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1. THE COHOMOLOGY OF UNRAMIFIED RAPOPORT–ZINK SPACES OF EL-TYPE AND HARRIS'S CONJECTURE

   https://doi.org/10.1017/S1474748020000535  

   Meli, Alexander Bertoloni (January 2021, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Abstract We study the l -adic cohomology of unramified Rapoport–Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $$\mathrm {GL_n}$$ and to show local–global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms $$\mathrm {Mant}_{b, \mu }$$ of Grothendieck groups of representations constructed from the cohomology of these spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin and others. Due to earlier work of Fargues and Shin we have a description of $$\mathrm {Mant}_{b, \mu }(\rho )$$ for $$\rho $$ a supercuspidal representation. In this paper, we give a conjectural formula for $$\mathrm {Mant}_{b, \mu }(\rho )$$ for $$\rho $$ an admissible representation and prove it when $$\rho $$ is essentially square-integrable. Our proof works for general $$\rho $$ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup. 

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2. The Kottwitz conjecture for unitary PEL-type Rapoport–Zink spaces

   https://doi.org/10.1515/crelle-2022-0077  

   Bertoloni Meli, Alexander; Nguyen, Kieu Hieu (January 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this paper we study the cohomology of PEL-type Rapoport–Zink spaces associated to unramified unitary similitude groups over ℚ p {\operatorname{\mathbb{Q}}_{p}} in an odd number of variables. We extend the results of Kaletha–Minguez–Shin–White and Mok to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapoport–Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider. 

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3. On the Kottwitz conjecture for local shtuka spaces

   https://doi.org/10.1017/fmp.2022.7  

   Hansen, David; Kaletha, Tasho; Weinstein, Jared (May 2022, Forum of mathematics)

   Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of GLn, the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands. 

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4. Coherent Springer theory and the categorical Deligne-Langlands correspondence

   https://doi.org/10.1007/s00222-023-01224-2  

   Ben-Zvi, David; Chen, Harrison; Helm, David; Nadler, David (February 2024, Inventiones mathematicae)

   Abstract Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant$$K$$ $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields$$F$$ $F$with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from$$K$$ $K$-theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, thecoherent Springer sheaf. As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of$$\mathrm{GL}_{n}(F)$$ ${\mathrm{GL}}_n\left(F\right)$into coherent sheaves on the stack of Langlands parameters. 

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5. Local parameters of supercuspidal representations

   https://doi.org/10.1017/fmp.2024.10  

   Gan, Wee Teck; Harris, Michael; Sawin, Will; Beuzart-Plessis, Raphaël (January 2024, Forum of Mathematics, Pi)

   Abstract For a connected reductive groupGover a nonarchimedean local fieldFof positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter$${\mathcal {L}}^{ss}(\pi )$$to each irreducible representation$$\pi $$. Our first result shows that the Genestier-Lafforgue parameter of a tempered$$\pi $$can be uniquely refined to a tempered L-parameter$${\mathcal {L}}(\pi )$$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of$${\mathcal {L}}^{ss}(\pi )$$for unramifiedGand supercuspidal$$\pi $$constructed by induction from an open compact (modulo center) subgroup. If$${\mathcal {L}}^{ss}(\pi )$$is pure in an appropriate sense, we show that$${\mathcal {L}}^{ss}(\pi )$$is ramified (unlessGis a torus). If the inducing subgroup is sufficiently small in a precise sense, we show$$\mathcal {L}^{ss}(\pi )$$is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is$${\mathbb {P}}^1$$and a simple application of Deligne’s Weil II. 

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