A twisted topological trace formula for Hecke operators and liftings from symplectic to general linear groups

Abstract

For the locally symmetric space X attached to an arithmetic subgroup of an algebraic group G of Q-rank r, we construct a compact manifoldX by gluing together 2 r copies of the Borel-Serre compactification of X. We apply the classical Lefschetz fixed point formula toX and get formulas for the traces of Hecke operators H acting on the cohomology of X. We allow twistings of H by outer automorphisms η of G. We stabilize this topological trace formula and compare it with the corresponding formula for an endoscopic group of the pair (G, η). As an application, we deduce a weak lifting theorem for the lifting of automorphic representations from Siegel modular groups to general linear groups.

Compatibility of the Fargues-Scholze and Gan-Takeda Local Langlands

“…We now would like to combine the analysis of sections 5.1 and 5.2 to deduce a strong multiplicity one result for G * = GSp 4 /F and certain inner forms. Our analysis is very similar to [CG15, Sections 10.5 and 10.6] and benefited from reading the proofs of [RW21, Proposition 10.1 and Theorem 11.4] in a paper of Rosner and Weissauer, where they prove a similar multiplicity one result using Weselmann's topological twisted trace formula [Wes12] instead of the simple twisted trace formula of Kottwitz-Shelstad. Let S st and S sc be disjoint finite sets of finite places.…”

Siegel modular varieties and the Eisenstein cohomology of PGL2g+1

“…Thereby we will prove their coincidence for γ and γ 1 : are the cardinalities of certain non empty subsets of H 1 ( η , ζ) as defined in [Wes12,2.24]. But since we assume that ζ is the trivial group, they all have to be 1.…”

“…This paper is a sequel of our paper [Wes12], where we developed a twisted topological trace formula and tried to understand liftings from symplectic to general linear groups. Here we want to analyse the lift from Sp 2g to PGL 2g+1 over the ground field Q in further detail, and we get a description of the image of this lift for the L 2 cohomology of Sp 2g (which is related to the intersection cohomology of the Shimura variety attached to GSp 2g ) in terms of the Eisenstein cohomology of the general linear group, whose building constituents are cuspidal representations of Levi groups.…”

“…Recall ( [Wes12,5.3. ]) that the roots β ∈ Φ(H, T H ) are of the form β = c(α)·S η (α), where S η (α) denotes the sum over all elements in the η-orbit of α ∈ Φ(G, T ) ⊂ X * (T ) and where c(α) ∈ {1, 2}.…”

Siegel Modular Varieties and the Eisenstein Cohomology of $\PGL_{2g+1}$

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We use the twisted topological trace formula developed in [Wes12] to understand liftings from symplectic to general linear groups. We analyse the lift from Sp 2g to PGL 2g+1 over the ground field Q in further detail, and we get a description of the image of this lift of the L 2 cohomology of Sp 2g (which is related to the intersection cohomology of the Shimura variety attached to GSp 2g ) in terms of the Eisenstein cohomology of the general linear group, whose building constituents are cuspidal representations of Levi groups.

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