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1. The test function conjecture for parahoric local models

   https://doi.org/10.1090/jams/955  

   Haines, Thomas; Richarz, Timo (January 2021, Journal of the American Mathematical Society)

   We prove the test function conjecture of Kottwitz and the first-named author for local models of Shimura varieties with parahoric level structure, and their analogues in equal characteristic. 

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2. Cellular pavings of fibers of convolution morphisms

   https://doi.org/10.46298/epiga.2024.12352  

   Haines, Thomas J (May 2025, Épijournal de Géométrie Algébrique)

   This article proves, in the case of split groups over arbitrary fields, that all fibers of convolution morphisms attached to parahoric affine flag varieties are paved by products of affine lines and affine lines minus a point. This applies in particular to the affine Grassmannian and to the convolution morphisms in the context of the geometric Satake correspondence. The second part of the article extends these results over $$\mathbb Z$$. Those in turn relate to the recent work of Cass-van den Hove-Scholbach on the geometric Satake equivalence for integral motives, and provide some alternative proofs for some of their results. Comment: 24 pages. Minor error corrected with the addition of Lemma 7.2. Lemma 7.3 added. Material on triviality of morphisms added to section 5. Minor changes in notation. Published version 

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3. Local types of (Γ,G)$$(\Gamma ,G)$$‐bundles and parahoric group schemes

   https://doi.org/10.1112/plms.12544  

   Damiolini, Chiara; Hong, Jiuzu (June 2023, Proceedings of the London Mathematical Society)

   Abstract Let be a simple algebraic group over an algebraically closed field . Let be a finite group acting on . We classify and compute the local types of ‐bundles on a smooth projective ‐curve in terms of the first nonabelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in . When , we prove that any generically simply connected parahoric Bruhat–Tits group scheme can arise from a ‐bundle. We also prove a local version of this theorem, that is, parahoric group schemes over the formal disc arise from constant group schemes via tamely ramified coverings. 

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4. The test function conjecture for local models of Weil-restricted groups

   https://doi.org/10.1112/S0010437X20007162  

   Haines, Thomas J.; Richarz, Timo (July 2020, Compositio Mathematica)

   We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $$p$$ -adic local fields with $$p\geqslant 5$$ . In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme. 

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5. Cohomological representations of parahoric subgroups

   https://doi.org/10.1090/ert/557  

   Chan, Charlotte; Ivanov, Alexander (January 2021, Representation Theory of the American Mathematical Society)

   We give a geometric construction of representations of parahoric subgroups P P of a reductive group G G over a local field which splits over an unramified extension. These representations correspond to characters θ \theta of unramified maximal tori and, when the torus is elliptic, are expected to give rise to supercuspidal representations of G G . We calculate the character of these P P -representations on a special class of regular semisimple elements of G G . Under a certain regularity condition on θ \theta , we prove that the associated P P -representations are irreducible. This generalizes a construction of Lusztig from the hyperspecial case to the setting of an arbitrary parahoric. 

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