Abstract
For a split reductive group$G$over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated$G$-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic$t$-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of$\ell$in the Langlands correspondence for function fields.
Highlights
(ii) We show that the intersection cohomology motive of the moduli stack of iterated G-shtukas with bounded modification and level structure is unconditionally defined, that is, without reference to the standard conjectures
Using DM( Vi ) = DM(Vi ), we further reduce to the case that X and Y are in AffSchfSt, in which case we know the desired properties of f ! by Synopsis 2.1.1, (iv) and (vi)
Suppose we are in the situation of Proposition 3.1.23: H ⊂ G is an inclusion of ordinary τ -sheaves of S-groups, X := (G/H )τ be the quotient of τ -sheaves which we assume to be a smooth finite type S-scheme equipped with a cellular Whitney–Tate stratification such that the base point S → X factors over X +
Summary
Following Soergel and Wendt [SW18, Definition 4.5] (who consider the case of finite type schemes over fields), the stratification ι : X + → X is called Whitney–Tate if ι∗ι∗1X+ ∈ DTM(X +), that is, (ι|Xv )∗(ι|Xw )∗1Xw is a Tate motive on Xv for each v, w ∈ W. A version of Theorem C for Witt vector (or p-adic) affine flag varieties is contained in the first arxiv version of this paper, and will be published elsewhere together with a motivic Satake equivalence in this context With this general set-up in hand, we construct in Section 6 the intersection cohomology motive of the moduli stack of bounded shtukas. : Roughly, after cutting down the situation to finite-dimensional stacks, the map inv is the map onto the local model, cf [Laf, Propositions 2.8, 2.9] This map is smooth, and pullback preserves the intersection complex (up to shift). We expect that the same result holds true for the intersection motives, cf. Remark 6.3.7
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