论文标题
更高的本地化和更高的分支法律
Higher localization and higher branching laws
论文作者
论文摘要
对于连接的还原组$ g $和仿射$ g $ -variety $ x $以上的数字，本地化函数将$ \ mathfrak {g} $ - 模块带到$ d_x $ -modules。我们将这种结构扩展到贝林森 - 吉茨堡（Beilinson-Ginzburg）引起的H-复合物的形式主义，将这种结构扩展到一个模棱两可的设置，并表明Harish-Chandra $（\ Mathfrak {G}，K）$ - 模块在$ x = h \ backslash g $上有常规的自然人组的$ subsets subssem specrouse specrouse specrouse spem subssem specrouse，相对Lie代数同源物和$ \ Mathrm {Ext} $ - $（\ Mathfrak {g}，K），K）$的分支空间 - 模块是根据刻度衍生的本地化来几何解释的。作为直接的后果，我们表明它们在相同的假设下是有限维度，并将Euler-Poincaré的特征与局部指数定理相关联；这恢复了M. Kitagawa最近结果的一部分。还包括有关与Schwartz同源性关系的示例和讨论。
For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over the complex numbers, the localization functor takes $\mathfrak{g}$-modules to $D_X$-modules. We extend this construction to an equivariant and derived setting using the formalism of h-complexes due to Beilinson-Ginzburg, and show that the localizations of Harish-Chandra $(\mathfrak{g}, K)$-modules onto $X = H \backslash G$ have regular holonomic cohomologies when $H, K \subset G$ are both spherical reductive subgroups. The relative Lie algebra homologies and $\mathrm{Ext}$-branching spaces for $(\mathfrak{g}, K)$-modules are interpreted geometrically in terms of equivariant derived localizations. As direct consequences, we show that they are finite-dimensional under the same assumptions, and relate Euler-Poincaré characteristics to local index theorem; this recovers parts of the recent results of M. Kitagawa. Examples and discussions on the relation to Schwartz homologies are also included.
