Wiki. “佐武等价” [佐武等价]

几何佐武等价

固定有限整体维数的 Noether 交换环 $k$. 取定连通约化代数群 $G$ 与极大环面 $T\subset G$. 考虑环路群 $LG$ 以及弧群 $L^+G$, 仿射 Grassmann 空间 $\mathrm{Gr}_G$ 定义为 $LG/L^+G$.

$\mathrm{Gr}_G$ 上的 $L^+G$-等变偏屈层的 Abel 范畴 $\mathsf{Perv}_{L^+G}(\mathrm{Gr}_G)$ 是等变导出范畴上某个偏屈 t-结构的心, 且继承了导出范畴上的卷积, 构成幺半范畴. 几何佐武等价指出这个幺半范畴等价于 Langlands 对偶群 $G^\vee$ 的表示范畴, $$ \mathsf{Perv}_{L^+G}(\mathrm{Gr}_G) \simeq \mathsf{Rep}(G^\vee). $$

Under the Satake isomorphism the classes of irreducible representations of $^LG$ go not to functions which correspond to constant sheaves on the orbits but to the irreducible perverse sheaves. This suggests that the Satake isomorphism itself may be elevated from the level of Grothendieck groups to the level of categories. — Frenkel, Recent Advances in the Langlands Program