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1. Coherent Springer theory and the categorical Deligne-Langlands correspondence

   https://doi.org/10.1007/s00222-023-01224-2  

   Ben-Zvi, David ; Chen, Harrison ; Helm, David ; Nadler, David ( February 2024 , Inventiones mathematicae)

   Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant$K$$K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields$F$$F$with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from$K$$K$-theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, thecoherent Springer sheaf. As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu).

   In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of$\mathrm{GL}_{n}(F)$${\mathrm{GL}}_n\left(F\right)$into coherent sheaves on the stack of Langlands parameters.

    

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2. Homological mirror symmetry for higher-dimensional pairs of pants

   https://doi.org/10.1112/S0010437X20007150  

   Lekili, Yankı ; Polishchuk, Alexander ( July 2020 , Compositio Mathematica)

   Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$ , for $k\geqslant n$ , with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ( $n$ -dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$ . By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$ -dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$ . We also prove similar equivalences for finite abelian covers of the $n$ -dimensional pair of pants. 

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3. 𝚤Hall algebra of the projective line and 𝑞-Onsager algebra

   https://doi.org/10.1090/tran/8798  

   Lu, Ming ; Ruan, Shiquan ; Wang, Weiqiang ( April 2023 , Transactions of the American Mathematical Society)

   The ı \imath Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of 1 1 -periodic complexes of coherent sheaves on the projective line. This ı \imath Hall algebra is shown to realize the universal q q -Onsager algebra (i.e., ı \imath quantum group of split affine A 1 A_1 type) in its Drinfeld type presentation. The ı \imath Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two ı \imath Hall algebras, explaining the isomorphism of the q q -Onsager algebra under the two presentations. 

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4. Calibrated representations of the double Dyck path algebra

   https://doi.org/10.1007/s00208-024-02937-2  

   González, Nicolle ; Gorsky, Eugene ; Simental, José ( August 2024 , Mathematische Annalen)

   The double Dyck path algebra$$\mathbb {A}_{q,t}$$$A_{q,t}$and its polynomial representation first arose as a key figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an equivalent algebra$$\mathbb {B}_{q,t}$$$B_{q,t}$was then given by the second author and Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In this article, we initiate the systematic study of the representation theory of the double Dyck path algebra$$\mathbb {B}_{q,t}$$$B_{q,t}$. We define a natural extension of this algebra and study its calibrated representations. We show that the polynomial representation is calibrated, and place it into a large family of calibrated representations constructed from posets satisfying certain conditions. We also define tensor products and duals of these representations, thus proving (under suitable conditions) the category of calibrated representations is generically monoidal. As an application, we prove that tensor powers of the polynomial representation can be constructed from the equivariant K-theory of parabolic Gieseker moduli spaces.

    

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5. STAR OPERATIONS FOR AFFINE HECKE ALGEBRAS

   Barbasch, Dan ; Ciubotaru, Dan ( July 2020 , Representation Theory, Automorphic Forms & Complex Geometry)

   null ; Yau, Shing-Tung (Ed.)

   Abstract. In this paper, we consider the star operations for (graded) affine Hecke algebras which preserve certain natural filtrations. We show that, up to inner conjugation, there are only two such star operations for the graded Hecke algebra: the first, denoted ⋆, corresponds to the usual star operation from reductive p-adic groups, and the second, denoted • can be regarded as the analogue of the compact star operation of a real group considered by [ALTV]. We explain how the star operation • appears naturally in the Iwahori- spherical setting of p-adic groups via the endomorphism algebras of Bernstein projectives. We also prove certain results about the signature of •-invariant forms and, in particular, about •-unitary simple modules. 

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