Abstract:Let $G$ be a reductive group and $L$ a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between $\operatorname {Ad}$-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) ${\mathscr {D}}$-modules on $G$ and $L$, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where $L=T$ is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra ${\mathscr {D}}$-module on ${G\times T}$. We show that this module is flat over ${\mathscr {D}}(T)$, which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of ${\mathscr {D}}$-modules.

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中文翻译：

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抛物线归纳法和 Harish-Chandra 𝒟-模块

摘要：令$G$为约简群，$L$为Levi子群。抛物线归纳和限制是 $\operatorname {Ad}$-等变派生类别的可构造滑轮或（不一定是完整的）${\mathscr {D}}$-在 $G$ 和 $L $，分别。Bezrukavnikov 和 Yom Din 通过推广 Lusztig 的经典结果证明了这些函子是精确的。在本文中，我们考虑一种特殊情况，其中 $L=T$ 是最大环面。我们根据 ${G\times T}$ 上的 Harish-Chandra ${\mathscr {D}}$-模给出抛物线归纳和限制的明确公式。我们证明这个模块在 ${\mathscr {D}}(T)$ 上是平坦的，这很容易暗示抛物线归纳和限制是 ${\mathscr {D}}$ 模块的相应阿贝尔类别之间的精确函子。

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