论文标题
重新访问混合几何形状
Revisiting mixed geometry
论文作者
论文摘要
我们从适用于任意的Artin堆栈的贝林森 - 金氏赛（Beilinson-Ginzburg-Soergel）的意义上，提供了“混合版本”或“分级升降机”的统一结构。特别是，我们获得了在几何表示理论和分类结中产生的许多类别的分级升力的一般结构。我们的新理论与$ \ overline {\ Mathbb {f}} _ q $ a + overline {\ mathbb {f}} _ q $对称的单体dg-category $ \ mathsf { $ \ Mathcal {y} $以及六函数的形式主义，一种不正当的$ t $结构，以及邦德科和帕克斯特洛的重量（或共同$ t $ - ）结构，与六函数的形式主义兼容，与六函数的形式主义，pressector $ t $结构，frobenius structures of frobenius structures us frobenius strone of the frobenius shove shea $ ell eell（混合）。 从经典上讲，混合版本仅在非常特殊的情况下是由于弗罗贝尼乌斯（Frobenius）的非偏不透性而构建的。我们的施工通过半简化Frobenius动作本身来避开了这个问题。但是，类别$ \ MATHSF {SHV} _ {\ MATHSF {gr}，C}（\ Mathcal {y}）$与先前可用时构造的人一致。例如，对于任何还原的组$ g $，带有固定对$ t \ subset b $的最大圆环和一个borel子组，我们具有同等的单型DG重量类别$ \ MATHSF {shv} _ { \mathsf{Ch}^b(\mathsf{SBim}_W)$, where $\mathsf{Ch}^b(\mathsf{SBim}_W)$ is the monoidal $\mathsf{DG}$-category of bounded chain complexes of Soergel bimodules and $W$ is the Weyl group of $G$.
We provide a uniform construction of "mixed versions" or "graded lifts" in the sense of Beilinson-Ginzburg-Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our new theory associates to each Artin stack of finite type $\mathcal{Y}$ over $\overline{\mathbb{F}}_q$ a symmetric monoidal DG-category $\mathsf{Shv}_{\mathsf{gr}, c}(\mathcal{Y})$ of constructible graded sheaves on $\mathcal{Y}$ along with the six-functor formalism, a perverse $t$-structure, and a weight (or co-$t$-)structure in the sense of Bondarko and Pauksztello, compatible with the six-functor formalism, perverse $t$-structures, and Frobenius weights on the category of (mixed) $\ell$-adic sheaves. Classically, mixed versions were only constructed in very special cases due to the non-semisimplicity of Frobenius. Our construction sidesteps this issue by semi-simplifying the Frobenius action itself. However, the category $\mathsf{Shv}_{\mathsf{gr}, c}(\mathcal{Y})$ agrees with those previously constructed when they are available. For example, for any reductive group $G$ with a fixed pair $T\subset B$ of a maximal torus and a Borel subgroup, we have an equivalence of monoidal DG weight categories $\mathsf{Shv}_{\mathsf{gr}, c}(B\backslash G/B) \simeq \mathsf{Ch}^b(\mathsf{SBim}_W)$, where $\mathsf{Ch}^b(\mathsf{SBim}_W)$ is the monoidal $\mathsf{DG}$-category of bounded chain complexes of Soergel bimodules and $W$ is the Weyl group of $G$.
