arXiv subjects

Alberto Mínguez

Publications and source records attributed to Alberto Mínguez.

8 recordsLinked to original sources

arXiv2026

Local Intertwining Relations and Co-tempered $A$-packets of Classical Groups

The local intertwining relation is an identity that gives precise information about the action of normalized intertwining operators on parabolically induced representations. We prove several instances of the local intertwining relation for quasi-split classical groups and the twisted general linear group, as they are required in the inductive proof of the endoscopic classification for quasi-split classical groups due to Arthur and Mok. In addition, we construct the co-tempered local $A$-packets by Aubert duality and verify their key properties by purely local means, which provide the seed cases needed as an input to the inductive proof. Together with further technical results that we establish, this makes the endoscopic classification conditional only on the validity of the twisted weighted fundamental lemma.

math.NT↗

arXiv2026

An algorithm for Aubert-Zelevinsky duality à la Mœglin-Waldspurger

Let $F$ be a locally compact non-Archimedean field of characteristic $0$, and let $G$ be either the split special orthogonal group $\mathrm{SO}_{2n+1}(F)$ or the symplectic group $\mathrm{Sp}_{2n}(F)$. The goal of this paper is to give an explicit description of the Aubert-Zelevinsky duality for $G$ in terms of Langlands parameters. We present a new algorithm, inspired by the Moeglin-Waldspurger algorithm for $\mathrm{GL}_n(F)$, which computes the dual Langlands data in a recursive and combinatorial way. Our method is simple enough to be carried out by hand and provides a practical tool for explicit computations. Interestingly, the algorithm was discovered with the help of machine learning tools, guiding us toward patterns that led to its formulation.

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arXiv2026

Ramanujan Complexes from Unitary Groups over Number Fields

In this article, we construct new families of Ramanujan complexes with local structure distinct from all previously known examples. Our approach is based on unitary groups over number fields, more specifically on what we call super-definite unitary groups, that is definite unitary groups that are anisotropic modulo their center at a finite place. These arise naturally as groups of units in central division algebras with involution of the second kind. Our first main result gives a general construction of infinite families of Ramanujan complexes associated with a super-definite unitary group $G$ over a totally real number field and a finite place $v_0$. The structure of the resulting complex is governed by the type of the Bruhat-Tits building at $v_0$. It includes new examples of type $A_n$ when $v_0$ is split, and novel families of type ${}^2\!A'_n$, ${}^2 \! A''_n$ (with $n$ even), $B$-$C_n$, ${}^2 \! B$-$C_n$ and $C$-$BC_n$ in the non-split case. This construction works uniformly across all ranks. Since much of the motivation for constructing expander complexes comes from computer science, we investigate the algorithmic explicitness of our construction in the latter part of the paper, and provide an example in rank 5 where it becomes fully explicit. In particular, this example yields golden gates for the real Lie group $PU(5)$.

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arXiv2024

Local transfer for quasi-split classical groups and congruences mod l

Let G be the group of rational points of a quasi-split p-adic special orthogonal, symplectic or unitary group for some odd prime number p. FollowingArthur and Mok, there are a positive integer N, a p-adic field E and a local functorial transfer from isomorphism classes of irreducible smooth complex representations of G to those of GL(N,E). By fixing a prime number l different from p and an isomorphism between the field of complex numbers and an algebraic closure of the field of l-adic numbers, we obtain a transfer map between representations with l-adic coefficients. Now consider a cuspidal irreducible l-adic representation pi of G: we can define its reduction mod l, which is a semi-simple smooth representation of G of finite length, with coefficients in a field of characteristic l. Let pi' be a cuspidal irreducible l-adic representation of G whose reduction mod l is isomorphic to that of pi. We prove that the transfers of pi and pi' have reductions mod l which may not be isomorphic, but which have isomorphic supercuspidal supports. When G is not the split special orthogonal group SO(2), we further prove that the reductions mod l of the transfers of pi and pi' share a unique common generic component.

math.RT↗

arXiv2024

On modular rigidity for ${\rm GL}_n$

Let $k$ be a global field and $\mathbb{A}_k$ be its ring of adeles. Let $\ell$ be a prime number and fix a field isomorphism from $\mathbb{C}$ to $\overline{\mathbb{Q}}_{\ell}$. Let $Π_1$ and $Π_2$ be cuspidal automorphic representations of ${\rm GL}_n(\mathbb{A}_k)$ for some integer $n\geq1$. In this paper, we study the following question: assuming that there is a finite set $S$ of places of $k$ containing all Archimedean places and all finite places above $\ell$ such that, for all $v\notin S$, the local components $Π_{1,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $Π_{2,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ are unramified and their Satake parameters are congruent mod $\ell$, are the local components $Π_{1,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $Π_{2,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ integral, and do their reductions mod $\ell$ share an irreducible factor for all non-Archimedean places $w$ not dividing $\ell$? We show that, under certain conditions on $Π_1$ and $Π_2$, the answer is yes. We also give a simple proof when $k$ is a function field.

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arXiv2021

The local Jacquet--Langlands correspondence and congruences modulo {\ell}

Let F be a non-Archimedean local field of residual characteristic p, and {\ell} be a prime number different from p. We consider the local Jacquet-Langlands correspondence between {\ell}-adic discrete series of GL(n,F) and an inner form GL(m,D). We show that it respects the relationship of congruence modulo {\ell}. More precisely, we show that two integral {\ell}-adic discrete series of GL(m,D) are congruent modulo {\ell} if and only if the same holds for their Jacquet-Langlands transfers to GL(m,D).

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arXiv2015

On parabolic induction on inner forms of the general linear group over a non-archimedean local field

We give new criteria for the irreducibility of parabolic induction on the general linear group and its inner forms over a local non-archimedean field. In particular, we give a necessary and sufficient condition when the inducing data is of the form $π\otimesσ$ where $π$ is a ladder representation and $σ$ is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual.

math.NT↗

arXiv2015

The $\ell$-modular Zelevinski involution

Let F be a non-Archimedean locally compact field with residual characteristic p, let G be an inner form of GL(n,F) for a positive integer n and let R be an algebraically closed field of characteristic different from p. When R has characteristic $\ell>0$, the image of an irreducible smooth R-representation $π$ of G by the Aubert involution need not be irreducible. We prove that this image (in the Grothendieck group of G) contains a unique irreducible term $π$* with the same cuspidal support as $π$. This defines an involution on the set of isomorphism classes of irreducible R-representations of G, that coincides with the Zelevinski involution when R is the field of complex numbers. The method we use also works for F a finite field of characteristic p, in which case we get a similar result.

math.RT↗