Langlands dual group
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Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians Open
In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton–Okounkov bodies for Grassmannians. We consider the Grassmannian $\\mathbb{X}=\\mathit{Gr}_{n-k}(\\mathbb{C}^{n})$ , as well as t…
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Geometrization of the local Langlands correspondence Open
Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues--Fontaine curve. In particular, we define a category of $\ell$-adic sheaves on the stack $\mathrm{Bun}_G$ of $G$-bundles on the Far…
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The local Langlands correspondence for inner forms of SL$$_{n}$$ Open
Let F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group SLn(F). It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for SLn(F)…
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Geometrization of the local Langlands correspondence: an overview Open
This article is an overview of the geometrization conjecture for the local Langlands correspondence formulated by the author.
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A `relative' local Langlands correspondence Open
For $E/F$ quadratic extension of local fields and $G$ a reductive algebraic group over $F$, the paper formulates a conjecture classifying irreducible admissible representations of $G(E)$ which carry a $G(F)$ invariant linear form, and the …
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Contragredient representations and characterizing the local Langlands correspondence Open
We consider the question: what is the contragredient in terms of L-homomorphisms? We conjecture that it corresponds to the Chevalley automorphism of the L-group, and prove this in the case of real groups. The proof uses a characterization …
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Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture Open
Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-struc…
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Affine Hecke algebras for Langlands parameters Open
It is well-known that affine Hecke algebras are very useful to describe the smooth representations of any connected reductive p-adic group G, in terms of the supercuspidal representations of its Levi subgroups. The goal of this paper is to…
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Coherent sheaves on the stack of Langlands parameters Open
We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known resu…
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Geometric Satake, Springer correspondence, and small representations II Open
For a split reductive group scheme over a commutative ring with Weyl group , there is an important functor defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representation…
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Quantum q-Langlands Correspondence Open
We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks of the quantum affine algebra and the deformed W-algebra associat…
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Moduli of Langlands parameters Open
Let F be a non-archimedean local field of residue characteristic p , let {\widehat G} be a split reductive group scheme over \mathbb{Z}[\frac{1}{p}] with an action of W_{F} , and let {}^{L}G denote the semidirect product \widehat{G}\rtimes…
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On Landau–Ginzburg models for quadrics and flat sections of Dubrovin connections Open
This paper proves a version of mirror symmetry expressing the (small) Dubrovin connection for even-dimensional quadrics in terms of a mirror-dual Landau–Ginzburg model View the MathML source(X?can,Wq). Here X?can is the complement of an an…
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Twisted orbital integrals and irreducible components of affine Deligne–Lusztig varieties Open
We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and $q$-analogues of the Kostant partition functio…
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Galois representations for general symplectic groups Open
We prove the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Stein…
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Irreducible components of extended eigenvarieties and interpolating Langlands functoriality Open
We study the basic geometry of a class of analytic adic spaces that arise in the study of the extended (or adic) eigenvarieties constructed by Andreatta--Iovita--Pilloni, Gulotta and the authors. We apply this to prove a general interpolat…
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Geometric Realization of the Local Langlands Correspondence for Representations of Conductor Three Open
We prove a realization of the local Langlands correspondence for two-dimensional representations of a Weil group of conductor three in the cohomology of Lubin–Tate curves by a purely local geometric method.
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On the notion of spectral decomposition in local geometric Langlands Open
The geometric Langlands program is distinguished in assigning spectral decompositions to all representations, not only the irreducible ones. However, it is not even clear what is meant by a spectral decomposition when one works with non-ab…
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Higher AGT correspondences, $\mathcal{W}$-algebras, and higher quantum geometric Langlands duality from M-theory Open
We further explore the implications of our framework in [arXiv:1301.1977,\narXiv:1309.4775], and physically derive, from the principle that the spacetime\nBPS spectra of string-dual M-theory compactifications ought to be equivalent,\n(i) a…
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Cluster duality and mirror symmetry for Grassmannians Open
In this article we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. For our $A$-model, we consider the Grassmannian $\m…
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Quantum K-theoretic geometric Satake: the case Open
The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group $G$ and the spherical perverse sheaves on the affine Grassmannian $Gr$ of its Langlands dual group. Bezrukavnikov and …
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Representations of Shifted Quantum Affine Algebras Open
We develop the representation theory of shifted quantum affine algebras $\mathcal {U}_\mu (\hat {\mathfrak {g}})$ and of their truncations, which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gau…
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Local Whittaker-Newforms for GSp(4) matching to Langlands parameters Open
We extend the local newform theory of B. Roberts and R. Schmidt for generic, irreducible, admissible representations of PGSp(4) to that for GSp(4). The newform matches to the Langlands parameter.
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Sheaves on Nilpotent Cones, Fourier Transform, and a Geometric Ringel Duality Open
Given the nilpotent cone of a complex reductive Lie algebra, we consider its equivariant constructible derived category of sheaves with coefficients in an arbitrary field. This category and its subcategory of perverse sheaves play an impor…
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Parameters and duality for the metaplectic geometric Langlands theory Open
We introduce the space of parameters for the metaplectic Langlands theory as *factorization gerbes* on the affine Grassmannian, and develop metaplectic Langlands duality in the incarnation of the metaplectic geometric Satake functor. We fo…
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Moment graphs in representation theory and geometry Open
This paper reviews the moment graph technique that allows to translate certain representation theoretic problems into geometric ones. For simplicity we restrict ourselves to the case of semisimple complex Lie algebras. In particular, we sh…
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W-algebras and Whittaker categories Open
Affine W-algebras are a somewhat complicated family of (topological) associative algebras associated with a semisimple Lie algebra, quantizing functions on the algebraic loop space of Kostant's slice. They have attracted a great deal of at…
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From geometric to function-theoretic Langlands (or how to invent shtukas) Open
This is an informal note that explains that the classical Langlands theory over function fields can be obtained from the geometric one by taking the trace of Frobenius. The operation of taking the trace of Frobenius takes place at the cate…
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Extended systems of Baxter Q-functions and fused flags I: simply-laced case Open
The spectrum of integrable models is often encoded in terms of commuting functions of a spectral parameter that satisfy functional relations. We propose to describe this commutative algebra in a covariant way by means of the extended Q-sys…
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Elliptic affine Hecke algebras and their representations Open
We apply equivariant elliptic cohomology to the Steinberg variety in Springer theory, and prove that the corresponding convolution algebra is isomorphic to the elliptic affine Hecke algebra constructed by Ginzburg-Kapranov-Vasserot. Under …