arXiv subjects

Maxim Gurevich

Publications and source records attributed to Maxim Gurevich.

At least 19 recordsLinked to original sources

arXiv2026

On reciprocal characters and the quantum affine Schur-Weyl duality

We identify the dominant part of the Frenkel-Reshetikhin $q$-character with a natural invariant arising from the Langlands/Zelevinsky parameterization for affine Hecke algebras. We introduce the reciprocal character of a module over a $GL_n$-type affine Hecke algebra, defined in terms of multiplicities within parabolic restriction. The main theorem claims that the reciprocal character matches, under quantum affine Schur--Weyl duality, with the dominant $q$-character for finite-dimensional modules over quantum affine algebras. This result gives a type $A$ realization of the Nakajima expectation that the dominant monomials in the $q$-character should play the role of monomial-basis coordinates in Lusztig's framework for finite quantum groups. Indeed, under the affine Hecke categorification of $U_q(\mathfrak{sl}_\infty)^+$, we prove that the reciprocal character is the specialization at $q=1$ of the coordinate map attached to a monomial basis. As a consequence, dominant $q$-character multiplicities for simple (or standard) modules are described by transition coefficients between monomial and canonical (or PBW) bases. Our methods rely on the development of explicit tableau-counting formulas for such dominant multiplicities, or equivalently for the reciprocal characters of standard modules over affine Hecke algebras.

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arXiv2024

On local character expansions for principal series representations of general linear groups

We obtain two explicit formulas for the full local character expansion of any irreducible representation of a p-adic general linear group in principal blocks. The first, generalizing previous work of the author on the Iwahori-spherical case, expresses the expansion in terms of dimensions of degenerate Whittaker models. The second gives a closed expression in terms of values of Kazhdan-Lusztig polynomials of a suitable permutation group.

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arXiv2024

Ramification of weak Arthur packets for p-adic groups

Weak Arthur packets have long been instrumental in the study of the unitary dual and automorphic spectrum of reductive Lie groups, and were recently introduced in the p-adic setting by Ciubotaru - Mason-Brown - Okada. For split odd orthogonal and symplectic p-adic groups, we explicitly determine the decomposition of weak Arthur packets into Arthur packets that arise from endoscopic transfer. We establish a characterization of the Arthur packets that partake in such decompositions by means of ramification properties of their constituents. A notion of weak sphericity for an irreducible representation is introduced: The property of containing fixed vectors with respect to a (not necessarily hyperspecial) maximal compact subgroup. We show that this property determines the weak Arthur packets in a precise sense. As steps towards this description, we explore alignments between Langlands-type reciprocities for finite and p-adic groups, and their dependence on the geometry of the unipotent locus of the dual Langlands group. Weak sphericity is shown to match with Lusztig's canonical quotient spaces that feature in the geometric theory for Weyl group representations, while the fine composition of weak Arthur packets is found to be governed by the partition of the unipotent locus into special pieces.

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arXiv2023

Parabolic recursions for Kazhdan-Lusztig polynomials and the hypercube decomposition

We employ general parabolic recursion methods to demonstrate the recently devised hypercube formula for Kazhdan-Lusztig polynomials of $S_n$, and establish its generalization to the full setting of a finite Coxeter system through algebraic proof. We introduce procedures for positive decompositions of $q$-derived Kazhdan-Lusztig polynomials within this setting, that utilize classical Hecke algebra positivity phenomena of Dyer-Lehrer and Grojnowski-Haiman. This leads to a distinct algorithmic approach to the subject, based on induction from a parabolic subgroup. We propose suitable weak variants of the combinatorial invariance conjecture and verify their validity for permutation groups.

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arXiv2022

A triangular system for local character expansions of Iwahori-spherical representations of general linear groups

For Iwahori-spherical representations of non-Archimedean general linear groups, Chan-Savin recently expressed the Whittaker functor as a restriction to an isotypic component of a finite Iwahori-Hecke algebra module. We generalize this method to describe principal degenerate Whittaker functors. Concurrently, we view Murnaghan's formula for the Harish-Chandra--Howe character as a Grothendieck group expansion of the same module. Comparing the two approaches through the lens of Zelevinsky's PSH-algebras, we obtain an explicit unitriangular transition matrix between coefficients of the character expansion and the principal degenerate Whittaker dimensions.

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arXiv2021

Graded Specht modules as Bernstein-Zelevinsky derivatives of the RSK model

We clarify the links between the graded Specht construction of modules over cyclotomic Hecke algebras and the RSK construction for quiver Hecke algebras of type A, that was recently imported from the setting of representations of p-adic groups. For that goal we develop a theory of crystal derivative operators on quiver Hecke algebra modules, that categorifies the Berenstein-Zelevinsky strings framework on quantum groups, and generalizes a graded variant of the classical Bernstein-Zelevinsky derivatives from the p-adic setting. Graded cyclotomic decomposition numbers are shown to be a special subfamily of the wider concept of RSK decomposition numbers.

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arXiv2021

Simple modules for quiver Hecke algebras and the Robinson-Schensted-Knuth correspondence

We formalize some known categorical equivalences to give a rigorous treatment of smooth representations of p-adic general linear groups, as ungraded modules over quiver Hecke algebras of type A. Graded variants of RSK-standard modules are constructed for quiver Hecke algebras. Exporting recent results from the p-adic setting, we describe an effective method for construction and classification of all simple modules as quotients of modules induced from maximal homogenous data. It is established that the products involved in the RSK construction fit the Kashiwara-Kim notion of normal sequences of real modules. We deduce that RSK-standard modules have simple heads, devise a formula for the shift of grading between RSK-standard and simple self-dual modules, and establish properties of their decomposition matrix, thus confirming expectations for p-adic groups raised in a work of the author with Lapid. Subsequent work will exhibit how the presently introduced RSK construction generalizes the much-studied Specht construction, when inflated from cyclotomic quotient algebras.

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arXiv2021

Quantum invariants for decomposition problems in type A rings of representations

We prove a combinatorial rule for a complete decomposition, in terms of Langlands parameters, for representations of p-adic $GL_n$ that appear as parabolic induction from a large family (ladder representations). Our rule obviates the need for computation of Kazhdan-Lusztig polynomials in these cases, and settles a conjecture posed by Lapid. These results are transferrable into various type A frameworks, such as the decomposition of convolution products of homogeneous KLR-algebra modules, or tensor products of snake modules over quantum affine algebras. The method of proof applies a quantization of the problem into a question on Lusztig's dual canonical basis and its embedding into a quantum shuffle algebra, while computing numeric invariants which are new to the p-adic setting.

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arXiv2020

Cyclic representations of general linear p-adic groups

Let $π_1,\ldots,π_k$ be smooth irreducible representations of $p$-adic general linear groups. We prove that the parabolic induction product $π_1\times\cdots\times π_k$ has a unique irreducible quotient whose Langlands parameter is the sum of the parameters of all factors (cyclicity property), assuming that the same property holds for each of the products $π_i\times π_j$ ($i<j$), and that for all but at most two representations $π_i\times π_i$ remains irreducible (square-irreducibility property). Our technique applies the recently devised Kashiwara-Kim notion of a normal sequence of modules for quiver Hecke algebras. Thus, a general cyclicity problem is reduced to the recent Lapid-Mínguez conjectures on the maximal parabolic case.

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arXiv2020

On restriction of unitarizable representations of general linear groups and the non-generic local Gan-Gross-Prasad conjecture

We prove one direction of a recently posed conjecture by Gan-Gross-Prasad, which predicts the branching laws that govern restriction from p-adic $GL_n$ to $GL_{n-1}$ of irreducible smooth representations within the Arthur-type class. We extend this prediction to the full class of unitarizable representations, by exhibiting a combinatorial relation that must be satisfied for any pair of irreducible representations, in which one appears as a quotient of the restriction of the other. We settle the full conjecture for the cases in which either one of the representations in the pair is generic. The method of proof involves a transfer of the problem, using the Bernstein decomposition and the quantum affine Schur-Weyl duality, into the realm of quantum affine algebras. This restatement of the problem allows for an application of the combined power of a result of Hernandez on cyclic modules together with the Lapid-Minguez criterion from the p-adic setting.

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arXiv2019

Robinson-Schensted-Knuth correspondence in the representation theory of the general linear group over a non-archimedean local field

We construct new "standard modules" for the representations of general linear groups over a local non-archimedean field. The construction uses a modified Robinson-Schensted-Knuth correspondence for Zelevinsky's multisegments. Typically, the new class categorifies the basis of Doubilet, Rota, and Stein for matrix polynomial rings, indexed by bitableaux. Hence, our main result provides a link between the dual canonical basis (coming from quantum groups) and the DRS basis.

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arXiv2018

An identity of parabolic Kazhdan-Lusztig polynomials arising from square-irreducible modules

We show a precise formula, in the form of a monomial, for certain families of parabolic Kazhdan-Lusztig polynomials of the symmetric group. The proof stems from results of Lapid-Minguez on irreducibility of products in the Bernstein-Zelevinski ring. By quantizing those results into a statement on quantum groups and their canonical bases, we obtain identities of coefficients of certain transition matrices that relate Kazhdan-Lusztig polynomials to their parabolic analogues. This affirms some basic cases of conjectures raised recently by Lapid.

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arXiv2016

Decomposition rules for the ring of representations of non-Archimedean $GL_n$

Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the sequence of p-adic groups $\{GL_n(F)\}_{n=0}^\infty$, with multiplication defined through parabolic induction. We study the problem of the decomposition of products of irreducible representations in $\mathcal{R}$. We obtain a necessary condition on irreducible factors of a given product by introducing a width invariant. Width $1$ representations form the previously studied class of ladder representations. We later focus on the case of a product of two ladder representations, for which we establish that all irreducible factors appear with multiplicity one. Finally, we propose a general rule for the composition series of a product of two ladder representations and prove its validity for cases in which the irreducible factors correspond to smooth Schubert varieties.

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arXiv2016

On two questions concerning representations distinguished by the Galois involution

Let E/F be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the smooth irreducible representations of GL(n,E) that are distinguished by its subgroup GL(n,F). One relates this class to representations which come as base change lifts from a quasi-split unitary group F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps we show that these two approaches are closely related. Using this observation, we are able to prove a statement relating base change and distinction for ladder representations. We then produce a wide family of examples in which the symmetry condition does not impose GL(n,F)-distinction, and thus exhibit the limitations of these two approaches.

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arXiv2016

A filtration on rings of representations of non-Archimedean $GL_n$

Let $F$ be a $p$-adic field. Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the groups $\{GL_n(F)\}_{n=0}^\infty$ taken together, with multiplication defined in the sense of parabolic induction. We introduce a width invariant for elements of $\mathcal{R}$ and show that it gives an increasing filtration on the ring. Irreducible representations of width $1$ are precisely those known as ladder representations. We thus obtain a necessary condition on irreducible factors of a product of two ladder representations. For such a product we further establish a multiplicity-one phenomenon, which was previously observed in special cases.

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arXiv2015

On a local conjecture of Jacquet, ladder representations and standard modules

Let $E/F$ be a quadratic extension of p-adic fields. We prove that every smooth irreducible ladder representation of the group $GL_n(E)$ which is contragredient to its own Galois conjugate, possesses the expected distinction properties relative to the subgroup $GL_n(F)$. This affirms a conjecture attributed to Jacquet for a large class of representations. Along the way, we prove a reformulation of the conjecture which concerns standard modules in place of irreducible representations.

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arXiv2015

A criterion for integrability of matrix coefficients with respect to a symmetric space

Let $G$ be a reductive group and $θ$ an involution on $G$, both defined over a $p$-adic field. We provide a criterion for $G^θ$-integrability of matrix coefficients of representations of $G$ in terms of their exponents along $θ$-stable parabolic subgroups. The group case reduces to Casselman's square-integrability criterion. As a consequence we assert that certain families of symmetric spaces are strongly tempered in the sense of Sakellaridis and Venkatesh. For some other families our result implies that matrix coefficients of all irreducible, discrete series representations are $G^θ$-integrable.

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arXiv2014

A Distributional Treatment of Relative Mirabolic Multiplicity One

We study the role of the mirabolic subgroup $P$ of $G=\mathbf{GL}_n(F)$ ($F$ a $p$-adic field) in smooth irreducible representations of $G$ that possess a non-zero invariant functional relative to a subgroup of the form $H_{k} = \mathbf{GL}_k(F)\times \mathbf{GL}_{n-k}(F)$. We show that if a non-zero $H_1$-invariant functional exists on a representation, then every $P\cap H_1$-invariant functional must equal to a scalar multiple of it. When $k>1$, we give a reduction of the same problem to a question about invariant distributions on the nilpotent cone of the tangent space of the symmetric space $G/H_k$. Some new distributional methods, which are suitable for a setting of non-reductive groups, are developed.

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