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1. Random and mean Lyapunov exponents for

   https://doi.org/10.1017/etds.2023.106  

   ARMENTANO, DIEGO; CHINTA, GAUTAM; SAHI, SIDDHARTHA; SHUB, MICHAEL (August 2024, Ergodic Theory and Dynamical Systems)

   Abstract We consider orthogonally invariant probability measures on$$\operatorname {\mathrm {GL}}_n(\mathbb {R})$$and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on$$\operatorname {\mathrm {GL}}_n(\mathbb {C})$$.Astérisque287(2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result. 

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2. Higher Orbital Integrals, Cyclic Cocycles and Noncommutative Geometry

   https://doi.org/10.1017/fms.2024.115  

   Song, Yanli; Tang, Xiang (January 2025, Forum of Mathematics, Sigma)

   Abstract LetGbe a linear real reductive Lie group. Orbital integrals define traces on the group algebra ofG. We introduce a construction of higher orbital integrals in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz algebra ofG. We analyze these higher orbital integrals via Fourier transform by expressing them as integrals on the tempered dual ofG. We obtain explicit formulas for the pairing between the higher orbital integrals and theK-theory of the reduced group$$C^{*}$$-algebra, and we discuss their application toK-theory. 

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3. MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL

   https://doi.org/10.1017/S1474748020000225  

   Le, Daniel; Morra, Stefano; Schraen, Benjamin (May 2020, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Let $$F$$ be a totally real field in which $$p$$ is unramified. Let $$\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $$v$$ above $$p$$ . Let $$\mathfrak{m}$$ be the corresponding Hecke eigensystem. We describe the $$\mathfrak{m}$$ -torsion in the $$\text{mod}\,p$$ cohomology of Shimura curves with full congruence level at $$v$$ as a $$\text{GL}_{2}(k_{v})$$ -representation. In particular, it only depends on $$\overline{r}|_{I_{F_{v}}}$$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $$\text{GL}_{2}(\mathbf{F}_{q})$$ -projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math.   200 (1) (2015), 1–96]. 

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4. An Euler system for GU(2, 1)

   https://doi.org/10.1007/s00208-021-02224-4  

   Loeffler, David; Skinner, Christopher; Zerbes, Sarah Livia (April 2022, Mathematische Annalen)

   Abstract We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of $${{\,\mathrm{GL}\,}}_3$$ GL 3 over imaginary quadratic fields, using the cohomology of Shimura varieties for $${\text {GU}}(2, 1)$$ GU ( 2 , 1 ) . 

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5. Set superpartitions and superspace duality modules

   https://doi.org/10.1017/fms.2022.90  

   Rhoades, Brendon; Wilson, Andrew Timothy (January 2022, Forum of Mathematics, Sigma)

   Abstract The superspace ring $$\Omega _n$$ is a rank n polynomial ring tensored with a rank n exterior algebra. Using an extension of the Vandermonde determinant to $$\Omega _n$$ , the authors previously defined a family of doubly graded quotients $${\mathbb {W}}_{n,k}$$ of $$\Omega _n$$ , which carry an action of the symmetric group $${\mathfrak {S}}_n$$ and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules $${\mathbb {W}}_{n,k}$$ in greater detail. We describe a monomial basis of $${\mathbb {W}}_{n,k}$$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called ordered set superpartitions . These are ordered set partitions $$(B_1 \mid \cdots \mid B_k)$$ of $$\{1,\dots ,n\}$$ in which the nonminimal elements of any block $$B_i$$ may be barred or unbarred. 

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