\documentclass[11pt]{amsart} \usepackage{amsmath,amssymb,mathrsfs,indentfirst} \usepackage[pdfusetitle,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \urlstyle{same} \renewcommand{\keywordsname}{Keywords} \newtheorem{maintheorem}{Theorem} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \setlength{\parindent}{15pt} \setlength{\emergencystretch}{1em} \raggedbottom \title[Singular supports of Lusztig sheaves]{The singular-support characterization of Lusztig sheaves} \author{Qiyue Tang} \date{November 8, 2026} \keywords{Lusztig sheaves, singular support, quiver varieties, Hochschild homology} \subjclass[2020]{Primary 16G20. Secondary 14F10, 17B37} \begin{document} \begin{abstract} We prove that for every finite loop-free quiver and every dimension vector, a simple equivariant perverse sheaf with complex coefficients on the quiver representation space is a Lusztig sheaf if and only if its singular support lies in Lusztig's nilpotent variety. \end{abstract} \maketitle \section*{Introduction} Ringel's Hall algebra construction relates the representation theory of Dynkin quivers to one half of a quantum group \cite{Ringel}. Multiplication counts subrepresentations with prescribed subobject and quotient. Lusztig combined this approach with the geometric construction of character sheaves. He used induction of constructible complexes along correspondences of quiver representations with invariant subspaces \cite[\S0.3 and \S9]{LusztigQuivers}. Iterating induction from the one-dimensional representations at the vertices gives proper maps from varieties of representations with invariant graded flags. The simple perverse constituents of their direct images realize the canonical basis. This construction applies to the symmetric Cartan datum of any finite loop-free quiver, with parallel arrows counted in its Cartan matrix. There is also a cotangent description of these sheaves. The cotangent bundle of a quiver representation space is itself a space of representations, obtained by adjoining a reverse to each arrow. The equations of the zero moment fiber are the preprojective relations. Imposing nilpotence on the doubled representation cuts out the Lagrangian variety used by Lusztig. In his 1990 paper, Lusztig proposed attaching to each irreducible component of this variety a simple equivariant perverse sheaf whose characteristic variety contains that component and is contained in the nilpotent variety. He proposed that this correspondence be a bijection for a bipartite graph \cite[\S10.1(a)]{LusztigII}. He also asked for a convolution realization of the quantum group and for orientation independence of the resulting basis, and illustrated the proposal with affine examples \cite[\S\S10.3, 10.5--10.6]{LusztigII}. His 1991 construction proves that every flag constituent has singular support in the nilpotent variety \cite[Corollary~13.6]{LusztigQuivers}. His example of the two-arrow Kronecker quiver in dimension \((2,2)\) has two flag constituents with full support but different local systems on a dense open subset, and their singular supports differ \cite[\S13.8]{LusztigQuivers}. Kashiwara and Saito identify the irreducible components with the crystal of the same canonical basis \cite[Theorem~5.3.2 and \S6.1]{KS}. Thus the number of components is known in each dimension vector. The recognition question asks whether the singular-support condition characterizes the flag constituents among all equivariant simple perverse sheaves. In Dynkin type, the representation space has finitely many orbits, each with connected stabilizer. The stabilizer of a representation is the group of units in its endomorphism algebra, an open subset of a vector space. The simple equivariant perverse sheaves are the intersection cohomology extensions of the constant local systems on those orbits. Gabriel's classification and the canonical-basis count give the same number of orbits and flag constituents \cite[Proposition~4.8]{Webster}. In affine type the regular representations occur in families. Lusztig's description of the flag constituents uses strata of regular representations and local systems obtained by permuting distinct homogeneous summands \cite[\S6]{LusztigAffine}. In the cyclic case he also characterized the sheaves attached to aperiodic nilpotent orbits by a simultaneous support condition on a sheaf and its Fourier transform \cite[\S2.13]{LusztigAntiorbital}. Hennecart proved the microlocal converse for affine quivers. His proof uses the stratification by preprojective, regular and preinjective summands and cyclic-quiver models near the nonhomogeneous tubes \cite[Theorem~1.1 and \S\S6--9]{Hennecart}. He formulates the recognition statement for all loop-free quivers in \cite[Conjecture~1.2]{Hennecart}. Webster gives a general reduction to a large framed quiver variety \cite[Proposition~4.8]{Webster}. The nilpotent singular-support condition becomes support on the fiber over the cone point. His proof then invokes a characteristic-cycle independence theorem attributed to Baranovsky and Ginzburg \cite[Theorem~4.9]{Webster}. Hennecart explains how results of Bezrukavnikov and Losev and of Losev supply the characteristic-cycle input in the affine case \cite[\S1.6]{Hennecart}. Baranovsky and Ginzburg subsequently stated the general independence result for finite-dimensional simple modules under derived localization in \cite[Corollary~11.4]{BG}, where the details of the underlying Chern character theorem are deferred to a separate paper. A different general proof by Fang was withdrawn \cite{Fang}. Let \(Q\) be a finite quiver with vertex set \(I\) and arrow set \(Q_1\). Write \(s(a)\) and \(t(a)\) for the tail and head of \(a\in Q_1\), and assume \(s(a)\ne t(a)\). For a dimension vector \(\mathbf v=(v_i)_{i\in I}\) of nonnegative integers, choose complex vector spaces \(V_i\) with \(\dim V_i=v_i\). The representation space and its base-change group are \[ E=\bigoplus_{a\in Q_1}\operatorname{Hom}(V_{s(a)},V_{t(a)}), \qquad G=\prod_{i\in I}\operatorname{GL}(V_i). \] We take \(\operatorname{GL}(0)=\{1\}\). For \(g=(g_i)\) and \(x=(x_a)\), the coordinate of \(gx\) at \(a\) is \(g_{t(a)}x_a g_{s(a)}^{-1}\). Let \(\overline Q\) be the doubled quiver, with one additional arrow \(a^*:t(a)\to s(a)\) for every \(a\in Q_1\). Write \(y_a:V_{t(a)}\to V_{s(a)}\) for the map on \(a^*\). The pairing \(\sum_a\operatorname{Tr}(y_a x_a)\), with trace on \(V_{s(a)}\), identifies \(y=(y_a)\) with the cotangent coordinate. Identify the dual of the Lie algebra of \(G\) with \(\bigoplus_i\operatorname{End}(V_i)\) by trace. The cotangent moment map \(\mu_Q:T^*E\to\bigoplus_i\operatorname{End}(V_i)\) is given by \[ (\mu_Q)_i(x,y)=\sum_{t(a)=i}x_a y_a-\sum_{s(a)=i}y_a x_a. \] Indeed, the infinitesimal action of \((z_i)_{i\in I}\) at \(x\) has coordinates \(z_{t(a)}x_a-x_a z_{s(a)}\). Pairing with \(y\) and cycling the trace gives this formula. Let \(\Lambda_{\mathbf v}\) be the subvariety of \(\mu_Q^{-1}(0)\) on which all sufficiently long doubled paths act by zero. Here a path acts by the composition of its arrow maps in traversal order. A complete graded flag \(L_\bullet\) in \(\bigoplus_i V_i\) is an increasing chain \(0=L_0\subset L_1\subset\cdots\subset L_{\sum_i v_i}=\bigoplus_i V_i\) of graded subspaces with one-dimensional successive quotients. The path condition is equivalent to the existence of such a flag with each doubled arrow carrying \(L_j\) into \(L_{j-1}\). To see this, let the term of index \(k\geq0\) of a decreasing filtration be the graded span of images of paths of length \(k\), including the identity paths at \(k=0\). These terms decrease, since each path image lies in the image of its terminal subpath of length one less. Every arrow maps the term of index \(k\) to that of index \(k+1\). A refinement of the filtration to a complete graded flag, read in increasing order, has the asserted property. Conversely a composition of \(\sum_i v_i\) arrows strictly lowers through the entire flag. For each sequence of vertices of the successive quotients, let the corresponding flag variety parametrize all complete graded flags of that type. The incidence variety of pairs \((x,L_\bullet)\) with \(x\) preserving the flag is a vector bundle over that flag variety. Its projection to \(E\) is proper because it is closed in the product of \(E\) with the projective flag variety. A \emph{Lusztig sheaf} is a simple \(G\)-equivariant perverse sheaf that occurs, up to cohomological shift, as a direct summand of the direct image of the constant complex under one of these projections. Allowing flag quotients of larger dimension at a single vertex gives the same simple constituents. In fact, every refinement within a single-vertex quotient is preserved, since the quiver has no loops. The forgetful map between the incidence varieties is therefore a relative product of full flag varieties. Its fibers are connected, and its direct image contains the constant complex as a summand by the decomposition theorem. All perverse sheaves below are algebraically constructible, of middle perversity, with complex coefficients. For such a sheaf \(\mathcal F\) we write \(SS(\mathcal F)\subseteq T^*E\) for its singular support, equivalently the characteristic variety of its regular holonomic module over the sheaf \(\mathcal D_E\) of algebraic differential operators under Riemann--Hilbert. \begin{maintheorem}\label{thm:main} For every finite loop-free quiver \(Q\) and dimension vector \(\mathbf v\), a simple \(G\)-equivariant perverse sheaf \(\mathcal F\) on \(E\) is a Lusztig sheaf if and only if \[ SS(\mathcal F)\subseteq\Lambda_{\mathbf v}. \] \end{maintheorem} We use the large-framing construction to compare the number of simple sheaves with nilpotent singular support to the number of components of the central fiber. Etingof and Schedler use zeroth Hochschild homology to bound the number of finite-dimensional simple modules of quantizations \cite[Corollary~1.11]{ESfinite}. Here the bound comes from a symplectic action of weight one. Its attracting strata reduce to cotangent bundles of smooth projective fixed components. We use the categorical cell decomposition of Bellamy, Dodd, McGerty and Nevins \cite{BBDM} to bound the zeroth Hochschild homology by the sum of the contributions from these components. Each contribution is one-dimensional by a calculation for twisted differential operators. The resulting bound equals the number of Lusztig sheaves. \section{Changing the orientation}\label{sec:orientation} \begin{lemma}\label{lem:orientation} Reversing one arrow gives a bijection of isomorphism classes of simple equivariant perverse sheaves with singular support in the respective nilpotent varieties. It carries Lusztig sheaves to Lusztig sheaves. \end{lemma} \begin{proof} Fix \(a\in Q_1\). The cotangent lift of dilation of the \(a\)-coordinate is \((x_a,y_a)\mapsto (u x_a,u^{-1}y_a)\) for \(u\in\mathbb C^*\). It preserves the moment equations. Each doubled path is rescaled by a nonzero scalar, so nilpotence is preserved. Since the acting torus is connected, it preserves each irreducible component of \(\Lambda_{\mathbf v}\). Each component is conic and Lagrangian \cite[Proposition~5.3.3]{KS}. For a conic Lagrangian in a cotangent bundle, contraction of the symplectic form with the fiber Euler field gives the tautological one-form, up to the sign convention for the symplectic form. The Euler field is tangent to the conic subvariety, so this one-form vanishes on its smooth locus. At a general smooth point where projection to the base is smooth onto its image, the cotangent coordinate thus annihilates the tangent space to that image. Equality of dimensions then identifies the component with the closure of this conormal bundle. Its projection is closed, since for a closed fiberwise conic subset the projection equals its intersection with the zero section, identified with \(E\). It is invariant under dilation of the \(a\)-coordinate. A perverse sheaf with singular support in \(\Lambda_{\mathbf v}\) has Lagrangian singular support, hence its singular support is a union of these components. Simplicity for a \(G\)-equivariant perverse sheaf implies simplicity of the underlying perverse sheaf. On an invariant smooth dense stratum of the support, an equivariant local system is a local system on the quotient stack, and the fundamental group of the stratum surjects onto that of the quotient stack because \(G\) is connected. Surjectivity also makes the forgetful functor on equivariant local systems fully faithful. On a common invariant smooth dense stratum of the support, an isomorphism of the underlying local systems therefore lifts to an equivariant isomorphism, which extends by intermediate extension. Thus forgetting equivariance distinguishes isomorphism classes of simple equivariant perverse sheaves. Now \cite[Corollary~D.2]{Hennecart} gives monodromicity for the \(a\)-dilation, meaning that the cohomology sheaves restrict to local systems on its orbits. Use the space of the other arrow maps as a base and the \(a\)-arrow space as the fiber of a trivial vector bundle. The relative Fourier--Sato transform is the pullback to the fiber product with the dual bundle, restricted to the closed half-space where the real part of the pairing is nonpositive, followed by direct image with proper supports to the dual bundle. We shift it by the complex rank of the bundle. It is a \(G\)-equivariant perverse equivalence on monodromic sheaves \cite[Appendix~D]{Hennecart}. On singular supports its action is \((x_a,y_a)\mapsto(y_a,-x_a)\), with the other coordinates unchanged. The moment contributions at \(t(a)\) and \(s(a)\) after reversal are \(x_a y_a\) and \(-y_a x_a\), respectively. The correspondence also preserves vanishing of long paths. Thus the transformed sheaf has singular support in the new nilpotent variety. Any simple equivariant perverse subobject has singular support contained in that of the transform by additivity and effectivity of characteristic cycles, so it too is monodromic. Applying the inverse perverse equivalence shows that the transform is simple. Applying the same argument to the inverse gives the bijection. Over a flag, the \(a\)-arrow maps preserving the flag have matrix entries only from later source positions to earlier target positions, in bases ordered by the graded flag. The two endpoints are distinct, so the source and target positions cannot coincide. The reverse entries paired with these entries by trace are exactly the entries forbidden by flag preservation for the reverse arrow. Thus the trace annihilator is the subspace of flag-preserving reverse maps. For a complex subspace of dimension \(d\) in a fiber of rank \(r\), the Fourier transform of its constant complex extended by zero is the constant complex on its annihilator, shifted by \(r-2d\). Indeed, in the half-space description the fiber over a point of the annihilator is the full real \(2d\)-dimensional vector space, while elsewhere it is a closed real half-space in that vector space, whose compactly supported cohomology is zero. The relative formula holds for the subbundles over the flag variety. Fourier transformation commutes with the proper projection forgetting the flag by base change and composition for direct images with proper supports. It therefore carries the flag complexes to the flag complexes for the reversed orientation, up to shifts. The equivalence on simple constituents gives the last assertion. \end{proof} Choose a total order on the vertices and reverse each arrow pointing from a larger to a smaller vertex. Successive applications of Lemma~\ref{lem:orientation} reduce Theorem~\ref{thm:main} to the resulting acyclic orientation. \section{Hochschild homology of twisted differential operators} Let \(F\) be a smooth complex variety, with structure sheaf \(\mathcal O_F\) and tangent sheaf \(T_F\). A Picard algebroid is a Lie algebroid \(\mathcal P\) with an exact sequence \(0\to\mathcal O_F\to\mathcal P\to T_F\to0\) of \(\mathcal O_F\)-modules. The projection to \(T_F\) is the anchor, the kernel is abelian, and the bracket with a kernel section \(f\) is the derivative of \(f\) by the anchor. Its enveloping algebra, with the section \(1\) of the kernel identified with the algebra unit, is a sheaf of twisted differential operators. Write \(\mathcal D_F^\tau\) for this sheaf, where \(\tau\) records the class of the Picard algebroid. Its order filtration gives degree zero to functions and degree one to lifts of vector fields. The Poincar\'e--Birkhoff--Witt theorem identifies the associated graded sheaf with \(\operatorname{Sym}_{\mathcal O_F}T_F\). We write \(\operatorname{Perf}(\mathcal D_F^\tau)\) for the dg category of perfect left modules, that is, complexes locally quasi-isomorphic to bounded complexes of finitely generated projective modules over \(\mathcal D_F^\tau\). Write \(HH(\mathcal C)\) for the ordinary Hochschild chain complex, with tensor products over \(\mathbb C\), of a differential graded category \(\mathcal C\), and \(HH_k(\mathcal C)\) for its homology in degree \(k\). For an algebra \(D\), the same notation denotes Hochschild homology of \(D\), identified by Morita invariance with that of \(\operatorname{Perf}(D)\). \begin{lemma}\label{cnt:tdo} If \(F\) is smooth, connected, and projective of dimension \(n\), then \[ HH_0\bigl(\operatorname{Perf}(\mathcal D_F^\tau)\bigr) \cong H^{2n}_{\mathrm{dR}}(F)\cong\mathbb C \] for every \(\mathcal D_F^\tau\). Here \(H^j_{\mathrm{dR}}\) denotes the degree-\(j\) hypercohomology of the algebraic de Rham complex. \end{lemma} \begin{proof} Let \(U\subset F\) be affine and set \(D=\Gamma(U,\mathcal D_F^\tau)\). Quasi-coherent left \(\mathcal D_F^\tau|_U\)-modules are equivalent by sections to left \(D\)-modules. The inverse localizes a \(D\)-module as an \(\mathcal O(U)\)-module. For \(f\in\mathcal O(U)\), a module element \(m\), and \(r\ge0\), a lift \(v\) of a vector field acts on the section \(m/f^r\) on the principal open \(f\ne0\) by \(v(m/f^r)=v(m)/f^r-rv(f)m/f^{r+1}\), where \(v(f)\) means the derivative of \(f\) by the image of \(v\) in \(T_F\). These actions satisfy the Picard-algebroid relations and glue, since the relations hold before localizing and the displayed rule determines their extensions. The order filtration \(F_pD\) is exhaustive, and we put \(F_pD=0\) for \(p<0\). Affine acyclicity and the Poincar\'e--Birkhoff--Witt theorem give \(\operatorname{gr}D=\mathcal O(T^*U)\). The symbol of the commutator in order one below the sum of the two orders is the cotangent Poisson bracket. On \(C_k(D)=D^{\otimes_{\mathbb C}(k+1)}\), use the increasing filtration spanned in degree \(p\) by tensors whose factors have total order at most \(p\). With \(q=k-p\), Hochschild--Kostant--Rosenberg identifies \(E^1_{p,q}\) with the regular \(k\)-forms on \(T^*U\) of weight \(p\) under cotangent fiber dilation. Let \(\varpi\) denote its Poisson bivector, and let \(\iota_\varpi\) be insertion of the bivector in a differential form, and \(d\) the de Rham differential. The first differential is \(\delta=\iota_\varpi d-d\iota_\varpi\). Choose filtered lifts \(\widetilde f_i\in D\) of homogeneous functions \(f_0,\ldots,f_k\) on \(T^*U\). Write \(S_k\) for the symmetric group on \(1,\ldots,k\), and \(\operatorname{sgn}\) for the permutation sign. The antisymmetrized chain is \[ \sum_{\sigma\in S_k}\operatorname{sgn}(\sigma)\, \widetilde f_0\otimes\widetilde f_{\sigma(1)}\otimes\cdots\otimes \widetilde f_{\sigma(k)}, \] Under the Hochschild--Kostant--Rosenberg map, normalized by dividing the wedge of differentials of a tensor by \(k!\), this chain represents \(f_0\,df_1\wedge\cdots\wedge df_k\). In its Hochschild boundary, multiplication terms cancel in pairs up to commutators. The terms pairing the coefficient with factor \(i\) have sign \((-1)^{i-1}\), and those pairing factors \(ip\) is a cycle, since its boundary has filtration at most \(p-r\). Every chain, including any primitive of a boundary, belongs to some finite filtration level. Hence this stable page computes the associated graded of the homology for the exhaustive filtration. There is only one possible filtration degree in each homological degree, and \(H_{\mathrm{dR}}^j(U)=0\) for \(j>n\) by affine acyclicity of differential forms. Thus \begin{equation}\label{cnt:tdo-affine} \begin{aligned} HH_k(D)&\cong H_{\mathrm{dR}}^{2n-k}(U) && (n\le k\le 2n),\\ HH_k(D)&=0 && (k2n). \end{aligned} \end{equation} All the maps in this calculation commute with restriction between affine opens, so the identifications in \eqref{cnt:tdo-affine} do as well. For descent, write \(D_{\mathrm{qc}}(\mathcal D_F^\tau|_V)\) for the derived category of left modules on an open \(V\subset F\) whose cohomology sheaves are quasi-coherent over \(\mathcal O_V\). For a quasi-compact open inclusion \(j:V'\hookrightarrow V\), restriction has fully faithful right adjoint \(\mathbf Rj_*\). Indeed, affine covers of the inverse images of affine opens under \(j\) compute the underlying direct image and preserve quasi-coherence, and \(j^*\mathbf Rj_*\) is the identity. Thus restriction is a localization with kernel the complexes supported on \(V\setminus V'\). The forgetful functor to \(D_{\mathrm{qc}}(\mathcal O_V)\) preserves coproducts, is conservative, and has left adjoint \(\mathcal D_F^\tau|_V\otimes_{\mathcal O_V}^{\mathbf L}(-)\). By adjunction, induction of perfect generators for \(D_{\mathrm{qc}}(\mathcal O_V)\) gives compact generators for \(D_{\mathrm{qc}}(\mathcal D_F^\tau|_V)\). Induction of perfect generators with support in \(V\setminus V'\) likewise generates the kernel and consists of objects compact in the ambient category \cite{TT}. The induced objects are locally perfect, and local perfectness is preserved by finite cones and retracts. Conversely, a locally perfect complex is compact by computing its derived Hom using a finite affine cover. Thus restriction of the perfect categories gives an exact dg sequence up to idempotent completion. If \(V=V_1\cup V_2\), the kernels of restriction from \(V\) to \(V_1\) and from \(V_2\) to \(V_1\cap V_2\) identify. Complexes supported on \(V\setminus V_1\) extend uniquely from \(V_2\), since they vanish on \(V_1\cap V_2\). The compact objects of the kernels identify as well. Applying Hochschild localization \cite[Theorem~1.5(c)]{Keller99} to these two dg sequences gives a homotopy Cartesian square for Hochschild complexes of the four opens. Iterating this square for a finite open cover gives descent by the finite alternating \v{C}ech diagram of its nonempty intersections. Embed \(F\) in projective space. Choose \(n+1\) hyperplanes with no common intersection on \(F\), by successively avoiding all components of the preceding intersection. Their complements \(U_0,\ldots,U_n\) cover \(F\). Each intersection is affine, since it is the principal projective open where the product of the corresponding linear forms is nonzero. In the Hochschild \v{C}ech spectral sequence the first page is \[ E_1^{a,-b} =\bigoplus_{0\le i_0<\cdots0\) and \(X\) has a \(\mathbb C^*\)-linearized relatively ample line bundle. Every good module supported on \(Y\) has finite length. The number of isomorphism classes of simple good modules supported on \(Y\) is at most \(\#\operatorname{Irr}(Y)\), the number of irreducible components of \(Y\). \end{theorem} \begin{proof} For \(x\in X\), its projected orbit extends across \(0\in\mathbb A^1\) with value \(0\in X_0\). The valuative criterion for \(\pi\) extends the orbit of \(x\) across \(0\), and uniqueness makes the limit fixed. The connected components \(F_i\) of \(X^{\mathbb C^*}\) are smooth and projective, since they lie in \(Y\). Each \(F_i\) is irreducible. At a point of \(F_i\), weight \(a\) of the tangent representation pairs nondegenerately with weight \(1-a\). Thus weight zero, the tangent space to \(F_i\), pairs with weight one. The negative weights pair with weights at least two. The rank of the negative tangent bundle at \(F_i\) is consequently \(\tfrac12\dim X-\dim F_i\). The Bia{\l}ynicki--Birula strata for the zero limit are \(C_i^+=\{x:\lim_{t\to0}t\cdot x\in F_i\}\). They are filtrable by closed unions of strata \cite[\S2]{BBDM}. The opposite strata \(C_i^-=\{x:\lim_{t\to\infty}t\cdot x\in F_i\}\) are affine bundles over \(F_i\) of rank \(\tfrac12\dim X-\dim F_i\). For the opposite action this follows by applying the projective Bia{\l}ynicki--Birula theorem in a smooth equivariant projective compactification. Such a compactification is obtained from an equivariant projective embedding using a power of the linearized ample line bundle and resolving the boundary equivariantly. Each proper \(F_i\) is a full fixed component in the compactification, and the invariant closed boundary cannot contain a point whose limit lies in \(F_i\). The opposite stratum is therefore already in \(X\). If \(x\in X\) admits an infinity limit, then \(t^d f(\pi(x))\) has a finite limit for each homogeneous function \(f\) of positive weight \(d\) on \(X_0\). Hence \(f(\pi(x))=0\), so \(\pi(x)=0\). Conversely, every point of \(Y\) has an infinity limit since \(Y\) is proper. Hence \(Y\) is the disjoint union of the \(C_i^-\). These are irreducible of the common dimension \(\tfrac12\dim X\). Their closures are the irreducible components of \(Y\). Two distinct strata cannot be dense in the same irreducible closure because each is open in its own closure. In particular, \[ \#\operatorname{Irr}(Y)=\#\{F_i\}. \] For the cell decomposition of \cite[\S2.2]{BBDM}, use the point action \(x\mapsto m_{t^{-1}}(x)\), so its attracting limit at infinity is our limit at zero. On sections and forms the corresponding action is pullback along the inverse point map, hence along \(m_t\). The positive form weight in \cite[Definition~1.1]{BBDM} is the weight for this representation on forms, as in the formula in their \S2.2. For tangent weights \(a,b\) of the inverse point action, the value of \(t\cdot\omega\) on the two vectors is \(t^{-a-b}\) times that of \(\omega\). Thus \(a+b=-1\) when they pair nontrivially. The weight of \(\hbar\) in the section algebra is \(+1\), in agreement with the Poisson bracket having degree \(-1\). Let \(S_i\) be the coisotropic reduction of \(C_i^+\), so the restricted two-form is the pullback of the symplectic form on \(S_i\). In this point-action convention, the tangent weights of \(S_i\) along \(F_i\) are the weights in \([-1,0]\) of \(T C_i^+|_{F_i}\), by \cite[Lemma~2.5]{BBDM}. They have total dimension \(2\dim F_i\) since weights \(0\) and \(-1\) pair nondegenerately. By \cite[Theorem~2.21 and Remark~2.26]{BBDM}, the reduced variety is \(T^*F_i\). Its good quantized modules identify with coherent modules over a sheaf of twisted differential operators \(\mathcal D_{F_i}^{\tau_i}\) \cite[Proposition~4.33 and Lemma~4.34]{BBDM}. Let \(U\) be the open union of the remaining attracting strata, with \(C_i^+\) closed in \(U\), and put \(U'=U\setminus C_i^+\). The limit of a point of \(U\) remains in its stratum's fixed component, so the attracting action is still defined on \(U\). The good modules on the quantization of \(S_i\) are equivalent to good modules on \(U\) supported on \(C_i^+\) \cite[Theorem~4.28]{BBDM}. We compare the bounded derived categories. On an adapted invariant affine chart of \(U\), write \(A\) for the sections of \(\mathcal A\), \(J\) for the left ideal quantizing the stratum, and \(B\) for the sections of the reduced quantization on the corresponding affine chart of \(S_i\), before inverting \(\hbar\). The tensor functor of the abelian equivalence followed by inclusion acts on a good module \(N\) on the reduced chart by \[ N\longmapsto (A/J)[\hbar^{-1}] \otimes_{B[\hbar^{-1}]} N . \] Here \((A/J)[\hbar^{-1}]\) is the left \(A[\hbar^{-1}]\), right \(B[\hbar^{-1}]\) bimodule of \cite[Theorem~4.17 and Corollary~4.18]{BBDM}. The algebraic right adjoint is Hom from this bimodule as a left module. The unit identifies \[ \operatorname{Hom}_{A[\hbar^{-1}]}\bigl( (A/J)[\hbar^{-1}],(A/J)[\hbar^{-1}]\otimes_{B[\hbar^{-1}]}N \bigr)\cong N , \] and the same expression with \(\operatorname{Ext}^j\) in place of Hom vanishes for \(j>0\), by \cite[Claim~4.23]{BBDM}. One obtains these localized statements from the lattice statements using a finite projective resolution of \(A/J\) and exactness of central localization. Finite global dimension and finite homogeneous generators on the affine charts are supplied by \cite[Lemmas~3.2 and 3.21]{BBDM}. The tensor functor is exact by faithful flatness of \(A/J\) over \(B\) \cite[Theorem~4.17]{BBDM}. To compute maps between two images, resolve the first input by finitely generated projectives over \(B[\hbar^{-1}]\). The terms after tensoring are direct summands of finite sums of the bimodule, so the Ext vanishing makes them acyclic for Hom to the second image. Applying the displayed unit term by term gives the derived Hom comparison. For equivariant modules one uses resolutions by finite sums of character shifts of the free modules and their projective summands. One takes weight zero in the Hom complexes. This is exact on rational representations and on the pro-rational lattices used here. Weight projections on the rational quotients modulo \(\hbar^j\) commute with transition maps, so they define projections on the complete lattices. Applying such a projection to a lift of a weight vector produces a lift of the same weight. A weight vector in a localized module is first multiplied by a power of \(\hbar\) to put it in a lattice. The charts in \cite[Lemmas~4.25--4.27]{BBDM} have the following compatibility. Their intersections are affine since \(U\) is separated. For the adapted cover, Lemma~4.25 identifies the part of each chart in \(C_i^+\) with the inverse image of its intersection with \(F_i\) under the limit map. The same equality holds for intersections of charts. The corresponding charts of \(S_i\) are cotangent bundles over these affine opens of \(F_i\), including on intersections. The bimodules and their Hom adjunctions restrict to those on the intersections, since they come from the sheaf quotient \(\mathcal A/\mathcal J\), where \(\mathcal J\) is the glued left ideal of \cite[Lemma~4.26]{BBDM}. For bounded good complexes the global derived Hom is computed by the finite alternating \v{C}ech total complex of local derived Homs on these charts. Indeed, on affine charts a finite projective resolution computes the internal derived Hom. Each term of the resulting Hom complex is a direct summand of a finite sum of copies of the target lattice and is acyclic for affine sections. Central localization preserves the calculation \cite[Proposition~3.11, Corollary~3.13, and Lemma~5.7]{BBDM}. The equivariant calculation takes weight zero on the same invariant chart intersections. Charts disjoint from \(C_i^+\) contribute zero to derived Hom between supported complexes. Thus the local comparisons give a quasi-isomorphism on the \v{C}ech total complexes. The functor is fully faithful on bounded derived categories. Every bounded complex with supported cohomology belongs to its image by the finite truncation triangles and the abelian equivalence. This is the single-stratum case of \cite[Lemma~5.3]{BBDM}. Let \(j:U'\hookrightarrow U\) be the open inclusion. Its restriction \(j^*\) on good modules is exact and has a fully faithful right adjoint \(j_*\) on Ind-categories \cite[Theorem~3.27 and Corollary~3.28]{BBDM}. A bounded good complex on \(U'\) lifts to a bounded good subcomplex of its termwise image under \(j_*\) by choosing good extensions of the terms and including their differential images \cite[Lemma~5.6]{BBDM}. A roof on \(U'\) between restrictions of two bounded good complexes on \(U\) lifts by this construction on its middle complex. For each endpoint take the sum of the image of the original endpoint and the image from the lifted middle, inside \(j_*\) of the restricted endpoint. The endpoint maps are isomorphisms after restriction, and the lifted denominator has supported cone. To check faithfulness, if the restriction of a roof is zero in the derived category, refine its restricted middle by a bounded quasi-isomorphism on which the restricted numerator is nullhomotopic. Lift its source as a bounded good subcomplex of its termwise right adjoint image. Enlarge the image of the original middle in the right adjoint image of its restriction by the image of this lift, then enlarge the endpoints by the images of the enlarged middle. Each enlargement restricts to an isomorphism, and the lifted refinement has supported cone. If \(f\) is the resulting numerator with domain \(K\), its restriction factors through \(\operatorname{Cone}(\mathrm{id}_{j^*K})\) by nullhomotopy. The map from that cone to the restricted endpoint adjoints to a chain map from \(\operatorname{Cone}(\mathrm{id}_K)\) to \(j_*\) of the restricted endpoint. Include its image in the endpoint extension. The map \(f\) now factors through the contractible cone. This last enlargement is again an isomorphism after restriction. Thus the bounded derived quotient by complexes with supported cohomology is the bounded good derived category on \(U'\). Write \(\operatorname{Perf}(\mathcal W|_U)\) for the dg category of perfect equivariant quantization complexes on \(U\). These are the bounded good complexes on each open \cite[Lemma~5.5]{BBDM}. The quotient and the derived supported comparison give a dg exact sequence, up to idempotent completion, \[ \operatorname{Perf}(\mathcal D_{F_i}^{\tau_i}) \longrightarrow \operatorname{Perf}(\mathcal W|_U) \longrightarrow \operatorname{Perf}(\mathcal W|_{U'}). \] Apply the Hochschild long exact sequence \cite[Theorem~1.5(c)]{Keller99} and remove the closed strata successively. Lemma~\ref{cnt:tdo} yields \begin{equation}\label{cnt:hh-bound} \dim_{\mathbb C}HH_0(\operatorname{Perf}(\mathcal W)) \ \le\ \sum_i\dim_{\mathbb C}HH_0 (\operatorname{Perf}(\mathcal D_{F_i}^{\tau_i})) \ =\ \#\operatorname{Irr}(Y). \end{equation} Replace a linearized relatively ample line bundle \(L\) by a relatively very ample power. The quantized line-bundle bimodule of \cite[Proposition~5.2]{BPW} gives a Morita equivalence from \(\mathcal W\) to the quantization whose period is translated by \(\hbar k c_1(L)\), for a positive integer \(k\). Here \(c_1(L)\) is the de Rham first Chern class of \(L\), and the period is the formal de Rham class classifying the quantization as in \cite[\S3.1 and Proposition~3.5]{BPW}. The bimodule lattice reduces modulo \(\hbar\) to \(L^k\) with the ordinary left and right \(\mathcal O_X\)-actions. Tensoring a module lattice by it therefore tensors the reduction with \(L^k\), preserving support. For all sufficiently large \(k\), invariant global sections give an abelian equivalence from good modules for the translated quantization \(\mathcal W'\) to finitely generated left modules over \[ R=\Gamma(X,\mathcal W')^{\mathbb C^*} \] by \cite[Corollary~5.17]{BPW}. Let \(\mathcal A'\) be the translated quantization lattice. For \(j\in\mathbb Z\), filter \(R\) by the subspaces \(\Gamma(X,\hbar^{-j}\mathcal A')^{\mathbb C^*}\). The quotient in order \(j\) is the weight-\(j\) part of \(\Gamma(X,\mathcal O_X)\) by coherent-cohomology vanishing and the invariant section calculation of \cite[\S3.3]{BPW}. Thus the associated graded ring is \(\mathcal O(X_0)\), since a proper birational map to a normal variety satisfies \(\pi_*\mathcal O_X=\mathcal O_{X_0}\). Negative orders vanish because the lattice is separated and the graded ring has no negative weights. Consequently \(R\) is Noetherian by lifting generators of graded ideals. Every \(R\)-module is a filtered union of finitely generated submodules, and those submodules are finitely presented by Noetherianity. Hence the abelian equivalence on good modules extends to an equivalence of the Ind-category with all \(R\)-modules. Its derived equivalence identifies \(\operatorname{Perf}(\mathcal W')\) with \(\operatorname{Perf}(R)\). In one direction, perfect quantization complexes are compact by \cite[Lemma~5.7]{BBDM}. In the other direction, a compact \(R\)-complex is represented by a bounded complex of finitely generated projectives, whose inverse image is bounded and good, hence perfect by \cite[Lemma~5.5]{BBDM}. Thus \(HH_0(R)=HH_0(\operatorname{Perf}(\mathcal W'))\), and \eqref{cnt:hh-bound} applies to \(\mathcal W'\). Let \(M\) be a good \(\mathcal W'\)-module supported on \(Y\) and choose an equivariant lattice \(P\subset M\). Write \(H=\Gamma(X,P)\). Multiplication by \(\hbar^j\) identifies \(P\) with \(\hbar^j P\), so the sections exact sequence gives \[ H/\hbar^jH\hookrightarrow\Gamma(X,P/\hbar^j P) \qquad (j\ge1). \] The target is finite-dimensional and rational for \(\mathbb C^*\). In fact \(P/\hbar^jP\) has successive quotients \(\hbar^l P/\hbar^{l+1}P\) for \(0\le l0\) choose a complex vector space \(W_i\) of dimension \(w_i\geq 2v_i+1\), and take \(W_i=0\) when \(v_i=0\). Put \[ E^w=E\oplus\bigoplus_{i\in I}\operatorname{Hom}(V_i,W_i). \] The base coordinates of \(T^*E^w\) are \((x,q)\), where \(q_i:V_i\to W_i\). Its dual coordinates are \((y,p)\), where \(p_i:W_i\to V_i\) is trace dual to \(q_i\). The action of \(g\in G\) sends \(q_i\) to \(q_i g_i^{-1}\) and \(p_i\) to \(g_i p_i\), and acts on \((x,y)\) by changes of bases at the vertices. Let \(\mathfrak g=\bigoplus_{i\in I}\operatorname{End}(V_i)\) be the Lie algebra of \(G\). Identify \(\mathfrak g^*\) with this direct sum via the trace pairings. The cotangent moment map \(\mu:T^*E^w\to\mathfrak g^*\) then has components \[ \mu_i(x,y,q,p)=(\mu_Q)_i(x,y)-p_iq_i . \] The framing term follows by pairing the infinitesimal change \(-q_i\zeta_i\), for \(\zeta_i\in\operatorname{End}(V_i)\), with \(p_i\). Call \((x,y,q,p)\) \emph{costable} if the only graded subspace preserved by all doubled arrows and contained in \(\bigoplus_i\ker q_i\) is zero. Write \(P\) for the costable locus in \(T^*E^w\). Let \[ X=(P\cap\mu^{-1}(0))/G,\qquad X_0=\operatorname{Spec}\mathbb C[\mu^{-1}(0)]^G, \] and let \(\pi:X\to X_0\) be the map to the affine quotient. \begin{lemma}\label{br:geometry} The map \(\mu\) is flat, and \(\pi\) is a projective symplectic resolution. The action \[ t\cdot(x,y,q,p)=(x,ty,q,tp),\qquad t\in\mathbb C^*, \] contracts \(X_0\) to the image of the origin and has symplectic weight one on \(X\). Also \(H^j(X,\mathcal O_X)=0\) for \(j>0\), and \(X\) has a relatively ample line bundle linearized for this action. \end{lemma} \begin{proof} For \(n\geq0\), intersect at each vertex \(i\) the kernels of the maps \(q_j b:V_i\to W_j\) over doubled paths \(b\) from \(i\) to \(j\) of length at most \(n\). Here \(b\) acts by composition of its arrows, and the length-zero path acts by the identity of \(V_i\). The direct sums of these kernels form a decreasing sequence. The term for \(n+1\) consists of the vectors in the term for \(n\) whose images under every doubled arrow lie in the term for \(n\). Equality of consecutive terms thus implies invariance under every doubled arrow. The sequence stabilizes by \(n=\sum_i v_i\) and its limit is the largest invariant graded subspace killed by all \(q_i\). Use the character \(\chi(g)=\prod_i\det(g_i)^{-1}\), with semistability defined by nonvanishing of a polynomial \(f\) satisfying \(f(gz)=\chi(g)^m f(z)\) for some integer \(m>0\). Costability says that the matrix obtained by stacking the finitely many maps \(q_j b\) at each \(i\) has full column rank. Choosing a nonzero maximal minor at each vertex gives, by taking their product, a nonvanishing semi-invariant of weight \(\chi\). A determinant on a zero-dimensional space is \(1\). Conversely, let \(L=\bigoplus_i L_i\) be a nonzero invariant subspace killed by \(q\). The one-parameter subgroup acting by \(t\) on \(L\) and by \(1\) on a graded complement has a limit on the point as \(t\to0\), since arrow blocks from \(L\) to the complement and framing blocks of \(q\) on \(L\) vanish, and the blocks of \(p\) have weights \(0\) or \(1\). Its value under \(\chi\) is \(t^{-\dim L}\), so any positive-weight semi-invariant vanishes at the point. This identifies \(P\) with the semistable open. The same criterion applies to the zero fiber, since restriction of polynomials to this invariant closed subscheme is surjective on each semi-invariant weight by reductivity. A stabilizer element \(g\) has \(\operatorname{im}(g-1)\) invariant and killed by \(q\), hence \(g=1\). No orbit in \(P\) has a smaller-dimensional orbit in its closure inside \(P\), so semistability equals stability. In addition, the open subsets obtained by choosing the nonvanishing minors above trivialize the quotient as a principal \(G\)-bundle in the Zariski topology. On such an open the selected rows at \(i\) give an isomorphism \(V_i\to\mathbb C^{v_i}\) that is multiplied on the right by \(g_i^{-1}\) under the action. After choosing an identification \(V_i\cong\mathbb C^{v_i}\), every orbit meets the locus where all these isomorphisms are identities in a unique point. The change of bases carrying a point to this locus is regular because the determinants are invertible on the open. The geometric invariant theory quotient \(P/G\) is separated, so \(G\times P\cong P\times_{P/G}P\) is closed in \(P\times P\). In particular the action on \(P\) is proper. Fix a value of \(\mu\) and fix the doubled-arrow coordinates. At vertex \(i\), the remaining equation is \(p_iq_i=B_i\), with \(B_i\in\operatorname{End}(V_i)\) determined by the fixed data. If \(b=\operatorname{rank}B_i\) and \(r=\operatorname{rank}q_i\), then \(r\geq b\) and \(\ker q_i\subseteq\ker B_i\). The possible kernels form a Grassmannian of dimension \((v_i-r)(r-b)\). Given the kernel, injections \(V_i/\ker q_i\to W_i\) form an open set of dimension \(w_i r\). The allowed maps \(p_i\) extend a fixed map on \(\operatorname{im}q_i\), so they form an affine space of dimension \(v_i(w_i-r)\). The deficit from the full-rank dimension \(2v_iw_i-v_i^2\) is therefore \[ (2v_iw_i-v_i^2) -\bigl((v_i-r)(r-b)+w_i r+v_i(w_i-r)\bigr) =(v_i-r)(w_i-v_i-r+b), \] which is at least \(2\) when \(r0\) because the symplectic volume form trivializes \(\omega_X\). Normality gives \(\pi_*\mathcal O_X=\mathcal O_{X_0}\). Since \(X_0\) is affine, \(H^j(X,\mathcal O_X)=0\) for \(j>0\). These direct images give rational singularities of \(X_0\), so duality identifies its canonical sheaf with \(\pi_*\omega_X\). The descended generator pulls back to the symplectic volume form, hence \(\pi\) is crepant. Over the smooth locus of \(X_0\), a divisorial exception would have positive discrepancy by the Jacobian formula. If a positive-dimensional fiber remained over that locus, it would contain a curve. A relatively ample divisor agrees there with the pullback of its pushforward Cartier divisor, since there are no exceptional divisors. Its degree on the curve would therefore be zero. This contradicts relative ampleness and gives an isomorphism over the smooth locus. The cotangent dilation scales the symplectic form by \(t\). Its weights on the coordinate ring are nonnegative, and the degree-zero invariants on the moment fiber are a quotient of \(\mathbb C[E^w]^G\) by reductivity. Choose negative integer weights for the vertices that increase strictly along every original arrow, using acyclicity. The central one-parameter subgroup of \(G\) with those weights contracts all of \(E^w\) to zero. On \(x_a\) the weight is the head weight minus the tail weight, and on \(q_i\) it is the negative of the weight at \(i\). Thus \(\mathbb C[E^w]^G=\mathbb C\), and the induced grading on \(X_0\) has only constants in degree zero. The corresponding limit is the image of the origin. \end{proof} Let \(Y=\pi^{-1}(0)\) with its reduced structure, where \(0\) denotes the image of the origin. \begin{lemma}\label{br:core} The inverse image of \(Y\) in \(P\cap\mu^{-1}(0)\), as a closed set, is \[ P\cap\left(\Lambda_{\mathbf v}\times \bigoplus_{i\in I}\operatorname{Hom}(V_i,W_i)\times\{p=0\}\right). \] Its quotient has \(\#\operatorname{Irr}(Y)=\#\operatorname{Irr}(\Lambda_{\mathbf v})\), where \(\operatorname{Irr}\) denotes the set of irreducible components. \end{lemma} \begin{proof} A point over \(0\) has the origin in its orbit closure. The affine Hilbert--Mumford criterion gives a one-parameter subgroup of \(G\) contracting the point to zero. Relative to the induced weight decomposition of the \(V_i\), a nonzero block of a doubled arrow must go from a smaller weight to a larger weight. The sum of the nonnegative weight spaces is invariant under the doubled arrows, and \(q\) kills it because the weight of a framing block of \(q\) is the negative of its source weight. Costability therefore makes all nonnegative weight spaces zero. A block of \(p_i\) has the weight of its target in \(V_i\), so \(p=0\). Strict increase of weights along every nonzero arrow block implies that all sufficiently long doubled paths vanish. The moment equations now reduce to \(\mu_Q=0\). Conversely, nilpotency gives a complete graded flag strictly lowered by every doubled arrow. Split the flag and give its successive graded lines strictly decreasing negative integer weights in the order of an increasing flag. A nonzero arrow block then raises weight, and the negativity of all weights contracts \(q\). With \(p=0\), the point contracts to the origin. The locus with all \(q_i\) injective meets the product of every component of \(\Lambda_{\mathbf v}\) with the framing space. Thus intersecting with \(P\) leaves one dense irreducible open in each such product. Their closures in the inverse image of \(Y\) are its distinct irreducible components. Each component is \(G\)-invariant because \(G\) is connected. The principal bundle quotient is open with orbit fibers, so saturated closed subsets have closed images and pull back from their images. In particular the component images are closed irreducible and preserve noncontainment. They are precisely the components of \(Y\). \end{proof} \section{Reduction on the injective framing locus}\label{br:reduction} For a smooth complex variety \(B\), write \(\mathcal D_B\) for its sheaf of algebraic differential operators. Its order filtration places functions in order zero and vector fields in order one. On \(E^w\) let \(\eta:\mathfrak g\to\Gamma(E^w,\mathcal D_{E^w})\) be the infinitesimal action on functions, obtained by differentiating \((g f)(z)=f(g^{-1}z)\). Thus the principal symbol of \(\eta(\zeta)\) is \(-\langle\mu,\zeta\rangle\). A coherent left \(\mathcal D_{E^w}\)-module is strongly \(G\)-equivariant if the derivative of its compatible algebraic \(G\)-action at \(\zeta\in\mathfrak g\) is left multiplication by \(\eta(\zeta)\). We use this condition also on microlocal modules with the same quantum moment map. Take the order Rees algebra \(\bigoplus_{m\geq0}\hbar^m F_m\mathcal D_{E^w}\), where \(F_m\) is the order-\(\leq m\) subsheaf, microlocalize on \(T^*E^w\), and complete in \(\hbar\). Denote the resulting deformation quantization by \(\mathcal A\). Its reduction modulo \(\hbar\) is \(\mathcal O_{T^*E^w}\), and \(\hbar\) has weight one for the cotangent dilation. A good order filtration of a coherent \(\mathcal D_{E^w}\)-module similarly gives a coherent \(\mathcal A\)-lattice whose reduction modulo \(\hbar\) is the sheaf associated to its graded module on \(T^*E^w\). Inverting \(\hbar\) gives its microlocalization. This construction and taking dilation invariants over the cotangent projection are inverse equivalences between coherent differential-operator modules and good equivariant microlocal modules \cite[Proposition~4.33]{BBDM}. The same equivalence applies over every smooth base \(B\) to the order quantization of \(T^*B\). In particular microlocalization is exact, and its support is the characteristic variety. For a strongly \(G\)-equivariant module on the affine space \(E^w\), use a finite-dimensional \(G\)-invariant subspace of generators for its good filtration. The resulting microlocalization is strongly \(G\)-equivariant, since the infinitesimal relation holds on the Rees module after inverting \(\hbar\). Write \(C=P\cap\mu^{-1}(0)\) and \(\psi:C\to X\) for the quotient map. The transfer lattice for Hamiltonian reduction at zero is the left-module quotient \[ \mathcal E_0=\mathcal A|_P\bigg/ \sum_{\zeta\in\mathfrak g}\mathcal A|_P\,\hbar\eta(\zeta). \] The quotient is supported on \(C\). In expressions involving \(\psi\), we view it and the Hom sheaves from it as sheaves on \(C\) via the closed inclusion \(C\hookrightarrow P\). The reduced quantization \(\mathcal A_X\) is the pushforward along \(\psi\) of the opposite endomorphism algebra of this lattice, which acts on the lattice from the right \cite[\S3.4]{BPW}. Its special fiber is \(\mathcal O_X\) by \cite[Proposition~2.8(i)]{KR}. Put \(\mathcal E=\mathcal E_0[\hbar^{-1}]\) and \(\mathcal W_X=\mathcal A_X[\hbar^{-1}]\). The free reduction theorem \cite[Proposition~2.8]{KR}, in the algebraic form of \cite[\S\S3.4 and 5.5]{BPW}, identifies good \(\mathcal W_X\)-modules with strongly \(G\)-equivariant good modules over \(\mathcal A|_P[\hbar^{-1}]\). The two functors are \[ \mathcal M\longmapsto \psi_*\mathcal H om_{\mathcal A|_P[\hbar^{-1}]}(\mathcal E,\mathcal M), \qquad \mathcal N\longmapsto \mathcal E\otimes_{\psi^{-1}\mathcal W_X}\psi^{-1}\mathcal N. \] The Hom in this formula consists of sections killed by every \(\eta(\zeta)\), hence invariant under the connected group \(G\). The equivalence is compatible with cotangent dilation and with restriction to invariant opens. The action on \(P\) is free and proper by Lemma~\ref{br:geometry}. The cotangent dilation preserves \(P\), and its degree-zero \(G\)-invariants on \(T^*E^w\) are constants. The global-section hypothesis in \cite[\S3.4]{BPW} holds because \(\pi_*\mathcal O_X=\mathcal O_{X_0}\). The complement of \(C\) in the normal zero fiber has codimension at least two, by the rank estimate in Lemma~\ref{br:geometry}. Thus regular semi-invariants on \(C\) extend uniquely to the zero fiber, giving the section comparison for powers of the character line in \cite[\S5.5]{BPW}. We verify the support formula for this equivalence. The symbols of the \(\hbar\eta(\zeta)\) for a basis \(\zeta\) of \(\mathfrak g\) form a regular sequence along \(C\) because \(d\mu\) is surjective there. They generate the unit ideal off \(C\). Use the complex with term \(\mathcal A|_P\otimes_{\mathbb C}\bigwedge^r\mathfrak g\) in homological degree \(r\). For \(\zeta_1,\ldots,\zeta_r\in\mathfrak g\), its differential on \(a\otimes\zeta_1\wedge\cdots\wedge\zeta_r\) is \[ \begin{aligned} &\sum_i(-1)^{i+1} a\hbar\eta(\zeta_i)\otimes \zeta_1\wedge\cdots\wedge\widehat{\zeta_i}\wedge\cdots\wedge\zeta_r\\ &\quad+\sum_{i