\section{Relative canonical induction}\label{sec:double}

We use the following construction for both the finite induction and the affine filtration. The algebra of a two-sided cell is the quotient of its downward-closed span by the span of the strictly smaller cells. These are spans of the original \(C_w\). The finite cell algebra in the next lemma has an integral asymptotic algebra \(J_I\) on the same indices. Its product uses the coefficient of degree \(A\) in the cell multiplication, where \(A\) is the common intrinsic \(a\)-value. Its unit is denoted by \(e_I\).

\begin{lemma}\label{lem:double-induction}
Let \(Q\) be a quotient of \(H\) by a two-sided ideal spanned by original Kazhdan--Lusztig elements. Suppose every coefficient of the product of two surviving original basis vectors in \(Q\) has degree at most \(A\). Let \(W_I\) be a finite standard parabolic with \({\rm P1}\)--\({\rm P15}\), and let \(\mathfrak c_I\) be a two-sided cell of intrinsic degree \(A\). Suppose the strictly smaller source cells have zero image in \(Q\). Let \(V_I\) be the image of
\[
 H\otimes_{H_I}(\mathcal A\otimes_{\mathbb Z} J_I)
         \otimes_{H_I}H\longrightarrow Q,\qquad
 h\otimes t_p\otimes k\longmapsto h C_p k \quad(p\in\mathfrak c_I).
\]
Here the source bimodule in the tensor product is the ordinary cell subquotient, with \(t_p\) identified as a vector with its original basis element. Then \(V_I\) is spanned over \(\mathcal A\) by constant integral vectors in the surviving \(C\)-coordinates. Over \(\mathbb Q\) their constant span carries commuting left and right actions with independent parameters. It contains the image of the source cell, and is stable under the degree-\(A\) multiplication operators in \(Q\).
\end{lemma}

\begin{proof}
Write \(\star\) for the product of \(J_I\) and \(\mu_v\) for the ordinary product in the cell subquotient, both on its symbol space. The exact source degree layer is a direct sum of source cell subquotients by P4 and P11 in \(W_I\). On the summand for \(\mathfrak c_I\), take the coefficient of degree \(A\) in the right parameter of P15, with the right input in the source cell. This says that the left action of each \(C_w\) in \(H_I\) commutes with right \(\star\)-multiplication. The right version follows by inversion. A linear map commuting with right multiplication in a unital algebra is left multiplication by its value on the unit. In particular, the left action of \(h\in H_I\) is left \(\star\)-multiplication by \(h\cdot e_I\), and the right action is right \(\star\)-multiplication by \(e_I\cdot h\), where the dots denote the ordinary actions. For vectors \(p,r\in\mathcal A\otimes J_I\),
first apply this fact to \(\mu_v(p,-)\), then apply its right-handed
version to \(\mu_v(-,e_I)\). This gives
\[
 \mu_v(p,r)=p\star d(v)\star r,\qquad d(v)=\mu_v(e_I,e_I).
\]
The coefficients of \(d(v)\) are bar invariant. Its degree-\(A\) coefficient is \(e_I\). Consequently right multiplication by \(d(v)\) for \(\star\) has matrix \(M(v)\) with \([v^A]M(v)=\operatorname{Id}\) and no higher coefficients.

Let \(X\) and \(Y=X^{-1}\) be the minimal representatives of \(W/W_I\) and \(W_I\backslash W\), respectively. In left induction write the relative canonical lift of \(N_x\otimes t_p\) in the form
\[
 \sum_{x'\in X}N_{x'}\otimes(a_{x',x}(v)\star t_p).
\]
The coefficients \(a_{x',x}(v)\) belong to \(\mathcal A\otimes J_I\). To see this, the induced bar on the coset filtration has blocks obtained from the ordinary left \(H_I\) action. Each block commutes with constant right multiplication for \(\star\). The solution of the triangular bar equation with strictly negative off-diagonal coefficients commutes with those same constant maps. Indeed applying such a map to the bar equation preserves the diagonal prescription and the strict negativity. Their difference is zero by taking its maximal coset and comparing its negative coefficients with their bar transforms. This proves right \(J_I\)-linearity of each correction block, giving the displayed expression. The corresponding right correction blocks are right multiplications by elements \(b_{y,y'}(v)\) of \(\mathcal A\otimes J_I\). The sums are finite for each input because each relative Bruhat interval is finite. The same proof works for an infinite ambient Coxeter group.

Relative canonical induction identifies the multiplication images of these lifts with the images of the original \(C_{xp}\) and \(C_{py}\) \cite[Proposition~3.2, Corollary~3.5, Theorem~3.6]{GeckRelative}. The omitted source terms lie in the strictly smaller source cells, whose images in \(Q\) are zero. For a vector \(r\in\mathcal A\otimes J_I\), write \(C_r\)
for its linear expansion in the source \(C\)-coordinates. Use the
\(\mathcal A\)-linear extension of \(\star\) to multiply the correction
vectors and source basis vectors in \(\mathcal A\otimes J_I\).
For \(x\in X\), \(p\in\mathfrak c_I\), and \(y\in Y\), put
\[
 Z_{x,p,y}=\sum_{x',y'}N_{x'}\, C_{\,a_{x',x}(v)\star t_p\star b_{y,y'}(v)}\,N_{y'}
 \quad\text{in } Q .
\]
Write \(C_{e_I,y}\) for the
right relative canonical lift of \(e_I\) at \(y\) followed by
multiplication. It is the constant integral linear combination of
the \(C_{ry}\) given by the \(t_r\)-coefficients of \(e_I\).
The source multiplication formula and associativity of \(\star\) give
\[
 C_{xp}\,C_{e_I,y}=\sum_{r\in\mathfrak c_I}M_{r,p}(v)Z_{x,r,y}.
\]
All original coefficients on the left have degree at most \(A\). If a coefficient in the finite vector of \(Z_{x,r,y}\) at any fixed original coordinate had maximal degree \(j>0\), the leading identity of \(M(v)\) would give a nonzero coefficient of degree \(A+j\) on the right. Therefore all coordinates of all \(Z\) have degree at most zero. The left side and \(M(v)\) are bar invariant. The leading identity makes \(M(v)\) invertible over \(\mathbb Q(v)\), so the coordinates of the \(Z\) are bar invariant and hence constant integers.

Use the same bimodule construction over \(\mathbb Q[v^{\pm1},u^{\pm1}]\) with \(H(v)\) on the left and \(H(u)\) on the right. On the middle space the left action uses \(h\cdot e_I\) at parameter \(v\) and the right action uses \(e_I\cdot h\) at parameter \(u\). These commute by associativity of \(\star\). In the basis obtained by the two relative corrections, the left matrices still depend only on \(v\) and the right matrices only on \(u\), since the corrections on the opposite side commute with each action. At \(u=v\) the map to \(Q\) has the constant columns \(Z\). These columns give the image of the double-induced comparison, since the relative corrections on each side are triangular with identity diagonal. The kernel of a constant matrix over \(\mathbb Q\) extends by scalars to its Laurent kernel. Equivariance on the diagonal makes that kernel invariant coefficient by coefficient under each action, so it is invariant with the parameters independent as well. This proves the commutation on \(V_I\). The column \(Z_{1,p,1}\) is \(C_p\). Stability under the degree-\(A\) operators follows by taking coefficients in the left and right ordinary actions.
\end{proof}