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1. Semisimplification of contragredient Lie algebras

   https://doi.org/10.2140/om.2024.1.211  

   Angiono, Iván; Plavnik, Julia; Sanmarco, Guillermo (January 2025, Orbita Mathematicae)

   We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic p with respect to the adjoint action of a Chevalley generator. In particular, we construct a root system for these algebras that arises as a parabolic restriction of the known root system for the classical Lie algebra. This gives a lattice grading with simple homogeneous components and a triangular decomposition for the semisimplified Lie algebra. We also obtain a non-degenerate invariant form that behaves well with the lattice grading. As an application, we exhibit concrete new examples of Lie algebras in the Verlinde category. 

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2. The companion section for classical groups

   https://doi.org/10.1142/S0129167X24410106  

   Hameister, Thomas; Ngô, Bao Châu (August 2024, International Journal of Mathematics)

   We use the companion matrix construction for [Formula: see text] to build canonical sections of the Chevalley map [Formula: see text] for classical groups [Formula: see text] as well as the group [Formula: see text]. To do so, we construct canonical tensors on the associated spectral covers. As an application, we make explicit lattice descriptions of affine Springer fibers and Hitchin fibers for classical groups and [Formula: see text]. 

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3. On the Restriction Maps of the Fourier and Fourier–Stieltjes Algebras Over Locally Compact Groupoids

   https://doi.org/10.1093/imrn/rnaf057  

   DeGaetani, Joseph; Ghandehari, Mahya (March 2025, International Mathematics Research Notices)

   Abstract The Fourier and Fourier–Stieltjes algebras over locally compact groupoids have been defined in a way that parallels their construction for groups. In this article, we extend the results on surjectivity or lack of surjectivity of the restriction map on the Fourier and Fourier–Stieltjes algebras of groups to the groupoid setting. In particular, we consider the maps that restrict the domain of these functions in the Fourier or Fourier–Stieltjes algebra of a groupoid to an isotropy subgroup. These maps are continuous contractive algebra homomorphisms. When the groupoid is étale, we show that the restriction map on the Fourier algebra is surjective. The restriction map on the Fourier–Stieltjes algebra is not surjective in general. We prove that for a transitive groupoid with a continuous section or a group bundle with discrete unit space, the restriction map on the Fourier–Stieltjes algebra is surjective. We further discuss the example of an HLS groupoid, and obtain a necessary condition for surjectivity of the restriction map in terms of property FD for groups, introduced by Lubotzky and Shalom. As a result, we present examples where the restriction map for the Fourier–Stieltjes algebra is not surjective. Finally, we use the surjectivity results to provide conditions for the lack of certain Banach algebraic properties, including the (weak) amenability and existence of a bounded approximate identity, in the Fourier algebra of étale groupoids. 

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4. Polar foliations on symmetric spaces and mean curvature flow

   https://doi.org/10.1515/crelle-2022-0045  

   Liu, Xiaobo; Radeschi, Marco (August 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined in [E. Heintze, X. Liu and C. Olmos,Isoparametric submanifolds and a Chevalley-type restriction theorem,Integrable systems, geometry, and topology,American Mathematical Society, Providence 2006, 151–190]. We will also prove a splitting theorem which, when leaves are compact, reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results in [X. Liu and C.-L. Terng,Ancient solutions to mean curvature flow for isoparametric submanifolds,Math. Ann. 378 2020, 1–2, 289–315] for mean curvature flows of isoparametric submanifolds in spheres. 

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5. A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

   https://doi.org/10.1007/s00029-024-00924-8  

   Lenart, Cristian; Naito, Satoshi; Sagaki, Daisuke (July 2024, Selecta Mathematica)

   NA (Ed.)

   We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory QK_{T}(G/B) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of QK_{T}(G/B). In type A_{n-1}, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory QK(SL_{n}/B); we also obtain very explicit information about the coefficients in the respective Chevalley formula. 

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