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r\rbrace _{0}$ (or, more generally, of $\widehat{P}\otimes V_\tau ^k$ or $S^k\widehat{P}$ , as in Definition
REF ) admits a resolution $\omega _0^*$ by injective complexes, where, with respect to an
$\alpha $ -stable flag of $\widehat{P}$ , each entry $\omega _{0,i}^*$ has the form $h^0\circ S^k F^*$ ,
where $F^*$ is an injective resolution of the complex $f$ appearing in the usual exact triangle
$0\longrightarrow {\cal O}_X(\xi _0-1)\longrightarrow {\cal O}_X^{\oplus n}\longrightarrow {\cal O}_X(\xi _0+1){\stackrel{f}{\longrightarrow }} {\cal O}_X\longrightarrow \cdots ,$
where
$\xi _0=E\cap C$ . Furthermore, $\omega _0^*$ has a nice morphism to ${\rm Hom}(\widehat{\eta }\boxtimes {\mathbf {1}}, {\mathbf {1}})\boxtimes V_\tau ^k$ which is a quasi-isomorphism
in the derived category. There is a similar construction of a nice complex $\widetilde{\omega }_0^*$ of ${\cal O}_U$ -modules, together with a
nice quasi-isomorphism $\widetilde{\omega }_0^*\longrightarrow {\rm Hom}^\bullet (\widetilde{\eta }\boxtimes \widehat{{\mathbf {1}}}, \widehat{{\mathbf {1}}})\boxtimes V_\tau ^k$ .
The above results allow us to make the following definition, which includes the special case in the proposition.
Definition 3.4
Let $V$ be a module for $\Gamma ^{\rm el}_+$ equipped with a homomorphism $\xi : M^0\rightarrow V$ . We
shall say that $V$ belongs to the category ${\cal M}^0_\mu (\Gamma ^{\rm el})$ if $V$ has finite length,
$\xi (M^0)$ is an irreducible $\Gamma ^{\rm el}_+$ -submodule of $V$ , $V$ admits a projective
presentation ${\cal P}\longrightarrow V\longrightarrow 0$ , such that the following conditions are satisfied. There is
an $\alpha $ -stable flag of $\widehat{P}$ satisfying the condition in Definition REF , and ${\cal P}$ is a filtered complex of finite type whose filtration quotients ${\cal F}{\cal P}_{i,i+1}$ admit representations on the associated graded $S^k\widehat{{\mathbf {1}}}$ such that, with respect to this flag of $\widehat{P}$ , ${\rm gr}^{\rm RH}_{\mu }{\cal F}{\cal P}_{i,i+1}\in \mathcal {RP}r_G(\lambda ^{i}, m)^{++}$ and $\dim {\rm Hom}^{\rm RH}({\rm gr}^{\rm RH}_{\mu }{\cal F}{\cal P}_{i,i+1}, \mathcal {RP}r_G(\mu )^{(0)})=0$ .
Furthermore, we require that $\dim \mathrm {Ext}^j_{\Gamma ^{\rm el}}(M^0,V)<\infty $ for all $j$ .
The proof of the following proposition is straightforward.
Proposition 3.5
Let $V$ belong to ${\cal M}^0_\mu (\Gamma ^{\rm el}_+)$ . Assume that $V^{\prime }\in \mathcal {M}^0_\mu (\Gamma ^{\rm el}_+)$ and $V^{\prime \prime }\in \mathcal {M}^0(\Gamma ^{\rm el}_+)$
such that $V^{\prime \prime }\hookrightarrow V$ and $V^{\prime \prime }\hookrightarrow V^{\prime }$ . Then we have a non-degenerate pairing of finite dimension
${}_V\langle V^{\prime }\rangle :=V^*_0\otimes _{\Lambda _0} V^{\prime }_0: {\rm Ext}^*_{\Gamma ^{\rm el}_+}(V^{\prime },V)\otimes {\rm Ext}^*_{\Gamma ^{\rm el}_+}(V,V^{\prime })\longrightarrow \mathbb {C}$
and there are exact sequences
$@C=20pt{ \cdots [r] &{}_V\langle V^{\prime \prime }\rangle [r]^-{\lambda } &{}_V\langle V\rangle [r] &{}_V\langle V/V^{\prime \prime }\rangle [r] & \cdots }\\@C=20pt{ \cdots [r] &{}_V\langle V^{\prime \prime }\rangle [r]^-{\lambda } &{}_V\langle V^{\prime }\rangle [r] &{}_V\langle V^{\prime }/V^{\prime \prime }\rangle [r] & \cdots }.$
Moreover, if $V$ has finite length as a $\Gamma ^{\rm el}_+$ -module, then the complex $V^*_0$ belongs
to the category $D^b({\rm mod}\ \Gamma ^{\rm el}_+)$ as a complex of left $\Gamma ^{\rm el}_+$ -modules.
Notice that $\widehat{P}$ with the highest term is the minimal projective cover of $M^0$ in $D^b({\rm mod}\ \Gamma ^{\rm el})$ .
Hence we have
${\rm Ext}^*_{\Gamma ^{\rm el}}(M^0, V)\cong H^*_V(M^0\otimes _{\Gamma ^{\rm el}}V^*_0).$
Here $V^*_0$ can be regarded as an object of $D^b({\rm mod}\ \Gamma ^{\rm el})$ since $V$ is assumed to have
finite length. Then Proposition REF follows from the spectral sequence (REF ), where we take $X= M^0$ and $Y^*=V^*_0$ .
## Quotients by cuspidal subgroups
It will be convenient to relate categories $D^b({\rm mod}\ \Gamma ^{\rm el})$ of categories of $G$ -perverse sheaves. It is possible to do this via $(\ref {tri})$ using the relative projectivity of the spectral varieties, but we prefer
to use the stronger Theorem REF , for which we need to generalize Theorem
REF .
We begin with a proof of the following useful result, which allows us to construct good quasi-isomorphisms using direct sums of projective modules.
Lemma 4.1
Suppose $X^1$ and $X^2$ are projective coherent $\widehat{\cal O}_N$ -modules and $W$ is an object
of $D^b(coh\ \widehat{{\cal O}}_N)$ satisfying ${\rm Ext}^1_{\widehat{{\cal O}}_N}(X^1,W)=0={\rm Ext}^1_{\widehat{{\cal O}}_N}(W,X^2)$ .
Suppose also that ${\cal P}^i_0\longrightarrow X^i_0$ are good quasi-isomorphisms, and that $W^{\prime }\longrightarrow W_0$ is an injective
complex with $W^{\prime }$ a filtered complex of finite type whose filtration
quotients ${\cal F}W^{\prime }_{a,b}$ have no non-trivial subquotients which are isomorphic to ${\rm gr}_{ab}^{\rm RH}\widehat{P}_{i,i+1}$ as $N$ -modules, for
all $i=0,1,\ldots $ and $a=1,2,\ldots .$ Then the morphisms
$X^1_0\otimes _{\widehat{{\cal O}}_N}W_0,\qquad W_0\otimes _{\widehat{{\cal O}}_N}X^2_0$
are quasi-isomorphisms of complexes of coherent $\widehat{{\cal O}}_U$ -modules.
The first assertion is easily seen by using the exact triangle
$@C=24pt{ H^1_U({\cal P}^1_0\otimes _{\widehat{{\cal O}}_N}W_0) [r]^-{\partial } &H^0_U({\cal F}W^{\prime }_{0,1}\otimes _{\widehat{{\cal O}}_N}X^1_0)[r] &H^0_U(X^1_0\otimes _{\widehat{{\cal O}}_N}W^{\prime }) }.$
The second assertion can be proved in the same way.
Now suppose that $X^1$ is a projective coherent $\widehat{\cal O}_N$ -module and let
${\cal P}^i_0\longrightarrow (X^1)^\vee _0$
be good quasi-isomorphisms. Furthermore, assume that
$V^\vee _0$ is a filtered complex of finite type whose filtration quotients ${\cal F}V^\vee _{a,b}$ admit representations on ${\rm gr}^{RH}_{ab}\widehat{P}_{i,i+1}$ such that the map
$H^{0}_{N}({\rm gr}^{RH}_{ab}\widehat{P}_{i,i+1}\otimes V^\vee _{a,b})\rightarrow H^{0}_{N}({\rm gr}^{RH}_{ab}\widehat{P}_{i,i+1}\otimes V^\vee _{0})$
is surjective for $i\le a$ , where $\widehat{P}$ is a minimal $\widehat{{\cal O}}_N$ -module with the highest term as in Definition REF . We define ${\cal T}^i_{V}\in D^{-,b}({\rm proj}\ \widehat{{\cal O}}_U)$ by the following mapping cone:
$@C=14pt{ {\cal T}^i_{V_0} : 0 [r] & {\cal P}^i_{0} [r] & {\cal P}^i_{0}\otimes _{\widehat{{\cal O}}_N} V^\vee _0 [r]^-{\alpha _{i}^V} &{\cal T}^i_{V_0}[r] & 0},$
where
$\alpha _{i}^V:{\cal P}^i_{0}\otimes _{\widehat{{\cal O}}_N}V^\vee _0\longrightarrow {\rm Cone}({\cal F}V^\vee _{0,1}\otimes _{\widehat{{\cal O}}_N}{\cal P}^i_0\longrightarrow {\cal P}^i_0\otimes _{\widehat{{\cal O}}_N}{\cal F}V^\vee _{0,1}).$
The following is an analogue of Theorem REF .
Proposition 4.2
There is an $i$ -independent isomorphism
$\phi ^V:\lim _{i\rightarrow \infty }{\cal T}^i_{V_0}\stackrel{\sim }{\longrightarrow }(X^1_0)\otimes _{{\cal O}_U}V.$
If $V$ is assumed to be a filtered complex of finite type whose filtration quotients ${\cal F}V_{a,b}$
have no non-trivial subquotients isomorphic to $\mathrm {gr}^{RH}_{ab}{\cal F}\widehat{P}_{i,i+1}$ as
$N$ -modules, then there is a similar isomorphism
$\phi ^{V^\vee }:\lim _{i\rightarrow \infty }{\cal T}^i_{(V^\vee )_0}\stackrel{\sim }{\longrightarrow }(X^1_0)^\vee \otimes _{{\cal O}_U}V^\vee .$
The first claim is obtained by applying
$H_{N}^\bullet (\widehat{^\vee _{N}\otimes _{\widehat{{\cal O}}_N}-)\ to the morphisms given in the proposition.Since \widehat{^\vee _{N}\otimes _{\widehat{{\cal O}}_N}{\cal P}^i_0 is a good quasi-isomorphism for each i, we have\begin{equation*}{\cal H}_N(\widehat{^\vee _{N}\otimes _{\widehat{{\cal O}}_N}(X^1)^\vee _0)=\widehat{^\vee _{N}\otimes _{\widehat{{\cal O}}_N}(X^1)^\vee =X^1^\vee }as is seen by taking a minimal resolution of (X^1)^\vee . Furthermore, the assumptions imply that{\cal H}_N(\widehat{^\vee _{N}\otimes _{\widehat{{\cal O}}_N}{\cal T}^i_{V_0})={\cal H}_N(\widehat{^\vee _{N}\otimes _{\widehat{{\cal O}}_N}{\cal P}^i_{0})}as graded modules. Hence we can regard X^1^\vee \otimes V as a filtration subquotient of\widehat{^\vee _{N}\otimes _{\widehat{{\cal O}}_N}{\cal T}^i_{V_0}. From Lemma \ref {lemma1567} it is clear thatthe H^1 term vanishes. On the other hand, the spectral sequence associated to the filtration{\cal F}V^\vee _{ab} is convergent and each of the filtration quotients of (\ref {eq1421}) has a trivialH^1 term. This proves that X^1^\vee \otimes V={\cal H}_N(\widehat{^\vee _{N}\otimes _{\widehat{{\cal O}}_N}(X^1)^\vee \otimes V), which establishes the isomorphism of \lim _{i\rightarrow \infty }{\cal T}^i_{V_0}and (X^1_0)\otimes V.\Box \medskip }The second part can be proved in the same way, using a minimal projective resolution of (X^1)^\vee and Lemma \ref {lemma1567}.\Box }}}}\section {The categories \mathcal {V}(\rho ) for G finite}}Consider the functor{\rm per}_{U}\longrightarrow D^b(coh\ \widehat{{\cal O}}_{U}),where per_{U} denotes the bounded derived category of perverse sheaves on U with constructible complex coefficients. Define the subcategories \mathcal {V}(\rho )\subset D^b(coh\ \widehat{{\cal O}}_{U}) as the essential images of the functors\begin{eqnarray*}@R=14pt{\Phi ^{\lambda }_{\rho }:D^b({\rm mod}\ \mathcal {RP}r_G(\lambda ,m))&\longrightarrow & {\rm per}_{U}\\@C=20pt{P&\longmapsto &\Phi ^{\lambda }_{\rho }(P):=j^*{\cal R}({\mathbb {G}}\times {\mathbb {G}},{\rm Gr}_\lambda ,P)}\\@R=14pt{\Phi ^{\lambda ,0}_{\rho }:D^b({\rm mod}\ \mathcal {RP}r_G(\lambda ,m)^{(0)})&\longrightarrow & {\rm per}_{U}\\@C=20pt{P&\longmapsto &\Phi ^{\lambda ,0}_{\rho }(P):=j^*{\cal R}({\mathbb {G}}\times {\mathbb {G}},{\rm Gr}^0_\lambda ,P)}\end{eqnarray*}with P=S^{r}P.\medskip }We will prove the following theorem, which is a GIT analogue of Theorem \ref {main1}for the group G. The proof is analogous.\end{equation*}\begin{thm} Assume G is finite. There is an equivalence of triangulatedcategoriesD^b(coh\ \widehat{{\cal O}}_{U})\cong {\mathcal {V}}(\rho ).Moreover, there is an equivalenceD^b({\rm mod}\ \Gamma ^{\rm el})\cong \bigsqcup _{\lambda \in \mathcal {L}}D^b({\rm mod}\ \mathcal {RP}r_G(\lambda ,m)^{(0)})that induces an isomorphism of each \mathcal {V}(\lambda ) with the essential imageof \bigsqcup _{k\in \mathbb {Z}} \Phi ^{\lambda ,0}_{\rho }(D^b({\rm mod}\ \mathcal {RP}r_G(\lambda ,m)^{(0)})[-k]).The equivalence respects all tensor products. Moreover, there exist compact generators for the categories \mathcal {RP}r_G(\lambda )^{(0)}\subset D^b({\rm mod}\ \mathcal {RP}r_G(\lambda )), hence compact generators for each \mathcal {V}(\lambda ),and hence also for D^b(coh\ \widehat{{\cal O}}_{U}).\end{thm}}\begin{lemma}We have\begin{equation}{\rm Ext}^*({\mathcal {V}(\rho )},D^b({\rm mod}\ \widehat{{\cal O}}_U))=0.\end{equation}\end{lemma}{\it Proof} This follows immediately from (\ref {extm}), the equivalence\cite {mac} and Proposition \ref {ind} (since \widehat{P} has the highestterm, it can be used to compute Ext_{N} as well as Ext_{\Gamma ^{\rm el}}).\Box \end{equation*}\begin{cor} Let V\in \mathcal {V}(\rho ) and X^1\in D^b(coh\ \widehat{{\cal O}}_N).The following natural morphisms are quasi-isomorphisms\begin{eqnarray*}&& {\rm Hom}_{\widehat{{\cal O}}_N}(X^1,\widehat{P})\otimes _{\widehat{{\cal O}}_U}V\longrightarrow {\rm Hom}(X^1,V);\\&& V^*\otimes _{\widehat{{\cal O}}_N}\widehat{P}\longrightarrow {\rm Hom}(V,{\widehat{P}}^*).\end{eqnarray*}\end{cor}}We omit the proof, which is a modification of the proof of Corollary \ref {c1}.$
Theorem can now be proved in the same way as Theorem REF using the above
two results.
The next result is a generalization of Proposition REF . We use the notation in §.
$\qquad $
Proposition 4.3
Assume $G$ is finite and fix $\lambda \in {\cal L}$ . For $\mu \in {\mathcal {L}}_\lambda $ , let
$A^r_{\mu }\in D^{b}({\rm mod}\ \widehat{{\cal O}}_{U})$ be an object, defined as the cone of
$&&R^{r}h_{1}\circ R^{r-1}h_{2}\circ \cdots \circ R^rh_{l}(\widehat{{\cal O}}_U\longrightarrow \widehat{{\cal O}}_{U_{l}^{\vee }})\\&&\qquad \quad +R^{r}h_{2}\circ R^{r-1}h_{3}\circ \cdots \circ R^rh_{l}(\widehat{{\cal O}}_U\longrightarrow \widehat{{\cal O}}_{U_{l}^{\vee }})\circ S_{\beta _{1}}^{1}\\&&\qquad \quad +\cdots \\&&\qquad \quad +R^{r}h_{l}(\widehat{{\cal O}}_U\longrightarrow \widehat{{\cal O}}_{U_{l}^{\vee }})\circ S_{\beta _{1}}^{1}\circ \cdots \circ S_{\beta _{l-1}}^{1}\qquad $
in $D^{b}({\rm mod}\ \widehat{{\cal O}}_{U})$ . Then we have $A^r_{\mu }\in {\mathcal {V}}(\lambda )$ .
Moreover, there is an object $A_{\mu }\in D^{b}({\rm mod}\ \widehat{{\cal O}}_{U})$ such that
${\cal M}^r_{\mu }:=A_{\mu }\otimes (A^r_{\mu })^\vee \in D^{b}({\rm mod}\ \mathcal {RP}r_{G}(\lambda , m)),$
where ${\cal M}^r_{\mu }$ is as in Proposition REF .
We omit the proof since it is quite similar to the proof of Proposition REF .
Corollary 4.4
The tensor product structure of the above theorem descends to
a tensor product structure on $D^b({\rm mod}\ \Gamma ^{\rm el})$ which is compatible with the product $\times $ on ${\rm Gr}^{\rm el}_r$ .
The proof is the same as that of Corollary REF .
## Deformation of {{formula:c111ae1d-b11e-40a6-b5de-f0b6f2cdae02}} and the proof of main theorem
We can now prove Theorem REF . The argument is a straightforward deformation argument based on
Theorem and Corollary REF over a suitable base scheme.
We now construct this base scheme, following .
First recall that the smooth affine variety ${\cal V}$ parametrizes nilpotent $N\times N$ matrices, for
$N = \dim _{k}V$ , and so that there is a diagram
${&{\cal V}[dl]_-{s}[d]^-{e}\\U\subset X&Y}$
which is isomorphic to the normal diagram $p:{X^1\rightarrow X}$ of ,
with $X = Y$ . By Theorem 1, the stack
${\mathbb {G}\backslash ({\cal V}\times _{Y} {\mathbb {G}})}= \lbrace (\eta ,g)\in {\cal V}\times _{Y}{\mathbb {G}};\ \eta g=0\rbrace $
is isomorphic to the quotient ${\rm Bun}_G$ of affine spaces by $G$ . The morphism $s\times {\mathbb {G}}$ on ${\rm Bun}_G$
is precisely the pullback of the morphism $p^*$ studied in that paper. The theorem below is now
proved precisely as in .
Theorem 5.1 There is a category equivalence
$\mathrm {F}: D^b_{{\rm cont}}(\mathcal {B}^0(\mathcal {O},{\rm Gr}^{\rm el})\otimes \mathbb {C})&\longrightarrow & D^{b}(\rm mod \ \Gamma ^{\rm el}_\mathcal {O}),$
in which $D^{b}(\rm mod \ \Gamma ^{\rm el})$ carries the tensor product structure given in Theorem REF (using Proposition REF ). The same equivalence preserves tensor product structure.
Similarly, there are equivalences
$\mathrm {F}: D^b_{{\rm cont}}(\mathcal {B}^0(\mathcal {O},{\rm Gr}_r)\otimes \mathbb {C})&\longrightarrow & D^{b}(\rm mod \ \mathcal {RP}r_{G,r,\mathcal {O}}),$
which is compatible with the tensor product structure given in Corollary REF .
Our main theorem, Theorem REF , follows from Theorem REF in the same way as Theorem REF follows from Theorem REF . We will now make this precise. We define
the morphisms
$\Phi _r^{\rho ,0}: D^b_{{\rm cont}}(\mathcal {B}^0(\mathcal {O},{\rm Gr}^0_{\lambda , r})\otimes \mathbb {C})\longrightarrow D^{b}(\rm mod \ \mathcal {RP}r_{G,r,\mathcal {O}}^{(0)}),$
just as in §REF , and we let $V_\lambda ^r$ be a lift to
$D^b(\rm mod \ \mathcal {RP}r_{G,r,\mathcal {O}}^{(0)})$ of $\mathcal {M}^r_\lambda $ , which exists by Corollary REF .
We define ${\mathcal {V}}(\lambda ,r)$ to be the essential image of the functor (REF ).
$\qquad $
(Proof of Theorem REF )
Part 1 follows from Theorem REF , and Parts 2 and 4 are straightforward since everything is an
equivalence. For Part 3, we need to prove that ${\mathcal {V}}(\rho ,m)$
has compact generators. From Theorem , it is clear that compact generators of the tensor product
category
${\mathcal {V}}(\lambda ,r_1)\boxtimes \cdots \boxtimes {\mathcal {V}}(\lambda ,r_k)$
for all $k,r$ and all $\lambda $ ,
will generate all of ${\mathcal {V}}(\rho ,m)$ . Thus we only have to prove this statement for a single $\lambda $ and $r$ .
The category ${\mathcal {V}}(\lambda , r)$ admits compact generators for the following reason. For an irreducible ${\cal RP}r_{G,r,\mathcal {O}}^{(0)}$ -module $L$ , we choose an $L^{\prime }\in {\mathcal {V}}(\lambda ,r)$ with ${\rm Ext}_{{\cal RP}r_{G,r,\mathcal {O}}}^{0}(L,L^{\prime })=k$ , and we define
${\cal T}_L:=L^{\prime }\otimes V^\vee _{\lambda }^r\in D^{-,b}(\mathcal {RP}r_{G,r,\mathcal {O}}^{(0)}).$
These ${\cal T}_L$ are compact objects in $D^{b}(\mathcal {RP}r_{G,r,\mathcal {O}})$ . Furthermore, if $P_L$ is a minimal projective module with simple quotient $L$ , we see from the standard spectral sequence, together with Theorem , that
${\rm Ext}^*_{\mathcal {RP}r_{G,r,\mathcal {O}}}(\sum _L{\cal T}_L, \bigoplus _LP_L^\vee )=0\quad {\rm and} \quad {\rm Hom}_{\mathcal {RP}r_{G,r,\mathcal {O}}}(\sum _L{\cal T}_L, \bigoplus _LP_L^\vee )=\mathbb {C},$
where in both cases the sum is over all isomorphism classes of irreducible ${\cal RP}r_{G,r,\mathcal {O}}^{(0)}$ -modules $L$ . It is now easy to see that the set $\lbrace {\cal T}_L\mid {\rm dim}\ L =1\rbrace $ generates $D^{b}(\mathcal {RP}r_{G,r,\mathcal {O}}^{(0)})$ . $\Box $
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