arXiv subjects

Robert Kurinczuk

Publications and source records attributed to Robert Kurinczuk.

At least 19 recordsLinked to original sources

arXiv2026

Block decompositions for $p$-adic classical groups and their inner forms

For an inner form $\mathrm{G}$ of a general linear group or classical group over a non-archimedean local field of odd residue characteristic, we decompose the category of smooth representations on $\mathbb{Z}[μ_{p^{\infty}},1/p]$-modules by endo-parameter. We prove that parabolic induction preserves these decompositions, and hence that it preserves endo-parameters. Moreover, we show that the decomposition by endo-parameter is the $\overline{\mathbb{Z}}[1/p]$-block decomposition; and, for $\mathrm{R}$ an integral domain, introduce a graph whose connected components parameterize the $\mathrm{R}$-blocks, in particular including the cases $\mathrm{R}=\overline{\mathbb{Z}}_{\ell}$ and $\mathrm{R}=\overline{\mathbb{F}}_\ell$ for $\ell\neq p$. From our description, we deduce that the $\overline{\mathbb{Z}_\ell}$-blocks and $\overline{\mathbb{F}_\ell}$-blocks of $\mathrm{G}$ are in natural bijection, as had long been expected. Our methods also apply to the trivial endo-parameter (i.e., the depth zero subcategory) of any connected reductive $p$-adic group, providing an alternative approach to results of Dat and Lanard in depth zero. Finally, under a technical assumption (known for inner forms of general linear groups) we reduce the $\mathrm{R}$-block decomposition of $\mathrm{G}$ to depth zero.

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arXiv2026

Godement-Jacquet gamma factors of distinguished representations of $\mathrm{GL}_n(\mathbb{F}_q)$

Let $k$ be a finite field of characteristic $p$. In the 1960s, Kondo attached non-abelian Gauss sums to irreducible $\mathbb{C}$-representations of $\mathrm{GL}_n(k)$, and computed them in terms of Green parameters. On the other hand, the Godement-Jacquet functional equation in which they occur was established by Macdonald in the 1980s. We first revisit Macdonald's and Kondo's results with a different perspective, in the process of generalizing their constructions to representations with coefficients in $\mathbb{Z}[\sqrt{p}^{-1},μ_p]$-algebras. Then, when $p$ is odd and $R$ is an algebraically closed field of characteristic different to $p$, our main result shows that the Godement-Jacquet gamma factor of a cuspidal irreducible $R$-representation, which is distinguished with respect to the subgroup fixed by a Galois or an inner involution, coincides with the sign of the associated period under the normalizer of this subgroup. Finally, we compute the gamma factors of these distinguished representations in terms of Green's and James' parametrizations of irreducible cuspidal $R$-representations.

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arXiv2026

Cuspidal endo-support and strong beta extensions

Let $G$ be an inner form of a general linear group or classical group over a non-archimedean local field of residual characteristic $p$, assumed odd in the classical case. We prove that every smooth representation of $G$ over an algebraically closed field $R$ of characteristic $\ell\neq p$ contains a maximal semisimple character, i.e., one for which the point in the building of the corresponding centralizer is a vertex. Further, for every endo-parameter adapted to $G$, we define its support, which leads also to the notion of cuspidal endo-support of an irreducible representation, and we relate this to its cuspidal support. We also introduce beta extensions for strong facets in the building of a centralizer, and show these are sufficient for the construction of types. These results are used in a subsequent paper to decompose the category of smooth $R$-representations of $G$.

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arXiv2026

Cuspidal $\ell$-modular representations of ${\rm GL}_n(F)$ distinguished by a Galois involution, II

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields with residual characteristic $p\neq2$, and $\ell$ be a prime number different from $p$. We classify those $\ell$-modular cuspidal irreducible representations of ${\rm GL}_n(F)$ which are ${\rm GL}_n(F_0)$-distinguished, that is, which carry a non-zero ${\rm GL}_n(F_0)$-invariant linear form. In the case when $\ell\neq2$, an $\ell$-modular cuspidal representation of ${\rm GL}_n(F)$ is ${\rm GL}_n(F_0)$-distinguished if and only if it lifts to a ${\rm GL}_n(F_0)$-distinguished cuspidal $\ell$-adic representation, whereas when $\ell=2$, it is ${\rm GL}_n(F_0)$-distinguished if and only if it is conjugate-self-dual.

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arXiv2025

Newforms in Cuspidal Representations

We consider newform vectors in cuspidal representations of $p$-adic general linear groups. We extend the theory from the complex setting to include~$\ell$-modular representations with~$\ell\neq p$, and prove that the conductor is compatible with congruences modulo~$\ell$ for (ramified) supercuspidal~$\ell$-modular representations and for depth zero cuspidals. In the complex and modular setting, we prove explicit formulae for depth zero and minimax cuspidal representations of integral depth, in Bushnell-Kutzko and Whittaker models.

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arXiv2024

Local Langlands in families: The banal case

We state a conjecture, local Langlands in families, connecting the centre of the category of smooth representations on $\mathbb{Z}[\sqrt{q}^{-1}]$-modules of a quasi-split $p$-adic group $\mathrm{G}$ (where $q$ is the cardinality of the residue field of the underlying local field), the ring of global functions on the stack of Langlands parameters for $\mathrm{G}$ over $\mathbb{Z}[\sqrt{q}^{-1}]$, and the endomorphisms of a Gelfand-Graev representation for $\mathrm{G}$. For a class of classical $p$-adic groups (symplectic, unitary, or split odd special orthogonal groups), we prove this conjecture after inverting an integer depending only on $\mathrm{G}$. Along the way, we show that the local Langlands correspondence for classical $p$-adic groups (1) preserves integrality of $\ell$-adic representations; (2) satisfies an "extended" (generic) packet conjecture; (3) is compatible with parabolic induction up to semisimplification (generalizing a result of Moussaoui), hence induces a semisimple local Langlands correspondence; and (4) the semisimple correspondence is compatible with automorphisms of $\mathbb{C}$ fixing $\sqrt{q}$.

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arXiv2024

Moduli of Langlands Parameters

Let $F$ be a nonarchimedean local field of residue characteristic $p$, let $\hat{G}$ be a split reductive group over $\mathbb{Z}[1/p]$ with an action of $W_F$, and let $^LG$ denote the semidirect product $\hat{G}\rtimes W_F$. We construct a moduli space of Langlands parameters $W_F \to {^LG}$, and show that it is locally of finite type and flat over $\mathbb{Z}[1/p]$, and that it is a reduced local complete intersection. We give parameterizations of the connected components and the irreducible components of the geometric fibers of this space, and parameterizations of the connected components of the total space over $\overline{\mathbb{Z}}[1/p]$ (under mild hypotheses) and over $\overline{\mathbb{Z}}_{\ell}$ for $\ell\neq p$. In each case, we show precisely how each connected component identifies with the "principal" connected component attached to a smaller split reductive group scheme. Finally we study the GIT quotient of this space by $\hat{G}$ and give a complete description of its fibers up to homeomorphism, and a complete description of its ring of functions after inverting an explicit finite set of primes depending only on $^LG$.

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arXiv2023

Cuspidal $\ell$-modular representations of $\mathrm{GL}_n(F)$ distinguished by a Galois involution

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$ with Galois automorphism $σ$, and let $R$ be an algebraically closed field of characteristic $\ell\notin\{0,p\}$. We reduce the classification of $\mathrm{GL}_n(F_0)$-distinguished cuspidal $R$-representations of $\mathrm{GL}_n(F)$ to the level $0$ setting. Moreover, under a parity condition, we give necessary conditions for a $σ$-selfdual cuspidal $R$-representation to be distinguished. Finally, we classify the distinguished cuspidal $\overline{\mathbb{F}}_{\ell}$-representations of $\mathrm{GL}_n(F)$ having a distinguished cuspidal lift to $\overline{\mathbb{Q}}_\ell$.

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arXiv2022

Finiteness for Hecke algebras of $p$-adic groups

Let $G$ be a reductive group over a non-archimedean local field $F$ of residue characteristic $p$. We prove that the Hecke algebras of $G(F)$ with coefficients in a ${\mathbb Z}_{\ell}$-algebra $R$ for $\ell$ not equal to $p$ are finitely generated modules over their centers, and that these centers are finitely generated $R$-algebras. Following Bernstein's original strategy, we then deduce that "second adjointness" holds for smooth representations of $G(F)$ with coefficients in any ring $R$ in which $p$ is invertible. These results had been conjectured for a long time. The crucial new tool that unlocks the problem is the Fargues-Scholze morphism between a certain "excursion algebra" defined on the Langlands parameters side and the Bernstein center of $G(F)$. Using this bridge, our main results are representation theoretic counterparts of the finiteness of certain morphisms between coarse moduli spaces of local Langlands parameters that we also prove here, which may be of independent interest

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arXiv2020

Endo-parameters for p-adic classical groups

For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal C-representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart's notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.

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arXiv2019

A characterization of the relation between two $\ell$-modular correspondences

Let $F$ be a non archimedean local field of residual characteristic $p$ and $\ell$ a prime number different from $p$. Let $\mathrm{V}$ denote Vignéras' $\ell$-modular local Langlands correspondence between irreducible $\ell$-modular representations of $\mathrm{GL}_n(F)$ and $n$-dimensional $\ell$-modular Deligne representations of the Weil group $\mathrm{W}_F$. In a previous work, enlarging the space of parameters to Deligne representations with non necessarily nilpotent operators, we proposed a modification of the correspondence of Vignéras into a correspondence $\mathrm{C}$ compatible with the formation of local constants in the generic case. In this note, following a remark of Alberto Mínguez, we characterize the modification $\mathrm{C}\circ \mathrm{V}^{-1}$ by a short list of natural properties.

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arXiv2019

Galois self-dual cuspidal types and Asai local factors

Let $F/F_{\mathsf{o}}$ be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and $σ$ be its non-trivial automorphism. We show that any $σ$-self-dual cuspidal representation of ${\rm GL}_n(F)$ contains a $σ$-self-dual Bushnell--Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai $L$-function of a ${\rm GL}_n(F_{\mathsf{o}})$-distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands--Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation.

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arXiv2019

Characterisation of the poles of the $\ell$-modular Asai $L$-factor

Let $E/F$ be a quadratic extension of non-archimedean local fields, and let $\ell$ be a prime number different from the residual characteristic of $F$. For a complex cuspidal representation $π$ of $GL(n,E)$, the Asai $L$-factor $L^+(X,π)$ has a pole at $X=1$ if and only if $π$ is $GL(n,F)$-distinguished. In this paper we solve the problem of characterising the occurrence of a pole at $X=1$ of $L^+(X,π)$ when $π$ is an $\ell$-modular cuspidal representation of $GL(n,E)$: we show that $L^+(X,π)$ has a pole at $X=1$ if and only if $π$ is a relatively banal distinguished representation; namely $π$ is $GL(n,F)$-distinguished but not $\vert\det(~ )|_{F}$-distinguished. This notion turns out to be an exact analogue for the symmetric space $GL(n,E)/GL(n,F)$ of M\' inguez and Sécherre's notion of banal cuspidal $\overline{\mathbb{F}}_\ell$-representation of $GL(n,F)$. Along the way we compute the Asai $L$-factor of all cuspidal $\ell$-modular representations of $GL(n,E)$ in terms of type theory, and prove new results concerning lifting and reduction modulo $\ell$ of distinguished cuspidal representations. Finally, we determine when the natural $GL(n,F)$-period on the Whittaker model of a distinguished cuspidal representation of $GL(n,E)$ is nonzero.

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arXiv2018

Extension of Whittaker functions and test vectors

We show that certain products of Whittaker functions and Schwartz functions on a general linear group extend to Whittaker functions on a larger general linear group. This generalizes results of Cogdell--Piatetski-Shapiro \cite{CPS} and Jacquet--Piatetski-Shapiro--Shalika \cite{JPSS83}. As a consequence, we prove that the Rankin--Selberg $L$-factor of the product of a discrete series representation and the Zelevinsky dual of a discrete series representation is given by a single Rankin--Selberg integral.

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arXiv2018

The $\ell$-modular local Langlands correspondence and local factors

Let $F$ be a non-archimedean local field of residual characteristic $p$, $\ell\neq p$ be a prime number, and $\mathrm{W}_F$ the Weil group of $F$. We classify the indecomposable $\mathrm{W}_F$-semisimple Deligne $\overline{\mathbb{F}_\ell}$-representations in terms of the irreducible $\overline{\mathbb{F}_\ell}$-representations of $\mathrm{W}_F$, and extend constructions of Artin-Deligne local factors to this setting. Finally, we define a variant of the $\ell$-modular local Langlands correspondence which satisfies a preservation of local factors statement for generic representations.

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arXiv2017

Test vectors for local cuspidal Rankin-Selberg integrals of GL(n), and reduction modulo $\ell$

Let $π_1,π_2$ be a pair of cuspidal complex, or $\ell$-adic, representations of the general linear group of rank $n$ over a non-archimedean local field $F$ of residual characteristic $p$, different to $\ell$. Whenever the local Rankin-Selberg $L$-factor $L(X,π_1,π_2)$ is nontrivial, we exhibit explicit test vectors in the Whittaker models of $π_1$ and $π_2$ such that the local Rankin-Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_F^n$ is equal to $L(X,π_1,π_2)$. We give an initial application of the test vectors to reduction modulo $\ell$ of $\ell$-adic $L$-factors.

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arXiv2015

Cuspidal $\ell$-modular representations of $p$-adic classical groups

For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial tool for extending the results here to arbitrary irreducible representations.

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arXiv2015

Rankin-Selberg local factors modulo $\ell$

After extending the theory of Rankin-Selberg local factors to pairs of $\ell$-modular representations of Whittaker type, of general linear groups over a non-archimedean local field, we study the reduction modulo $\ell$ of $\ell$-adic local factors and their relation to these $\ell$-modular local factors. While the $\ell$-modular local $γ$-factor we associate to such a pair turns out to always coincide with the reduction modulo $\ell$ of the $\ell$-adic $γ$-factor of any Whittaker lifts of this pair, the local $L$-factor exhibits a more interesting behaviour; always dividing the reduction modulo-$\ell$ of the $\ell$-adic $L$-factor of any Whittaker lifts, but with the possibility of a strict division occurring. In our main results, we completely describe $\ell$-modular $L$-factors in the generic case. We obtain two simple to state nice formulae: Let $π,π'$ be generic $\ell$-modular representations; then, writing $π_b,π'_b$ for their banal parts, we have \[L(X,π,π')=L(X,π_b,π_b').\] Using this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that \[L(X,π,π')=\mathbf{GCD}(r_{\ell}(L(X,τ,τ'))),\] where the divisor is over all integral generic $\ell$-adic representations $τ$ and $τ'$ which contain $π$ and $π'$, respectively, as subquotients after reduction modulo $\ell$.

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