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1. Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

   https://doi.org/10.1007/s00029-021-00654-1  

   Sahi, Siddhartha; Stokman, Jasper V.; Venkateswaran, Vidya (July 2021, Selecta Mathematica)

   null (Ed.)

   Abstract We construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters $$g_i$$ g i , and which we refer to as metaplectic representations. We realize these representations as quotients of certain parabolically induced modules, and we apply the method of Baxterization (localization) to obtain actions of corresponding Weyl groups on rational functions on the torus. Our construction both generalizes and provides a conceptual proof of earlier results of Chinta, Gunnells, and Puskas, which had depended on a crucial computer verification. A key motivation is that when the parameters $$g_i$$ g i are specialized to certain Gauss sums, the resulting representation and its localization arise naturally in the consideration of p -parts of Weyl group multiple Dirichlet series. In this special case, similar results have been previously obtained in the literature by the study of Iwahori Whittaker functions for principal series of metaplectic covers of reductive p -adic groups. However this technique is not available for generic parameters $$g_i$$ g i . It turns out that the metaplectic representations can be extended to the double affine Hecke algebra, where they share many important properties with Cherednik’s basic polynomial representation, which they generalize. This allows us to introduce families of metaplectic polynomials, which depend on the $$g_i$$ g i , and which generalize Macdonald polynomials. In this paper we discuss in some detail the situation for type A , which is of considerable interest in algebraic combinatorics. We postpone some of the proofs, as well as a discussion of other types, to the sequel. 

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2. The Gelfand–Graev representation of classical groups in terms of Hecke algebras

   https://doi.org/10.4153/S0008414X2200030X  

   Bakić, Petar; Savin, Gordan (July 2022, Canadian Journal of Mathematics)

   Abstract Let G be a p -adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra—the endomorphism algebra of a pro-generator of the given component. Using Heiermann’s construction of these algebras, we describe the Bernstein components of the Gelfand–Graev representation for $$G=\mathrm {SO}(2n+1)$$ , $$\mathrm {Sp}(2n)$$ , and $$\mathrm {O}(2n)$$ . 

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3. Braided Frobenius algebras from certain Hopf algebras

   https://doi.org/10.1142/S0219498823500123  

   Saito, Masahico; Zappala, Emanuele (September 2021, Journal of Algebra and Its Applications)

   A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation [Formula: see text], that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations. 

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4. Springer correspondence for the split symmetric pair in type

   https://doi.org/10.1112/S0010437X18007443  

   Chen, Tsao-Hsien; Vilonen, Kari; Xue, Ting (November 2018, Compositio Mathematica)

   In this paper we establish Springer correspondence for the symmetric pair $$(\text{SL}(N),\text{SO}(N))$$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at $q=-1$ . Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at $q=-1$ arise in geometry. 

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5. Simple 𝑝-adic Lie groups with abelian Lie algebras

   https://doi.org/10.1515/crelle-2024-0030  

   Caprace, Pierre-Emmanuel; Minasyan, Ashot; Osin, Denis (June 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract For each prime 𝑝 and each positive integer 𝑑, we construct the first examples of second countable, topologically simple 𝑝-adic Lie groups of dimension 𝑑 whose Lie algebras are abelian.This answers several questions of Glöckner and Caprace–Monod.The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups. 

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