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Abstract We geometrize the modpSatake isomorphism of Herzig and Henniart–Vignéras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirković–Vilonen cycles, for the Satake transform of an arbitrary parahoric modpHecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric modpHecke algebra. Our methods involve the constant term functors inspired from the geometric Langlands program, and we also treat the case of reductive groups in equal characteristic. We expect this to be a first step toward a geometrization of a modpLocal Langlands Correspondence. 

Abstract We prove the closure ordering conjecture on the local 𝐿-parameters of representations in local Arthur packets of ${\mathrm{G}}_n={\mathrm{Sp}}_{2n},{\mathrm{SO}}_{2n+1}$\mathrm{G}_{n}=\mathrm{Sp}_{2n},\mathrm{SO}_{2n+1}over a non-Archimedean local field of characteristic zero.Precisely, given any representation 𝜋 in a local Arthur packet ${\mathrm{\Pi }}_{\psi }$\Pi_{\psi}, the closure of the local 𝐿-parameter of 𝜋 in the Vogan variety must contain the local 𝐿-parameter corresponding to 𝜓.This conjecture reveals a geometric nature of local Arthur packets and is inspired by the work of Adams, Barbasch and Vogan, and the work of Cunningham, Fiori, Moussaoui, Mracek and Xu, on ABV-packets.As an application, for general quasi-split connected reductive groups, we show that the closure ordering conjecture implies the enhanced Shahidi conjecture, under certain reasonable assumptions.This provides a framework towards the enhanced Shahidi conjecture in general.We verify these assumptions for ${\mathrm{G}}_n$\mathrm{G}_{n}, hence give a new proof of the enhanced Shahidi conjecture.Finally, we show that local Arthur packets cannot be fully contained in other ones, which is in contrast to the situation over Archimedean local fields and is of independent interest. 

Abstract Let $$k$$ be a field, let $$H \subset G$$ be (possibly disconnected) reductive groups over $$k$$, and let $$\Gamma $$ be a finitely generated group. Vinberg and Martin have shown that the induced morphism $$\underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , H)//H \to \underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , G)//G$$ is finite. In this note, we generalize this result (with a significantly different proof) by replacing $$k$$ with an arbitrary locally Noetherian scheme, answering a question of Dat. Along the way, we use Bruhat–Tits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings. 

Abstract We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies $\mathrm{\Lambda }\left(X^{m,n}\right)\subset \mathcal{E}$\Lambda(X^{m{,}n})\subset\mathcal{E}of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann.As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial.In particular, we obtain a formula for ${\nabla }^m{s}_{\lambda }$\nabla^{m}s_{\lambda}which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one. 

Abstract We construct a moduli space $$\textsf {LP}_{G}$$ of $$\operatorname {SL}_{2}$$-parameters over $${\mathbb {Q}}$$, and show that it has good geometric properties (e.g., explicitly parametrized geometric connected components and smoothness). We construct a Jacobson–Morozov morphism$$\textsf {JM}\colon \textsf {LP}_{G}\to \textsf {WDP}_{G}$$ (where $$\textsf {WDP}_{G}$$ is the moduli space of Weil–Deligne parameters considered by several other authors). We show that $$\textsf {JM}$$ is an isomorphism over a dense open of $$\textsf {WDP}_{G}$$, that it induces an isomorphism between the discrete loci $$\textsf {LP}^{\textrm {disc}}_{G}\to \textsf {WDP}_{G}^{\textrm {disc}}$$, and that for any $${\mathbb {Q}}$$-algebra $$A$$ it induces a bijection between Frobenius semi-simple equivalence classes in $$\textsf {LP}_{G}(A)$$ and Frobenius semi-simple equivalence classes in $$\textsf {WDP}_{G}(A)$$ with constant (up to conjugacy) monodromy operator. 

Abstract We show that an orthogonal root number of a temperedL-parameter $$\varphi $$ $\phi$decomposes as the product of two other numbers: the orthogonal root number of the principal parameter and the value on a central involution of Langlands’s central character for $$\varphi $$ $\phi$. The formula resolves a conjecture of Gross and Reeder and computes root numbers of Weil–Deligne representations arising in a conjectural description of the Plancherel measure. 

In this paper, we prove the local converse theorem for split even special orthogonal groups over a non-Archimedean local field of characteristic $p\ne <\#comment/>2$. This is the only case left on local converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. We apply a new idea by considering a certain sum of partial Bessel functions to overcome this difficulty. As a direct application, we obtain a weak rigidity theorem for irreducible generic cuspidal representations of split even special orthogonal groups. 

We study generalizations of Schur functors from categories consisting of flags of vector spaces. We give different descriptions of the category of such functors in terms of representations of certain combinatorial categories and infinite rank groups, and we apply these descriptions to study polynomial representations and representation stability of parabolic subgroups of general linear groups. 

We prove a version of the Lefschetz hyperplane theorem for fppf cohomology with coefficients in any finite commutative group scheme over the ground field. As consequences, we establish new Lefschetz results for the Picard scheme.