Middle perverse t structure glues and has simple intersection complexes

The equivariant t-structure. For $A\in D_{G,c}^b(X;k)$ define $A\in {}^p{D}_G^{\le a}$ if and only if the restriction $A_X$ lies in the ordinary aisle ${}^p{D}^{\le a}(X;k)$, and ${}^p{D}_G^{\ge a}$ dually. By [F1] this is equivalent to $Q_N(A)[s_N]\in {}^p{D}^{\le a}(X_N;k)$ for every sufficiently large $N$, so the definition is independent of the resolution. The pair $({}^p{D}_G^{\le 0},{}^p{D}_G^{\ge 0})$ satisfies the t-structure axioms because the defining conditions are those of a t-structure pointwise: closure under shifts and extensions follows from the corresponding ordinary closures and the conservativity of restriction, orthogonality $\mathrm{Hom}({}^p{D}_G^{\le 0},{}^p{D}_G^{\ge 1})=0$ follows from the ordinary orthogonality transported through the fully faithful Hom comparison of [F1], and truncation triangles exist by the lifting clause of [F1]. The structure is bounded because the ordinary one is and restriction to $X$ is conservative. This proves the t-structure and detection assertions.

Gluing and independence of refinement. The inequalities of the Statement are checked stratumwise and are therefore independent of compatible refinement by the ordinary clause of [F3]; gluing across an algebraic open/closed decomposition uses the localization triangles of [F2] and the ordinary gluing theorem: for a $G$-invariant open immersion $j$ and its closed complement $i$, the Čech-style recollement conditions $j^{\ast }{i}_{\ast }=0$, the triangles $j_{!}{j}^{\ast }\to \mathrm{i}\mathrm{d}\to i_{\ast }{i}^{\ast }$ and $i_{\ast }{i}^{!}\to \mathrm{i}\mathrm{d}\to j_{\ast }{j}^{\ast }$ hold in $D_{G,c}^b$ by [F2], and the transport of step 1.1 reduces the glued t-structure statement to the ordinary one.

Heart and finite length. Kernel and cokernel of a morphism of the heart are the perverse cohomology objects in degrees $-1$ and $0$ of its cone, computed by the truncation functors of step 1.1; hence the heart is abelian. Restriction to $X$ is perverse exact and conservative on the heart by [F1], so a strictly increasing or decreasing chain of subobjects in the equivariant heart gives a strictly increasing or decreasing chain in the ordinary heart; the ordinary heart has finite length by [F3], so the equivariant heart does. For heart objects $A,B$ and $m<0$, $B[m]$ lies in ${}^p{D}_G^{\ge 1}$ while $A$ lies in ${}^p{D}_G^{\le 0}$, so orthogonality gives $\mathrm{Hom}(A,B[m])=0$.

Local systems and intermediate extension. The group $G$ is path connected: elementary matrices and nonzero diagonal matrices are joined to the identity, and Gaussian elimination expresses every invertible complex matrix as their product. Let $L$ be an equivariant local system on an invariant smooth open $U$, with action isomorphism $\varphi :{\mathrm{pr}}_U^{\ast }L\simeq a^{\ast }L$ on $G\times U$. For any ordinary sub-local-system $M\subset L$, the sub-local-systems $\varphi ({\mathrm{pr}}_U^{\ast }M)$ and $a^{\ast }M$ agree on $\{e\}\times U$ by the identity condition. Every point $(g,x)$ is joined to $(e,x)$ by a path in $G\times U$; parallel transport therefore makes these sub-local-systems agree everywhere. Restricting $\varphi$ supplies equivariant data on $M$, and the identity and cocycle equations restrict as well. In the action-nerve model this is a Cartesian sub-sheaf, hence an equivariant local system. Thus $L$ is simple equivariantly if and only if it is simple ordinarily. For irreducible invariant $Z$ and invariant dense smooth $j:U\hookrightarrow Z$ of dimension $d$, a simple $L$ gives the simple ordinary perverse object $L[d]$ by [F3], and exact conservative restriction [F1] makes it simple in the equivariant heart too. Define $j_{!\ast }(L[d])=\mathrm{im}({}^p{H}^0{j}_{!}L[d]\to {}^p{H}^{0}R{j}_{\ast }L[d])$. The first term has no boundary quotient and the second no boundary subobject, by the localization adjunctions [F2], so the image has neither. Any extension with neither property is this image: its adjunction maps from the first term and into the second have boundary cokernel and kernel, hence are epic and monic. A subobject of this image restricts either to zero or to all of $L[d]$; the boundary subobject or resulting boundary quotient must then vanish. Closed pushforward $i_{Z\ast }$ is exact and fully faithful on hearts by the same localization and detection, so ${\mathrm{IC}}_Z(L)=i_{Z\ast }{j}_{!\ast }(L[d])$ is simple.

Classification of simple objects. Let $A$ be a nonzero simple equivariant heart object, and let $T$ be the closed support of $A_X$. Localization identifies $A$ with closed pushforward of a heart object on $T$, which we again call $A$. Equivariance preserves $T$ and its smooth local-system locus. Its finitely many irreducible components are individually invariant because connected $G$ cannot permute them nontrivially. In the smooth locus of each component away from all component intersections, algebraic constructibility gives a dense algebraic open on which the cohomology of $A_X$ is locally constant. Its $G$-saturation is still an algebraic open in this locus, and equivariance makes the cohomology locally constant there. These saturations form disjoint invariant smooth opens $W_i$; set $W={⨆}_i{W}_i$. Write $h:W\hookrightarrow T$. The perverse inequalities give $h^{\ast }A={⨁}_i{L}_i[\dim W_i]$, with every $L_i$ nonzero. The left heart adjunction supplies a map ${}^p{H}^0{h}_{!}(L_i[\dim W_i])\to A$ whose restriction is the inclusion of this summand. Its image is nonzero, hence all of $A$ by simplicity. Exact open restriction then forces this summand to equal $h^{\ast }A$, excluding every other support component. Thus $T=Z$ is irreducible. On its invariant dense smooth open $U$, write $j^{\ast }A=L[\dim Z]$. A nonzero proper equivariant sub-local-system $M\subset L$ would similarly give an image subobject of $A$ restricting to $M[\dim Z]$, contradicting simplicity. Hence $L$ is simple equivariantly, and therefore ordinarily by step 3.1. Simplicity of $A$ and $j^{\ast }A\mathrm{\ne }0$ exclude boundary subobjects and quotients. The characterization of step 3.1 identifies $A$ with the intermediate extension, and closed pushforward gives precisely ${\mathrm{IC}}_Z(L)$ on $X$.

Endomorphism rings. Let $S$ be a simple heart object and $f:S\to S$ a nonzero endomorphism. Its kernel is a subobject of $S$ and hence zero, and its image is a nonzero subobject and hence all of $S$; thus $f$ is invertible. Therefore $\mathrm{End}(S)$ is a division $k$-algebra, and it is not asserted to equal $k$. This completes the proof.

Conclusion. Steps 1.1 and 2.1 give the t-structure, its independence of refinement and gluing; steps 2.2, 3.1, 4.1 and 5.1 give the heart, its finite length, the negative-Hom vanishing and the intersection-complex classification with division endomorphism rings. AC enters as recorded in [A1], and no semisimplicity or characteristic restriction beyond characteristic zero is used.