Jiawei Yang's Mathematics Homepage

Let ψ be a real Arthur parameter and let ψ₂ be the unipotent parameter in a fixed Jordan decomposition. Adams, Ionov, Mason-Brown, and Vogan proved that the microlocal packet of ψ is contained in the support of the two-step Jordan induction of the packet of ψ₂, and conjectured equality. We prove the reverse inclusion. The argument first passes to a sufficiently positive translate, where the relevant connected components of the ABV spaces have a common flag-variety model. Perverse dévissage reduces the problem to a singular-support implication, which follows from the incidence calculation underlying the AIMV microstalk comparison. Coherent continuation for individual packet members then returns the result to the original parameter.

We prove Steinness over Q̆ₚ for the basic local Shimura tower attached to G = GLₙ, μ = (1, 1, 0ⁿ⁻²), with basic Newton slope 2/n, for every odd n ≥ 3, at every finite level and for every prime p. The proof proceeds by constructing global analytic functions from the crystalline period map, proving compactness of their sublevel sets via a perfectoid normalization of Tate lattices, and showing that the resulting function map to affine space is finite. The key geometric inputs are the Fargues-Fontaine realization of the universal cover, a normalized determinant on the rank-two locus, and an integral PEL realization over W(F̄ₚ) used to produce bounded global Hodge generators. The integral constructions include the prime p = 2.

Let F/ℚₚ be a finite extension and let C be a fixed complete algebraically closed field containing F̆. For 0 < d < n, let W_{n,d} denote the v-sheaf parametrizing quotient bundles 𝒪(1/n) ↠ ℰ on the relative Fargues-Fontaine curve whose geometric fibres are isomorphic to 𝒪(1/d). We prove that W_{n,d} is represented by a perfectoid space. This is Conjecture 5.4 in [HM].