论文标题
连贯的施普林格理论和分类deligne-langlands对应
Coherent Springer theory and the categorical Deligne-Langlands correspondence
论文作者
论文摘要
卡兹丹（Kazhdan）和卢斯蒂格（Lusztig）确定了史坦伯格（Steinberg Iwahori固定矢量。我们将来自派生的代数几何形状的技术从$ k $ - 理论传递给Hochschild同源性，从而将$ \ Mathcal {h} $与一个连贯的edeaft骨上的相干纱堆在Unipotent Langlands参数（Coherent Langlands参数）上，Cooherent Springer Sheeaf中识别出$ \ Mathcal {H} $。结果，$ \ mathcal {h} $ - 模块的派生类别被实现为该堆栈上连贯的束带的完整子类别，证实了当地兰兰兹当地的强烈形式的期望（包括最近的猜想，包括Fargues-Scholze，Sholmann和Zhu）。 在通用线性组的情况下，我们的结果使我们能够将本地的Langlands分类不可减至的表示：我们构建了$ \ mathrm {gl} _n（f）$的平滑表示的派生类别的完整嵌入到Langlands parameters的堆栈中的相干安全带中。
Kazhdan and Lusztig identified the affine Hecke algebra $\mathcal{H}$ with an equivariant $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields $F$ with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from $K$-theory to Hochschild homology and thereby identify $\mathcal{H}$ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf. As a result the derived category of $\mathcal{H}$-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of $\mathrm{GL}_n(F)$ into coherent sheaves on the stack of Langlands parameters.
