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  pdftitle={On properties of Plancherel algebras},
  pdfauthor={Shurui Liu},
  pdfsubject={Flatness, factorization, and commutativity of Plancherel algebras},
  pdfkeywords={Plancherel algebra, relative Langlands duality,
    geometric Satake, loop rotation, hyperspherical variety,
    factorization algebra}
}

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\title{On properties of Plancherel algebras}
\author{Shurui Liu}
\address{Stanford University}
\email{srliu@stanford.edu}
\date{}
\subjclass[2020]{Primary 22E57; Secondary 14F08, 14M27, 18N40}
\keywords{Plancherel algebra, geometric Satake, loop rotation,
hyperspherical variety, factorization algebra}

\begin{document}

\begin{abstract}
We proved the predictions by  Ben-Zvi, Sakellaridis,
and Venkatesh that the Plancherel algebra $\mathrm{PL}_{X}$ is commutative and its loop-rotated version $\mathrm{PL}_{X,\hbar}$ is flat over $k[\hbar]$.
\end{abstract}

\maketitle
\tableofcontents

% Begin introduction.tex
\section{Introduction}
\label{sec:introduction}

The Plancherel algebra records the local multiplicities of the
spherical Hecke action on a variety with reductive group action.
In the relative Langlands framework of Ben-Zvi, Sakellaridis, and
Venkatesh \cite{BZSV}, its expected spectral description relates
these multiplicities to a dual Hamiltonian space. For hyperspherical
cotangent spaces, their conjectures predict two fundamental
properties: commutativity of the unrotated algebra
\cite[Conjecture~8.1.8]{BZSV} and flatness of loop-rotated algera over $k[\hbar]$ \cite[Conjecture~8.5.2]{BZSV}. These predictions are restated by Liu and Wang in
\cite[Conjecture~4.36(1)--(2)]{LW}.

\subsection{Main results}

Let \(F=\overline{\mathbb F}_q\), and let \(k\) be a
characteristic-zero \(\ell\)-adic coefficient field, with
\(\ell\ne\operatorname{char}F\). We work in the sheaf theory and super linear algebra conventions of \cite{LW}. Let \(G\) be a
connected reductive group over \(F\), and write \(\check G\)
for its Langlands dual over \(k\). Put
\[
\begin{gathered}
  \cO=F[[t]],\qquad \cK=F((t)),\qquad \Trot=\mathbb G_m,\\
  R=H^*(B\Trot,k)=k[\hbar],\qquad |\hbar|=2.
\end{gathered}
\]
Loop rotation acts by \(t\mapsto at\).
For a smooth affine \(G\)-variety \(X\), let
\(\PL_{X,\hbar}\) and \(\PL_X\) denote its rotated and
unrotated Plancherel algebras. We regard these as algebras in
the presentable derived category \(\cC\) of rational super
\(\check G\)-representations, using convention
of \cite[\S4.3.3]{LW}. Their precise definition, including their
multiplication, is recalled in Section~\ref{subsec:algebra}.

\begin{theorem}[Flatness]
\label{thm:main}
Let \(H\subset G\) be a connected reductive subgroup, let
\(Y\) be a finite-dimensional \(H\)-representation, and set
\(X=G\times^H Y\). Then \(\PL_{X,\hbar}\) is flat over
\(R\). If \(k\) is a splitting coefficient field for
\(\check G\), its graded cohomology is a free \(R\)-module,
and there is a noncanonical quasi-isomorphism of underlying
\(R\)-module complexes
\begin{equation}
\label{eq:main-splitting}
  \PL_{X,\hbar}\simeq\bigoplus_{\alpha\in I}R[-d_\alpha].
\end{equation}
\end{theorem}

Here flatness refers to the graded cohomology over the parameter
ring.  It implies that multiplication by \(\hbar\)
is injective on cohomology and that derived specialization has
no additional Tor contribution. The splitting in
\eqref{eq:main-splitting} concerns modules and does not assert
formality of the algebra.

Taking \(Y=0\) gives every homogeneous space \(G/H\) with
\(H\) connected reductive. Taking \(H=G\) gives every
finite-dimensional \(G\)-representation.

\begin{theorem}[Commutativity]
\label{thm:commutativity}
Let \(X\) be a smooth affine \(G\)-variety. There is a unital
commutative differential graded algebra \(B_X\) in rational
super \(\check G\)-representations and an equivalence
\[
  \PL_X\simeq B_X
  \qquad\text{in }\Alg_{E_1}(\cC).
\]
In particular, the equivalence preserves the
internal-endomorphism multiplication, unit, and
\(\check G\)-action.
\end{theorem}

Commutativity uses the symmetry
\((-1)^{rs+\epsilon\eta}\) on elements of cohomological
degrees \(r,s\) and super parities \(\epsilon,\eta\).
Theorem~\ref{thm:commutativity} asserts the existence of a
commutative dg model; it does not identify that model with its
cohomology algebra or assert that the choice is canonical.

\begin{corollary}[Hyperspherical cotangent spaces]
\label{cor:hyperspherical}
Let \(X\) be smooth affine, and suppose that \(T^*X\), with
its canonical Hamiltonian \(G\)-action, is hyperspherical with
the untwisted distinguished polarization and characteristic
restrictions of Section~\ref{subsec:structure}. Then
\(\PL_{X,\hbar}\) is flat over \(k[\hbar]\), and
\(\PL_X\) is commutative.
Thus the flatness and commutativity predictions of
\cite[Conjectures~8.1.8 and~8.5.2]{BZSV} hold.
\end{corollary}

The geometric input for the corollary is the presentation
\(X=G\times^H Y\) supplied by the polarized hyperspherical
datum. We recall this input in Section~\ref{subsec:structure},
including its interpretation in positive characteristic.

\subsection{Methods and relation to previous work}

For an irreducible representation \(V\) of \(\check G\),
let \(\cT_V\) be its Satake complex and let \(\delta_X\)
be the constant sheaf on \(X_\cO\), extended to \(X_\cK\).
The corresponding rotated multiplicity is
\[
  M_{V,\hbar}
  =\RHom_{\Sat_{X,\hbar}}
       (\cT_V*\delta_X,\delta_X).
\]
For \(X=G\times^H Y\), the equivalence
\([X/G]\simeq[Y/H]\) expresses this complex through the
space of triples for \((H,Y)\). On bounded Hecke support,
the space has a finite model stratified by vector bundles
over \(H_\cO\)-orbits in \(\Gr_H\). The Satake parity
theorem gives a single parity on all strata. The rotation
stabilizers have even equivariant cohomology, free over
\(R\), and the resulting localization sequences split.

This extends the filtration argument of
Braverman--Finkelberg--Nakajima \cite[Lemmas~2.5--2.6]{BFN}
to Satake coefficients. The identification of the
multiplicities is based on the finite-model calculation of
Ben-Zvi--Sakellaridis--Venkatesh
\cite[Proposition~8.3.1]{BZSV}. We record both its
rotation equivariance and the ordinary Verdier duality
normalization needed for the parity argument.

For commutativity, we use the factorizable Hecke module of
\cite[Proposition~16.2.8]{BZSV} together with the coefficient
factorization categories of \cite{Lys}. Relative rigidity
allows the internal endomorphisms of the basic object to
commute with diagonal restriction and exterior factorization.
Taking the total tensor product of the coefficient labels
produces a factorization associative family of
\(\cC\)-valued sheaves. Finite-object Verdier duality
compares its singleton algebra with \(\PL_X\), through
an opposite algebra and the Cartan pullback.

The geometric factorization construction is an input to this
argument. The introduction to \cite[\S16]{BZSV} notes that
some technical details of that formalism are not verified there,
and Proposition~16.2.8 is presented without a detailed proof.
Our commutativity result depends on this input; we do not
reconstruct the underlying sheaf theory of factorizable loop
spaces here.

The passage from the geometric finite-field setting to
topological factorization takes place in the categories
generated by closed coordinate diagonals. We compare these
categories, with their six-functor diagrams, across
characteristic and with Betti sheaves. A compact-support
construction then gives a unital \(E_3\)-enhancement whose
specified associative restriction is the required
internal-endomorphism algebra. This is the
fusion--convolution mechanism of \cite{Noc,HA}, applied
with the coefficient action and units retained.
Finally, relative formality of \(E_1\to E_3\)
\cite[Theorem~D\(^{\prime}\)]{FW} supplies the commutative
dg model. Relative formality is needed here to control
the given associative product.

\subsection{Organization}

Section~\ref{sec:conventions} fixes the conventions and
geometric input. Sections~\ref{sec:parity}--\ref{sec:freeness}
prove Theorem~\ref{thm:main}. Sections~\ref{sec:factorization}
and~\ref{sec:comparison} construct and compare the
factorization family. Section~\ref{sec:commutativity}
proves Theorem~\ref{thm:commutativity} and
Corollary~\ref{cor:hyperspherical}.
All numbered references to preprints use the versions
specified in the bibliography.
% End introduction.tex

% Begin conventions.tex
\section{Conventions and geometric input}
\label{sec:conventions}

\subsection{Sheaves, coefficients, and signs}
\label{subsec:coefficients}

We use the geometric sheaf theory of \cite[\S\S2.2--2.4 and 4.3]{LW}.
For a connected reductive group \(J\), write
\[
  J_\cO=J(F[[t]]),\qquad J_\cK=J(F((t))),\qquad
  \Gr_J=J_\cK/J_\cO;
\]
these expressions also denote the corresponding arc and loop functors.
We use left actions. Passing from the right-action notation of
\cite{LW} to this notation does not change the constructions.

Let \(\cV=D(k\text{-supervect})\), and let
\(\cC=\Rep_k(\check G)\) denote the presentable derived category
of rational super representations. The symmetry on homogeneous
objects of cohomological degrees \(r,s\) and super parities
\(\epsilon,\eta\) has sign
\begin{equation}
\label{eq:symmetry}
  (-1)^{rs+\epsilon\eta}.
\end{equation}
All commutative and opposite algebras use this symmetry.
Tensor products of coefficient and sheaf categories are balanced
over \(\cV\). In particular, adjoining the finite parity grading
is part of the coefficient theory; the six operations preserve this
grading and exterior products use \eqref{eq:symmetry}.
Mapping objects in the multiplicity formulas are enriched over
\(\cV\), so they retain both super parities. Their even parts
are the ordinary \(k\)-valued mapping complexes.
We abbreviate universally locally acyclic by ULA; its
categorical formulation is recalled in the proof of
Lemma~\ref{lem:factorization-end}.

Cohomological shifts satisfy
\(\mathcal H^i(\cF[r])=\mathcal H^{i+r}(\cF)\).
For finite Satake labels we use the super normalization of
\cite{LW}, denoted by \(\cT_V\). We write
\(\langle d\rangle=\Pi^d[d](d/2)\) for the simultaneous
parity shift, cohomological shift, and Tate twist; the half twists
are chosen as in \cite{LW}. Over a splitting coefficient field,
for a simple Satake object write
\(\cT_V=\cS_V\otimes\mathcal L_V\), where \(\cS_V\)
is the ordinary perverse intersection complex and \(\mathcal L_V\)
is the invertible coefficient line encoding the fixed super parity
and Tate twist. It is placed in cohomological degree zero.
We keep this line explicit when passing from the ordinary IC
calculation to the normalized Plancherel multiplicity.

The parameter \(\hbar\) is the degree-two class of the standard
character of \(\Trot\). All rotation-equivariant identifications
respect this class. On each finite model, arc-group equivariance
is computed using a sufficiently deep finite jet quotient; the
omitted congruence kernel is pro-unipotent.

\subsection{The Plancherel multiplication}
\label{subsec:algebra}

Let \(i:X_\cO\hookrightarrow X_\cK\), and set
\(\delta_X=i_*\kk_{X_\cO}\), with \(G_\cO\)-equivariance.
Write \(\Sat_X\) for the corresponding equivariant
*-sheaf category on \(X_\cK\), and \(\Sat_{X,\hbar}\)
for its loop-rotation-equivariant version. The Satake realization
functor from \(\cC\) acts on these categories by convolution.
For a finite representation \(V\), the unrotated algebra is
characterized by
\begin{equation}
\label{eq:plancherel-definition}
  \RHom_{\cC}(V,\PL_X)
  \simeq\RHom_{\Sat_X}(\cT_V*\delta_X,\delta_X).
\end{equation}
The rotated analogue uses
\(G_\cO\rtimes\Trot\)-equivariance and is \(R\)-linear.
Equivalently, one first forms the internal endomorphism object
in the derived Satake category and then applies the right adjoint
to the Satake realization functor, as in
\cite[\S4.3.3]{LW}.
This uses a monoidal functor from the derived category of the
Satake heart; it does not require that functor to be fully faithful
on derived mapping complexes.

As in \cite[\S4.3.3]{LW}, forgetting loop rotation gives
derived specialization
\begin{equation}
\label{eq:derived-specialization}
  \PL_{X,\hbar}\otimes_R^{\mathbf L}k\simeq\PL_X.
\end{equation}
Here \(R\) denotes the polynomial dg model of the cochains on
\(B\Trot\), with zero differential and the specified class
\(\hbar\); the universal first Chern class gives this model.
Consequently, the freeness theorem gives
\(H^*(\PL_X)\simeq H^*(\PL_{X,\hbar})/\hbar\).

The product is composition of evaluations. If
\(a:\cT_V*\delta_X\to\delta_X[r]\) and
\(b:\cT_W*\delta_X\to\delta_X[s]\), their product is the
composite
\begin{equation}
\label{eq:composition-product}
\begin{split}
  \cT_{V\otimes W}*\delta_X
  &\simeq\cT_V*(\cT_W*\delta_X)
    \xrightarrow{\,\cT_V*b\,}(\cT_V*\delta_X)[s]\\
  &\xrightarrow{\,a[s]\,}\delta_X[r+s].
\end{split}
\end{equation}
The identity of \(\delta_X\) is the unit. Thus
\eqref{eq:plancherel-definition} specifies an associative algebra,
including its multiplication and unit.

We distinguish this constant basic object from the dualizing
basic object \(\delta_X^!\) in the !-sheaf theory of \cite{BZSV}.
The latter gives an internal-endomorphism algebra \(A_X^!\).
Their comparison, as associative algebras, is established in
Lemma~\ref{lem:duality-comparison}.

\subsection{The induced presentation}
\label{subsec:structure}

We use distinguished polarizations in the sense of
\cite[\S3.7]{BZSV}, with the cotangent conventions recalled in
\cite[\S2.6]{LW}. In particular, \(X\) is smooth affine and
the Hamiltonian structure on \(T^*X\) is the canonical one.
The geometric input for Corollary~\ref{cor:hyperspherical} is the
presentation
\begin{equation}
\label{eq:induced}
  X\simeq G\times^H Y,
\end{equation}
where \(H\subset G\) is connected reductive and \(Y\) is a
finite-dimensional \(H\)-representation. We use left actions, so
\(G\times^H Y\) denotes the quotient by
\((g,y)\cdot h=(gh,h^{-1}y)\).
This is the same presentation as \(Y\times^H G\) in the
right-action notation of \cite[\S2.6]{LW}.

In characteristic zero the structural results behind
\eqref{eq:induced} are
\cite[Theorem~3.6.1 and \S3.7]{BZSV}.
In the finite-field setting, we use the definition by structural
data, with the characteristic restrictions imposed there; see \cite[\S2.4 and Definitions~3.9.3, 3.9.5]{BZSV}.
Here an untwisted distinguished polarization is supplied by
an untwisted polarized datum over the characteristic-zero base
ring and its reduction. In particular, the untwisted condition
is imposed on the datum, before reduction. We recall why this
construction has the form \eqref{eq:induced}.

The general polarized construction has
\(X=Y\times^{HU}G\), with a Whittaker character on \(U\);
see \cite[(3.33)]{BZSV}.
In Hamiltonian induction, moment values restrict, after conjugation,
to this character on \(\mathfrak u\).
A nonzero character excludes zero from the image of the moment map.
Over characteristic zero, the canonical moment map of \(T^*X\)
vanishes on the
zero section. The character must therefore vanish. In the even
\(\mathfrak{sl}_2\) construction,
\cite[Lemma~3.7.3]{BZSV} then forces the datum to be trivial,
so that \(U=1\). The untwisted polarized datum therefore has
trivial Whittaker part, and the same is true after base change
to the geometric finite field. The polarized formula reduces
to \eqref{eq:induced}.
Connectedness of \(H\) is part of the polarized hyperspherical
structure; see \cite[Proposition~3.7.4 and Definition~3.9.3]{BZSV}.
Thus the finite-field use of \eqref{eq:induced} comes from these
structural data, without an application of the characteristic-zero
slice theorem.

\subsection{Satake parity}
\label{subsec:satake-parity}

Let \(J\) be connected reductive and let \(\nu\) be a dominant
coweight. Normalize the intersection complex
\(\IC_\nu\) of \(\overline{\Gr}_J^\nu\) to be perverse.
For every geometric point \(\xi\), the Satake parity theorem states
that
\begin{equation}
\label{eq:satake-parity}
  \mathcal H^j(i_\xi^*\IC_\nu)=0
  \quad\text{unless}\quad
  j\equiv\dim\Gr_J^\nu\pmod 2.
\end{equation}
The restriction to each \(J_\cO\)-orbit has constant cohomology
sheaves, and the assertion holds for the usual
loop-rotation-equivariant lift.

See \cite[\S11(c)]{Lus} and
\cite[\S5.6, (5.6.2) and the ensuing stalk formula]{Zhu}.
In the latter formula, the possible stalk degrees are
\(2i-\langle2\rho_J,\nu\rangle\), which gives
\eqref{eq:satake-parity} in the \'etale setting.
The characteristic-zero assumption here is on the coefficients.
It is not an assumption on the geometric ground field.
% End conventions.tex

% Begin parity.tex
\section{Equivariant parity and finite filtrations}
\label{sec:parity}

For an algebraic group \(A\) acting on a finite-type scheme \(Z\)
and \(\cF\in D_A^b(Z)\), set
\begin{equation}
\label{eq:dual-cohomology}
  \mathsf C_A(Z,\cF)
  :=R\Gamma_A(Z,\bD_Z\cF)
  =\RHom_A(\cF,\omega_Z).
\end{equation}
Here \(\bD_Z\) is ordinary Verdier duality on \(Z\), equipped
with its induced equivariant structure. It is not Verdier duality
on the quotient stack. The definition applies to an arbitrary
constructible complex, without a local constancy assumption on
\(Z\).

\begin{lemma}[Equivariant splitting on an orbit]
\label{lem:orbit-splitting}
Let \(O=A/K\), where \(K\) is connected, and suppose that
\(H^{\mathrm{odd}}(BK,k)=0\).
Let \(\mathcal E\in D_A^b(O)\) have constant cohomology sheaves
concentrated in degrees congruent to \(p\) modulo \(2\).
If \(W_j\) denotes its degree-\(j\) stalk cohomology, there is a
noncanonical equivariant decomposition
\begin{equation}
\label{eq:orbit-splitting}
  \mathcal E\simeq
  \bigoplus_{j\equiv p\ (2)}\underline{W_j}_O[-j].
\end{equation}
\end{lemma}

\begin{proof}
The quotient stack \([O/A]\) is \(BK\).
An equivariant constructible local system on \(O\) corresponds
to a representation of the component group of \(K\).
Since \(K\) is connected, these local systems are constant.

We split the cohomological truncation triangles by induction on
the number of nonzero cohomology sheaves. Let \(m\) be the
largest degree with \(W_m\ne0\).
After splitting the lower truncation, the connecting morphism has
components in
\[
  \Hom_{D(BK)}(W_m[-m],W_j[-j+1])
  =\Hom_k(W_m,W_j)\otimes_k H^{m-j+1}(BK,k).
\]
Both \(m\) and \(j\) have parity \(p\), so \(m-j+1\) is odd.
The displayed group vanishes. The triangle therefore splits off
\(W_m[-m]\), completing the induction.
\end{proof}

For the rest of this section, put
\(A=H_\cO\rtimes\Trot\), where \(H\) is connected reductive.
Choose a maximal torus and a positive root system of \(H\).
For a dominant coweight \(\lambda\), write
\[
  O_\lambda=H_\cO t^\lambda H_\cO/H_\cO
  \subset\Gr_H,\qquad
  L_\lambda=Z_H(\lambda).
\]

\begin{lemma}[Rotation stabilizers]
\label{lem:stabilizer}
The stabilizer \(K_\lambda\) of \(t^\lambda H_\cO\) in \(A\)
is connected and has a split quotient
\begin{equation}
\label{eq:stabilizer-quotient}
  K_\lambda\longrightarrow L_\lambda\times\Trot
\end{equation}
with pro-unipotent kernel. Consequently there is an isomorphism
of \(R\)-algebras
\begin{equation}
\label{eq:stabilizer-cohomology}
  H^*(BK_\lambda,k)
  \simeq H^*(BL_\lambda,k)\otimes_k R.
\end{equation}
The factor \(H^*(BL_\lambda,k)\) is concentrated in even
degrees.
\end{lemma}

\begin{proof}
Pure rotation fixes the coset \(t^\lambda H_\cO\), since
\((at)^\lambda=t^\lambda\lambda(a)\). Thus
\[
  K_\lambda=
  (H_\cO\cap t^\lambda H_\cO t^{-\lambda})
  \rtimes\Trot.
\]
Evaluation at \(t=0\) maps the first factor onto
\(P_{-\lambda}\), with pro-unipotent kernel.
Indeed, the coordinate of the root subgroup for \(\alpha\)
must be divisible by
\(t^{\max(0,\langle\alpha,\lambda\rangle)}\).
The Levi quotient of \(P_{-\lambda}\) is \(L_\lambda\).
Rotation acts trivially after evaluation, and constant loops in
\(L_\lambda\) split this quotient and commute with rotation.
This proves \eqref{eq:stabilizer-quotient} and the connectedness
assertion.

In the sheaf theory under consideration, a unipotent group has
no positive-degree classifying-stack cohomology.
Moreover, with characteristic-zero coefficients,
\(H^*(BL_\lambda,k)\) is the Weyl-invariant polynomial algebra
on the degree-two cohomology of a maximal torus.
It is therefore even. These facts give
\eqref{eq:stabilizer-cohomology}.
The splitting respects projection to \(\Trot\), so the
isomorphism is over the specified ring \(R\).
The calculation may be made in a sufficiently deep finite jet
quotient, whose discarded congruence kernel is unipotent.
\end{proof}

\begin{lemma}[Freeness from a finite filtration]
\label{lem:filtration}
Let \(Z\) be a finite-type \(A\)-scheme with a finite filtration
by invariant closed subschemes. Suppose that each stratum \(E_i\)
is an \(A\)-equivariant vector bundle over some \(O_\lambda\).
Let \(\cF\in D_A^b(Z)\).
Assume that \(\cF|_{E_i}\) is pulled back from a complex on
\(O_\lambda\) satisfying Lemma~\ref{lem:orbit-splitting},
with the same parity \(p\) for every stratum.
Then \(H^*(\mathsf C_A(Z,\cF))\) is a graded free
\(R\)-module, concentrated in degrees congruent to \(p\)
modulo \(2\).
\end{lemma}

\begin{proof}
Let \(d_i=\dim E_i\). By equivariant splitting and smooth
Verdier duality,
\[
  \bD_{E_i}(\cF|_{E_i})
  \simeq
  \bigoplus_{j\equiv p\ (2)}
  W_{ij}^{\vee}\otimes_k\kk_{E_i}[j+2d_i](d_i).
\]
Equivariant homotopy invariance for \(E_i\to O_\lambda\)
identifies \(H_A^*(E_i,k)\) with
\(H^*(BK_\lambda,k)\).
Lemma~\ref{lem:stabilizer} therefore proves the desired
freeness and parity for each stratum.

Consider one step of the filtration, with closed inclusion
\(i:Z'\hookrightarrow Z''\) and open complement
\(j:E\hookrightarrow Z''\). Verdier duality gives
\[
\begin{aligned}
  i^!\bD_{Z''}(\cF|_{Z''})&\simeq\bD_{Z'}(\cF|_{Z'}),\\
  j^*\bD_{Z''}(\cF|_{Z''})&\simeq\bD_E(\cF|_E).
\end{aligned}
\]
The localization triangle hence takes the form
\begin{equation}
\label{eq:localization}
\begin{split}
  \mathsf C_A(Z',\cF|_{Z'})
  &\longrightarrow\mathsf C_A(Z'',\cF|_{Z''})\\
  &\longrightarrow\mathsf C_A(E,\cF|_E)
  \xrightarrow{+1}.
\end{split}
\end{equation}
By induction, the cohomology of both outer terms is concentrated
in parity \(p\). All connecting maps change degree by one and
thus vanish.
We obtain a short exact sequence of graded \(R\)-modules.
Its quotient is graded free, so homogeneous lifts of a homogeneous
basis split the sequence.
Induction on the filtration proves the lemma.
\end{proof}

\begin{remark}
\label{rem:duality}
The use of \(\bD_Z\cF\) in
\eqref{eq:dual-cohomology} is what makes
\eqref{eq:localization} compatible with ordinary restrictions of
the coefficient complex. The argument does not require \(\cF\)
to be locally constant on all of \(Z\).
\end{remark}
% End parity.tex

% Begin multiplicities.tex
\section{A finite model for Plancherel multiplicities}
\label{sec:finite-model}

Assume \(X=G\times^H Y\) as in
Theorem~\ref{thm:main}, and first take a splitting coefficient
field for \(\check G\).
For \(V\in\Irr(\check G)\), let \(\cS_V=\IC_V\) be the
ordinary perverse Satake sheaf, so that
\(\cT_V=\cS_V\otimes\mathcal L_V\) as in
Section~\ref{subsec:coefficients}. With equivariance understood, set
\[
  \delta_X=i_*\kk_{X_\cO},
  \qquad i:X_\cO\hookrightarrow X_\cK,
\]
and define
\begin{equation}
\label{eq:multiplicity}
  N_{V,\hbar}
  :=\RHom_{\Sat_{X,\hbar}}(\cS_V*\delta_X,\delta_X).
\end{equation}
The actual \(V\)-multiplicity uses the fully normalized kernel:
\begin{equation}
\label{eq:normalized-multiplicity}
\begin{split}
  M_{V,\hbar}
  &:=\RHom_{\Sat_{X,\hbar}}(\cT_V*\delta_X,\delta_X)\\
  &\simeq \mathcal L_V^\vee\otimes_k N_{V,\hbar}.
\end{split}
\end{equation}
Thus freeness of \(N_{V,\hbar}\) is equivalent to freeness
of the normalized multiplicity \(M_{V,\hbar}\).
The kernel of \(\Aut(D)\to\Trot\), for the formal disk
\(D=\operatorname{Spec}F[[t]]\), is pro-unipotent.
Thus \eqref{eq:multiplicity} may be computed with
\(G_\cO\rtimes\Trot\)-equivariance; this replacement preserves
\(\hbar\).

Consider the space of triples
\begin{equation}
\label{eq:triples}
  \cR_{H,Y}
  =\bigl\{(hH_\cO,v)\in\Gr_H\times Y_\cO:
                  h^{-1}v\in Y_\cO\bigr\},
\end{equation}
and the map \(q:\cR_{H,Y}\to\Gr_G\) induced by
\(H\hookrightarrow G\).

\begin{lemma}[Finite multiplicity model]
\label{lem:finite-model}
Let \(S\subset\Gr_H\) be the reduced inverse image of
\(\supp\cS_V\) under \(\Gr_H\to\Gr_G\).
Then \(S\) is a finite union of \(H_\cO\)-orbits.
For all sufficiently large \(N\), the formula
\begin{equation}
\label{eq:finite-model}
  Z_N=
  \bigl\{(hH_\cO,\bar v)\in S\times(Y_\cO/t^NY_\cO):
                  v\in hY_\cO\bigr\}
\end{equation}
defines a finite-type scheme.
Writing \(q_N:Z_N\to\Gr_G\) for the induced map and
\(n=\dim Y\), there is an \(R\)-linear identification
\begin{equation}
\label{eq:normalized-model}
  N_{V,\hbar}\simeq
  R\Gamma_{H_\cO\rtimes\Trot}
     (Z_N,\bD_{Z_N}q_N^*\cS_V)[-2Nn](-Nn).
\end{equation}
These normalized complexes are independent of the choice of
sufficiently large \(N\).
\end{lemma}

\begin{proof}
We first identify the correspondence and its coefficient complex,
and then verify the finite bounds and the normalization.

\smallskip
\noindent\emph{The correspondence.}
Put \(\mathscr Q_X=[X_\cO/G_\cO]\).
The double-regular relative Grassmannian is the fiber product
\begin{equation}
\label{eq:double-regular}
  \mathscr Z_X
  :=\mathscr Q_X
       \times_{[X_\cK/G_\cK]}
       \mathscr Q_X.
\end{equation}
It classifies two \(G\)-bundles on the disk with regular
\(X\)-sections and an isomorphism on the punctured disk that
respects the sections.
The equivalence \([X/G]\simeq[Y/H]\) sends a
\(G\)-bundle with an \(X\)-section to its \(H\)-reduction
with a \(Y\)-section. Applying this description on the disk
and punctured disk gives
\begin{equation}
\label{eq:triples-quotient}
  \mathscr Z_X\simeq[\cR_{H,Y}/H_\cO].
\end{equation}
In these coordinates, the \(G\)-Hecke map is induced by
viewing \(h\in H_\cK\) as an element of \(G_\cK\).
The identifications are identities of moduli stacks and therefore
descend from trivializations of the bundles. They also commute
with loop rotation.

\smallskip
\noindent\emph{The coefficient complex.}
Let \(p:\mathscr Z_X\to\mathscr Q_X\) be the projection
to the target section. The convolution correspondence and
adjunction identify \eqref{eq:multiplicity} with
\begin{equation}
\label{eq:adjunction-model}
  \RHom(q^*\cS_V,p^!\kk_{\mathscr Q_X})
\end{equation}
on \(\mathscr Z_X\), with rotation equivariance.
Here \(q^*\cS_V\) denotes pullback of the equivariant Satake
kernel by the Hecke map.

For clarity, the adjunction calculation can be made at a finite
truncation. Restrict the source of the convolution correspondence
to the regular-arc locus and call the resulting one-sided
correspondence \(E\). Let
\(a:E\to[X_\cK/G_\cO]\) be its target map and
\(\widetilde q\) its Hecke map.
Base change along \(i:\mathscr Q_X\to[X_\cK/G_\cO]\)
gives a Cartesian square
\[
\begin{tikzcd}[column sep=large,row sep=large]
  \mathscr Z_X \arrow[r,"j"] \arrow[d,"p"']
    & E \arrow[d,"a"]\\
  \mathscr Q_X \arrow[r,"i"']
    & {[X_\cK/G_\cO]}.
\end{tikzcd}
\]
Since the source sheaf is constant, convolution is
\(a_!\widetilde q^*\cS_V\). Consequently,
\begin{equation}
\label{eq:adjunction-calculation}
\begin{aligned}
  \RHom(a_!\widetilde q^*\cS_V,i_*\kk_{\mathscr Q_X})
  &\simeq
     \RHom(i^*a_!\widetilde q^*\cS_V,\kk_{\mathscr Q_X})\\
  &\simeq
     \RHom(p_!j^*\widetilde q^*\cS_V,\kk_{\mathscr Q_X})\\
  &\simeq
     \RHom(q^*\cS_V,p^!\kk_{\mathscr Q_X}).
\end{aligned}
\end{equation}
In particular, the coefficient is the ordinary pullback of the
Satake kernel, even when it is not locally constant.

Restricting to bounded Hecke support and finite arc truncations
makes these the usual six-functor operations.
Their compatibility with the infinite-dimensional definition is
the finite-model calculation of
\cite[Proposition~8.3.1, (8.11), and \S8.3.6]{BZSV}.
Every map and every adjunction or base-change isomorphism in
\eqref{eq:adjunction-calculation} is rotation equivariant,
so the identification is over \(R\).

\smallskip
\noindent\emph{Finite bounds.}
Choose a faithful \(G\)-representation \(W\).
The support of \(\cS_V\) is finite dimensional, so the associated
lattices admit a uniform bound
\[
  t^bW_\cO\subset gW_\cO\subset t^{-b}W_\cO
\]
for some \(b\).
If \(t^\lambda\in\Gr_H\) maps into this support, then
every weight \(\chi\) of \(W|_H\) satisfies
\(|\langle\chi,\lambda\rangle|\le b\).
Faithfulness implies that these weights span the rational
character space of a maximal torus of \(H\).
Hence only finitely many dominant coweights \(\lambda\) can
occur.
The closed inverse image \(S\) is thus a finite union of
spherical strata and is of finite type.
Passing to its reduced structure does not change the
constructible sheaf category.

Choose \(N\) such that
\(t^NY_\cO\subset hY_\cO\) for every \(hH_\cO\in S\);
the finite list of coweights makes this possible.
The condition in \eqref{eq:finite-model} is then independent
of the representative \(v\) of \(\bar v\).
It imposes finitely many linear conditions on the jet of \(v\),
and gives the finite-type scheme \(Z_N\).
For sufficiently large \(m\ge N\), the action on this model
factors through the jet group
\(H_m=H(\cO/t^m)\).

\smallskip
\noindent\emph{Duality and normalization.}
Under \eqref{eq:triples-quotient}, the target
\(\mathscr Q_X\) becomes \([Y_\cO/H_\cO]\).
Write \(Y_N=Y_\cO/t^NY_\cO\).
Its finite model \([Y_N/H_m]\) is smooth of dimension
\(Nn-\dim H_m\).
Thus the term \(p^!\kk\) in \eqref{eq:adjunction-model}
is the dualizing complex of the source stack, shifted and
twisted by minus twice, and minus once, this dimension.

On the other hand, the pullback of the stack dualizing complex
from \([Z_N/H_m]\) to \(Z_N\) is
\(\omega_{Z_N}[-2\dim H_m](-\dim H_m)\).
The two group-dimension contributions cancel, leaving exactly
\[
  \bD_{Z_N}q_N^*\cS_V[-2Nn](-Nn)
\]
inside equivariant cohomology.
Including \(\Trot\) in both quotients cancels its dimension
in the same way.
This proves \eqref{eq:normalized-model}, with ordinary
Verdier duality on \(Z_N\).

\smallskip
\noindent\emph{Independence of the truncation.}
For \(N\) sufficiently large, the map
\(Z_{N+1}\to Z_N\) is an affine bundle of rank \(n\),
and \(q_{N+1}^*\cS_V\) is the ordinary pullback of
\(q_N^*\cS_V\).
Smooth Verdier duality contributes \([2n](n)\).
This is canceled by the change in the normalization
\([-2Nn](-Nn)\).
Equivariant homotopy invariance therefore identifies the
normalized complexes at consecutive levels.
Increasing \(m\) changes only unipotent equivariance and
has no effect. This proves the asserted independence and
completes the proof.
\end{proof}

\begin{remark}
\label{rem:normalizations}
Changing the normalization of the Satake action changes
\eqref{eq:normalized-model} by a uniform shift and twist
depending on \(V\), which does not affect freeness.
We use only the module identification of
\cite[Proposition~8.3.1]{BZSV}.
The comparison of algebra multiplications discussed in
\cite[Remark~8.3.2]{BZSV} is not needed.
\end{remark}
% End multiplicities.tex

% Begin freeness.tex
\section{Freeness over the rotation parameter}
\label{sec:freeness}

\begin{proposition}
\label{prop:multiplicity-free}
Under the hypotheses of Theorem~\ref{thm:main}, and over a
splitting coefficient field, the graded \(R\)-module
\(H^*(M_{V,\hbar})\) is free for every
\(V\in\Irr(\check G)\).
\end{proposition}

\begin{proof}
Use the finite model of Lemma~\ref{lem:finite-model}.
For an \(H_\cO\)-orbit \(O_\lambda\subset S\), let
\(Z_{N,\lambda}\) denote its inverse image in \(Z_N\).
The map \(Z_{N,\lambda}\to O_\lambda\) is a vector
bundle with fiber
\begin{equation}
\label{eq:stratum-fiber}
  (Y_\cO\cap t^\lambda Y_\cO)/t^NY_\cO.
\end{equation}
Indeed, the stabilizer acts linearly on this space, and the
bundle is obtained by translating it along \(O_\lambda\).
This description is equivariant under
\(H_\cO\rtimes\Trot\).
If \(Y=\bigoplus_\chi Y_\chi\) is the weight decomposition,
then its rank is
\begin{equation}
\label{eq:stratum-rank}
  Nn-\sum_\chi
      \max(\langle\chi,\lambda\rangle,0)\dim Y_\chi.
\end{equation}
This is the lattice-intersection calculation of
\cite[Lemma~2.2]{BFN}.
We use a fixed target lattice in \eqref{eq:finite-model};
using the source lattice instead reverses the sign of
\(\lambda\) in \eqref{eq:stratum-rank}.

Let \(\nu\) be the highest coweight corresponding to \(V\),
and put
\[
  p_V=\langle2\rho_G,\nu\rangle\bmod 2.
\]
The map \(O_\lambda\to\Gr_G\) lands in a single
\(G_\cO\)-orbit.
By \eqref{eq:satake-parity}, the pullback of \(\cS_V\)
to \(O_\lambda\) has cohomology only in parity \(p_V\).
Its equivariant cohomology sheaves are constant, since its
stabilizer is connected by Lemma~\ref{lem:stabilizer}.
That lemma also gives the vanishing of odd stabilizer
cohomology.
Lemma~\ref{lem:orbit-splitting} consequently splits this
pullback equivariantly.
Pulling the decomposition back along
\(Z_{N,\lambda}\to O_\lambda\) verifies the coefficient
hypothesis of Lemma~\ref{lem:filtration}, with parity
\(p_V\) independent of \(\lambda\).

The finite collection of orbits in \(S\) is closed under
specialization. Order the orbits compatibly with closure and
take inverse images in \(Z_N\) of the resulting closed
unions.
This gives a finite invariant closed filtration with strata
\(Z_{N,\lambda}\).
Lemma~\ref{lem:filtration} shows that
\[
  H^*R\Gamma_{H_\cO\rtimes\Trot}
     (Z_N,\bD_{Z_N}q_N^*\cS_V)
\]
is graded free over \(R\).
The uniform shift and twist in
\eqref{eq:normalized-model} preserve freeness, which proves
freeness of \(N_{V,\hbar}\). Tensoring with the invertible
line \(\mathcal L_V^\vee\) in
\eqref{eq:normalized-multiplicity} proves the proposition for
the actual Plancherel multiplicity \(M_{V,\hbar}\).
\end{proof}

\begin{remark}
The constant-coefficient filtration of
\cite[Lemmas~2.5--2.6]{BFN} provides the geometric pattern
for the proof.
For Satake coefficients, the additional ingredients are
equivariant splitting of the restricted kernel, parity
independent of the stratum, and the duality identity that
produces \eqref{eq:localization}.
\end{remark}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
First suppose that \(k\) is a splitting coefficient field
for \(\check G\).
Since \(\check G\) is reductive in characteristic zero,
its rational representations are semisimple. The same argument
applies separately in the two super parities.
The inner-Hom characterization gives, as \(R\)-module
complexes with \(\check G\)-action,
\begin{equation}
\label{eq:isotypic}
  \PL_{X,\hbar}\simeq
  \bigoplus_{V\in\Irr(\check G)}
       V\otimes_k M_{V,\hbar}.
\end{equation}
Here \(V\) ranges over ordinary irreducible representations,
regarded as even; its multiplicity is a complex of super vector
spaces. Thus both super parities are included in the multiplicity,
without counting each \(V\) twice. These are the multiplicities
in \eqref{eq:normalized-multiplicity}, obtained by the right-adjoint
construction of \cite[\S4.3.3]{LW}.
Equivalently, test the underlying object against finite
representations and their parity reversals.
The parameter \(\hbar\) has trivial \(\check G\)-action
and acts in the multiplicity factors.

By Proposition~\ref{prop:multiplicity-free}, tensoring
with \(V\) and taking the direct sum in \eqref{eq:isotypic}
gives
\begin{equation}
\label{eq:free-cohomology}
  H^*(\PL_{X,\hbar})
  \quad\text{graded free over }R.
\end{equation}
Choose a homogeneous \(R\)-basis \(\{e_\alpha\}\) and,
in a dg \(R\)-module model, choose a cocycle representing
each basis element.
Sending the generator of
\(R[-\deg e_\alpha]\) to the chosen cocycle gives an
\(R\)-linear map
\[
  \bigoplus_\alpha R[-\deg e_\alpha]
  \longrightarrow \PL_{X,\hbar}
\]
that induces the isomorphism \eqref{eq:free-cohomology}.
It is therefore a quasi-isomorphism.
This is an assertion about the underlying module complex;
no compatibility with multiplication is required.
It gives flatness, injectivity of multiplication by
\(\hbar\), and the absence of additional Tor in
specialization at \(\hbar=0\).

For an arbitrary coefficient field \(k\), extend to a
splitting field \(k'\).
The finite-model construction
\eqref{eq:normalized-model} commutes with extension of
coefficients. Flatness then descends along the faithfully
flat map \(k[\hbar]\to k'[\hbar]\).
\end{proof}
% End freeness.tex

% Begin factorization.tex
\section{Factorization and internal endomorphisms}
\label{sec:factorization}

We now assume only that \(X\) is smooth and affine. The purpose of
this section is to construct a factorization associative family with
singleton algebra \(A_X^!\), and to compare that algebra with
\(\PL_X\). Throughout this section, categorical tensor products
are balanced over \(\cV\).

\subsection{The coefficient factorization category}

Take \(\Sigma=\mathbb A^1_F\), and put
\(R_I=\Shv(\Sigma^I;\cV)\), with exceptional tensor product
\(\otimes^!\) and unit \(\omega_{\Sigma^I}\). The notation
\(R_I\) refers to a sheaf category and is distinct from the
parameter ring \(R=k[\hbar]\).
We use the closed-diagonal construction of \cite[\S\S2.1--2.2
and 2.5]{Lys}. It applies to Ind-constructible \(\ell\)-adic
sheaves. Its formulation for a projective curve gives the present
construction by restriction from \(\mathbb P^1\).

For a finite nonempty set \(I\), let \(\Tw(I)\) have objects
\(s=(I\twoheadrightarrow J\twoheadrightarrow K)\).
A morphism from \(s_1\) to \(s_2\) consists of compatible
surjections \(J_1\twoheadrightarrow J_2\) and
\(K_2\twoheadrightarrow K_1\). To \(s\) associate
\[
  D_s=\cC^{\otimes J}\otimes\Shv(\Sigma^K),
\]
viewed as an \(R_I\)-module category supported on the closed
diagonal \(\Delta_K:\Sigma^K\hookrightarrow\Sigma^I\).
The transition functor tensors coefficient labels along
\(J_1\to J_2\) and then pushes forward along
\(\Sigma^{K_1}\hookrightarrow\Sigma^{K_2}\). Define
\begin{equation}
\label{eq:coefficient-category}
  \cC_I=\colim_{s\in\Tw(I)}D_s
\end{equation}
in \(R_I\)-module categories.

The construction has coherent diagonal and factorization equivalences
\begin{equation}
\label{eq:coefficient-factorization}
  \Delta_\alpha^!\cC_I\simeq\cC_J,
  \qquad
  \cC_I|_{U_\beta}\simeq
    \left(\boxtimes_b\cC_{I_b}\right)|_{U_\beta}.
\end{equation}
Here \(\alpha:I\twoheadrightarrow J\), and \(U_\beta\)
is the locus where coordinates in different blocks \(I_b\)
are disjoint. Pullback of a module category means extension of
scalars through exceptional pullback on base sheaves.
For the equivalences and the pointwise symmetric monoidal
structures, see \cite[\S\S2.1.11--2.1.18, 2.2.9--2.2.17,
and 2.5]{Lys}.

\begin{lemma}[Total tensor product of labels]
\label{lem:total-tensor}
There are continuous, base-linear, strong symmetric monoidal functors
\begin{equation}
\label{eq:total-tensor}
  Q_I:\cC_I\longrightarrow\Shv(\Sigma^I)\otimes\cC
\end{equation}
compatible with exceptional diagonal restriction, factorization,
permutations, and insertion of unit-labelled points. For a singleton,
\(Q_I\) is the identity under the natural identification.
\end{lemma}

\begin{proof}
Let \(m_J:\cC^{\otimes J}\to\cC\) be total tensor product.
On the term \(D_s\), define
\begin{equation}
\label{eq:total-tensor-term}
  Q_s=m_J\otimes\Delta_{K,!}.
\end{equation}
These functors form a cocone: coefficient transitions use
\(m_{J_2}m_{J_1/J_2}=m_{J_1}\), and geometric transitions
use functoriality of proper pushforward. The proper projection
formula gives base linearity, so the colimit defines \(Q_I\).

For a diagonal map \(f:\Sigma^{I'}\to\Sigma^I\), push out
the index diagram along \(I\to I'\), obtaining
\(I'\to J'\to K'\). Then
\(\Sigma^{K'}=\Sigma^K\times_{\Sigma^I}\Sigma^{I'}\),
and proper exceptional base change gives
\begin{equation}
\label{eq:total-tensor-base-change}
  f^!\Delta_{K,!}\simeq\Delta_{K',!}f_K^!.
\end{equation}
After contracting coefficient labels, their total tensor product
is still \(m_J\). These are the pushout-index maps defining
\eqref{eq:coefficient-factorization}; the cofinality equivalences
therefore identify \(f^!Q_I\) with \(Q_{I'}f^!\).

For monoidality, take indices
\(s_a=(I\to J_a\to K_a)\), for \(a=1,2\), and write
\(\Sigma^{K_0}=\Sigma^{K_1}\times_{\Sigma^I}\Sigma^{K_2}\).
If \(r_a:\Sigma^{K_0}\to\Sigma^{K_a}\), then
\begin{equation}
\label{eq:common-support}
  \Delta_{K_1,!}F_1\otimes^!\Delta_{K_2,!}F_2
  \simeq\Delta_{K_0,!}
       (r_1^!F_1\otimes^!r_2^!F_2).
\end{equation}
The product in \eqref{eq:coefficient-category} uses this
common-support diagram and the coefficient slots of
\(J_1\amalg J_2\). Total tensor product gives
\(m_{J_1}(-)\otimes m_{J_2}(-)\), proving monoidality on
the colimit terms and hence on \(\cC_I\). Associativity and
symmetry follow from the corresponding six-functor constraints
and the symmetry of \(\cC\).

The unit is represented by unit labels and
\(\omega_{\Sigma^I}\) in the term \(I=J=K\), and maps
to the target unit. Over a disjoint-block locus, the indices
and their tensor products split blockwise. Inserting points
inserts unit labels. These descriptions also give all the
refinement compatibilities.
\end{proof}

The functors \(Q_I\) retain the rational \(\check G\)-action
and the super symmetry. The use of \(\cC_I\) is essential:
forming internal Homs in a fixed \(\Rep(\check G^I)\) and
then restricting diagonally would introduce coinduction at
collisions and would not give \eqref{eq:total-tensor-base-change}.

\subsection{The factorizable Satake action}
\label{subsec:factorizable-satake}

Factorizable constructible Satake gives coherent base-linear
monoidal functors from \(\cC_I\) to the spherical Hecke
categories over \(\Sigma^I\). They have the diagonal
compatibilities of \eqref{eq:coefficient-factorization},
preserve exterior factorization, and send the unit to the
trivial modification; see \cite[Proposition~4.1.15 and
\S\S7.3.8--7.3.17]{Lys}. The exterior factorization
functors in the constructible target are fully faithful.

We fix the normalization as follows. On finite labels use
\(\cT_V\), including its super parity. On \(\Sigma^K\),
normalize the ULA multifusion family to have unit
\(\omega_{\Sigma^K}\): if \(P_{V,K}\) is the
super-normalized global perverse family, use
\(P_{V,K}\langle|K|\rangle\). The multifusion and
refinement identities of \cite[Proposition~5.4.2 and
Remark~5.4.12, (5.4.13)--(5.4.14)]{Zhu} then identify
exceptional restriction at a collision with tensoring the
labels, with this normalization already included.
In particular, for unit labels the normalized perverse family is
\(\Pi^{|K|}k_{\Sigma^K}[|K|](|K|/2)\), so the additional
\(\langle|K|\rangle\) gives precisely \(\omega_{\Sigma^K}\).
The factorization symmetry includes the central-parity correction
of \cite[\S7.3.17]{Lys}; in the super convention fixed above,
this is part of the symmetry \eqref{eq:symmetry}.

On a term \((I\to J\to K)\) of
\eqref{eq:coefficient-category}, the action functor sends
finite labels and a base sheaf \(F\) to the closed-diagonal
pushforward of
\begin{equation}
\label{eq:satake-family}
  \cT_{(\bigotimes_{j\mapsto k}V_j)_{k\in K},K}
      \otimes_{R_K}F.
\end{equation}
Associativity handles coefficient transitions; proper
projection formula and fusion restriction handle geometric
transitions. The common-support calculation
\eqref{eq:common-support}, together with convolution of
Satake families, proves pointwise monoidality. On disjoint
loci the construction is exterior product. The finite-label
fusion constraints are determined by intermediate extension
from the disjoint locus, as in \cite[\S7.3.16]{Lys}; this
gives their coherence with convolution in the specified
super normalization.

We use the ind-finite, or renormalized, Hecke category of
\cite[Definition~B.6.1]{BZSV}. Finite labels and
constructible base sheaves give kernels of bounded
Beilinson--Drinfeld Schubert support, descending to finite
jet quotients. They lie in the small finite-sheaf category
whose Ind completion defines the renormalized category.
By \cite[Lemma~2.6.12]{Lys}, the localized finite labels
compactly generate \(\cC_I\). Defining the action on
these labels and extending continuously gives the
renormalized action with the same constraints. For finitely
many kernels, a sufficiently thick sum of graph divisors
provides a common jet bound.

The second geometric input is
\cite[Proposition~16.2.8]{BZSV}. For smooth affine \(X\),
it supplies a faithful unital factorization module
\(\cM_I\) for the Hecke categories, with dualizing basic
object \(\delta_I=\delta_I^!\). Its ULA condition may be
read over \(R_I\): the relative rigidity of the Hecke category
and the equivalence of the two ULA conditions are
\cite[Lemma~16.2.6 and Proposition~B.9.7(2)]{BZSV}.
Here faithfulness means that exterior product on disjoint
loci is fully faithful. We use the algebraic assertion,
which includes the \'etale theory.

\begin{lemma}[Factorization of internal endomorphisms]
\label{lem:factorization-end}
The functor
\[
  T_I:\cC_I\longrightarrow\cM_I,
  \qquad c\longmapsto\Sat_I(c)*\delta_I
\]
has a continuous base-linear right adjoint \(T_I^R\).
The internal-endomorphism algebras
\begin{equation}
\label{eq:internal-end}
  B_I=T_I^R(\delta_I)
\end{equation}
form a unital Cartesian factorization associative algebra.
Its singleton fiber is \(A_X^!\), with the composition product.
\end{lemma}

\begin{proof}
Tensor product of rational representations has a continuous
right adjoint given by coinduction along the diagonal, with
its projection formula. The right adjoint to the unit
\(\cV\to\cC\) is taking invariants, and is continuous
because \(\check G\) is reductive in characteristic zero.
These adjunctions make \(\cC\) relatively rigid over
\(\cV\), also with super coefficients. By
\cite[Corollary~2.7.7 and Appendix~G]{Lys}, \(\cC_I\)
is relatively rigid over \(R_I\).

The categorical ULA property is the existence of a continuous
linear right adjoint to action on the given object
\cite[Definition~B.9.4]{BZSV}. For a category relatively
rigid over \(R_I\), being ULA over \(R_I\) is equivalent
to being ULA over that category
\cite[Proposition~B.9.7(2)]{BZSV}. Applied to \(\delta_I\),
this gives a continuous \(\cC_I\)-linear right adjoint,
and in particular a continuous \(R_I\)-linear right
adjoint. Composition of evaluations defines the unital
associative product on \(B_I\).

Extension of scalars along a diagonal preserves the adjunction
\(T_I\dashv T_I^R\). By the geometric and coefficient
identifications, its left member becomes \(T_J\).
Uniqueness of right adjoints, with their counits, gives
\begin{equation}
\label{eq:end-diagonal}
  \Delta^!B_I\simeq B_J
\end{equation}
as associative algebras. The continuity and base linearity
above ensure that this is an extension of the full adjunction.

On a disjoint-block locus, let
\(u:\boxtimes_b\cM_{I_b}\to\cM_I\) be exterior product.
There
\[
  T_I=u\circ\boxtimes_bT_{I_b},\qquad
  \delta_I=u(\boxtimes_b\delta_{I_b}).
\]
Full faithfulness of \(u\), followed by the tensor product
of the continuous adjunctions, identifies the evaluation
Hom functor with that represented by
\(\boxtimes_bB_{I_b}\). Consequently,
\begin{equation}
\label{eq:end-factorization}
  B_I|_{U_\beta}\simeq
       (\boxtimes_bB_{I_b})|_{U_\beta}.
\end{equation}
Evaluations and their compositions agree under this
identification. The coherence of
\eqref{eq:end-diagonal}--\eqref{eq:end-factorization}
follows from the same adjunction and factorization diagrams.

Insertion of an identity modification preserves the basic
object. Applying it to evaluations gives the unit-insertion
maps; on a separated additional point it inserts the identity
endomorphism. These maps agree with the associative unit.
At a singleton the algebra is therefore characterized by
\begin{equation}
\label{eq:shriek-end}
  \RHom_{\cC}(V,A_X^!)
  \simeq\RHom_{\cM^!}
        (\cT_V\star_!\delta_X^!,\delta_X^!),
\end{equation}
with multiplication given by composition of evaluations.
\end{proof}

\subsection{Comparison of the basic objects}

\begin{lemma}[Finite-object duality]
\label{lem:duality-comparison}
Let \(\sigma\) be the symmetric monoidal Cartan pullback
on \(\cC\) in the chosen Satake convention. There is an
equivalence of unital associative algebras
\begin{equation}
\label{eq:opposite-comparison}
  \PL_X\simeq\sigma^{-1}\bigl((A_X^!)^{\mathrm{op}}\bigr).
\end{equation}
\end{lemma}

\begin{proof}
Use contravariant Verdier duality on finite objects in the
!- and *-sheaf theories. At finite levels, ordinary Verdier
duality exchanges exceptional and ordinary pullback and
commutes with proper pushforward. These identities are
compatible with pro-smooth arc-tail transitions and bounded
ind-proper support transitions. They also apply to finite
jet quotient presentations and their equivariant descent
diagrams. This is the finite-object duality of
\cite[\S7.2.3 and \S\S B.7.1--B.7.2]{BZSV}.
The functor from renormalized sheaves to the usual
descent-defined sheaf category is fully faithful on the
small finite-sheaf category
\cite[Definition~B.6.1]{BZSV}; see also
\cite[Example~4.17]{LW}. Mapping complexes between these
objects can therefore be computed at finite levels.

Only the basic object and its convolutions by finitely many
finite Satake kernels enter the calculation. The ULA
property over the coefficient category makes the basic
object compact in the ind-finite category. Action by a
finite Satake kernel has continuous right adjoint given
by its rigid dual, and hence preserves compactness.
These objects have finite Schubert bounds and finite jet
presentations as in \cite[\S\S8.2.4 and 8.3.6]{BZSV}.
Duality carries \(\delta_X^!\) to the constant basic
object \(\delta_X^*=\delta_X\). This is a duality on
the finite objects used here; the comparison of complete
loop-space categories discussed in
\cite[Remark~7.5.4]{BZSV} is a separate assertion.

Orient the bounded Hecke correspondence as in
\eqref{eq:composition-product}. Verdier duality exchanges
exceptional pullback and tensor product with ordinary
pullback and tensor product. The pushforward is proper
on the Satake support and is self-dual. Thus
\begin{equation}
\label{eq:convolution-duality}
  \bD(\cT_V\star_!F)
  \simeq(\bD\cT_V)\star_*\bD F.
\end{equation}
The normalized Satake duality is
\begin{equation}
\label{eq:satake-duality}
  \bD\cT_V\simeq\cT_{\sigma(V^\vee)}.
\end{equation}
Equivalently, \(\sigma\) is the composite of the
contravariant tensor dualities on finite representations
and on the normalized Satake heart. Normalized Schubert
intersection complexes are Verdier self-dual, and their
ULA fusion gives the tensor and symmetry constraints in
\eqref{eq:satake-duality}, including the super parity.
Extending from finite labels gives a symmetric monoidal
autoequivalence of \(\cC\).

Applying duality to \eqref{eq:plancherel-definition},
then rigid adjunction for the Satake kernel, yields
\begin{equation}
\label{eq:duality-multiplicities}
\begin{aligned}
  \RHom_{\cC}(V,\PL_X)
  &\simeq\RHom_{\cM^!}
       (\delta_X^!,\cT_{\sigma(V^\vee)}\star_!\delta_X^!)\\
  &\simeq\RHom_{\cM^!}
       (\cT_{\sigma(V)}\star_!\delta_X^!,\delta_X^!)\\
  &\simeq\RHom_{\cC}(\sigma(V),A_X^!).
\end{aligned}
\end{equation}
Finite representations and their parity reversals generate
\(\cC\), so this identifies the underlying objects in
\eqref{eq:opposite-comparison}.

To check multiplication, transpose an evaluation by
\(\bD\) and take its rigid mate. This is the map in
\eqref{eq:duality-multiplicities}. Transposition reverses
a composite of evaluations. Taking rigid mates moves the
dual label to the input; symmetric interchange identifies
the reversed tensor order with the opposite multiplication,
with sign \eqref{eq:symmetry}. The same calculation for
any finite sequence of evaluations respects associativity
coherences, and the identity evaluation transposes to the
identity. Hence \eqref{eq:opposite-comparison} is an
equivalence of unital \(E_1\)-algebras.
\end{proof}

\subsection{The coefficient-valued family}

Apply Lemma~\ref{lem:total-tensor} to
Lemma~\ref{lem:factorization-end}, and put
\begin{equation}
\label{eq:coefficient-family}
  F_I=Q_I(B_I)\in\Shv(\Sigma^I)\otimes\cC.
\end{equation}
This is a unital factorization family with compatible
associative products. Translation and the coordinate
\(t-x\) trivialize its unary construction, giving
\begin{equation}
\label{eq:unary-family}
  F_{\{1\}}=\omega_{\mathbb A^1}\otimes A_X^!.
\end{equation}
For a diagonal stratum whose blocks are indexed by \(J\),
equations \eqref{eq:end-diagonal} and
\eqref{eq:end-factorization} identify the exceptional
restriction with
\begin{equation}
\label{eq:stratum-family}
  \omega_{\Conf_J(\mathbb A^1)}\otimes(A_X^!)^{\otimes J}.
\end{equation}
Here \(\Conf_J\) is the ordered configuration space of
distinct \(J\)-labelled points. In particular, these
exceptional restrictions have constant coefficients.
% End factorization.tex

% Begin comparison.tex
\section{Comparison for sheaves generated by diagonals}
\label{sec:comparison}

To extract topological operations from
\eqref{eq:coefficient-family}, we compare a specific class of
sheaves across characteristic and with Betti sheaves. The comparison
retains the full factorization diagrams and their associative
products.

Let \(p=\operatorname{char}F\), and let \(S\) be the strict
henselization of \(\Spec\mathbb Z_{(p)}\). Choose a geometric
generic point and an embedding of its algebraic closure into
\(\mathbb C\). Choose compatible trivializations of the
prime-to-\(p\) Tate lines over \(S\). We use the resulting
comparison only geometrically, without a Frobenius-equivariance
assertion.

For a finite set \(I\), let \(\cD_I(S)\) be the Ind completion
of the thick category generated by
\[
  e_L=i_{L*}k_L,
  \qquad L\subset\mathbb A^I_S
  \text{ a closed coordinate diagonal}.
\]
The diagonal for the discrete partition is the entire space.
Use the analogous notation for the two geometric fibers and
for Betti sheaves on \(\mathbb C^I\).

\begin{lemma}[Diagonal comparison]
\label{lem:diagonal-comparison}
Restriction to either geometric fiber and the
characteristic-zero \'etale--Betti comparison give equivalences
on these generated categories. In all arities together, they
intertwine exterior products, exceptional tensor products,
diagonal exceptional pullbacks and proper pushforwards,
permutations, partial-disjoint open restrictions and extensions,
and exceptional pullbacks along coordinate projections.
They preserve the exceptional fiber at the origin and all
composition and base-change coherences. The same assertions
hold after extension of coefficients to \(\cC\).
\end{lemma}

\begin{proof}
For closed diagonals \(L,M\), put \(N=L\cap M\) and
\(c=\operatorname{codim}_M N\). Proper exceptional base
change and purity for a regular linear immersion give
\begin{equation}
\label{eq:diagonal-homs}
  \RHom(e_L,e_M)
  \simeq R\Gamma(N,k)[-2c](-c)
  \simeq k[-2c](-c).
\end{equation}
The last step is affine-space homotopy invariance over the
strictly henselian local base. In this linear situation,
purity also follows by iterating codimension-one localization
and the Kummer class. Intersections of coordinate diagonals
are smooth over \(S\), and restriction preserves their
fundamental classes and codimensions. Thus the actual
restriction maps on \eqref{eq:diagonal-homs} are
quasi-isomorphisms. This proves full faithfulness on the
generators, hence on their thick closure and Ind completion.
Every target generator is in the image. The
characteristic-zero \'etale--Betti comparison preserves the
same fundamental classes and gives the Betti equivalence.

For compatibility, use the common diagram of functors over
\(S\). Exterior product and proper diagonal pushforward
preserve the generators. If
\(d:D\hookrightarrow\mathbb A^I\) is a diagonal, then
\begin{equation}
\label{eq:diagonal-pullback}
  d^!e_L\simeq
  i_{L\cap D,D*}k[-2c'](-c'),
  \qquad c'=\operatorname{codim}_L(L\cap D).
\end{equation}
The same formula applies at the origin. Exceptional
pullback along a coordinate projection is smooth pullback
with its relative dimension shift. Exceptional tensor
product is exterior product followed by diagonal exceptional
pullback. Over \(S\), use the relative dualizing
normalization, with unit \(k[2|I|](|I|)\); this restricts
to the dualizing unit on each fiber. All finite generators
are ULA over \(S\), and relative Verdier duality preserves
their thick closure.

If \(j:U\hookrightarrow\mathbb A^I\) is a
partial-disjoint open, its complement is a finite union of
coordinate diagonals. The finite closed-cover \v Cech
resolution and localization express \(j_!j^*e_L\) in
the thick diagonal category. Verdier duality gives the same
assertion for \(j_*j^*e_L\). These constructions compare
the adjunctions and localization triangles as well.
Finite cohomological dimension allows the formulas to
commute with arbitrary coefficient complexes and Ind
extensions. All transformations come from the common
six-functor diagram, so their higher composition
compatibilities are preserved. Tensoring the diagram with
\(\cC\) proves the coefficient-valued assertion.
\end{proof}

\begin{lemma}[A membership criterion]
\label{lem:diagonal-membership}
A coefficient-valued sheaf whose exceptional restrictions
to all diagonal strata are constant complexes belongs to
\(\cD_I\otimes\cC\).
\end{lemma}

\begin{proof}
Let \(u:S_\pi\hookrightarrow L_\pi\) be a diagonal
stratum in its closed diagonal closure. The finite
localization construction in the preceding proof expresses
\(u_!k\) in diagonal generators, and Verdier duality
does the same for \(u_*k\). Tensoring with any constant
coefficient complex gives the assertion for that stratum,
with any dualizing normalization.

For a sheaf \(F\), use the localization triangle
\[
  i_*i^!F\longrightarrow F\longrightarrow j_*j^*F
    \xrightarrow{+1},
\]
where \(j\) is the union of the top open strata and
\(i\) is its closed complement. Induction on the finite
number of strata proves the claim. This argument uses the
exceptional restrictions in the hypothesis.
\end{proof}

By \eqref{eq:stratum-family}, these lemmas apply to
\(F_I\). They transport the family, its associative
products, and its Cartesian, factorization, and unit
diagrams to Betti sheaves
\begin{equation}
\label{eq:betti-family}
  F_I^{\mathrm B}\in\Shv(\mathbb C^I)\otimes\cC.
\end{equation}
The singleton exceptional fiber is still \(A_X^!\),
with the composition product of \eqref{eq:shriek-end}.
% End comparison.tex

% Begin commutativity.tex
\section{Fusion, convolution, and commutativity}
\label{sec:commutativity}

\subsection{Extracting the topological algebra}

We give the compact-support construction explicitly to
track the associative product and unit. The topological
Ran space below parametrizes finite nonempty subsets.

\begin{lemma}[Fusion and convolution]
\label{lem:extraction}
Let \(F=(F_I)\) be a \(\cC\)-valued unital Cartesian
factorization associative family on powers of \(\mathbb C\),
belonging to the categories generated by coordinate diagonals.
Suppose that its unary object is
\(\omega_{\mathbb C}\otimes A\), with its given
associative product, and its exceptional fiber at a
configuration of \(m\) distinct points is \(A^{\otimes m}\).
Then \(A\), with that product, is the specified associative
restriction of a unital \(E_3\)-algebra in \(\cC\).
\end{lemma}

\begin{proof}
Write \(q_I:\mathbb C^I\to\Ran(\mathbb C)\).
Let \(\mathsf{Fin}_{\mathrm{surj}}\) be the category of
finite nonempty sets and surjections. Define
\begin{equation}
\label{eq:ran-cosheaf}
  K_F(U)=\colim_{I\in\mathsf{Fin}_{\mathrm{surj}}^{\mathrm{op}}}
      R\Gamma_c(q_I^{-1}U,F_I).
\end{equation}
The transition maps are induced by the counits
\[
  \Delta_!F_J=\Delta_!\Delta^!F_I\longrightarrow F_I.
\]
On each finite-dimensional power, compactly supported
sections give the covariant Verdier construction from
!-sheaves to cosheaves. This construction takes no linear
dual of the coefficient object \(A\).

\smallskip
\noindent\emph{Contraction on disk sectors.}
Choose disjoint round disks \(D_1,\ldots,D_m\), with
centers \(z_1,\ldots,z_m\). Consider the independent
basic open
\[
  U=\Ran(D_1)\star\cdots\star\Ran(D_m),
\]
whose configurations meet every disk. Its preimage in
\(\mathbb C^I\) is the disjoint union of sectors
\(W_\eta=\prod_aD_a^{\eta^{-1}(a)}\), indexed by
surjections \(\eta:I\twoheadrightarrow[m]\).
Let \(z_\eta\) be the tuple of their centers. The
closed-point counit induces
\begin{equation}
\label{eq:contraction}
  i_{z_\eta}^!F_I\longrightarrow R\Gamma_c(W_\eta,F_I).
\end{equation}
This is an equivalence on the entire category generated
by diagonals. Indeed, for a generator \(e_L\), both
sides vanish if \(L\) identifies points in different
disks. Otherwise \(L\cap W_\eta\) is a product of
disks containing its center, and both sides are
\(k[-2\dim_{\mathbb C}L]\); the map is the
orientation-class isomorphism. Both functors are
continuous, so the calculation extends to arbitrary
coefficient complexes and Ind objects.

The Cartesian and factorization identifications make the
left-hand side \(A^{\otimes m}\). Under these
identifications, the transition counits in
\eqref{eq:ran-cosheaf} are identity maps. The indexing
category of pairs \((I,\eta:I\twoheadrightarrow[m])\),
with opposite-surjection transitions, has initial object
\(([m],\mathrm{id})\). Its nerve is contractible, so
\begin{equation}
\label{eq:ran-basic-value}
  K_F(U)\simeq A^{\otimes m}.
\end{equation}
Inclusions of basic opens preserving the clusters induce
equivalences: on generators, they preserve the same
orientation class. Exterior factorization and
compact-support K\"unneth give
\(K_F(U\star V)\simeq K_F(U)\otimes K_F(V)\)
for independent opens, with refinement coherences.

\smallskip
\noindent\emph{Descent.}
Each \(q_I^{-1}U\) is an open subset of the
finite-dimensional manifold \(\mathbb C^I\).
The coefficient-valued compact-support cosheaf at this
level satisfies hyperdescent for unbounded complexes.
Pulling a hypercover of a Ran open back to each finite
level, and exchanging its colimit with the colimit in
\eqref{eq:ran-cosheaf}, proves hyperdescent for \(K_F\).
A uniform bound on the finite-level dimensions is not
required. This is the finite-level-to-Ran argument of
\cite[Proposition~4.5, Theorem~4.6, and Appendix~B.31--B.33]{Noc};
here the finite-level input also follows from the
contraction calculation and finite covering dimension.

\smallskip
\noindent\emph{The associative product.}
Compactly supported sections have the colax comparison
\begin{equation}
\label{eq:colax-comparison}
  R\Gamma_c(W,F\otimes^!G)
    \longrightarrow R\Gamma_c(W,F)\otimes R\Gamma_c(W,G),
\end{equation}
induced by the proper diagonal counit. On a basic disk
sector it is an equivalence. Naturality of
\eqref{eq:contraction} gives the commutative diagram
\begin{equation}
\label{eq:contraction-monoidality}
\begin{tikzcd}[column sep=small,row sep=large]
  i_z^!(F\otimes^!G) \arrow[r,"\sim"] \arrow[d,"\sim"']
    & i_z^!F\otimes i_z^!G \arrow[d,"\sim"]\\
  R\Gamma_c(W,F\otimes^!G) \arrow[r]
    & R\Gamma_c(W,F)\otimes R\Gamma_c(W,G).
\end{tikzcd}
\end{equation}
The vertical maps are the contraction equivalences.
After the Ran colimit the same conclusion follows from
\eqref{eq:ran-basic-value}. Inverting this canonical
colax equivalence makes extraction strong symmetric
monoidal on basic opens. Its associativity, symmetry,
and compatibility with inclusions are inherited from
the counits. In particular,
\[
  R\Gamma_c(D,\omega_D\otimes A)\simeq A
\]
on one disk, with its original product and without an
additional shift.

\smallskip
\noindent\emph{Units and additivity.}
The constructible-factorization-cosheaf theorem
\cite[Theorem~5.5.4.10]{HA}, applied first with target
\(\cC\), gives an \(E_2^{\mathrm{nu}}\)-algebra.
By \eqref{eq:contraction-monoidality}, its disk
operations preserve the associative products, with
all their coherences. Hence the action lifts to the
category of unital associative algebras in \(\cC\).
The supplied insertion of an identity-labelled point
gives a quasi-unit: fusing it with an occupied disk
acts as the identity, by the unit-insertion
compatibilities of the original family. This is the
associative identity endomorphism, so it is compatible
with the associative structure.

The quasi-unit theorem \cite[Theorem~5.4.4.5]{HA}
therefore gives a unital \(E_2\)-algebra in unital
associative algebras. Dunn additivity
\cite[Theorem~5.1.2.2]{HA} identifies it with a unital
\(E_3\)-algebra. The product calculation above identifies
its specified associative restriction with \(A\).
\end{proof}

Apply Lemma~\ref{lem:extraction} to
\eqref{eq:betti-family}. Its hypotheses follow from
\eqref{eq:end-diagonal}--\eqref{eq:end-factorization}
and \eqref{eq:unary-family}--\eqref{eq:stratum-family},
transported by Lemma~\ref{lem:diagonal-comparison}.
We obtain a unital \(E_3\)-algebra in rational super
\(\check G\)-representations whose specified
\(E_1\)-restriction is \(A_X^!\).

\subsection{Relative formality}

\begin{lemma}[A commutative model]
\label{lem:relative-formality}
Over a characteristic-zero field, the associative
restriction of a unital \(E_3\)-algebra in complexes of
rational super representations of a reductive group admits
a unital commutative dg model.
\end{lemma}

\begin{proof}
The rational relative formality theorem of
Fresse--Willwacher \cite[Theorem~D\(^{\prime}\)]{FW}
identifies, for \(n\ge3\), the arrow of unitary chain
operads
\[
  C_*(D_1^+;\mathbb Q)\longrightarrow C_*(D_n^+;\mathbb Q)
\]
with the arrow
\begin{equation}
\label{eq:formal-operad-arrow}
  \Ass^+\longrightarrow\Pois_n^+
\end{equation}
through a zigzag of weak equivalences of operad arrows.
The latter factors as
\(\Ass^+\to\Com^+\to\Pois_n^+\).
Thus the theorem compares the inclusion as well as the
two operads. The formulation using augmented
\(\Lambda\)-operads encodes the zero-arity operation
and input-deletion maps; by \cite[Introduction and
Theorem~B\(^{\prime}\)]{FW}, it is equivalent to the
unitary formulation with arity-zero value \(\mathbb Q\).
It requires no augmentation from an algebra to \(\mathbb Q\).

Tensor this relative diagram with \(k\) and take \(n=3\).
Exactness of scalar extension preserves its weak
equivalences. Give each operation complex trivial group
action and even super parity. Transport of algebra
structures along this diagram takes place inside the
symmetric monoidal category of rational super
representations. If \(A\) is the specified associative
restriction of the original \(E_3\)-algebra, it gives
a \(\Pois_3^+\)-algebra \(B\) and an equivalence
\begin{equation}
\label{eq:associative-formality}
  A\simeq B|_{\Ass^+}
  \qquad\text{in }\Alg_{E_1}(\cC).
\end{equation}
Restriction along \(\Com^+\to\Pois_3^+\) gives
\(B\) its unital commutative multiplication.

Characteristic-zero operadic rectification provides a
strict dg presentation: rational representations of a
reductive group are semisimple, tensor product is exact,
and finite-group coinvariants are exact. These properties
apply to unbounded complexes with the super symmetry.
The relative formality diagram ensures that
\eqref{eq:associative-formality} preserves the original
product and unit. Since transport takes place in the
representation category, it also preserves the group
action.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:commutativity}]
Lemmas~\ref{lem:total-tensor} and~\ref{lem:factorization-end}
construct the unital factorization associative family
\eqref{eq:coefficient-family}, with singleton algebra
\(A_X^!\). By \eqref{eq:stratum-family} and
Lemmas~\ref{lem:diagonal-comparison}--\ref{lem:diagonal-membership},
its complete structure transports to Betti sheaves.
Lemma~\ref{lem:extraction} enhances the composition
algebra \(A_X^!\) to a unital \(E_3\)-algebra.
By Lemma~\ref{lem:relative-formality}, there is
a unital commutative dg algebra \(B\) and an
associative equivalence \(A_X^!\simeq B\).

By Lemma~\ref{lem:duality-comparison},
\[
  \PL_X\simeq\sigma^{-1}\bigl((A_X^!)^{\mathrm{op}}\bigr)
  \simeq\sigma^{-1}(B^{\mathrm{op}})
  =\sigma^{-1}(B).
\]
A commutative algebra equals its opposite, and
\(\sigma^{-1}\) is symmetric monoidal. Taking
\(B_X=\sigma^{-1}(B)\) proves the theorem, with
the product, unit, rational \(\check G\)-action,
and super symmetry specified in
\eqref{eq:plancherel-definition}--\eqref{eq:composition-product}.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:hyperspherical}]
In the stated conventions, \(X\) is smooth and affine.
Section~\ref{subsec:structure} gives
\(X\simeq G\times^H Y\), with \(H\) connected
reductive and \(Y\) an \(H\)-representation.
Theorem~\ref{thm:main} proves flatness of
\(\PL_{X,\hbar}\), and
Theorem~\ref{thm:commutativity} gives the commutative
dg model of \(\PL_X\).
\end{proof}
% End commutativity.tex

% Bibliography embedded from the amsalpha BibTeX output.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{BZSV24}

\bibitem[BFN18]{BFN}
Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima, \emph{Towards a
  mathematical definition of {Coulomb} branches of $3$-dimensional {$\mathcal
  N=4$} gauge theories, {II}}, Adv. Theor. Math. Phys. \textbf{22} (2018),
  no.~5, 1071--1147, \href{https://arxiv.org/abs/1601.03586}{arXiv:1601.03586}.

\bibitem[BZSV24]{BZSV}
David Ben-Zvi, Yiannis Sakellaridis, and Akshay Venkatesh, \emph{Relative
  {Langlands} duality}, 2024,
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\bibitem[FW20]{FW}
Benoit Fresse and Thomas Willwacher, \emph{The intrinsic formality of
  {$E_n$}-operads}, J. Eur. Math. Soc. (JEMS) \textbf{22} (2020), no.~7,
  2047--2133, Numbering follows
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\bibitem[Lur17]{HA}
Jacob Lurie, \emph{Higher algebra}, 2017, September 2017 version,
  \href{https://www.math.ias.edu/~lurie/papers/HA.pdf}{available from the
  author's website}.

\bibitem[Lus83]{Lus}
George Lusztig, \emph{Singularities, character formulas, and a $q$-analog of
  weight multiplicities}, Ast\'{e}risque \textbf{101--102} (1983), 208--229,
  \href{https://www.numdam.org/item/AST_1983__101-102__208_0/}{Numdam}.

\bibitem[LW26]{LW}
Shurui Liu and Zeyu Wang, \emph{Higher period integrals and derivatives of
  {$L$}-functions}, 2026,
  \href{https://arxiv.org/abs/2504.00275v3}{arXiv:2504.00275v3}; first posted
  in 2025.

\bibitem[Lys24]{Lys}
Sergey Lysenko, \emph{Note on factorization categories}, 2024, Revised July
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\bibitem[Noc20]{Noc}
Guglielmo Nocera, \emph{A model for the {$E_3$} fusion-convolution product of
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\bibitem[Zhu17]{Zhu}
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\end{thebibliography}

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