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1. Rational homotopy type and nilpotency of mapping spaces between Quaternionic projective spaces

   https://doi.org/10.15673/pigc.v16i2.2648  

   Abebaw, Tilahun; Gatsinzi, Jean-Baptiste; Yeruk, Smegnsh Demelash (June 2024, Proceedings of the International Geometry Center)

   The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space's algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models. 

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2. A geometric formula for multiplicities of 𝐾-types of tempered representations

   https://doi.org/10.1090/tran/7857  

   Hochs, Peter; Song, Yanli; Yu, Shilin (December 2019, Transactions of the American Mathematical Society)

   Let G G be a connected, linear, real reductive Lie group with compact centre. Let K > G K>G be compact. Under a condition on K K , which holds in particular if K K is maximal compact, we give a geometric expression for the multiplicities of the K K -types of any tempered representation (in fact, any standard representation) π \pi of G G . This expression is in the spirit of Kirillov’s orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of π | K \pi |_K obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of SU ( p , 1 ) \textrm {SU}(p,1) , SO 0 ( p , 1 ) \textrm {SO}_0(p,1) , and SO 0 ( 2 , 2 ) \textrm {SO}_0(2,2) restrict multiplicity freely to maximal compact subgroups. 

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3. Geometric $$L$$-packets of Howe-unramified toral supercuspidal representations

   https://doi.org/10.4171/JEMS/1396  

   Chan, Charlotte; Oi, Masao (March 2025, Journal of the European Mathematical Society)

   We show thatL-packets of toral supercuspidal representations arising from unramified maximal tori ofp-adic groups are realized by Deligne–Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra’s characterization of real discrete series representations by their characters on regular elements of compact maximal tori, a characterization which Langlands relied on in his construction ofL-packets of these representations. In parallel to the real case, we characterize the members of Kaletha’s toralL-packets by their characters on sufficiently regular elements of elliptic maximal tori. 

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4. COMPATIBLE SYSTEMS OF GALOIS REPRESENTATIONS ASSOCIATED TO THE EXCEPTIONAL GROUP

   https://doi.org/10.1017/fms.2018.24  

   BOXER, GEORGE; CALEGARI, FRANK; EMERTON, MATTHEW; LEVIN, BRANDON; MADAPUSI PERA, KEERTHI; PATRIKIS, STEFAN (January 2019, Forum of Mathematics, Sigma)

   We construct, over any CM field, compatible systems of $$l$$ -adic Galois representations that appear in the cohomology of algebraic varieties and have (for all $$l$$ ) algebraic monodromy groups equal to the exceptional group of type $$E_{6}$$ . 

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5. Arakelov–Milnor inequalities and maximal variations of Hodge structure

   https://doi.org/10.1112/S0010437X23007157  

   Biquard, Olivier; Collier, Brian; García-Prada, Oscar; Toledo, Domingo (May 2023, Compositio Mathematica)

   In this paper we study the $$\mathbb {C}^*$$ -fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. An important result of this paper is a bound on this invariant which generalizes the Milnor–Wood inequality for a Hodge bundle in the Hermitian case, and is analogous to the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case. 

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