Proof. The inequality \(i\le j\) follows from property [fil32bounds]. For the inequality \(j\le 2i\), we only need to consider the case \(i\le d_{\gamma}\) for otherwise since \(j\le 2d_{\gamma}\) and it holds automatically. When \(i\le d_{\gamma}\) and \(\mathrm{Gr}^{P}_{i}V^{j}\ne0\), property [HL] implies \(\mathrm{Gr}^{P}_{2d_{\gamma}-i}V^{j+2d_{\gamma}-2i}\ne0\). In particular, \(j+2d_{\gamma}-2i\le 2d_{\gamma}\), i.e., \(j\le 2i\). ◻

Proof. In [10], the same statement for the affine Springer fibers in the affine Grassmannian \(\mathrm{Gr}\) is proved. That proof can be adapted to our situation with one small change of [10], which should now state: for \(h\in G(F)\) and \(\gamma=\gamma(a)\), we have \[\label{adj32Lie32J} \mathrm{Ad}(h^{-1})\gamma\in \mathrm{Lie}\ \mathbf{I}\iff \mathrm{Ad}(h^{-1})\mathrm{Lie}\ (J_{a})\subset \mathrm{Lie}\ \mathbf{I}.\tag{6}\] The direction \(\Leftarrow\) is trivial. To see \(\Rightarrow\), let \(\gamma'=\mathrm{Ad}(h^{-1})\gamma\in \mathrm{Lie}\ \mathbf{I}\). Then \(\mathrm{Ad}(h^{-1})\) induces a map of group schemes \(\beta: J_{a}\to G_{\gamma'}\) over \(\mathcal{O}_{F}\). Here \(G_{\gamma'}\subset G_{\mathcal{O}_{F}}\), and its special fiber is the centralizer \(C_{G}(\overline{\gamma'})\), where \(\overline{\gamma'}=\gamma'\mod t\), which lies in \(\mathfrak{b}\) by assumption. By [11], the special fiber of \(\beta\) lands in \(C_{B}(\overline{\gamma'})\). In particular, \(\mathrm{Ad}(h^{-1})J_{a}\subset \mathbf{I}\), hence \(\mathrm{Ad}(h^{-1})\mathrm{Lie}\ (J_{a})\subset \mathrm{Lie}\ \mathbf{I}\). This proves 6 . The rest of the argument is the same as the proof of [10]. ◻

Proof. (1) is clear from the properties of \(g\). (3) holds because left translation by \(g\) acts by identity on \(\mathrm{H}^{*}({\mathrm{Fl}_{\gamma}})\) (for \(LG\) is connected). For (2), we need to match the actions of each affine simple reflection \(s\in W_{\mathrm{aff}}\) under \(\mathrm{Ad}(g)^{*}\). Let \(\mathrm{Fl}_{\mathbf{P}_{s}, \gamma}\subset \mathrm{Fl}_{\mathbf{P}_{s}}=LG/\mathbf{P}_{s}\) be the parahoric version of the affine Springer fiber, where \(\mathbf{P}_{s}\) is the standard parahoric subgroup of \(LG\) whose Levi quotient \(L_{s}\) has roots \(\pm\alpha_{s}\). We have a commutative diagram in which both squares are Cartesian \[\label{two32Flg} \xymatrix{\mathrm{Fl}_{\gamma_{0}}\ar[r]^-{\mathrm{Ad}(g)}\ar[d]^{\nu^{\gamma_{0}}_{s}} & \mathrm{Fl}_{\gamma}\ar[d]^{\nu^{\gamma}_{s}}\ar[r]^-{\mathrm{ev}_{s,\gamma}} & [\mathfrak{b}_{s}/B_{s}]\ar[d]^{\nu^{\natural}_{s}}\\ \mathrm{Fl}_{\mathbf{P}_{s}, \gamma_{0}}\ar[r]^-{\mathrm{Ad}(g)} & \mathrm{Fl}_{\mathbf{P}_{s}, \gamma} \ar[r]^-{\mathrm{ev}_{\mathbf{P}_{s}, \gamma}} & [\mathfrak{l}_{s}/L_{s}]}\tag{7}\] Here \(B_{s}\subset L_{s}\) is the Borel subgroup given by the image of \(\mathbf{I}\subset \mathbf{P}_{s}\) in \(L_{s}\), and \(\mathfrak{b}_{s}\subset \mathfrak{l}_{s}\) are the Lie algebras of \(B_{s}\) and \(L_{s}\). The map \(\nu^{\natural}_{s}\) is the Grothendieck alteration for \(\mathfrak{l}_{s}\). The map \(\mathrm{ev}_{s,\gamma}\) (resp. \(\mathrm{ev}_{\mathbf{P}_{s},\gamma}\)) sends \(g\mathbf{I}\in \mathrm{Fl}_{\gamma}\) (resp. \(g\mathbf{P}_{s}\in \mathrm{Fl}_{\mathbf{P}_{s},\gamma}\)) to the image of \(\mathrm{Ad}(g^{-1})\gamma\in \mathrm{Lie}\ \mathbf{I}\) under the projection \(\mathrm{Lie}\ \mathbf{I}\to \mathfrak{b}_{s}\) (resp. the image of \(\mathrm{Ad}(g^{-1})\gamma\in \mathrm{Lie}\ \mathbf{P}_{s}\) under the projection \(\mathrm{Lie}\ \mathbf{P}_{s}\to \mathfrak{l}_{s}\)). The action of \(s\) on both \(\mathrm{H}^{*}({\mathrm{Fl}_{\gamma}})\) and \(\mathrm{H}^{*}({\mathrm{Fl}_{\gamma_{0}}})\) is induced from the Springer action on \(\mathbf{R}\nu^{\natural}_{s*}\overline{\mathbb{Q}}_\ell\) by proper base change. Therefore \(\mathrm{Ad}(g)^{*}\) on cohomology is \(s\)-equivariant. ◻

Proof. We relax the assumption on \(G\) to allow it to be reductive.

Let \(\mathcal{O}_{x,N}=\mathcal{O}_{x}/\mathfrak{m}_{x}^{N}\) and \[\mathfrak{c}_{x,N}=R_{\mathcal{O}_{x,N}/k}\mathfrak{c}\] the Weil restriction of \(\mathfrak{c}\). It is an affine space of dimension \(Nr\). Using the trivialization of \(\mathcal{L}|_{D_{x}}\), we have a map given by the Taylor expansion of \(a\in \mathcal{A}\) at \(x\) to the \(N\)th order \[e_{x,N}: \mathcal{A}\to \mathfrak{c}_{x,N}.\] For \(y\notin x\) we similarly define an evaluation map \(e_{y,2}:\mathcal{A}\to \mathfrak{c}_{y,2}\). Consider the combined map \[e_{(x,N);(y,2)}: \mathcal{A}\to \mathfrak{c}_{x,N}\times\mathfrak{c}_{y,2}\] We claim that \(e_{(x,N);(y,2)}\) is surjective and is an affine space fibration if \(\deg\mathcal{L}\ge 2g+N+1\). Indeed, choosing homogeneous generators \(f_{1},\cdots, f_{r}\in k[\mathfrak{g}]^{G}\) of degree \(d_{1},\cdots, d_{r}\), we identify \(\mathcal{A}\) with \(\prod_{i=1}^{r}\mathrm{H}^{0}({X, \mathcal{L}^{\otimes d_{i}}})\). The map \(e_{(x,N);(y,2)}\) is the product over \(1\le i\le r\) of the restriction maps \[\mathrm{H}^{0}({X, \mathcal{L}^{\otimes d_{i}}})\to \mathcal{L}^{\otimes d_{i}}\otimes_{\mathcal{O}_{X}}(\mathcal{O}_{x}/\mathfrak{m}_{x}^{N}\oplus\mathcal{O}_{y}/\mathfrak{m}_{y}^{2}).\] It is surjective if \(\mathrm{H}^{1}({X, \mathcal{L}^{\otimes d_{i}}(-Nx-2y)})=0\), which is guaranteed by our assumption on \(\deg\mathcal{L}\) and \(d_{i}\ge1\). This proves that \(e_{(x,N);(y,2)}\) is surjective, and its fibers are torsors under \(\prod_{i=1}^{r}\mathrm{H}^{0}({X, \mathcal{L}^{\otimes d_{i}}(-Nx-2y)})\), which is an affine space.

Let \(\mathcal{A}_{a_{0}}=e_{x,N}^{-1}(a_{0})\subset \mathcal{A}\), an affine subspace of dimension \(\dim\mathcal{A}-\dim\mathfrak{c}_{x,N}=\dim\mathcal{A}-Nr\). We claim that \(\mathcal{A}^{\mathrm{ell}}_{a_{0}}:=\mathcal{A}_{a_{0}}\cap \mathcal{A}^{\mathrm{ell}}\ne \varnothing\). Indeed, for a proper Levi subgroup \(M\) of \(G\), let \(\mathcal{A}_{M}\) be the Hitchin base for \(M\) using the same line bundle \(\mathcal{L}\). By [10], \(\dim\mathcal{A}_{M}\le \dim\mathcal{A}-\deg \mathcal{L}\), which by our assumption on \(\deg\mathcal{L}\) is less than \(\dim\mathcal{A}-Nr=\dim\mathcal{A}_{a_{0}}\). Since \(\mathcal{A}^{\mathrm{ell}}_{a_{0}}\subset \mathcal{A}_{a_{0}}\) is the complement \(\mathcal{A}_{a_{0}}\setminus(\cup_{M}\mathcal{A}_{M})\) where the union is through a finite set of proper Levi subgroups \(M\), it is non-empty.

On the other hand, consider the closed subscheme \(\mathfrak{B}\subset \mathcal{A}_{a_{0}}\times (X-\{x\})\) of \((a,y)\) such that \(a:X\to \mathfrak{c}/\mathbb{G}_m\) intersects the discriminant divisor \(\mathfrak{D}/\mathbb{G}_m\) with multiplicity \(\ge2\) at \(y\). For fixed \(x\ne y\in X(k)\), the fiber \(\mathfrak{B}_{y}\) is the preimage in \(\mathcal{A}_{a_{0}}\) of a closed subscheme \(\mathfrak{c}^{\mathrm{bad}}_{y,2}\subset \mathfrak{c}_{y,2}\) under \(e_{y,2}\), and \(\mathfrak{c}^{\mathrm{bad}}_{y,2}\) has codimension \(\ge2\) in \(\mathfrak{c}_{y,2}\). By the smoothness of \(e_{(x,N);(y,2)}\), \(\dim\mathfrak{B}_{y}\le \dim \mathcal{A}_{a_{0}}-2\). Therefore, \(\dim\mathfrak{B}\le \dim\mathcal{A}_{a_{0}}-1\). Let \(\mathcal{A}_{a_{0}}^{\mathrm{bad}}\) be the closure of the image of the projection \(\mathfrak{B}\to \mathcal{A}_{a_{0}}\), then \(\dim \mathcal{A}_{a_{0}}^{\mathrm{bad}}\le \dim\mathcal{A}_{a_{0}}-1\). The open complement \(\mathcal{A}_{a_{0}}-\mathcal{A}_{a_{0}}^{\mathrm{bad}}\) consists of points that are good away from \(x\).

Putting things together, we conclude that \(\mathcal{A}^{\mathrm{ell}}_{a_{0}}\setminus\mathcal{A}_{a_{0}}^{\mathrm{bad}}\) is open dense in \(\mathcal{A}_{a_{0}}\), and any \(k\)-point there satisfies the requirements of the lemma. ◻

Proof. For \(y\in X(k)\), after choosing a trivialization of \(\mathcal{L}|_{\mathrm{Spec}\ \mathcal{O}_{y}}\), we may identify \(a|_{\mathrm{Spec}\ \mathcal{O}_{y}}\) with an element of \(\mathfrak{c}(\mathcal{O}_{y})\). Let \(\gamma_{y}(a)\in \mathfrak{g}(\mathcal{O}_{y})\) be the corresponding point in the Kostant section. The product formula [11] (following [10]) states that there is a canonical homeomorphism \[\mathcal{P}_{a}\times^{\mathcal{P}_{a,x}\times \prod_{y\ne x}\mathcal{P}_{a,y}}(\mathrm{Fl}_{\gamma}\times \prod_{y\ne x} \mathrm{Gr}_{\gamma_{y}(a)})\to \mathcal{M}_{a}.\] The product is over the finitely many points where \(R_{a,y}\) is nontrivial. Since \(R_{a,y}\) is finite for \(y\ne x\), \(\mathrm{Gr}_{\gamma_{y}(a)}\) is \(0\)-dimensional, hence a \(\mathcal{P}_{a,y}\)-torsor, which allows us to simplify the above formula to the desired form 13 . ◻

Proof. There is a spectral sequence with \(E_{2}\)-page \(E_{2}^{i,j}=\mathrm{H}^{i}({\Lambda_{\gamma}, \mathrm{H}^{j}({\mathrm{Fl}_{\gamma}})})\) convergent to \(\mathrm{H}^{i+j}({\Lambda_\gamma\backslash\mathrm{Fl}_\gamma})\). The differentials are strictly compatible with the weight filtrations. Passing to the pure part, namely replacing \(E_{2}^{i,j}\) by \(\mathrm{Gr}^{W}_{i+j}E_{2}^{i,j}\), we have \(\mathrm{Gr}^{W}_{i+j}E_{2}^{i,j}=0\) unless \(i=0\), in which case \(\mathrm{Gr}^{W}_{j}E_{2}^{0,j}=\mathrm{H}^{j}({\mathrm{Fl}_{\gamma}})_{\mathrm{pur}}^{\Lambda_{\gamma}}\). Therefore the pure part of the spectral sequence degenerates at \(E_{2}\) with the only nonzero entries \(\mathrm{H}^{j}({\mathrm{Fl}_{\gamma}})_{\mathrm{pur}}^{\Lambda_{\gamma}}\) at \((0,j)\). This implies that the pullback along \(\mathrm{Fl}_{\gamma}\to\Lambda_\gamma\backslash\mathrm{Fl}_\gamma\) induces a canonical isomorphism \(\mathrm{H}^{j}({\Lambda_\gamma\backslash\mathrm{Fl}_\gamma})_{\mathrm{pur}}\stackrel{\sim}{\to}\mathrm{H}^{j}({\mathrm{Fl}_{\gamma}})^{\Lambda_{\gamma}}_{\mathrm{pur}}\) for all \(j\in \mathbb{Z}\). ◻

Proof. The \(\mathcal{P}'_{a}\)-torsor \(\pi\) gives a spectral sequence with \(E_{2}\)-page \(E_{2}^{i,j}=\mathrm{H}^{i}({\Lambda_\gamma\backslash\mathrm{Fl}_\gamma, \mathbf{R}^{j}\pi_{*}\overline{\mathbb{Q}}_\ell})\) convergent to \(\mathrm{H}^{i+j}({M_{a}})\cong \mathrm{H}^{i+j}({\mathcal{M}_{a}})\). Note that \(\mathbf{R}^{j}\pi_{*}\overline{\mathbb{Q}}_\ell\) is a local system with stalks isomorphic to \(\mathrm{H}^{j}({\mathcal{P}'_{a}})\). Passing to invariants under \(\pi_{0}(\mathcal{P}'_{a})=\pi_{0}(\mathcal{P}_{a})\), the local system \(\mathbf{R}^{j}\pi_{*}\overline{\mathbb{Q}}_\ell\) become the trivial local system with stalk \(\mathrm{H}^{j}({A_{a}})\). Further passing to the pure part, we see that there is a spectral sequence with \(E_{2}\)-page \(E_{2}^{i,j}=\mathrm{H}^{i}({\Lambda_\gamma\backslash\mathrm{Fl}_\gamma})_{\mathrm{pur}}\otimes\mathrm{H}^{j}({A_{a}})\cong H^{i}_{\gamma,\mathrm{pur}}\otimes\mathrm{H}^{j}({A_{a}})\) convergent to \(\mathrm{H}^{i+j}({\mathcal{M}_{a}})^{\pi_{0}(\mathcal{P}_{a})}_{\mathrm{pur}}=\mathrm{H}^{i+j}\mathcal{K}_{a,\mathrm{pur}}\). Purity implies that the spectral sequence degenerates at \(E_{2}\). In particular, this gives a decreasing filtration \(F^{i}\mathrm{H}^{n}\mathcal{K}_{a,\mathrm{pur}}\) on each cohomology group \(\mathrm{H}^{n}\mathcal{K}_{a,\mathrm{pur}}\) with associated graded \(\mathrm{Gr}^{i}_{F}\mathrm{H}^{n}\mathcal{K}_{a,\mathrm{pur}}=H^{i}_{\gamma,\mathrm{pur}}\otimes\mathrm{H}^{n-i}({A_{a}})\). Taking the direct sum over \(n\), we get a decreasing filtration \(F^{i}\mathcal{K}_{a,\mathrm{pur}}\) on \(\mathcal{K}_{a,\mathrm{pur}}\) with associated graded \(\mathrm{Gr}^{i}_{F}\mathcal{K}_{a,\mathrm{pur}}=H^{i}_{\gamma,\mathrm{pur}}\otimes\mathrm{H}^{*}({A_{a}})[-i]\). The action of \(\mathrm{H}_{*}({A_{a}})\) preserves this filtration and induced on \(\mathrm{Gr}^{i}_{F}\) the cap product action on the \(\mathrm{H}^{*}({A_{a}})\)-factor.

The canonical map \(\tau\) is obtained by truncating to the top-degree cohomology sheaf of \(\mathbf{R}\pi_{*}\overline{\mathbb{Q}}_\ell\), therefore it factors through a canonical map \(\overline{\tau}: \mathcal{K}_{a,\mathrm{pur}}\otimes_{\mathrm{H}_{*}({A_{a}})}\overline{\mathbb{Q}}_\ell\to H_{\gamma,\mathrm{pur}}[-2\overline{d}](-\overline{d})\). Passing to the associated graded for the filtration \(F\) on \(\mathcal{K}_{a,\mathrm{pur}}\), each \(\mathrm{Gr}^{i}_{F}\mathcal{K}_{a,\mathrm{pur}}\) is a free \(\mathrm{H}_{*}({A_{a}})\)-module, and \(\overline{\tau}\) induces an isomorphism \(\mathrm{Gr}^{i}_{F}\mathcal{K}_{a,\mathrm{pur}}\otimes_{\mathrm{H}_{*}({A_{a}})}\overline{\mathbb{Q}}_\ell\cong H^{i}_{\gamma,\mathrm{pur}}[-i-2\overline{d}](-\overline{d})\). Therefore \(\overline{\tau}\) itself is an isomorphism. ◻

Proof. We need to show that \(\tau\) is equivariant for each affine simple reflection \(s\in W_{\mathrm{aff}}\). Let \(\mathbf{P}_{s}\supset \mathbf{I}\) be the standard parahoric subgroup of \(LG\) whose reductive quotient \(L_{s}\) has roots \(\pm\alpha_{s}\). We have the parahoric analog \(\mathcal{M}_{\mathbf{P}_{s}}\) of \(\mathcal{M}\) (simply changing the \(B\)-reduction at \(x\) to a \(\mathbf{P}_{s}\)-level structure) and its fiber \(\mathcal{M}_{\mathbf{P}_{s},a}\) over \(a\in \mathcal{A}(k)\). See [11] for more details. Similarly we have the local analog \(\mathrm{Fl}_{\mathbf{P}_{s},\gamma}\), the affine Springer fiber in the affine partial flag variety \(\mathrm{Fl}_{\mathbf{P}_{s}}=LG/\mathbf{P}_{s}\). The product formula has a parahoric analog, a homeomorphism \(M_{\mathbf{P}_{s},a}:=\mathcal{P}'_{a}\times^{\Lambda_{\gamma}}\mathrm{Fl}_{\mathbf{P}_{s},\gamma}\to \mathcal{M}_{\mathbf{P}_{s}, a}\). We have a commutative diagram in which both squares are Cartesian \[\xymatrix{M_{a}\ar[r]^-{\pi}\ar[d]^{\nu_{s}^{a}} & \Lambda_{\gamma}\backslash\mathrm{Fl}_{\gamma}\ar[d]^{\nu^{\gamma}_{s}}\ar[r]^-{\mathrm{ev}_{s}^{\gamma}} & [\mathfrak{b}_{s}/B_{s}]\ar[d]^{\nu^{\natural}_{s}}\\ M_{\mathbf{P}_{s}, a}\ar[r]^-{\pi_{s}} & \Lambda_{\gamma}\backslash\mathrm{Fl}_{\mathbf{P}_{s}, \gamma} \ar[r]^-{\mathrm{ev}_{\mathbf{P}_{s}, \gamma}} & [\mathfrak{l}_{s}/L_{s}]}\] The right half of the diagram is the same as that of 7 . The Springer action of \(s\) on \(\mathbf{R}\nu^{\natural}_{s*}\overline{\mathbb{Q}}_\ell\) induces an action on \(\mathbf{R}\nu^{\gamma}_{s*}\overline{\mathbb{Q}}_\ell\) and \(\mathbf{R}\nu^{a}_{s*}\overline{\mathbb{Q}}_\ell\) by proper base change. By construction, the \(s\)-actions on \(\mathrm{H}^{*}({\mathcal{M}_{a}})\cong \mathrm{H}^{*}({M_{a}})\) and on \(\mathrm{H}^{*}({\Lambda_{\gamma}\backslash\mathrm{Fl}_{\gamma}})\) are induced from the \(s\)-actions on \(\mathbf{R}\nu^{\gamma}_{s*}\overline{\mathbb{Q}}_\ell\) and \(\mathbf{R}\nu^{a}_{s*}\overline{\mathbb{Q}}_\ell\). We conclude that the Gysin map \(\tau\) for the projection \(\pi: M_{a}\to \Lambda_{\gamma}\backslash\mathrm{Fl}_{\gamma}\) is \(s\)-equivariant. ◻

Proof. From the construction we have a surjection of cohomologically graded vector spaces \(\mathrm{Gr}^{P}_{i+2\overline{d}}\mathcal{K}_{a,\mathrm{pur}}\twoheadrightarrow\mathrm{Gr}^{P}_{i}H_{\gamma,\mathrm{pur}}[-2\overline{d}](-\overline{d})\) for all \(i\). Therefore 16 in question is surjective. To show 16 is an isomorphism, it suffices to compare the total dimension. By the freeness result in [p:perv32Gr32free], the left side of 16 has dimension equal to \(\dim\mathrm{Gr}^{P}_{*}\mathcal{K}_{a,\mathrm{pur}}/\dim \mathrm{H}_{*}({A_{a}})\), which is the same as \(\dim \mathcal{K}_{a,\mathrm{pur}}/\dim \mathrm{H}_{*}({A_{a}})\). By Lemma 7, \(\dim \mathcal{K}_{a,\mathrm{pur}}/\dim \mathrm{H}_{*}({A_{a}})=\dim H_{\gamma,\mathrm{pur}}\), which is the same as \(\dim \mathrm{Gr}^{P}_{*}H_{\gamma,\mathrm{pur}}\). Thus the two sides of 16 have the same total dimension. The lemma is proved. ◻

Proof. We check the defining properties for the perverse filtration.

[comp32grading] holds by construction.

[fil32bounds] By 8 , the range of the perverse filtration \(P_{\bullet}\mathcal{K}_{a}\) is the interval \([0,2d]\). Therefore \(P_{2d}\mathcal{K}_{a}=\mathcal{K}_{a}\) hence \(P_{2d}\mathcal{K}_{a,\mathrm{pur}}=\mathcal{K}_{a,\mathrm{pur}}\). This implies \(P_{2d_{\gamma}}H_{\gamma,\mathrm{pur}}=H_{\gamma,\mathrm{pur}}\), hence \(\mathrm{Gr}^{P}_{i}H_{\gamma,\mathrm{pur}}=0\) for \(i>2d_{\gamma}\). By Lemma 12, we have \(\mathrm{Gr}^{P}_{i}H_{\gamma,\mathrm{pur}}=0\) for \(i<0\). Since \(P_{i'}H_{\gamma,\mathrm{pur}}=0\) for \(i'\) sufficiently negative, this implies \(P_{-1}H_{\gamma,\mathrm{pur}}=0\).

We now show that \(P_{i}H^{i}_{\gamma,\mathrm{pur}}=H^{i}_{\gamma,\mathrm{pur}}\). By construction, it suffices to show that \(P_{i}(\mathrm{H}^{i}\mathcal{K}_{a,\mathrm{pur}})=\mathrm{H}^{i}\mathcal{K}_{a,\mathrm{pur}}\), or even stronger \(P_{i}(\mathrm{H}^{i}\mathcal{K}_{a})=\mathrm{H}^{i}\mathcal{K}_{a}\), or \(\mathrm{Gr}^{P}_{i'}(\mathrm{H}^{i}\mathcal{K}_{a})=0\) for \(i'>i\). In other words, we need to show that \(\mathrm{Gr}^{P}_{i'}\mathcal{K}_{a}\) is in cohomological degrees \(\ge i'\). By 8 , \(\mathrm{Gr}^{P}_{i'}\mathcal{K}_{a}\) is the stalk of a shifted perverse sheaf \({}^{p}\mathcal{K}^{\dim \mathcal{A}+i'}[-\dim \mathcal{A}-i']\) at \(a\). By Theorem 4, \({}^{p}\mathcal{K}^{\dim \mathcal{A}+i'}\) has full support. Therefore, all stalks of \({}^{p}\mathcal{K}^{\dim \mathcal{A}+i'}\) lie in degrees \(\ge -\dim \mathcal{A}\), and hence all stalks of \({}^{p}\mathcal{K}^{\dim \mathcal{A}+i'}[-\dim \mathcal{A}-i']\) lie in degrees \(\ge i'\).

[PW] By Lemma 8 and the definition of \(P_{i}H_{\gamma,\mathrm{pur}}\) as the image of perverse filtration pieces of \(\mathcal{K}_{a,\mathrm{pur}}\), it suffices to show that \(P_{i}\mathcal{K}_{a,\mathrm{pur}}\), or \(P_{i}\mathcal{K}_{a}\), is stable under \(W_{\mathrm{aff}}\). However, this is clear because \(W_{\mathrm{aff}}\) acts on the complex \(\mathcal{K}\), and it has to preserve the perverse truncations \({}^{p}\tau_{\le i}\mathcal{K}\).

[HL] We postpone the proof to Proposition 1.

[Chern32half] Each \(\lambda\in \mathbb{X}^*(T)\) gives a \(G\)-equivariant line bundles \(\mathcal{L}_{\lambda}\) on \(\mathcal{B}\). We need to show that the cup product by \(c_{1}(\mathcal{L}_{\lambda})\) on \(H_{\gamma,\mathrm{pur}}\) sends \(P_{i}H_{\gamma,\mathrm{pur}}\) to \(P_{i+1}H_{\gamma,\mathrm{pur}}\).

We have maps \[M_{a}\xrightarrow{\pi} \Lambda_{\gamma}\backslash\mathrm{Fl}_{\gamma}\to \mathbb{B}T\] taking \((\mathcal{E}, \mathcal{E}^{B}_{x}, \varphi)\) to the \(T\)-torsor \(\mathcal{E}^{B}_{x}/N\), and this map factors through \(\Lambda_{\gamma}\backslash\mathrm{Fl}_{\gamma}\). The \(G\)-equivariant line bundle \(\mathcal{L}_{\lambda}\) on \(\mathcal{B}\) descends to \(\mathbb{B}T\); we use the same notation \(\mathcal{L}_{\lambda}\) to denote its pullback to \(\Lambda_{\gamma}\backslash\mathrm{Fl}_{\gamma}\) and to \(\mathcal{M}_{a}\). It is clear that the map \(\tau\) is equivariant under the cup product by \(c_{1}(\mathcal{L}_{\lambda})\). Therefore, by the definition of the perverse filtration on \(H_{\gamma,\mathrm{pur}}\), it suffices to show that \(c_{1}(\mathcal{L}_{\lambda})\) sends \(P_{ i}\mathcal{K}_{a,\mathrm{pur}}\) to \(P_{ i+1}\mathcal{K}_{a,\mathrm{pur}}\). By [12], \(c_{1}(\mathcal{L}_{\lambda})\) sends \(P_{ i}\mathcal{K}_{a}\) to \(P_{ i+1}\mathcal{K}_{a}\), hence the same is true for the pure part. ◻

Proof. With the globalization data \((X,\mathcal{L}, x, a)\) satisfying the statement of Lemma 4, we transport the filtration \(P_{\bullet}H_{\gamma,\mathrm{pur}}\) to \(H_{\gamma_{0},\mathrm{pur}}\) under the isomorphism \(H_{\gamma_{0},\mathrm{pur}}\cong H_{\gamma,\mathrm{pur}}\) as in 9 . By Lemma 3, this isomorphism is equivariant under the actions of \(\mathrm{Sym}(\mathrm{H}^{2}({\mathrm{Fl}}))\) and \(W_{\mathrm{aff}}\), Theorem 5 implies that \(H_{\gamma_{0},\mathrm{pur}}\) also satisfies all the axioms for a perverse filtration. ◻

Proof. The map \(s_{\gamma}\) is obtained by modifying a fixed \(G\)-Higgs \((\mathcal{E}_{0},\varphi_{0})\in \mathcal{M}^{\mathrm{Hit}}_{a}\) at \(x\). By construction, it fits into a commutative diagram \[\xymatrix{ \mathrm{Fl}_{\gamma}\ar[d]_{\omega_{\gamma}} \ar[r]^{s_{\gamma}} & \mathcal{M}_{a}\ar[d]^{\omega_{a}}\\ \mathrm{Gr}_{G,\mathcal{E}_{0}}\ar[r]^{u_{x}} & \mathrm{Bun}_{G} }\] Here \(\mathrm{Gr}_{G,\mathcal{E}_{0}}\) is the moduli space classifying a \(G\)-bundle on \(X\) together with an isomorphism with \(\mathcal{E}_{0}\) over \(X-\{x\}\). Choosing a trivialization of \(\mathcal{E}_{0}\) on the formal disk \(D_{x}\) around \(x\) identifies \(\mathrm{Gr}_{G,\mathcal{E}_{0}}\) with \(\mathrm{Gr}_{G}\), such that \(\mathcal{E}_{0}|_{D_{x}}\) corresponds to the base point of \(\mathrm{Gr}_{G}\). The vertical maps are the forgetful maps. The desired statement follows from the canonical isomorphism of line bundles on \(\mathrm{Gr}_{G}\) \[u_{x}^{*}\mathbb{L}\cong\mathbb{L}_{\mathrm{Gr}}\otimes\det\mathbf{R}\Gamma(X,\mathcal{E}_{0}\times^{G}V)\] which follows from definition (the second tensor factor on the right side is a constant line). ◻

Proof. Let \(\mathcal{A}^{x}\subset \mathcal{A}\) be the open subscheme consisting of those \(a:X\to \mathfrak{c}/\mathbb{G}_m\) such that \(a(x)\in \mathfrak{c}^{\mathrm{rs}}/\mathbb{G}_m\). By our assumption on the degree of \(\mathcal{L}\), the evaluation map \(\mathrm{ev}_{x}: \mathcal{A}\to \mathfrak{c}/\mathbb{G}_m\) at \(x\) is surjective, hence \(\mathcal{A}^{x}\) is open dense in \(\mathcal{A}^{x}\). In particular, \(\mathcal{A}^{\mathrm{ell},x}=\mathcal{A}^{\mathrm{ell}}\cap \mathcal{A}^{x}\) is open dense in \(\mathcal{A}^{\mathrm{ell}}\).

By Theorem 4, each simple constituent of \({}^{p}\mathcal{K}^{i}\) has full support, to show that 17 is an isomorphism, it suffices to show that it is an isomorphism when restricted to \(\mathcal{A}^{\mathrm{ell},x}\). ².

The map \(\nu: \mathcal{M}\to \mathcal{M}^{\mathrm{Hit}}\) restricts to a map \(\nu^{\mathrm{ell}, x}: \mathcal{M}|_{\mathcal{A}^{\mathrm{ell},x}}\to \mathcal{M}^{\mathrm{Hit}}|_{\mathcal{A}^{\mathrm{ell},x}}\) that is a \(W\)-torsor (since the choices of \(B\)-reductions compatible with \(\varphi(x)\), which is regular semisimple, form a \(W\)-torsor). In particular, \(\nu^{\mathrm{ell},x}\) is finite. Since \(\mathbb{L}_{\mathrm{Hit}}\) is anti-ample on \(\mathcal{M}^{\mathrm{Hit}}|_{\mathcal{A}^{\mathrm{ell},x}}\), its pullback to \(\mathcal{M}|_{\mathcal{A}^{\mathrm{ell},x}}\) is also anti-ample. The isomorphism then follows from the relative hard Lefschetz theorem [13]. ◻

Proof. Define Laurent polynomials \[\begin{align} &&\Phi(t)=\sum_{i\in \mathbb{Z}}(\dim \mathrm{Gr}^{P}_{d_{\gamma}+i}H_{\gamma,\mathrm{pur}}) t^{i}\\ &&\Psi(t)=\sum_{i\in \mathbb{Z}}(\dim \mathrm{Gr}^{P}_{d+i}K_{a,\mathrm{pur}}) t^{i}\\ &&A(t)=\sum_{i\in \mathbb{Z}}(\dim \mathrm{H}_{\overline{d}-i}({A_{a}}))t^{i}=t^{-\overline{d}}(1+t)^{2\overline{d}}. \end{align}\] By the freeness statement of Proposition 1 and 16 , we have \[\Psi(t)=\Phi(t)A(t).\] Now \(\Psi(t)\) satisfies \(\Psi(t)=\Psi(t^{-1})\) by Lemma 11. Clearly \(A(t)=A(t^{-1})\). We conclude that \(\Phi(t)=\Phi(t^{-1})\), which is what we need. ◻

Proof. Consider the projection \(u: A_{a}\times \mathrm{Fl}_{\gamma}\to \mathcal{M}_{a}\), and \(\widetilde{\tau}^{j}: \mathrm{H}^{2\overline{d}+j}({A_{a}\times \mathrm{Fl}_{\gamma}})\to \mathrm{H}^{2\overline{d}}({A_{a}})\otimes\mathrm{H}^{j}({\mathrm{Fl}_{\gamma}})\) be the projection from the Künneth formula. By Lemma 10, we have \[u^{*}\eta_{\mathcal{M}}\in 1\otimes\eta+ \mathrm{H}^{1}({A_{a}})\otimes\mathrm{H}^{1}({\mathrm{Fl}_{\gamma}})+\mathrm{H}^{2}({A_{a}})\otimes 1\] under the Künneth decomposition of \(\mathrm{H}^{2}({A_{a}\times \mathrm{Fl}_{\gamma}})\). From this we get a commutative diagram \[\label{diag32Kunneth} \xymatrix{\mathrm{H}^{2\overline{d}+j}({A_{a}\times \mathrm{Fl}_{\gamma}})_{\mathrm{pur}} \ar[d]_{\widetilde{\tau}^{j}}\ar[r]^-{\cup u^{*}\eta_{\mathcal{M}}} & \mathrm{H}^{2\overline{d}+j+2}({A_{a}\times \mathrm{Fl}_{\gamma}})_{\mathrm{pur}}(1)\ar[d]^{\widetilde{\tau}^{j+2}}\\ \mathrm{H}^{2\overline{d}}({A_{a}})\otimes\mathrm{H}^{j}({\mathrm{Fl}_{\gamma}})_{\mathrm{pur}}\ar[r]^-{\cup (1\otimes\eta)} & \mathrm{H}^{2\overline{d}}({A_{a}})\otimes\mathrm{H}^{j+2}({\mathrm{Fl}_{\gamma}})_{\mathrm{pur}}(1)}\tag{19}\]

The pullback map \(u^{*}\) induces a map from diagram 18 to diagram 19 . For the bottom row in 19 , we identify \(\mathrm{H}^{2\overline{d}}({A_{a}})\) with \(\overline{\mathbb{Q}}_\ell(-\overline{d})\), so that the map from the bottom row of 18 can be identified with the inclusion of \(H^{j}_{\gamma,\mathrm{pur}}=\mathrm{H}^{j}({\mathrm{Fl}_{\gamma}})^{\Lambda_{\gamma}}_{\mathrm{pur}}\hookrightarrow\mathrm{H}^{j}({\mathrm{Fl}_{\gamma}})_{\mathrm{pur}}\). The cube formed by the map from 18 to 19 has all the faces commutative except possibly for the face 18 . By the injectivity of the lower right corners, the remaining face has to commute, i.e., 18 is commutative. ◻

Proof. By Lemma 12, \(\dim \mathrm{Gr}^{P}_{d_{\gamma}-i}H_{\gamma,\mathrm{pur}}= \dim \mathrm{Gr}^{P}_{d_{\gamma}+i}H_{\gamma,\mathrm{pur}}\), therefore it suffices to show that ?? is surjective.

Iterating Lemma 13, we get that for any \(i,j\ge0\) a commutative diagram \[\xymatrix{\mathrm{H}^{2\overline{d}+j}\mathcal{K}_{a,\mathrm{pur}} \ar[d]_{\tau^{j}}\ar[r]^-{\cup \eta^{i}_{\mathcal{M}}} & \mathrm{H}^{2\overline{d}+j+2i}\mathcal{K}_{a,\mathrm{pur}}(i)\ar[d]^{\tau^{j+2i}}\\ H^{j}_{\gamma,\mathrm{pur}}(-\overline{d})\ar[r]^-{\cup \eta^{i}} & H^{j+2i}_{\gamma,\mathrm{pur}}(-\overline{d}+i)}\] By definition, \(\tau^{j}\) is compatible with the filtrations \(P_{\bullet}\) on \(\mathcal{K}_{a,\mathrm{pur}}\) and on \(H_{\gamma,\mathrm{pur}}\) up to an index shift by \(2\overline{d}\). Passing to the associated graded, we have a commutative diagram \[\label{GrP32eta32local32global} \xymatrix{\mathrm{Gr}^{P}_{2\overline{d}+d_{\gamma}-i}\mathrm{H}^{2\overline{d}+j}\mathcal{K}_{a,\mathrm{pur}} \ar[d]_{\mathrm{Gr}^{P}_{d_{\gamma}-i}\tau^{j}}\ar[r]^-{\cup \eta^{i}_{\mathcal{M}}} & \mathrm{Gr}^{P}_{2\overline{d}+d_{\gamma}+i}\mathrm{H}^{2\overline{d}+j+2i}\mathcal{K}_{a,\mathrm{pur}}(i)\ar[d]^{\mathrm{Gr}^{P}_{d_{\gamma}+i}\tau^{j+2i}}\\ \mathrm{Gr}^{P}_{d_{\gamma}-i}H^{j}_{\gamma,\mathrm{pur}}(-\overline{d})\ar[r]^-{\cup \eta^{i}} & \mathrm{Gr}^{P}_{d_{\gamma}+i}H^{j+2i}_{\gamma,\mathrm{pur}}(-\overline{d}+i)}\tag{20}\] The upper row in 20 is surjective by the hard Lefschetz property for \(\mathcal{K}\) (hence for \(\mathcal{K}_{a,\mathrm{pur}}\)) proved in Lemma 11: because the mid-point of the degrees \(2\overline{d}+d_{\gamma}-i\) and \(2\overline{d}+d_{\gamma}+i\) is \(2\overline{d}+d_{\gamma}\ge \overline{d}+d_{\gamma}=d\) (the middle index for the filtration \(P_{\bullet}\mathcal{K}_{a}\)). By construction, the vertical maps \(\mathrm{Gr}^{P}_{d_{\gamma}-i}\tau^{j}\) and \(\mathrm{Gr}^{P}_{d_{\gamma}+i}\tau^{j+2i}\) are surjective. Therefore the bottom row in 20 is also surjective. This finishes the proof. ◻

Proof. The ind-scheme \(\mathrm{Fl}_{\gamma}\) has a stratification \(\mathrm{Fl}_{\gamma,\mathcal{O}}\) indexed by \(\mathcal{O}\in \mathrm{RT}(\gamma)\). Let \(\widehat{\mathrm{Gr}}_{\gamma,e}\to \mathrm{Gr}_{\gamma,\mathcal{O}}\) be the total space of the \(G_{e}\)-torsor classified by the evaluation map \(\mathrm{Gr}_{\gamma,\mathcal{O}}\to \mathcal{O}/G\cong \{e\}/G_{e}\). In other words, \(\widehat{\mathrm{Gr}}_{\gamma,e}=\{g\in LG/(L^{+}G)^{1}|\mathrm{Ad}(g^{-1})\gamma\in e+t\mathfrak{g}[\![t]\!]\}\). Then the multiplication map \(LG/(L^{+}G)^{1}\times G/B\to LG/\mathbf{I}=\mathrm{Fl}\) restricts to a map \(\widehat{\mathrm{Gr}}_{\gamma,e}\times \mathcal{B}_{e}\to \mathrm{Fl}_{\gamma,\mathcal{O}}\), inducing an isomorphism \[\mathrm{Fl}_{\gamma,\mathcal{O}}\cong (\widehat{\mathrm{Gr}}_{\gamma,e}\times \mathcal{B}_{e})/G_{e}.\] Note that \(\widehat{\mathrm{Gr}}_{\gamma,e}\to \widetilde{\mathrm{Gr}}_{\gamma,e}\) is a \(G_{e}^{\circ}\)-torsor, therefore \(\dim\widehat{\mathrm{Gr}}_{\gamma,e}\le d_{\gamma}-d_{\mathcal{O}}+\dim G_{e}=d_{\gamma}+d_{\mathcal{O}}+r\) (where \(r\) is the rank of \(G\)), with equality if and only if \(\mathcal{O}\) is \(\gamma\)-relevant. Taking top homology we get \[\mathrm{H}^{\mathrm{BM}}_{2d_{\gamma}}({\mathrm{Fl}_{\gamma,\mathcal{O}}})\cong (\mathrm{H}^{\mathrm{BM}}_{2(d_{\gamma}+d_{\mathcal{O}}+r)}({\widehat{\mathrm{Gr}}_{\gamma, e}})\otimes\mathrm{H}_{2d_{\mathcal{O}}}({\mathcal{B}_{e}}))_{A_{e}}.\] The \(G_{e}^{\circ}\)-torsor \(\widehat{\mathrm{Gr}}_{\gamma,e}\to \widetilde{\mathrm{Gr}}_{\gamma,e}\) induces an \(A_{\gamma}\times A_{e}\)-equivariant isomorphism of top homology groups, therefore we can rewrite the above as \[\label{top32hom32Fl32gO} \mathrm{H}^{\mathrm{BM}}_{2d_{\gamma}}({\mathrm{Fl}_{\gamma,\mathcal{O}}})\cong (\mathrm{H}^{\mathrm{BM}}_{2d_{\gamma}-2d_{\mathcal{O}}}({\widetilde{\mathrm{Gr}}_{\gamma, e}})\otimes\mathrm{H}_{2d_{\mathcal{O}}}({\mathcal{B}_{e}}))_{A_{e}}.\tag{22}\] Finally we have a canonical decomposition \[\mathrm{H}^{\mathrm{BM}}_{2d_{\gamma}}({\mathrm{Fl}_{\gamma}})=\bigoplus_{\mathcal{O}\in \mathrm{RT}_{\mathrm{rel}}(\gamma)}\mathrm{H}^{\mathrm{BM}}_{2d_{\gamma}}({\mathrm{Fl}_{\gamma,\mathcal{O}}})\] because both the left side and each term on the right are spanned by the fundamental classes of top-dimensional irreducible components. Moreover, this decomposition is equivariant under \(A_{\gamma}\times W\). Combined with 22 we get 21 . ◻

Proof. In the decomposition ?? , all simple \(W\)-modules isomorphic to \(E_{\mathcal{O}}\cong\mathrm{H}_{2d_{\mathcal{O}}}({\mathcal{B}_{e}})_{A_{e}}\) has to come from the summand corresponding to \(\mathcal{O}\), because by Springer correspondence, for another nilpotent orbits \(\mathcal{O}'\notin \mathcal{O}\) and \(e'\in \mathcal{O}'\), \(\mathrm{H}_{2d_{\mathcal{O}'}}({\mathcal{B}_{e'}})\) does not contain subquotients isomorphic to \(E_{\mathcal{O}}\) as a \(W\)-module. The multiplicity space of \(E_{\mathcal{O}}\) in \(\mathrm{H}_{2d_{\gamma}}({\mathrm{Fl}_{\gamma}})_{A_{\gamma}}\), which is the dual of \(E_{\gamma}\), is \(\mathrm{H}^{\mathrm{BM}}_{2d_{\gamma}-2d_{\mathcal{O}}}({\widetilde{\mathrm{Gr}}_{\gamma, e}})_{A_{\gamma}\times A_{e}}\cong \mathrm{H}^{\mathrm{BM}}_{2d_{\gamma}-2d_{\mathcal{O}}}({\mathrm{Gr}_{\gamma, \mathcal{O}}})_{A_{\gamma}}\), using that \(\widetilde{\mathrm{Gr}}_{\gamma,e}\to \mathrm{Gr}_{\gamma,\mathcal{O}}\) is an \(A_{e}\)-torsor. ◻

Proof. By properties [Chern32half] and [HL] of a perverse filtration on \(H_{\gamma,\mathrm{pur}}\), \(\varphi_{\gamma}\) sends \(\mathrm{Sym}^{j}(\mathrm{H}^{2}({\mathcal{B}}))\eta^{i}\) to \(P_{j+2i}H^{2j+2i}_{\gamma,\mathrm{pur}}\). Therefore \(\varphi_{\gamma}(P_{i}S)\subset P_{i}H_{\gamma,\mathrm{pur}}\).

To show strictness, we may restrict to each cohomological degree \(2n\), \(0\le n\le d_{\gamma}\). Let \(0\ne \xi=\sum_{j=0}^{n}\theta_{j}\eta^{n-j}\in \mathrm{Sym}^{n}(\mathrm{H}^{2}({\mathrm{Fl}}))\), where \(\theta_{j}\in \mathrm{Sym}^{j}(\mathrm{H}^{2}({\mathcal{B}}))\). Suppose \(\varphi_{\gamma}(\xi)\in P_{i}H^{2n}_{\gamma,\mathrm{pur}}\), we need to show that there exists \(\xi'\in P_{i}S\) such that \(\varphi_{\gamma}(\xi)=\varphi_{\gamma}(\xi')\). We take \(\xi'=\sum_{j\ge 2n-i}\theta_{j}\eta^{n-j}\), which clearly belongs to \(P_{i}S\). We need to show that \(\varphi_{\gamma}(\xi-\xi')=\sum_{j<2n-i}\varphi_{\gamma}(\theta_{j})\eta^{n-j}=0\). Let \(j_{0}\) be the smallest integer \(j\) such that \(\varphi_{\gamma}(\theta_{j})\ne 0\) (as an element in \(H^{2j}_{\gamma,\mathrm{pur}}\)). Then \(\varphi_{\gamma}(\xi)\subset P_{2n-j_{0}}H^{2n}_{\gamma,\mathrm{pur}}\). Projecting to \(\mathrm{Gr}^{P}_{2n-j_{0}}H^{2n}_{\gamma,\mathrm{pur}}\), only the term \(\theta_{j_{0}}\eta^{n-j_{0}}\) in \(\xi\) contributes. Therefore the image of \(\varphi_{\gamma}(\xi)\) in \(\mathrm{Gr}^{P}_{2n-j_{0}}H^{2n}_{\gamma,\mathrm{pur}}\) is the same as the image of \(\varphi_{\gamma}(\theta_{j_{0}})\) under the map \[\cup \eta^{n-j_{0}}: \mathrm{Gr}^{P}_{j_{0}}H^{2j_{0}}_{\gamma,\mathrm{pur}}\to \mathrm{Gr}^{P}_{2n-j_{0}}H^{2n}_{\gamma,\mathrm{pur}}.\] Since \(n\le d_{\gamma}\), the above map is injective by property [HL] of the perverse filtration. Now \(\varphi_{\gamma}(\theta_{j_{0}})\in P_{j_{0}}H^{2j_{0}}_{\gamma,\mathrm{pur}}\stackrel{\sim}{\to}\mathrm{Gr}^{P}_{j_{0}}H^{2j_{0}}_{\gamma,\mathrm{pur}}\) is nonzero by assumption, hence the image of \(\varphi_{\gamma}(\xi)\) in \(\mathrm{Gr}^{P}_{2n-j_{0}}H^{2n}_{\gamma,\mathrm{pur}}\) is nonzero. Since we assume \(\varphi_{\gamma}(\xi)\in P_{i}H^{2n}_{\gamma,\mathrm{pur}}\), this forces \(2n-j_{0}\le i\), i.e., \(j_{0}\ge 2n-i\), which implies \(\varphi_{\gamma}(\xi)=\varphi_{\gamma}(\xi')\). ◻

Proof. Recall the image \({}'E_{\gamma}=\mathrm{Im}(\varphi^{d_{\gamma}}_{\gamma})\subset E_{\gamma}\) is equipped with the induced perverse filtration from \(E_{\gamma}\). By the strictness proved in Lemma 15, we have a canonical isomorphism of graded vector spaces \[\label{Gr32E39} \mathrm{Gr}^{P}_{\bullet}{}'E_{\gamma}\cong \mathrm{Im}(\mathrm{Gr}^{P}_{\bullet}S^{d_{\gamma}}\to \mathrm{Gr}^{P}_{\bullet}E_{\gamma}).\tag{26}\] By 24 , \[\mathrm{Gr}^{P}_{2d_{\gamma}-j}S^{d_{\gamma}}=\mathrm{Sym}^{j}(\mathrm{H}^{2}({\mathcal{B}}))\otimes\eta^{d_{\gamma}-j}.\] Let \[\psi^{j}_{\gamma}:\mathrm{Sym}^{j}(\mathrm{H}^{2}({\mathcal{B}}))\to P_{j}H^{2j}_{\gamma,\mathrm{pur}}=\mathrm{Gr}^{P}_{j} H^{2j}_{\gamma,\mathrm{pur}}\] be the restriction of \(\varphi^{j}_{\gamma}\). We have a commutative diagram \[\xymatrix{\mathrm{Sym}^{j}(\mathrm{H}^{2}({\mathcal{B}}))\ar[d]_{\psi^{j}_{\gamma}}\ar[r]^-{\eta^{d_{\gamma}-j}}_-{\sim} & \mathrm{Sym}^{j}(\mathrm{H}^{2}({\mathcal{B}}))\otimes\eta^{d_{\gamma}-j}\ar[d]^{\mathrm{Gr}^{P}_{2d_{\gamma}-j}\varphi_{\gamma}^{d_{\gamma}}}\\ \mathrm{Gr}^{P}_{j}H^{2j}_{\gamma,\mathrm{pur}}\ar[r]^-{\eta^{d_{\gamma}-j}} & \mathrm{Gr}^{P}_{2d_{\gamma}-j}E_{\gamma}}\] where the bottom horizontal arrow is an isomorphism by the property [HL] of a perverse filtration. Therefore, \(\eta^{d_{\gamma}-j}\) induces a \(W\)-equivariant isomorphism \[\mathrm{Im}(\psi^{j}_{\gamma})\cong \mathrm{Im}(\mathrm{Gr}^{P}_{2d_{\gamma}-j}\varphi^{d_{\gamma}}_{\gamma})\stackrel{\eqref{Gr32E39}}{=}\mathrm{Gr}^{P}_{2d_{\gamma}-j}{}'E_{\gamma}.\] It remains to construct a \(W\)-equivariant surjection \[\mathrm{Im}(\psi^{j}_{\gamma})\twoheadrightarrow{}'H^{2j}_{\mathcal{O}}.\]

Pick a point \(x=gL^{+}G\in \mathrm{Gr}_{\gamma,\mathcal{O}}\), and let \(e\in \mathcal{O}\) be the reduction of \(\mathrm{Ad}(g)^{-1}\gamma\) mod \(t\). Then the fiber \(\pi_\gamma^{-1}(x)\subset \mathrm{Fl}_\gamma\) is isomorphic to \(\mathcal{B}_e\) (well-defined up to \(G_e\)). Restriction along the inclusion \(\kappa_x: \mathcal{B}_e\cong \pi_\gamma^{-1}(x)\hookrightarrow\mathrm{Fl}_\gamma\) gives a \(W\)-equivariant map \[\kappa_x^*: \mathrm{H}^{*}({\mathrm{Fl}_\gamma})\to \mathrm{H}^{*}({\mathcal{B}_e})\] and it factors through \(\mathrm{H}^{*}({\mathrm{Fl}_\gamma})_{\mathrm{pur}}\) because \(\mathrm{H}^{*}({\mathcal{B}_e})\) is pure by Springer [18]. Composing \(\psi^{j}_{\gamma}\) with \(\kappa_{x}^{*}\) we get a map of \(W\)-modules \[\label{res32Flg32to32Be32inv} \mathrm{Sym}^{j}(\mathrm{H}^{2}({\mathcal{B}}))\twoheadrightarrow\mathrm{H}^{2j}({\mathcal{B}})\to P_jH^{2j}_{\gamma,\mathrm{pur}}\to \mathrm{H}^{2j}({\mathcal{B}_e})\tag{27}\] which is the restriction along the inclusion \(i_e: \mathcal{B}_e\hookrightarrow\mathcal{B}\). Therefore we get a surjection of \(W\)-modules \[\label{Im32phj} \mathrm{Im}(\psi^{j}_\gamma)\twoheadrightarrow\mathrm{Im}(i_{e}^{*}: \mathrm{H}^{2j}({\mathcal{B}})\to \mathrm{H}^{2j}({\mathcal{B}_e}))={}'H^{2j}_{\mathcal{O}},\tag{28}\] as desired. Since \('H^{2j}_{\mathcal{O}}\) as a quotient of \(\mathrm{Sym}^{j}(\mathrm{H}^{2}({\mathcal{B}}))\) does not depend on the choice of \(e\in \mathcal{O}\), the map 28 does not depend on the choice of \(x\in\mathrm{Gr}_{\gamma,\mathcal{O}}\) either, hence the canonicity of the map. ◻

Proof. Let \(\mathcal{O}\in \mathrm{RT}(\gamma)\). By Theorem 7, \(E_{\mathcal{O}}\) is a subquotient of \(E_{\gamma}\) as a \(W\)-module. By Corollary 1, \(\mathcal{O}\) is \(\gamma\)-relevant. ◻

Proof. Let \(\mathcal{O}\) be a nilpotent orbit such that \(\mathrm{Gr}_{\gamma,\mathcal{O}}\ne\varnothing\), and let \(\mathcal{O}'\) be another nilpotent orbit such that \(\overline{\mathcal{O}'}\supset\mathcal{O}\). By Lemma 1, it suffices to show that \(E_{\mathcal{O}'}\) appears in \(E_{\gamma}\) as a \(W\)-module. By Theorem 7, it suffices to show that \({}'H_{\mathcal{O}}\) contains \(E_{\mathcal{O}'}\) as a \(W\)-module.

Let \(e\in \mathcal{O}\) and \(e'\in \mathcal{O}'\). Let \(d\) and \(d'\) be the dimensions of \(\mathcal{B}_{e}\) and \(\mathcal{B}_{e'}\). Consider the Springer map \(\pi: \widetilde{\mathcal{N}}\to \mathcal{N}\). Then \(i^{*}_{e}\) is the cospecialization map from the stalk of \(\pi_{*}\overline{\mathbb{Q}}_\ell\) at \(0\) to the stalk at \(e\). We may compose this with the further cospecialization map to the stalk at \(e'\) and get \[i^{*}_{e'}: \mathrm{H}^{*}({\mathcal{B}})\xrightarrow{i^{*}_{e}} \mathrm{H}^{*}({\mathcal{B}_{e}})\xrightarrow{s} \mathrm{H}^{*}({\mathcal{B}_{e'}})\] Taking degree \(2d'\) we get \[i^{*}_{e'}: \mathrm{H}^{2d'}({\mathcal{B}})\xrightarrow{i^{*}_{e}} \mathrm{H}^{2d'}({\mathcal{B}_{e}})\xrightarrow{s} \mathrm{H}^{2d'}({\mathcal{B}_{e'}})\cong E_{\mathcal{O}'}\oplus (\cdots)\] We know that the image of \(i^{*}_{e'}\) in degree \(2d'\) is precisely \(E_{\mathcal{O}'}\). Therefore \({}'H^{2d'}_{\mathcal{O}}=\mathrm{Im}(i^{*}_{e})\cap \mathrm{H}^{2d'}({\mathcal{B}_{e}})\) maps surjectively to \(E_{\mathcal{O}'}\). ◻

Proof. (1) (2) The fact \(t^{-1}\gamma\) is integral is equivalent that the adjoint orbit of \(\gamma\) intersects \(t\mathfrak{g}[\![t]\!]\), i.e., \(\mathrm{RT}(\gamma)\) contains the zero orbit. By Theorem 7, \(\mathrm{Gr}^{P}_{\bullet}{}'E_{\gamma}\) maps surjectively onto \(\mathrm{Im}(\mathrm{H}^{*}({\mathcal{B}})\to \mathrm{H}^{*}({\mathcal{B}}))=\mathrm{H}^{*}({\mathcal{B}})\) as a \(W\)-module. Now \(\dim \mathrm{H}^{*}({\mathcal{B}})=|W|\) and \(\dim E_{\gamma}=|W|\) by Theorem 8. Therefore, we must have \({}'E_{\gamma}=E_{\gamma}\), i.e., \(\varphi^{d_{\gamma}}_{\gamma}\) is surjective. Moreover, \(\theta_{\gamma,\{0\}}\) has to be an isomorphism for dimension reasons.

(3) Write \(V=\mathrm{H}^{2}({\mathcal{B}})\) and \(N=\dim \mathcal{B}\). Since \(t^{-1}\gamma\) is integral, we have \(d_{\gamma}\ge N\). We identify \(\mathrm{Sym}^{d_{\gamma}}(V)\) with \(\oplus_{j=0}^{N}\mathrm{Sym}^{j}(V)\otimes\eta^{d_{\gamma}-j}\). Let \(I=\ker(\mathrm{Sym}(V)\to \mathrm{H}^{*}({\mathcal{B}}))\) and \(I^{j}=I\cap \mathrm{Sym}^{j}(V)\) be the homogeneous pieces of \(I\). Since the map \(\varphi_{\gamma}\) factors through \(\mathrm{H}^{*}({\mathcal{B}})\otimes\mathbb{Q}_{\ell}[\eta]\), we see that \(\oplus_{j=0}^{N}I^{j}\otimes\eta^{d_{\gamma}-j}\subset \ker(\varphi^{2d_{\gamma}}_{\gamma})\). Since \(\dim E_{\gamma}=|W|\), for dimension reasons, we must have the equality \(\ker(\varphi^{2d_{\gamma}}_{\gamma})=\bigoplus_{j=0}^{N}I^{j}\otimes\eta^{d_{\gamma}-j}\). This identifies \(E_{\gamma}\) with the following quotient of \(\mathrm{Sym}^{d_{\gamma}}(V\oplus\overline{\mathbb{Q}}_\ell\eta)\) \[\label{Eg32as32quot} E_{\gamma}\cong \mathrm{Sym}^{d_{\gamma}}(V\oplus\overline{\mathbb{Q}}_\ell\eta)/\left(\bigoplus_{j=0}^{N}I^{j}\otimes\eta^{d_{\gamma}-j}\right)\cong \bigoplus_{j=0}^{N}\mathrm{H}^{2j}({\mathcal{B}})\otimes\eta^{d_{\gamma}-j} \cong \mathrm{Sym}(V)/I.\tag{29}\] This gives the required splitting of the perverse filtration.

(4)The canonical action of \(W_{\mathrm{aff}}\) on \(\mathrm{H}^{*}({\mathrm{Fl}})\) when restricted to \(\mathrm{H}^{2}({\mathrm{Fl}})\cong V\oplus\overline{\mathbb{Q}}_\ell\eta\) takes the following form: \(W\) acts on \(V\) by the reflection representation and acts trivially on \(\eta\); \(\lambda\in \mathbb{X}_*(T)\) acts trivially on \(V\) and sends \(\eta\) to \(\eta+\iota_{\eta}(\lambda)\), where \(\iota_{\eta}: \mathbb{X}_*(T)\to V\cong \mathbb{X}^*(T)_{\overline{\mathbb{Q}}_\ell}\) is induced from the \(W\)-invariant bilinear form on \(\mathbb{X}_*(T)\) given by \(\eta\). This \(W_{\mathrm{aff}}\)-action induces a \(W_{\mathrm{aff}}\)-action on the quotient \(\mathrm{Sym}(V)/I\) of \(\mathrm{Sym}(\mathrm{H}^{2}({\mathrm{Fl}}))\), making 29 \(W_{\mathrm{aff}}\)-equivariant.

View \(\mathrm{Sym}(V)\) as the ring of differential operators on the polynomial ring \(\overline{\mathbb{Q}}_\ell[V]\cong \mathrm{Sym}(V^{*})\) that are invariant under translations. Consider the graded-perfect pairing \[\mathrm{Sym}(V)\times \overline{\mathbb{Q}}_\ell[V]\to \overline{\mathbb{Q}}_\ell\] defined by \((D, f)=(Df)(0)\), where \(D\in \mathrm{Sym}(V)\) and \(f\in \overline{\mathbb{Q}}_\ell[V]\). Recall the space of \(W\)-harmonic polynomials \(\mathcal{H}\subset \overline{\mathbb{Q}}_\ell[V]\) consists of those annihilated by the \(W\)-invariant differential operators \(\mathrm{Sym}^{>0}(V)^{W}\), which generate \(I\). Therefore, restricting the second argument of the above pairing to \(\mathcal{H}\), the pairing factors through a perfect pairing \[\label{pairing32harmonic} \mathrm{Sym}(V)/I\times \mathcal{H}\to \overline{\mathbb{Q}}_\ell\tag{30}\] Viewing \(V\) as the affine space \(V+\eta\subset V\oplus\overline{\mathbb{Q}}_\ell\eta\), the \(W_{\mathrm{aff}}\)-action on the latter restricts to an action of \(W_{\mathrm{aff}}\) on \(V\) by affine linear transformations: \(W\) acts by the reflection representation and \(\lambda\in \mathbb{X}_*(T)\) acts by translation by \(\iota_{\eta}(\lambda)\). From the construction, the pairing 30 is \(W_{\mathrm{aff}}\)-equivariant. By the \(W_{\mathrm{aff}}\)-equivariant isomorphism 29 , we conclude that \(E_{\gamma}\) is dual to \(\mathcal{H}\) as \(W_{\mathrm{aff}}\)-modules. ◻

Proof. The factorization pattern of \(f\) implies that the adjoint orbit of \(\gamma\) meets a Levi subgroup \(M\subset G\) whose block sizes are given by \(\{\deg f_{i}\}_{i\in I}\). We may assume \(\gamma\in \mathfrak{m}(\!(t)\!)\). By [17], the minimal reduction type \(\mathcal{O}\) of \(\gamma\) viewed as an element of \(\mathfrak{g}(\!(t)\!)\) is the \(G\)-orbit of the minimal reduction type \(\mathcal{O}_{M}\) of \(\gamma\) viewed as an element of \(\mathfrak{m}(\!(t)\!)\). This allows us to reduce to the case where \(f\) is irreducible, i.e., \(\gamma\) is elliptic.

We compute the minimal reduction type \(\mathcal{O}\) assuming \(\gamma\) is elliptic. Let \(E=F[x]/(f(x))\), a degree \(n\) field extension of \(F\). Let \(v=\frac{d}{n}\) be the valuation of a root of \(f\), so that \((d,n)=1\).

If \(v\ge1\), then \(t^{-1}\gamma\) is integral. The minimal reduction type is the zero orbit, and its Jordan type is the trivial partition, which is the same as \(\lambda_{v}\) by definition.

Consider the case \(v<1\). In [17] we defined the notion of the skeleton of \(\mathrm{Gr}_{G}\) with respect to a maximal \(F\)-torus \(T\) in \(G\): it is the fixed point locus of the neutral component of \(LT\) on \(\mathrm{Gr}_{G}\). In our case, since \(G_{\gamma}\) is a maximal torus of Coxeter type, by [17], the skeleton of \(\mathrm{Gr}_{G}\) with respect to \(G_{\gamma}\) is a single point. If we identify \(\mathrm{Gr}_{G}\) with the moduli space of \(\mathcal{O}_{F}\)-lattices in \(E\) that have the same volume as \(\mathcal{O}_{E}\), then the lattice \(\mathcal{O}_{E}\) itself is the unique point in the skeleton. By [17], \(\mathcal{O}\) is the orbit of the nilpotent endomorphism \(\overline{\gamma}: \mathcal{O}_{E}/(t)\to \mathcal{O}_{E}/(t)\) (the reduction of multiplication by \(\gamma\)). Choosing a uniformizer \(t^{1/n}\) of \(E\) and using the \(k\)-basis \(\{1,t^{1/n},\cdots, t^{(n-1)/n}\}\) of \(\mathcal{O}_{E}/(t)\), we see that the Jordan type of \(\overline{\gamma}\) is the most balanced partition \(\lambda\) of \(n\) with \(d\) parts, i.e., \(\lambda_{v}\). ◻

Proof. In view of the surjectivity of \(\theta_{\gamma,\mathcal{O}}\) proved in Theorem 7, it suffices to show that \(\dim E_{\gamma}=\dim H_{\mathcal{O}}\).

We first reduce the general case to the case where \(\gamma\) is elliptic. Changing \(\gamma\) in its adjoint orbit if necessary, there is a Levi subgroup \(M\subset G\) whose block sizes are the degrees \(\{\deg f_{i}\}\) of irreducible factors of the characteristic polynomial of \(\gamma\), such that \(\gamma\in \mathfrak{m}(\!(t)\!)\) and is elliptic there. By Lemma 17, \(\dim E_{G,\gamma}=|W/W_{M}|\dim E_{M,\gamma}\). On the other hand, by [17], the minimal reduction type \(\mathcal{O}\) of \(\gamma\) viewed as an element of \(\mathfrak{g}(\!(t)\!)\) is the \(G\)-orbit of the minimal reduction type \(\mathcal{O}_{M}\) of \(\gamma\) viewed as an element of \(\mathfrak{m}(\!(t)\!)\). By 31 , we have \(\dim H_{\mathcal{O}}=|W/W_{M}|\dim H_{\mathcal{O}_{M}}\). It then suffices to show that \(\dim E_{M,\gamma} = \dim H_{\mathcal{O}_{M}}\). The neutral component of \(\mathrm{Fl}_{M,\gamma}\) is isomorphic to a product of \(\mathrm{Fl}_{\mathrm{SL}_{n_{i}},\gamma_{i}}\) where \(n_{i}=\deg f_{i}\) are the block sizes of \(M\), and \(\gamma_{i}\in \mathfrak{m}_{i}(\!(t)\!)\) has irreducible characteristic polynomial \(f_{i}\). We thus reduce to the case where \(\gamma\) is elliptic, which we now assume.

Let \(v\) be the valuation of any root of \(f(x)\), the characteristic polynomial of \(\gamma\). By Lemma 16, the minimal reduction type \(\mathcal{O}\) of \(\gamma\) has Jordan type \(\lambda_{v}\). The result of Kivinen and Tsai [5] says that \(\dim E_{\gamma}=|W/W_{\lambda_{v}}|\) in this case. On the other hand, 32 implies \(\dim H_{\mathcal{O}}=|W/W_{\lambda_{v}}|\). The equality \(\dim E_{\gamma}=\dim H_{\mathcal{O}}\) is proved. ◻

Proof. Choose a parabolic subgroup \(P\subset G\) containing \(M\) as a Levi factor. We may assume \(P\) is standard, i.e., \(P\supset B\). Let \(N_{P}\) be the unipotent radical of \(P\). For each connected component \((LM)^{\omega}\) of \(LM\), where \(\omega\in \pi_{1}(M)\), and each \(w\in [W_{M}\backslash W]\) (minimal length representative with respect to the length function on \(W\) given by \(B\)), consider the semi-infinite orbit \(\Sigma^{\omega}_{w}:=(LN_{P})(LM)^{\omega} w \mathbf{I}/\mathbf{I}\subset \mathrm{Fl}\). The intersection \(\mathrm{Fl}_{G,\gamma}\cap \Sigma^{\omega}_{w}\) has a canonical map \(q^{\omega}_{w}: \mathrm{Fl}_{G,\gamma}\cap \Sigma^{\omega}_{w}\to \mathrm{Fl}_{M,\gamma}\cap \mathrm{Fl}^{\omega}_{M}\) (where \(\mathrm{Fl}^{\omega}_{M}=(LM)^{\omega}/\mathbf{I}_{M}\)) by sending \(umw\mathbf{I}\) to \(m\mathbf{I}_{M}\). The same argument as [1] shows that \(q^{\omega}_{w}\) is an iterated affine space fibration, and that \(\mathrm{Fl}_{G,\gamma}\cap \Sigma^{\omega}_{w}\to \mathrm{Fl}_{M,\gamma}\) has the same dimension as \(\mathrm{Fl}_{G,\gamma}\). Let \(\Sigma_{w}=(LP) w\mathbf{I}/\mathbf{I}\subset \mathrm{Fl}\), which is the union of \(\Sigma^{\omega}_{w}\) for varying \(\omega\in \pi_{1}(M)\). The maps \(q^{\omega}_{w}\) induce a bijection between the sets of irreducible components \(\mathfrak{q}_{w}: \mathrm{Irr}(\mathrm{Fl}_{G,\gamma}\cap \Sigma_{w})\stackrel{\sim}{\to}\mathrm{Irr}(\mathrm{Fl}_{M,\gamma})\) that is equivariant under the translation actions by \(A_{\gamma}\). It then induces a bijection on \(A_{\gamma}\)-orbits \(\overline{\mathfrak{q}}_{w}: \mathrm{Irr}(\mathrm{Fl}_{G,\gamma}\cap \Sigma_{w})/A_{\gamma}\stackrel{\sim}{\to}\mathrm{Irr}(\mathrm{Fl}_{M,\gamma})/A_{\gamma}\). Now \(\{\Sigma_{w}\}_{w\in [W_{M}\backslash W]}\) form a partition of \(\mathrm{Fl}_{G}\), we conclude that \(|\mathrm{Irr}(\mathrm{Fl}_{G,\gamma})/A_{\gamma}|=|W/W_{M}|\cdot|\mathrm{Irr}(\mathrm{Fl}_{M,\gamma})/A_{\gamma}|\), which then implies 33 . ◻