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Erez Lapid

Publications and source records attributed to Erez Lapid.

At least 19 recordsLinked to original sources
arXiv2026

A hereditary theorem for rigid and pseudorigid components of Lusztig's nilpotent varieties in type $A$

For irreducible components $C_\sigma$ of Lusztig's nilpotent varieties of type $A$ with graded dimension $(1,2,\dots,n,n-1,\dots,1)$ arising from permutations $\sigma\in S_n$, we characterize the property that $C_\sigma\oplus C_\sigma$ is an irreducible component combinatorially in terms of $\sigma$. The permutations that occur, which we call \emph{pseudosmooth}, are described by a recursion on direct sums and deleting suitable corners, whose terminal cases are the permutations obtained from \[ 3412,\quad 4231,\quad 35142,\quad 42513,\quad 45312,\quad 426153,\quad 463152,\quad 526413 \] by inflating the entries into consecutive decreasing blocks. The main new input is a hereditary property valid for decompositions of multisegments with disjoint extreme points.

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arXiv2026

A greedy open-orbit criterion for solvable algebraic group actions, with applications to Lusztig's nilpotent varieties

Let $G$ be a connected solvable algebraic group acting rationally on a finite-dimensional vector space $U$. Using a $G$-stable complete flag, we formulate a successive-quotient procedure that decides whether $U$ contains an open $G$-orbit. When the procedure succeeds, it constructs an open-orbit vector of minimum support cardinality and determines the image of a generic stabilizer in the maximal torus quotient. We also give an infinitesimal version detecting open separable orbits. We apply the criterion to the action of $Aut_Q(M)$ on $Ext^1_Q(M,M)^*$ where $M$ is a multiplicity-free representation of a Dynkin quiver. Rigidity of the corresponding component of Lusztig's nilpotent variety is thereby reduced to a rank test together with an acyclicity condition on a graph of active extension coordinates; the connected components of the resulting forest determine the generic indecomposable decomposition. For equioriented type $A$ this yields an explicit algorithm for a family of multisegments encoded by incidence matrices, including nonregular examples with repeated beginnings or ends.

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arXiv2024

On the convergence of zeta functions of prehomogeneous vector spaces

We prove a general convergence result for zeta functions of prehomogeneous vector spaces extending results of H. Saito, F. Sato and Yukie. Our analysis points to certain subspaces which yield boundary terms. We study it further in the setup arising from nilpotent orbits. In certain cases we determine the residue at the rightmost pole of the zeta function.

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arXiv2022

Some perspectives on Eisenstein series

We review some topics in the analytic theory of Eisenstein series, including meromorphic continuation, $L^2$-spectral expansion and Fourier coefficients. We also discuss some open problems.

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arXiv2022

A binary operation on irreducible components of Lusztig's nilpotent varieties {II}: applications and conjectures for representations of $GL_n$ over a non-archimedean local field

In the first part of the paper we defined and studied a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. For type $A$ we conjecture, following Geiss and Schröer, that this operation is compatible with taking the socle of parabolic induction of representations of general linear groups over a local non-archimedean field, at least when one of the irreducible components is rigid. We verify this conjecture in special cases.

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arXiv2022

On the meromorphic continuation of Eisenstein series

Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a "soft" proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg's later proofs.

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arXiv2020

Some combinatorial results on smooth permutations

We show that any smooth permutation $σ\in S_n$ is characterized by the set ${\mathbf{C}}(σ)$ of transpositions and $3$-cycles in the Bruhat interval $(S_n)_{\leqσ}$, and that $σ$ is the product (in a certain order) of the transpositions in ${\mathbf{C}}(σ)$. We also characterize the image of the map $σ\mapsto{\mathbf{C}}(σ)$. As an application, we show that $σ$ is smooth if and only if the intersection of $(S_n)_{\leqσ}$ with every conjugate of a parabolic subgroup of $S_n$ admits a maximum. This also gives another approach for enumerating smooth permutations and subclasses thereof. Finally, we relate covexillary permutations to smooth ones and rephrase the results in terms of the (co)essential set in the sense of Fulton.

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arXiv2020

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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arXiv2019

Conjectures and results about parabolic induction of representations of $GL_n(F)$

In 1980 Zelevinsky introduced commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss--Leclerc--Schröer condition that occurs in the conjectural characterization of $\square$-irreducible representations. We verify some special cases of the new conjecture and check that the geometric and representation-theoretic conditions are compatible in various ways.

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arXiv2019

On the remainder term of the Weyl law for congruence subgroups of Chevalley groups

Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(\mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen their result by giving a power saving estimate for the remainder term.

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arXiv2018

Conjectures about certain parabolic Kazhdan--Lusztig polynomials

Irreducibility results for parabolic induction of representations of the general linear group over a local non-archimedean field can be formulated in terms of Kazhdan--Lusztig polynomials of type $A$. Spurred by these results and some computer calculations, we conjecture that certain alternating sums of Kazhdan--Lusztig polynomials known as parabolic Kazhdan--Lusztig polynomials satisfy properties analogous to those of the ordinary ones.

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arXiv2017

Geometric conditions for $\square$-irreducibility of certain representations of the general linear group over a non-archimedean local field

Let $π$ be an irreducible, complex, smooth representation of $GL_n$ over a local non-archimedean (skew) field. Assuming $π$ has regular Zelevinsky parameters, we give a geometric necessary and sufficient criterion for the irreducibility of the parabolic induction of $π\otimesπ$ to $GL_{2n}$. The latter irreducibility property is the $p$-adic analogue of a special case of the notion of "real representations" introduced by Leclerc and studied recently by Kang-Kashiwara-Kim-Oh (in the context of KLR or quantum affine algebras). Our criterion is in terms of singularities of Schubert varieties of type $A$ and admits a simple combinatorial description. It is also equivalent to a condition studied by Geiss-Leclerc-Schröer.

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