[Theorem 4.7 and Corollary 4.23]{CK}. The discrete-series exponents for short parameter \(u\) are the box exponents \(u+j-i\) of a partition, up to signed permutation. This convention also applies to \(u<0\), by transposing the partition and changing the signs of the coordinates.

We describe the canonical finite type in the full-sign-twisted convention of Section~\ref{fac:degree}. For a signed-permutation factor with \(2u\notin\mathbb Z\), delete a first row or first column of the discrete-series partition at a largest absolute box exponent, or delete a balanced string of largest radius, comparing these radii in a single list. Record a deleted row in the upper partition and a deleted column in the lower partition, changing \(u\) by \(-1\) or \(+1\), respectively. A string of size \(k\) adds upper and lower rows of lengths \(C(k,u)\) and \(k-C(k,u)\), leaving \(u\) fixed. This gives the canonical label in the chamber, which occurs in the factor.

For \(u>0\) with \(4u\) odd, the real-central tempered representations and their canonical finite types are described by the Spin generalized Springer correspondence \cite[Theorems~4.12 and~4.15 and Proposition~4.19]{CK}. For a real-central tempered module the central character determines the semisimple middle element of the nilpotent's \(\mathfrak{sl}_2\)-triple \cite[Theorem~1.21]{LusztigCuspidalIII}. In the cuspidal Levi of \cite[Section~2.13(e)]{LusztigCuspidalI}, each relative coordinate acts on a two-dimensional block and its dual. In the coordinate scale of \cite[Section~4.4]{CK}, a real exponent \(r\) contributes scalar weights \(4r,-4r\) on these two spaces, in addition to their regular \(\mathfrak{sl}_2\) weights \(1,-1\). Thus the four weights contributed by \(r\) are \(\pm(4r+1),\pm(4r-1)\), in addition to the fixed cuspidal part of the ambient space. For the discrete-series box exponents this gives the odd-part construction preceding \cite[Algorithm~4.17]{CK}. For a string \(R_k\), the weights \(4r+1,4r-1\) form the sequence \(2k-1,2k-3,\ldots,1-2k\). Their negatives give the same sequence, so the string contributes two \(\mathfrak{sl}_2\) summands of dimension \(2k\). Consequently its Jordan contribution consists of two blocks of length \(2k\).

The odd-part branch of \cite[Algorithm~4.14]{CK} is the row and column rule by Claim~E there. This computation applies with even pairs present because the odd branch uses only the deleted odd part and the defect, and every even pair contributes zero to the defect. The even-pair branch at size \(2k\), with \(u=h+1/4\) or \(u=h+3/4\) for an integer \(h\geq0\), has upper part \(\lfloor k/2\rfloor-h\) or \(\lceil k/2\rceil-h-1\), respectively. Indeed the defects in the two cases are \(-h\) and \(h+1\). These are the two nearest-integer formulas for \(k/2-u\). The Jordan part \(2k\) is ordered by its radius \((k-1)/2\), since \(2k=4((k-1)/2)+2\). The largest remaining cuspidal odd part, when it exists, is \(4u-2\), whereas a nonempty discrete-series partition gives an odd part of size at least \(4u+2\). Thus when a cuspidal part is deleted, the discrete-series remainder is empty. If a string remains, then \(2k<4u-2\) and \(u>1\). Substituting \(4u-2\) and the defect in Algorithm~4.14, the odd branch records a zero and changes \(u\) to \(u-1>(k-1)/2\). Its upper length remains zero in the clipped rule. After a sign change of \(u\) the corresponding statement uses the color exchange. This proves the rule at the geometric quarter point. The full finite characters and their canonical labels are constant on each open half-integer chamber by \cite[Corollary 4.20]{CK}. The deletion rule is constant in the chamber. A tie between absolute box exponents with opposite signs requires \(2u\in\mathbb Z\), and a tie with a string radius requires \(u\) to be a half-integer. The row and column deletions shift \(u\) by integers, and the floors in \(C(k,u)\) also change only at half-integers. Negative \(u\) follows by the short-sign twist, transposition of the discrete-series partition, and exchange of the two finite colors.

Scale the numerical labels along their ray to one with the unitary phase fixed. The finite parabolic Hecke algebras remain semisimple, so the primitive central idempotents indexed by ordinary finite types have continuous coefficients. Their ranks on the scaling family are constant. In the limit, the character is induced from \(\mathbb Z^n\rtimes (W_0)_z\), with \(\mathbb Z^n\) acting by \(z\) and \((W_0)_z\) acting by the restriction of the graded factors \cite[Corollary~2.2]{CiubotaruOpdam}\cite[(4.10) for discrete series]{CKK}. Mackey restriction of this induced character therefore computes the finite parabolic multiplicities before scaling.

For the explicit phase factor in Mackey restriction, let \(p_k\) be the vector with first \(k\) entries \(1/2\) and the rest zero. The facet subgroup \(W_k\) consists of \((p_k-wp_k,w)\) for \(w\) in the two-block signed-permutation subgroup. A coordinate sign change in the first block contributes its translation coordinate one, whereas permutations in either block have translation part zero. Thus, on \(-1\) coordinates in the first block, the restriction twists by the character \((-1)^{\text{number of sign changes}}\). It exchanges the two finite colors in the left block. On \(+1\) coordinates and on permutations inside the other phase groups there is no such factor. In particular, each distribution of the coordinates of a phase among the two blocks supplies a Mackey summand with its actual restriction multiplicities. The full sign of a facet is the determinant of its projection to \(W_0\), so the simultaneous full-sign twist used above commutes with these restriction and induction operations.

When \(b,c,b+c,b-c\notin\mathbb Z\) in normalized units, apply Lemma~\ref{fac:extension} to the two discrete-series partitions and all strings. The subsequence at each phase \(+1,-1\) is its canonical deletion order, since deletions outside that phase preserve its short parameter and partition. In each phase color the two allocated partitions add rowwise to the canonical partition. For phase \(-1\) this uses the lower-upper and upper-lower sums in \eqref{fac:packet-box}, as the first block exchanges the colors. The Littlewood--Richardson coefficient of the rowwise sum is one, by the Cartan constituent. For a string in a phase other than \(+1,-1\), let \(h\) be the number of entries allocated to one block. After the full-sign twist its \(\mathfrak S_h\) constituent is trivial. Every bipartition of \(h\) with at most one row in each color restricts to a representation containing the trivial \(\mathfrak S_h\)-type: restriction gives the induction of the trivial types of the two Young subgroup factors. Frobenius reciprocity therefore gives each of the allocated two-row pieces in the corresponding \(B_h\) induction. Combine the pieces at each endpoint by taking the sorted union of their rows in each color, a Littlewood--Richardson constituent of multiplicity one by transposing the Cartan constituent. Condition \eqref{fac:ordering} already puts those rows in partition order. This gives the required constituent on the facet whose first block size is the sum of the left row lengths. Its spectral degree is \(A_R\) by the recursions of Section~\ref{fac:degree}.

\subsection{Specialization at critical short parameters}
\label{fac:critical}

For a real-central graded signed-permutation factor of short parameter \(u>0\) with \(2u\) integral, the finite characters of the irreducible tempered representations, after the common full-sign twist, form a nonnegative unitriangular matrix indexed by the finite types. This is the generalized Springer character matrix. The identification at ratio \(u\) is given by \cite[proof of Theorem 4.12]{Kato} and \cite[Theorems 1.17 and 1.21]{LusztigCuspidalIII}. In the Springer character matrix the same-orbit block is the identity, and the other types lie on strictly larger orbits in the closure order, by \cite[Theorems 3.1 and 3.5]{Kato}. The same statement for a negative short parameter follows by the short-sign twist.

At \(u=0\) in rank \(r\geq2\), the graded algebra is the equal-parameter type \(D_r\) graded algebra crossed with the order-two outer action. In the graded Bernstein presentation the short reflection now acts on the polynomial algebra by conjugation without a correction term, and conjugating the long reflection at the end of the chain gives the second fork reflection of \(D_r\). Here \(D_2=A_1\times A_1\). For the type \(D_r\) factor use the tempered classification in the ordinary Springer series, namely the series for the torus cuspidal datum \cite[Theorems~1.17 and~1.21]{LusztigCuspidalIII}. Its character matrix has the same unitriangularity by \cite[Theorems~3.1 and~3.5]{Kato}. The outer involution preserves the orbit closure order and acts on the canonical labels by transport of Springer fibers and their local systems. Clifford theory for the order-two group transports this matrix to a unitriangular matrix of type \(B_r\). A pair of exchanged type \(D_r\) modules induces to one simple module and their canonical types induce to one simple finite type, with multiplicity one. A fixed module has two extensions. Its canonical finite type has multiplicity one on the \(D_r\) subgroup, so its isotypic space is preserved by the outside involution and affords one extension of that type. Twisting by the quotient sign interchanges the two extensions of the module and also the two extensions of its canonical type. The other types in either case come from strictly larger orbits. This gives each type \(B_r\) label once, including the labels obtained from the very-even pair of type \(D_r\). In rank one the zero-parameter graded algebra is \(\mathbb C[x]\rtimes\{\pm1\}\), with the involution sending \(x\) to \(-x\). Real-central temperedness forces the real eigenvalue of \(x\) to be zero, since the two opposite eigenvalues would otherwise violate the tempered inequalities. At zero the two simple modules have the two one-dimensional finite labels. The rank-zero assertion is the one-dimensional trivial case.

Fix a critical short parameter \(u\) and one adjacent open chamber. Index the complete lists of irreducible real-central tempered modules in the chamber and at \(u\) by their canonical finite types. In the two matrices \(U,V\), an entry is the multiplicity of the column's finite type in the row's module. These matrices are nonnegative and unitriangular in their respective label orders.

For \(u\geq0\), realize the generic graded factors on the positive phase of affine algebras with both endpoint weights equal to the varying short parameter, taking the positive chamber when \(u=0\). Fix the numerical base greater than one. The normalized affine simple generators are self-adjoint in the tempered representations, and their two possible eigenvalues stay bounded as the weights converge. Thus, for each generic label, the generator matrices in orthonormal bases converge simultaneously along a subsequence. The defining relations persist in the limit. The Bernstein elements are products of standard generators and their inverses, which also converge because the generator eigenvalues stay bounded away from zero. The tempered inequalities on eigenvalue moduli of dominant translations persist, and the limiting central character is real by continuity of the eigenvalues. The limit is unitary and splits into real-central tempered simples at \(u\). Ranks of the finite primitive central idempotents are preserved. By the scaling identification, the same finite-character identity holds on the graded side. For \(u<0\), or the negative chamber adjacent to zero, apply the short-sign twist.

Let \(T\) record the multiplicities of the critical irreducibles in these limits, with rows indexed by generic modules and columns by critical modules. Then \(U=TV\). In the label order of \(U\), the nonnegative entries of \(T\) have triangular support because \(V\) has diagonal entries one. Both \(U\) and \(V\) have determinant one, as simultaneous permutations of their rows and columns preserve the determinant. Therefore \(T\) has determinant one and each of its nonnegative integral diagonal entries is one. This shows that from either adjacent chamber the generic factor with the prescribed critical label has a limit containing that critical factor. The generic factor's central character is scalar on every central Bernstein element. The continuity of those elements therefore gives the same limiting central character on all summands. The box exponents and balanced strings of the chosen generic label have the same limits along any approach in that chamber.

\begin{proof}[Proof of Proposition~\ref{prop:facets}]
For \(b,c>0\), perturb the two phase centers independently in the region \(m>|d|\). At a critical center choose the factor with the prescribed canonical label from an adjacent chamber. At a noncritical center keep its label in its chamber. Keep all other phase factors fixed, and take \(b,c,b+c,b-c\) nonintegral along the perturbation. This includes \(b=c\), where the negative phase uses the short-parameter-zero argument. At \(c=0\) the two centers coincide at \(b/2\). If this is critical, take \((b+\epsilon/2,\epsilon)\) with small \(\epsilon>0\), putting the positive phase in the upper adjacent chamber and the negative phase in the lower one. If it is noncritical, keep both centers in the same chamber. At the target \(c=0\) the affine endpoint label is one, so the scaling and finite restriction formulas still apply.

Pass to a subsequence on which the attaining facet type and the branching choices are fixed. The branching calculation uses the same canonical finite types of the prescribed target factors and the same type \(A\) strings, so it gives that facet type in the prescribed irreducible representation. Formula \eqref{fac:symbol} is continuous in the endpoint weights, and \eqref{fac:root-degree} is continuous in the weights and exponents. The limiting equality gives the required degree after multiplication by \(a\).
\end{proof}