论文标题
几何Eisenstein系列，交织操作员和Shin的平均公式
Geometric Eisenstein Series, Intertwining Operators, and Shin's Averaging Formula
论文作者
论文摘要
Braverman-Gaitsgory和Laumon在“几何兰兰兹”计划中，构建了几何Eisenstein函子，该函数将Eisenstein系列的经典结构几何化。 Fargues和Scholze最近通过在Fargues-Fontaine曲线上发生的几何Langlands对应关系建造了当地Langlands对应的一般候选人。我们将一些几何艾森斯坦系列的理论带到了Fargues Fontaine的环境中。也就是说，给定一个准切片连接的还连接的还原组$ g/\ mathbb {q} _ {p} $，简单地连接的派生的group和maximal $ t $，我们在$ bun_ {t $ bunves $上构建了eisenstein functor $ \ mathrm {neis}（neis}（neis} $ sheaves on sheaves on sheaves on sheaves on sheaves on sheaves on sheaves。我们表明，给定一个足够漂亮的$ l $ - 参数$ ϕ_ {t}：w _ {\ mathbb {q} _ {p}}} \ rightarrow \ rightarrow \ phantom {}^{l} {l} $ \ mathrm {neis}（ - ）$ to $ bun_ {t} $ on $ bun_ {t} $ on $ bun_ {t} $ on $ necke eigensheaf $ \ mathcal {s} _ {t} $ in $ nation_ {t} $。我们表明，$ \ mathrm {neis}（\ Mathcal {s} _ {ϕ_ {t {t}}）$与Verdier二元性很好地相互作用，并且，假设Fargues-Scholze的兼容性与适当的langlands相应形式的相当不错的形式的langlands cormantion cormant corment of eigig of stall of stall of stall of stall of stall of stall of stall of stall of stall of eig stall of eig s the eig s eigig of stall of eig eig eig stall。 $χ$附加到$ ϕ_ {t} $。这有几个令人惊讶的后果。首先，它恢复了具有合理系数的局部Shimura品种的共同体的平均SHIN公式的特殊情况，并将其推广到非微分案例。其次，在参数$ ϕ $足够好的情况下，它可以完善平均公式，从而为共同体学位提供了明确的公式，即某些抛物线归纳量放入，并且该精制公式即使具有扭转系数也可以保留。
In the geometric Langlands program over function fields, Braverman-Gaitsgory and Laumon constructed geometric Eisenstein functors which geometrize the classical construction of Eisenstein series. Fargues and Scholze very recently constructed a general candidate for the local Langlands correspondence, via a geometric Langlands correspondence occurring over the Fargues-Fontaine curve. We carry some of the theory of geometric Eisenstein series over to the Fargues-Fontaine setting. Namely, given a quasi-split connected reductive group $G/\mathbb{Q}_{p}$ with simply connected derived group and maximal torus $T$, we construct an Eisenstein functor $\mathrm{nEis}(-)$, which takes sheaves on $Bun_{T}$ to sheaves on $Bun_{G}$. We show that, given a sufficiently nice $L$-parameter $ϕ_{T}: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}T$, there is a Hecke eigensheaf on $Bun_{G}$ with eigenvalue $ϕ$, given by applying $\mathrm{nEis}(-)$ to the Hecke eigensheaf $\mathcal{S}_{ϕ_{T}}$ on $Bun_{T}$ attached to $ϕ_{T}$ by Fargues and Zou. We show that $\mathrm{nEis}(\mathcal{S}_{ϕ_{T}})$ interacts well with Verdier duality, and, assuming compatibility of the Fargues-Scholze correspondence with a suitably nice form of the local Langlands correspondence, provide an explicit formula for the stalks of the eigensheaf in terms of parabolic inductions of the character $χ$ attached to $ϕ_{T}$. This has several surprising consequences. First, it recovers special cases of an averaging formula of Shin for the cohomology of local Shimura varieties with rational coefficients, and generalizes it to the non-minuscule case. Second, it refines the averaging formula in the cases where the parameter $ϕ$ is sufficiently nice, giving an explicit formula for the degrees of cohomology that certain parabolic inductions sit in, and this refined formula holds even with torsion coefficients.
