\section{The original affine Kazhdan--Lusztig filtration}\label{sec:affine-kl}

For a finite standard parabolic let \(s_\sigma(v)\) be the Schur element of its simple character \(\sigma\) for the standard trace. Thus the coefficient of that character in the trace is \(s_\sigma(v)^{-1}\). Its positive-canonical spectral degree is \(A(\sigma)=\frac12\deg_v s_\sigma(v)\). The finite spectral description in Lemma~\ref{lem:finite-spectral} identifies each finite \(a\)-level with the corresponding sum of generic matrix blocks. On a block of degree \(A(\sigma)\), convergence of its normalized matrices to zero for an element and its adjoint implies convergence to zero in any unitary model. Indeed the squared Hilbert--Schmidt norm there is the character of the product with its adjoint, computed also as the trace of the product of the two normalized balanced matrices. The required finite parabolic theorems follow from Theorem~\ref{thm:finite-b} and the equal-parameter theory \cite[\S15]{Lusztig}. Indeed the factors in \(\widetilde C_n\) are of types \(A\) and \(B\), and those in \(\widetilde B_n\) can also be of type \(D\), with equal parameters on each \(D\) factor. The product argument is Lemma~\ref{fin:products}.

Let \(\pi_{\alpha,v,z}\) have degree \(A=A_R\) and fix a finite standard parabolic \(W_I\). Every constituent \(\sigma\) of its restriction satisfies \(A(\sigma)\leq A\). To see this, finite Schur orthogonality shows that in an orthonormal basis
\[
 \sum_{w\in W_I}\|\sigma(N_w)\|_{\rm HS}^2=(\dim\sigma)^2 s_\sigma(v),
\]
where \(\|\cdot\|_{\rm HS}\) is the Hilbert--Schmidt norm. The compact matrix bound for each of the finitely many \(N_w\) gives the inequality. This applies at every unitary point and to every constituent of the induced representation because the representation is unitary and the orthogonal projections onto its constituents have norm one. The finite-facet attainment results show that every prescribed irreducible tempered component has such a finite constituent with \(A(\sigma)=A\). In particular all spectral degrees are integers. We use these witnesses to generate the whole degree-\(A\) polynomial algebra by finite cell units.

\begin{lemma}\label{lem:fullness}
In the degree-\(A\) summand of \(J_{\mathbb C}=\mathbb C\otimes_{\mathbb Z}J\), the ordinary two-sided ideals generated by the images of the asymptotic unit elements of all finite source cells of intrinsic degree \(A\) sum to the whole summand.
\end{lemma}

\begin{proof}
For a finite source cell \(\mathfrak c\) of degree \(A\) and a family of degree \(A\), its lower boundary has intrinsic degree \(>A\), by finite P4 and P11, and hence acts by zero. Its normalized leading action defines a homomorphism from its finite asymptotic algebra to \(\mathcal B_\alpha\). On restriction to the finite parabolic, a simple \(\sigma\) with \(A(\sigma)<A\) has zero leading action at degree \(A\). The finite asymptotic algebra of degree \(A\) is the sum of its two-sided cell ideals by finite P8. Each of its simple matrix blocks belongs to exactly one of those ideals, so the unit of a finite cell acts as the identity on that cell's blocks and as zero on the others. On a constituent of degree \(A\), the normalized lift of that unit (the same linear combination of \(v^{-A}C_w\)) converges to the indicated operator in norm. To see this, subtract the indicated scalar operator and apply the character formula for the squared norm to the difference. The lift is self-adjoint by finite P6, and its balanced limit is given by Lemma~\ref{lem:finite-spectral}. The unit image \(E_{\mathfrak c}(z)\) is therefore a Laurent polynomial idempotent. Its action is nonzero on an irreducible tempered component containing an attaining \(\sigma\).

At a unitary point \(z\), let \(G_z\) be the subgroup of \(G_\alpha\) fixing \(z\). The exact and limiting intertwiners determine continuous projective representations of \(G_z\) for large \(v\) including the limit. Divide each matrix by a continuous \(d_\alpha\)-th root of its determinant, with the roots chosen continuously up to the limit. The resulting cocycle has values in the finite group of \(d_\alpha\)-th roots of unity, by taking determinants in the projective multiplication law. It is constant. The normalized matrices thus represent a fixed finite central extension of \(G_z\), with isotypic projectors of constant ranks. The exact stabilizer theorem identifies its active blocks with the irreducible tempered components. The exact isotypic projectors commute with the finite parabolic central idempotents, since the intertwiners at \(z\) commute with the Hecke action. These central idempotents are rational in the parameter and continuous on the interval of large \(v\). The ranks of their products with the exact isotypic projectors are constant on this interval. A nonzero product means that the associated tempered constituent contains the corresponding finite simple. Choose an attaining simple for that constituent and use its source cell. The rank of the source unit on that isotypic space is positive in the limit: the normalized lift differs in norm by \(o(1)\) from the sum of the finite central idempotents for the simple blocks of the source cell, by the preceding norm calculation. The isotypic projector itself converges to its limiting projector. Their limiting product has positive rank, since the exact orthogonal projection products have positive constant rank. This proves coverage of every simple block of \(\operatorname{Mat}_{d_\alpha}(\mathbb C)^{G_z}\) on the unitary torus.

We extend this statement to all complex points of \(T_\alpha\). For a subgroup \(K\leq G_\alpha\), each connected component of its fixed locus is a translate of a subtorus with dense unitary points. In fact write a point as \(z_0\exp(x)\) with \(z_0\) unitary and \(x\) a real logarithmic vector. The affine monomial fixed equations separate into equations for \(z_0\) and linear equations for \(x\). Thus \(z_0\exp(tx)\) stays fixed for \(0\leq t\leq1\), and the unitary points are Zariski dense in the component. If a point has exact stabilizer \(K\), removing the other fixed loci leaves a nonempty Zariski-open set in its component and still leaves dense unitary points.

On such a component adjoin \(d_\alpha\)-th roots of the invertible Laurent functions \(\det U_g(z)\), \(g\in K\). This gives a finite torus cover, possibly disconnected. Each component has dense unitary points since the root equations are monomial and the determinant values at unitary points have modulus one. On a connected component of the cover the determinant-normalized cocycle is constant. Each finite-group isotypic projector is regular there. Its product with \(E_{\mathfrak c}\) is again an idempotent because the source image commutes with the stabilizer. Its trace is an integer-valued regular function and hence constant. A source hitting a given block at a unitary point of exact stabilizer \(K\) thus hits the same block throughout the component of the cover. This proves the required coverage at every complex point.

For each family, write \(R_\alpha=\mathbb C[T_\alpha]\) and \(Z_\alpha=R_\alpha^{G_\alpha}\). Evaluation at \(z\) maps \(\mathcal B_\alpha\) onto \(\operatorname{Mat}_{d_\alpha}(\mathbb C)^{G_z}\), by interpolation on the finite orbit of \(z\) followed by averaging. The algebra \(\mathcal B_\alpha\) is finite over \(Z_\alpha\). For the maximal ideal \(\mathfrak m\) of \(Z_\alpha\) corresponding to the orbit, averaging commutes with reduction modulo \(\mathfrak m\): the average of \(\sum_i a_i M_i\), with \(a_i\in\mathfrak m\), is \(\sum_i a_i\) times the respective averages of \(M_i\). The fiber is therefore the invariant matrix algebra over \(R_\alpha/\mathfrak mR_\alpha\). This ring is Artinian and has the orbit as its support, since invariant functions separate finite orbits by interpolation and averaging. Its quotient by its nilradical is the product of the residue fields on the orbit. Evaluation on those residue fields is equivariant. Its kernel in the fiber is nilpotent, since a product of sufficiently many matrices with entries in the nilradical is zero. Every simple fiber module therefore factors through the evaluated stabilizer algebra. If the ideal generated by the source idempotents were proper, the quotient would be a nonzero finite \(Z_\alpha\)-module. Localization at a maximal ideal in its support and Nakayama's lemma give a nonzero quotient fiber. That finite-dimensional algebra has a simple module, on which all source idempotents vanish. This contradicts the coverage. The same conclusion holds for the finite direct sum of all the families at degree \(A\), since its central family identities belong to that algebra.
\end{proof}

\begin{proposition}\label{prop:kl-filtration}
For each spectral degree \(A\), the span of \(C_w\) with \(w\in\bigcup_{B\geq A}S_B\) is an original two-sided ideal. Every family of degree below \(A\) annihilates this ideal for sufficiently large real \(v\). On the quotient of consecutive ideals, the left and right actions commute with independent parameters. Moreover \(a(w)=A\) for \(w\in S_A\), and the actual degree-\(A\) product of original basis vectors is the product of the corresponding leading algebra of Proposition~\ref{prop:leading}.
\end{proposition}

\begin{proof}
Descend through the finite list of degrees. Assume the ideal for levels \(>A\) has been constructed, with exact annihilation by all families of degrees \(\leq A\). At the largest degree this ideal is zero. In the quotient, take the positive form given by the truncated Plancherel integral over the remaining families. On any fixed finite set of original surviving \(C_w\) its Gram matrix tends to the identity, because the leading Fourier image of each such vector has support in the remaining levels. This uses the degree-zero triangular transition between leading standard and canonical vectors.

For fixed \(x,y\), on a family of degree \(B\leq A\) the product matrix has norm \(O(v^{2B})\) uniformly on the compact torus. Its squared norm integrated against the family measure is \(O(v^{4B}v^{-2B})=O(v^{2B})\). Summing gives \(O(v^{2A})\) for the squared norm of the quotient product. The product has finite original support. The Gram limit on that support forces every coefficient of this product in the quotient to have degree at most \(A\). For a surviving index \(z\), the coefficient at degree \(A\) is
\[
 [v^A]h_{x,y,z}
   =\sum_{\alpha:\,A_R=A}\int_{T_{\alpha,u}}
       {\rm Tr}\bigl(c_{\alpha,x}(\xi)c_{\alpha,y}(\xi)c_{\alpha,z}(\xi)^*\bigr)\,
                      m_{\alpha,0}(\xi)\,d\xi .
\]
Indeed one divides the original finite-coordinate expansion by \(v^A\) before taking the limit, so the Gram off-diagonal terms vanish. In the trace pairing of the product with \(C_z\) on a family of degree \(B\), the three matrices contribute at most \(v^{3B}\), the measure contributes \(v^{-2B}\), and the division by \(v^A\) leaves a factor \(v^{B-A}\). Thus the families of smaller degree have zero limit. By the coordinate cutoffs the right side can be nonzero only when \(x,y,z\in S_A\).

For \(p\) in any finite standard parabolic, its leading family matrix can be nonzero only at degree \(a_I(p)\), its intrinsic finite \(a\)-value. In the finite orthogonal decomposition of a restriction, only a constituent with \(A(\sigma)=A_R\) can contribute at degree \(A_R\). In such a constituent the finite leading matrices on \(N_p\), equivalently \(C_p\), vanish unless its degree equals \(a_I(p)\), by the finite spectral description. The corresponding leading matrix for the adjoint also vanishes, since \(a_I(p^{-1})=a_I(p)\) by finite P14. The squared-norm character formula above gives the vanishing in the compact unitary model. The total Fourier norm of \(p\) is one, so it belongs to \(S_{a_I(p)}\).

For a finite source cell of intrinsic degree \(A\), its strictly lower source cells annihilate all remaining families, by their intrinsic degree and the finite restriction bound. Their original indices therefore lie in the higher ideal. The source cell survives in the quotient, by the preceding paragraph. Lemma~\ref{lem:double-induction} gives a constant source image \(V_I\) with commuting independent actions. It annihilates the smaller families exactly. In fact the entire source cell acts by zero there, since the restriction contains only finite characters of smaller degree. A constant vector in the remaining quotient that annihilates the smaller families has no coordinates in their levels, by its leading Fourier image. It follows that \(V_I\) is contained in the constant span of \(S_A\).

The source image contains the finite cell unit and is stable under both original actions by Lemma~\ref{lem:double-induction}. Its span is constant in the parameter, so each Laurent coefficient of either action preserves it. In particular the coefficient formula shows stability under multiplication on both sides by the degree-\(A\) leading algebra. Lemma~\ref{lem:fullness} shows that the sum of these images after complexification is the whole span of \(S_A\). All columns defining these images have integer coordinates. Thus over \(\mathbb Q\) their sum is already the coordinate space, as follows by extending the quotient by this sum to \(\mathbb C\). The rational invariance implies integral coordinate invariance because the original Hecke action has integral Laurent coefficients. Independent-parameter commutation holds on the entire layer, since it holds on each summand of a spanning sum. Exact lower annihilation passes to each original \(C_w\) in that layer. This proves the induction.

For \(w\in S_A\), the quotient bound gives \(a(w)\leq A\). The degree-\(A\) coefficient formula and multiplication by the unit of that leading level show that some coefficient with output \(w\) attains degree \(A\). Thus \(a(w)=A\) and the asserted product follows.
\end{proof}