On the Kottwitz conjecture for local shtuka spaces

Kaletha, Tasho; Hansen, David J.; Weinstein, Jared

doi:10.1017/fmp.2022.7

Cited by 9 publications

(2 citation statements)

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“…A refined endoscopic datum e = (H, H, s, η) of G and a Levi subgroup M ⊂ G can be upgraded to the structure of an embedded endoscopic datum in potentially many non-equivalent ways and these are parametrized by a set D(M, e) ∼ = W ( H)\W (M, H)/W ( M ), where W (M, H) is defined to be the subset of the Weyl group 13 We note that the degree shift built into the Satake sheaf Sµ causes a sign in the above formula that does not appear in much of the literature on this topic. This is explained in Remark 2.4.4 of [HKW21]. For GL n , this is known in the trivial endoscopic case for all representations by [Shi12].…”

Section: Geometric Eisenstein Series Intertwining Operators and Shin'...mentioning

confidence: 95%

Geometric Eisenstein Series, Intertwining Operators, and Shin's Averaging Formula

Hamann¹

2022

Preprint

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In the geometric Langlands program over function fields, Braverman-Gaitsgory [BG02] and Laumon [Lau90] constructed geometric Eisenstein functors which geometrize the classical construction of Eisenstein series over function fields by replacing the Eisenstein series with an automorphic sheaf on the moduli stack BunG of G-bundles of the smooth projective curve defined by the function field. These sheaves are Hecke Eigensheaves, encode the completed Eisenstein series under the function-sheaf dictionary, and satisfy a functional equation with respect to the action of the Weyl group. Fargues and Scholze [FS21] very recently constructed a general candidate for the local Langlands correspondence, via a geometric Langlands correspondence occurring over the Fargues-Fontaine curve. In this note, 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/Qp with simply connected derived group and maximal torus T , we construct a geometric Eisenstein functor nEis(−), which takes sheaves on the moduli stack BunT to sheaves on the moduli stack BunG. We show that, given a sufficiently nice L-parameter φT : W Qp → L T with induced parameter φ given by composing with the natural embedding L T → L G, there is a Hecke eigensheaf on BunG with eigenvalue φ, given by applying nEis(−) to the Hecke eigensheaf S φ T on BunT attached to φT constructed by Fargues [Far16; Far20] and Zou [Zou22]. We show that the object nEis(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 φT . This explicit description of the eigensheaf has several surprising consequences. First, it recovers special cases of an averaging formula of Shin [Shi12; Ber21] for the cohomology of local Shimura varieties with rational coefficients, and generalizes it to the non-minuscule case. Moreover, 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. The sufficiently nice condition on φT is related to zeros and poles of the intertwining operators attached to the normalized parabolic induction i G B (χ), and we analogously see that the refined averaging formula actually gives a geometric construction of such intertwiners.

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Section: Geometric Eisenstein Series Intertwining Operators and Shin'...mentioning

confidence: 95%

Geometric Eisenstein Series, Intertwining Operators, and Shin's Averaging Formula

Hamann¹

2022

Preprint

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“…Corollary 5.6 is used in the work of Hansen, Kaletha, and Weinstein (see [HKW22, Proposition 5.6.2]).…”

Section: The Case Of Group Actionsmentioning

confidence: 99%

Local terms for transversal intersections

Varshavsky¹

2023

Compositio Math.

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The goal of this note is to show that in the case of ‘transversal intersections’ the ‘true local terms’ appearing in the Lefschetz trace formula are equal to the ‘naive local terms’. To prove the result, we extend the strategy used in our previous work, where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of an étale sheaf to the normal cone is monodromic and the assertion that local terms are ‘constant in families’. As an application, we get a generalization of the Deligne–Lusztig trace formula.

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