Quiver Full Flag Projective Semisimplicity

For each vertex $i$, forgetting the repeated colors identifies the flag data of colors in $V_i$ with a single full flag of $V_i$, so ${\mathcal{F}}_{\nu }$ is identified with ${\prod }_i\mathrm{Flag}(V_i)$: a graded increasing flag with successive colors ${\nu }_1,\dots ,{\nu }_m$ is exactly a choice, for every occurrence of $i$ in $\nu$, of a strictly increasing chain of subspaces of $V_i$ ending at $V_i$. The full flag variety of a $\beta$-dimensional space $W$ is the closed incidence locus of successive inclusions of tautological subspaces inside ${\prod }_{d=1}^{\beta -1}\mathrm{Gr}(d,W)$: over the product of Grassmannians the composite of the tautological monomorphism $S_d\hookrightarrow W$ with the tautological quotient $W\twoheadrightarrow W\mathrm{/}{S}_{d^{\mathrm{\prime }}}$ for $d<d^{\mathrm{\prime }}$ vanishes exactly on the pairs of subspaces with $S_d\subseteq S_{d^{\mathrm{\prime }}}$, a closed condition. Each Grassmannian is projective by [F2] and [F3], finite products of nonempty projective varieties are projective by [F4], so $\mathrm{Flag}(W)$ is a nonempty projective algebraic set. It is irreducible: $\mathrm{GL}(W)$, the nonvanishing-determinant open of a matrix affine space, is irreducible and maps surjectively to the flag set by transporting a fixed flag, since every flag admits an adapted basis. Thus it is a projective variety; empty products and $\beta \in \{0,1\}$ give a point.

Successively choosing a line in the remaining quotient exhibits $\mathrm{Flag}(W)$ as an iterated projective-space bundle with relative dimensions $\beta -1,\beta -2,\dots ,1$: the step adding the next subspace has fibre the projective space of lines in the quotient by the previously chosen subspace, and the standard charts of [F2] trivialize each step algebraically. Hence $\mathrm{Flag}(W)$ is smooth of dimension $\beta (\beta -1)\mathrm{/}2$, and the chart computations give algebraic local bases adapted to the tautological flag, in which the flag is the span of the first $r$ basis vectors. Applied factorwise this makes ${\mathcal{F}}_{\nu }={\prod }_i\mathrm{Flag}(V_i)$ a smooth projective variety of dimension ${\sum }_i{\beta }_i({\beta }_i-1)\mathrm{/}2$, covered by charts carrying an adapted local basis of $V$.

On such an adapted-basis chart, label each basis vector $e_p$ by its occurrence $p$ in $\nu$, so $F^r$ is spanned by the $e_p$ with $p\le r$. Stability says that the coefficient of $e_q$ in $x_h{e}_p$ vanishes whenever $q>p$; a loopless arrow has distinct source and target colors, so $q=p$ never occurs, and the free coefficients are exactly the pairs with ${\nu }_p=i$, ${\nu }_q=j$, $q<p$ for an arrow $h:i\to j$, in the number $r_{\nu }={\sum }_{h:i\to j}\mathrm{\#}\{(p,q):{\nu }_p=i,{\nu }_q=j,q<p\}$ of the Statement. These defining equations are linear in the representation coordinates with $r_{\nu }$ independent free coordinates, so on the chart the incidence is the product of the chart with a linear subspace of $E_V$; changes of adapted basis preserve the conditions and act linearly on the coordinates, so the local kernels glue to a vector subbundle $W_{\nu }\subseteq E_V\times {\mathcal{F}}_{\nu }$ with locally free quotient, whose total space is exactly $X_{\nu }$. Consequently $\dim X_{\nu }=r_{\nu }+{\sum }_i{\beta }_i({\beta }_i-1)\mathrm{/}2$ by step 2.1, and the action of $g\in G_V$ on matrices and on flags by translation preserves the incidence equations, so $W_{\nu }$ is $G_V$-equivariant.

By step 1.1 the flag variety ${\mathcal{F}}_{\nu }$ is a closed subvariety of a product of projective spaces, hence of some ${\mathbb{P}}^N$, and $E_V\times {\mathcal{F}}_{\nu }$ is closed in the relative projective space ${\mathbb{P}}_{E_V}^N$. Since $X_{\nu }\subseteq E_V\times {\mathcal{F}}_{\nu }$ is closed by step 3.1, the map ${\pi }_{\nu }$ is a closed immersion followed by that projection, hence projective, and proper by [F5], which consumes the assumed AC as recorded in [A1]. No paving or semismallness hypothesis enters this argument.

Definition of the compactification: ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(U)=\mathrm{Proj}\mathrm{Sym}(U^{\vee })$ is the instance of the quotient convention of [F1] for the dual bundle, so on a local frame it is again a projective-space bundle with homogeneous coordinates the coefficients of a vector. The inclusion $W_{\nu }\oplus \mathcal{O}\hookrightarrow (E_V\oplus \mathbb{C})\times {\mathcal{F}}_{\nu }$ of vector bundles over ${\mathcal{F}}_{\nu }$ induces a closed immersion of line projectivizations ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(W_{\nu }\oplus \mathcal{O})\hookrightarrow {\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(E_V\oplus \mathbb{C})\times {\mathcal{F}}_{\nu }$ over ${\mathcal{F}}_{\nu }$, because on a local frame the equations are the same homogeneous linear equations with the extra coordinate $t$ unconstrained. Composing with the projection to ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(E_V\oplus \mathbb{C})$ gives the $G_V$-equivariant map ${\bar{\pi }}_{\nu }$, which is projective; its source is projective by its displayed closed embedding into the product of projective varieties [F4], and smooth because [F1] gives local products of the smooth base ${\mathcal{F}}_{\nu }$ with ${\mathbb{P}}^{r_{\nu }}$, including $r_{\nu }=0$. The affine chart $t\mathrm{\ne }0$ of the target consists of the lines spanned by vectors $(w,1)$ with $w\in E_V$, and its inverse image under the closed immersion consists of the $(w,1)$ with $w\in W_{\nu ,F}$, so as a set and scheme-theoretically it is exactly $X_{\nu }$, with ${\bar{\pi }}_{\nu }(w,F)=w={\pi }_{\nu }(w,F)$ there.

For the finite associated version fix $N_i\ge {\beta }_i$ and put $B_N={\prod }_{{\beta }_i>0}\mathrm{Gr}({\beta }_i,N_i)$ with tautological subbundles $S_i\subseteq {\mathcal{O}}^{N_i}$ for ${\beta }_i>0$, setting $S_i=0$ for ${\beta }_i=0$, and $E_N={⨁}_{h:i\to j}\mathcal{H}om(S_i,S_j)$ over $B_N$. Let ${\mathcal{F}}_{\nu ,N}$ be the relative full flag variety over $B_N$ whose points over $(S_i{)}_i$ are the increasing full graded flags of ${⨁}_i{S}_i$ with successive colors ${\nu }_r$, recording the terminal ${\beta }_i$-plane $S_i\subseteq {\mathbb{C}}^{N_i}$. Recording all intermediate subspaces and the terminal $S_i$ realizes this variety as the product, over ${\beta }_i>0$, of the closed inclusion loci in ${\prod }_{d=1}^{{\beta }_i}\mathrm{Gr}(d,N_i)$; hence it is projective by [F3] and [F4]. Over a Grassmannian frame chart it is a product of that chart with ${\prod }_i\mathrm{Flag}({\mathbb{C}}^{{\beta }_i})$, so by [F2] and step 2.1 it is smooth and irreducible, of dimension $s_N+f$, where $s_N={\sum }_i{\beta }_i(N_i-{\beta }_i)$ and $f={\sum }_i{\beta }_i({\beta }_i-1)\mathrm{/}2$. The adapted-basis computation of step 3.1 applies verbatim to $E_N$ and to the chains inside the $S_i$, so the associated incidence is the total space of a vector subbundle $W_{\nu ,N}\subseteq E_N{\times }_{B_N}{\mathcal{F}}_{\nu ,N}$ of the same rank $r_{\nu }$.

Absolute projectivity of the target needs no general compactification theorem. Arrows with a zero-dimensional endpoint contribute the zero Hom bundle and may be omitted. Put $D={⨂}_{{\beta }_i>0}\det S_i$ (the trivial line for the empty product); for each remaining arrow, $S_i^{\vee }\otimes \det S_i\cong {\bigwedge }^{{\beta }_i-1}{S}_i$, so $\mathcal{H}om(S_i,S_j)\otimes D\cong {\bigwedge }^{{\beta }_i-1}{S}_i\otimes S_j\otimes {⨂}_{l\mathrm{\ne }i}\det S_l$, and $D$ itself is a tensor product of the $\det S_l$. Each $S_i\subseteq {\mathcal{O}}^{N_i}$ is a subbundle with locally free quotient, hence so are its exterior powers, and tensor products of such subbundles are again subbundles of finite trivial bundles with locally free quotients; therefore $(E_N\oplus \mathcal{O})\otimes D$ is a subbundle of a finite trivial bundle ${\mathcal{O}}^M$. Since $D$ is a line bundle, twisting identifies ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}((E_N\oplus \mathcal{O})\otimes D)={\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(E_N\oplus \mathcal{O})$, which is thus a closed subvariety of the smooth projective $B_N\times {\mathbb{P}}^{M-1}$ by [F1] and [F4], hence projective. It is smooth by its projective-space-bundle charts over smooth $B_N$, not merely by closedness.

The incidence inclusion of step 3.1 embeds ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(W_{\nu ,N}\oplus \mathcal{O})$ closed in ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(E_N\oplus \mathcal{O}){\times }_{B_N}{\mathcal{F}}_{\nu ,N}$ over $B_N$, exactly as in step 4.2 with base $B_N$. The fiber product is closed in the product of the two smooth projective varieties, being the inverse image of the closed diagonal of $B_N$, so the source is projective by step 5.1, and smooth by its projective-space-bundle charts over the smooth ${\mathcal{F}}_{\nu ,N}$ of step 4.3. The induced map ${\bar{\pi }}_{\nu ,N}$ is projective: its graph is closed in the product of its projective source with the target, since the target has closed diagonal, and a projective embedding of the source gives a closed immersion of the graph into a relative projective space over the target. Thus it is a projective map between smooth projective varieties. Its inverse image of the affine chart $t\mathrm{\ne }0$ is, by the same unique-representative argument as in step 4.2, the total space of $W_{\nu ,N}$ over ${\mathcal{F}}_{\nu ,N}$, mapped to $E_N$ by $(w,b,F)\mapsto w$, which is the associated incidence map. The empty dimension, zero arrow and arbitrary finite multiplicity cases are included throughout, since they only remove factors or arrows from the constructions above.

There are finitely many color words $\nu$ of content $\beta$. If $\beta =0$ then $G_V$ is trivial, $E_V=0$, each $X_{\nu }$ and $L_{\nu }=k[0]$ is the simple equivariant perverse sheaf $k$, so assume some ${\beta }_i>0$ and fix once the finite identifications $V_i\cong {\mathbb{C}}^{{\beta }_i}$. For each $\nu$, the constant sheaf $k_{X_{\nu }}$ with its trivial equivariant structure is a bounded Cartesian constructible object of $D_{G,c}^b(X_{\nu };k)$ by [F3], and ${\pi }_{\nu }$ is $G_V$-equivariant by [F1], so $L_{\nu }=R{\pi }_{\nu !}k[d_{\nu }]$ is an object of $D_{G,c}^b(E_V;k)$ by [F4]. For $x\in E_V$ the fiber ${\pi }_{\nu }^{-1}(x)$ is a closed subvariety of ${\mathcal{F}}_{\nu }$ of complex dimension at most $f={\sum }_i{\beta }_i({\beta }_i-1)\mathrm{/}2$ by [F1], forget equivariance and apply ordinary proper base change, using the restriction-compatible construction of [F4], to the ordinary inclusion of $x$; this gives $(R{\pi }_{\nu !}k{)}_x\simeq R{\mathrm{\Gamma }}_c({\pi }_{\nu }^{-1}(x);k)$, and the fibre is compact because ${\pi }_{\nu }$ is projective; no uniform all-sheaf dimension hypothesis is inferred from the constant-coefficient calculation; the compact-support cohomology of the fiber lies in degrees $[0,2f]$ by [F2]. Hence $L_{\nu }$ has ordinary amplitude $[-d_{\nu },2f-d_{\nu }]$, and the finite sum $K={⨁}_{\nu }{L}_{\nu }$ has ordinary amplitude $[b,a]$ with $b=-{\max }_{\nu }{d}_{\nu }$ and $a=2f-{\min }_{\nu }{d}_{\nu }$. Choose integers $N_i\ge {\beta }_i+1$ with $r_N=2{\min }_{{\beta }_i>0}(N_i-{\beta }_i)\ge a-b$; such a tuple exists because the bounds are finite, and then $r_N\ge 2f$, the width of every single $L_{\nu }$.

Fix the chosen tuple $N$, put $U_N={\prod }_{{\beta }_i>0}\mathrm{Inj}({\mathbb{C}}^{{\beta }_i},{\mathbb{C}}^{N_i})$, and identify the two models of the same quotient: the frame model $X_N=(U_N\times E_V)\mathrm{/}{G}_V$ of [F5] is the total space of ${⨁}_{h:i\to j}\mathcal{H}om(S_i,S_j)$ over $B_N$ with tautological subbundles $S_i$, and $(U_N\times X_{\nu })\mathrm{/}{G}_V$ is the incidence model $X_{\nu ,N}$ of [F1], both because a frame trivializes $S_i\cong {\mathbb{C}}^{{\beta }_i}$ and transports representations and flags. Write $q_N:U_N\times E_V\to E_N$ and ${\mathrm{pr}}_{E_V}:U_N\times E_V\to E_V$ for the two projections and $q_{\nu }:U_N\times X_{\nu }\to X_{\nu ,N}$; the quotient square comparing $q_N,q_{\nu }$ with the equivariant map $\mathrm{i}\mathrm{d}\times {\pi }_{\nu }$ is Cartesian because both $G_V$-actions on $U_N\times E_V$ and $U_N\times X_{\nu }$ are free (the action on $U_N$ is free by [F5]), and the induced map of quotients is ${\pi }_{\nu ,N}$ by [F1]. By the descent equivalence of [F6], $Q_N$ is characterized by $q_N^{\ast }{Q}_N(A)\cong {\mathrm{pr}}_{E_V}^{\ast }A$, and the principal-bundle Cech nerve carries the descent equivalence; ordinary $q_N^{\ast }$ is conservative, while descent-compatible morphisms are recovered through that equivalence. Applying the unconditional exchange $u^{\ast }R{f}_{!}\simeq R{f}_{!}^{\mathrm{\prime }}{v}^{\ast }$ of [F4] to the Cartesian squares of the two quotient maps gives ${\mathrm{pr}}_{E_V}^{\ast }(R{\pi }_{\nu !}k[d_{\nu }])\cong R(\mathrm{i}\mathrm{d}\times {\pi }_{\nu }{)}_{!}k[d_{\nu }]$ and $q_N^{\ast }(R{\pi }_{\nu ,N!}k[d_{\nu }])\cong R(\mathrm{i}\mathrm{d}\times {\pi }_{\nu }{)}_{!}k[d_{\nu }]$, where $\mathrm{i}\mathrm{d}\times {\pi }_{\nu }:U_N\times X_{\nu }\to U_N\times E_V$ is the common base change. Combining with the characterizing identity gives $q_N^{\ast }{Q}_N(L_{\nu })\cong q_N^{\ast }(R{\pi }_{\nu ,N!}k[d_{\nu }])$, and both canonical base-change identifications satisfy the same Cartesian descent data by their naturality; hence the Cech-nerve descent equivalence yields $Q_N(L_{\nu })\cong R{\pi }_{\nu ,N!}k[d_{\nu }]$, and hence $Q_N(K)[s_N]\cong {⨁}_{\nu }R{\pi }_{\nu ,N!}k[d_{\nu }+s_N]$ with ${\dim }_{\mathbb{C}}{X}_{\nu ,N}=d_{\nu }+s_N$ by [F1].

For each $\nu$ the map ${\bar{\pi }}_{\nu ,N}:{\bar{X}}_{\nu ,N}\to {\bar{E}}_N$ of [F1] is a projective morphism of complex projective varieties with ${\bar{X}}_{\nu ,N}$ smooth of pure dimension $d_{\nu }+s_N$, so it is proper by [F12], and ${\bar{X}}_{\nu ,N}$ restricts over the chart $E_N$ to $X_{\nu ,N}$ with restricted map ${\pi }_{\nu ,N}$. By [F10] applied to ${\bar{\pi }}_{\nu ,N}$ there is a finite decomposition $R{\bar{\pi }}_{\nu ,N\ast }k[d_{\nu }+s_N]\cong {⨁}_i{S}_i[m_i]$ into shifts of simple ordinary perverse sheaves on ${\bar{E}}_N$; applying $j^{\ast }$ for the open immersion $j:E_N\hookrightarrow {\bar{E}}_N$ and using $R{\bar{\pi }}_{\nu ,N\ast }=R{\bar{\pi }}_{\nu ,N!}$ of [F4] together with the exchange $j^{\ast }R{\bar{\pi }}_{\nu ,N!}\simeq R{\pi }_{\nu ,N!}{j}^{\mathrm{\prime }\ast }$ along the Cartesian chart square gives $R{\pi }_{\nu ,N!}k[d_{\nu }+s_N]\cong {⨁}_i(j^{\ast }{S}_i)[m_i]$. Open restriction is perverse exact by [F8], so each $j^{\ast }{S}_i$ is perverse, and it is either zero or simple: if $Z$ is the irreducible support of $S_i$ with $Z\cap E_N=\mathrm{\varnothing }$ then $j^{\ast }{S}_i=0$; otherwise $Z\cap E_N$ is a nonempty dense open of $Z$, the intermediate extension $S_i={\mathrm{IC}}_Z(L)$ restricts to ${\mathrm{IC}}_{Z\cap E_N}(L{\mathrm{\mid }}_{Z^{sm}\cap E_N})$, because the intermediate extension is computed as the image of ${}^p{H}^0(j_{!}L)\to {}^p{H}^0(R{j}_{\ast }L)$ and the open immersion commutes with extension by zero and with pushforward along the same restriction, and that restricted local system is simple because $Z^{sm}\cap E_N\hookrightarrow Z^{sm}$ is a nonempty Zariski open inclusion of a connected smooth complex algebraic variety, hence surjective on fundamental groups by [F11]. Therefore each $R{\pi }_{\nu ,N!}k[d_{\nu }+s_N]$, and with step 2.1 each $Q_N(L_{\nu })[s_N]$, is a finite direct sum of shifts of simple ordinary perverse sheaves on $E_N$.

Fix $\nu$ and write $L=L_{\nu }$, of ordinary amplitude $[-d_{\nu },2f-d_{\nu }]$ with $2f\le r_N$ by step 1.1. The finite decomposition of $Q_N(L)[s_N]$ into shifts of simple ordinary perverse sheaves from step 3.1 determines pairwise orthogonal idempotents $e_i$ in $\mathrm{End}(Q_N(L)[s_N])=\mathrm{End}(Q_{N}L)$ with sum the identity, the projectors onto the displayed summands. Since $r_N\ge 2f$, the degree-zero Hom comparison of [F5], restated for the actual category in [F6], is a ring isomorphism ${\mathrm{End}}_G(L)\cong {\mathrm{End}}_{E_N}(Q_{N}L)$, so these idempotents and all their relations lift uniquely to ${\mathrm{End}}_G(L)$; by the idempotent completeness of $D_{G,c}^b(E_V;k)$ established in [F6], the lifted idempotents split inside that actual category, producing $G_V$-equivariant summands $T_i$ of $L$ with $Q_N(T_i)[s_N]\cong S_i[m_i]$ and ordinary amplitude within $[-d_{\nu },2f-d_{\nu }]$.

For each summand $T_i$ of step 4.1 the model $Q_N(T_i)[s_N]$ is concentrated in perverse degree $-m_i$. By [F7] the normalized functor $Q_N(-)[s_N]$ is perverse exact and conservative and preserves and reflects perverse concentration and simplicity at the stabilization bound; hence $T_i$ is concentrated in perverse degree $-m_i$, say $T_i=U_i[m_i]$ with $U_i$ in the equivariant perverse heart of [F9], and $U_i$ is simple because simplicity is reflected. Thus $L={⨁}_i{T}_i$ is a finite direct sum of shifts of simple $G_V$-equivariant perverse sheaves, repeating this argument for each of the finitely many words $\nu$ proves the individual clause for every $L_{\nu }$ at the same tuple $N$.

Since there are finitely many words and each $L_{\nu }$ is a finite direct sum of shifts of simple equivariant perverse sheaves, the finite sum $K={⨁}_{\nu }{L}_{\nu }$ is again such a finite direct sum, i.e. semisimple in the actual bounded equivariant constructible category. The argument used only the amplitude bound of step 1.1, the model identification of step 2.1, the compact smooth-source theorem [F10] for the projective maps ${\bar{\pi }}_{\nu ,N}$ and its restriction to the open chart, the ordinary perverse restriction facts of [F8], the fundamental-group lemma [F11] and the normalization [F7]; no semismallness of ${\pi }_{\nu }$, no finite-ADE hypothesis and no affine paving of any fiber is assumed or used, and the empty dimension vector, zero arrows, zero-rank subbundles and arbitrary finite multiplicities are covered by steps 1.1 and 3.1. AC is consumed exactly as recorded in [A1].