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1. Elliptic zastava

   https://doi.org/10.1090/jag/803  

   Finkelberg, Michael; Matviichuk, Mykola; Polishchuk, Alexander (April 2023, Journal of Algebraic Geometry)

   We study the elliptic zastava spaces, their versions (twisted, Coulomb, Mirković local spaces, reduced) and relations with monowalls moduli spaces and Feigin-Odesskiĭ moduli spaces of G G -bundles with parabolic structure on an elliptic curve. 

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2. Pseudo-locality for a coupled Ricci flow

   Guo, Bin; Huang, Zhijie; Phong, Duong H. (July 2018, Communications in analysis and geometry)

   Let (M,g,\phi) be a solution to the Ricci flow coupled with the heat equation for a scalar field \phi. We show that a complete, \kappa-noncollapsed solution (M,g,\phi) to this coupled Ricci flow with a Type I singularity at time T<\infty will converge to a non-trivial Ricci soliton after parabolic rescaling, if the base point is Type I singular. A key ingredient is a version of Perelman pseudo-locality for the coupled Ricci flow. 

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3. Pure braid group actions on category $$\mathcal{O}$$ modules

   https://doi.org/10.4310/PAMQ.2024.v20.n1.a3  

   Appel, Andrea; Toledano Laredo, Valerio (January 2024, Pure and Applied Mathematics Quarterly)

   Let be a symmetrisable Kac–Moody algebra and Uhg its quantised enveloping algebra. Answering a question of P. Etingof, we prove that the quantum Weyl group operators of Uhg give rise to a canonical action of the pure braid group of g on any category O (not necessarily integrable) module. By relying on our recent results, we show that this action describes the monodromy of the rational Casimir connection on the Uhg-module corresponding to V. We also extend these results to yield equivalent representations of parabolic pure braid groups on parabolic category for Uhg and g. 

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4. A reduction principle for Fourier coefficients of automorphic forms

   https://doi.org/10.1007/s00209-021-02784-w  

   Gourevitch, Dmitry; Gustafsson, Henrik P.; Kleinschmidt, Axel; Persson, Daniel; Sahi, Siddhartha (March 2022, Mathematische Zeitschrift)

   Abstract We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients. 

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5. Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

   https://doi.org/10.4153/S0008414X20000711  

   Gourevitch, Dmitry; Gustafsson, Henrik P.; Kleinschmidt, Axel; Persson, Daniel; Sahi, Siddhartha (February 2022, Canadian Journal of Mathematics)

   Abstract In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let $$\pi $$ be a minimal or next-to-minimal automorphic representation of G . We prove that any $$\eta \in \pi $$ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on $$\operatorname {GL}_n$$ . We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type $$D_5$$ and $$E_8$$ with a view toward applications to scattering amplitudes in string theory. 

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