Smith's conjecture on perverse sheaves and the Smith operator

Let $X$ be a complex algebraic variety equipped with a $\boldsymbol{\varpi}$-action, and let $P$ be a $\boldsymbol{\varpi}$-equivariant perverse sheaf of $K$-vector spaces on $X$. The fixed-point variety is denoted by $X^{\varpi}$, and the Smith operator is

$$\mathbf{Psm}:D^b_{\varpi}(X;K)\to \operatorname{Perf}(X^{\varpi};\mathcal{T}).$$

Write $\otimes_K\mathcal{T}$ for the natural functor from $D^b(X^{\varpi};K)$ to $\operatorname{Perf}(X^{\varpi};\mathcal{T})$ described in the source. Smith's conjecture. There exist perverse sheaves of $K$-vector spaces $P_1,\ldots,P_n$ on $X^{\varpi}$ and integers $a_1,\ldots,a_n$ such that

$$\mathbf{Psm}(P)\cong (P_1[a_1]\oplus\cdots\oplus P_n[a_n])\otimes_K\mathcal{T}$$

in $\operatorname{Perf}(X^{\varpi};\mathcal{T})$. This conjecture asserts that the Smith operator interacts with the perverse $t$-structure by producing, up to the Tate-periodic coefficient object, a finite direct sum of shifts of perverse sheaves on the fixed-point variety.