arXiv subjects

Sabyasachi Dhar

Publications and source records attributed to Sabyasachi Dhar.

9 recordsLinked to original sources

arXiv2026

On Tate cohomology and base change of representations of $D^\times$

Let $F$ be a non-Archimedean local field with residue characteristic $p$, and let $D$ be a central $F$-division algebra of degree $d$. Let $E/F$ be a finite Galois extension of prime degree $\ell$, where $\ell \ne p$ and $\ell$ does not divide $d$. Set $D_E=D\otimes_F E$. Let $\mathcal{K}$ be the maximal unramified extension of $\mathbb{Q}_\ell$ in $\overline{\mathbb{Q}}_\ell$. In this article, we explicitly compute the Tate cohomology groups of absolutely irreducible, integral, depth-zero, $\mathcal{K}$-representations of $D_E^\times$ in the context of local base change lifting, which verifies a conjecture of Treumann--Venkatesh on mod-$\ell$ functoriality.

math.RT↗

arXiv2026

Asai Gamma Factors and Distinction in families

Let $F$ be a finite extension of $\mathbb{Q}_p$ and let $E$ be a quadratic extension of $F$. A representation $(π,V)$ of ${\rm GL}_n(E)$ is said to be ${\rm GL}_n(F)$-distinguished if there exists a non-zero linear functional $ϕ$ on $V$ such that $ϕ(π(h)v) = ϕ(v)$ for all $h \in {\rm GL}_n(F)$ and $v \in V$. In this article, we study the notion of ${\rm GL}_n(F)$-distinguished representations for $R[{\rm GL}_n(E)]$ modules of Whittaker type, where $R$ is a Noetherian algebra over the ring of Witt vectors of $\overline{\mathbb{F}}_\ell$ with $\ell \ne p$. We first derive a functional equation, which gives the existence of the Asai $γ$-factors associated with $R[{\rm GL}_n(E)]$ modules of Whittaker type. We then provide a necessary condition for cuspidal $R[{\rm GL}_n(E)]$ modules of Whittaker type to be Whittaker ${\rm GL}_n(F)$-distinguished, expressed in terms of their Asai $γ$-factors.

math.RT↗

arXiv2025

A note on finiteness of Tate cohomology groups

Let $G$ be a reductive algebraic group defined over a non-Archimedean local field $F$ of residue characteristic $p$. Let $σ$ be an automorphism of $G$ of order $\ell$ -- a prime number -- with $\ell\neq p$. Let $Π$ be a finite length $\overline{\mathbb{F}}_\ell$-representation of $G(F)\rtimes \langleσ\rangle$. We show that the Tate cohomology $\widehat{H}^i(\langleσ\rangle, Π)$ is a finite length representation of $G^σ(F)$. We give an application to genericity of these Tate cohomology spaces.

math.RT↗

arXiv2025

Jacquet modules of Tate cohomology and base change lifting

Let $G$ be a connected reductive group defined over a non-Archimedean local field $F$ of residue characteristic $p$. Let $\ell$ be a prime number distinct from $p$. Let $E$ be a cyclic Galois extension of $F$ with $[E:F]=\ell$. Let $Π$ be a finite length $\overline{\mathbb{F}}_\ell$-representation (or an $\ell$-modular representation) of $G(E)\rtimes {\rm Gal}(E/F)$. In this context, we prove a conjecture of Treumann and Venkatesh which predicts that the Tate cohomology groups $\widehat{H}^i({\rm Gal}(E/F), Π)$ are finite length representations of $G(F)$. We discuss the explicit computation of these Tate cohomology groups when $G$ is ${\rm GL}_n$ and $Π$ is obtained as a base change lifting of a depth-zero cuspidal representation of ${\rm GL}_n(F)$. The primary novelty from our previous work is that we treat the case where $Π$ is possibly non-cuspidal. We also study the ${\rm Gal}(\mathbb{F}_{q^\ell}/\mathbb{F}_q)$-Tate cohomology groups of the mod-$\ell$ reduction of the unipotent cuspidal representation of ${\rm Sp}_4(\mathbb{F}_{q^\ell})$.

math.RT↗

arXiv2025

Kazhdan isomorphism over families and integrality under close local fields

Let $G$ be a split connected reductive group defined over $\mathbb{Z}$. Let $F$ be a locally compact non-Archimedean field with residue characteristic $p$. For a locally compact non-Archimedean field $F'$ that is sufficiently close to $F$, D.Kazhdan establishes an isomorphism between the Hecke algebras $\mathcal{H}(G(F),K_m)$ and $\mathcal{H}(G(F'),K_m')$ with coefficients in $\mathbb{C}$, where $K_m$ (resp. $K_m'$) is the $m$-th congruence subgroup of $G(F)$ (resp. $G(F')$). This result is generalised to arbitrary connected reductive algebraic groups by R.Ganapathy. In this article, we extend the result further where the coefficient ring of the Hecke algebras is considered to be more general, namely Noetherian $\mathbb{Z}_l$-algebras with $l\ne p$. Then we use this isomorphism to prove certain compatibility result in the context of $l$-adic representation theory.

math.RT↗

arXiv2025

Families over the integral Bernstein Center and Tate cohomology of local Base change lifts for GL(n, F)

Let $p$ and $l$ be distinct odd primes, and let $F$ be a $p$-adic field. Let $π$ be a generic smooth integral representation of ${\rm GL}_n(F)$ over an $\overline{\mathbb{Q}}_l$-vector space. Let $E$ be a finite Galois extension of $F$ with $[E:F]=l$. Let $Π$ be the base change lift of $π$ to the group ${\rm GL}_n(E)$. Let $\mathbb{W}^0(Π, ψ_E)$ be the lattice of $\overline{\mathbb{Z}}_l$-valued functions in the Whittaker model of $Π$, with respect to a standard ${\rm Gal}(E/F)$-equivaraint additive character $ψ_E:E\rightarrow \overline{\mathbb{Q}}_l^\times$. We show that the unique generic sub-quotient of the zero-th Tate cohomology group of $\mathbb{W}^0(Π, ψ_E)$ is isomorphic to the Frobenius twist of the unique generic sub-quotient of the mod-$l$ reduction of $π$. We first prove a version of this result for a family of smooth generic representations of ${\rm GL}_n(E)$ over the integral Bernstein center of ${\rm GL}_n(F)$. Our methods use the theory of Rankin-selberg convolutions and simple identities of local $γ$-factors. The results of this article remove the hypothesis that $l$ does not divide the pro-order of ${\rm GL}_{n-1}(F)$ in our previous work.

math.NT↗

arXiv2024

Tate cohomology and local base change of generic representations of ${\rm GL}_3$ -- non-banal case

Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a finite Galois extension of $F$ with degree of extension $l$, where $l$ and $p$ are distinct odd primes. Let $π_F$ be an integral, $l$-adic generic representation of ${\rm GL}_3(F)$, and let $π_E$ be the base change lifting of $π_F$ to ${\rm GL}_3(E)$. Let $J_l(π_F)$ (resp. $J_l(π_E)$) be the unique generic sub-quotient of the mod-$l$ reduction of $π_F$ (resp. $π_E$). In this article, using the local converse theorem over local Artinian $\overline{\mathbb{F}}_l$-algebras, we prove that the Frobenius twist of $J_l(π_F)$ is isomorphic to the Tate cohomology group $\widehat{H}^0({\rm Gal}(E/F),J_l(π_E))$. The result of this article removes the hypothesis that the prime $l$ does not divide the pro-order of ${\rm GL}_2(F)$.

math.RT↗

arXiv2024

Compatibility of Kazhdan and Brauer homomorphism

Let $G$ be a connected split reductive group defined over $\mathbb{Z}$. Let $F$ and $F'$ be two non-Archimedean $m$-close local fields, where $m$ is a positive integer. D.Kazhdan gave an isomorphism between the Hecke algebras ${\rm Kaz}_m^F :\mathcal{H}\big(G(F),K_F\big) \rightarrow \mathcal{H}\big(G(F'),K_{F'}\big)$, where $K_F$ and $K_{F'}$ are the $m$-th usual congruence subgroups of $G(F)$ and $G(F')$ respectively. On the other hand, if $σ$ is an automorphism of $G$ of prime order $l$, then we have Brauer homomorphism ${\rm Br}:\mathcal{H}(G(F),U(F))\rightarrow \mathcal{H}(G^σ(F),U^σ(F))$, where $U(F)$ and $U^σ(F)$ are compact open subgroups of $G(F)$ and $G^σ(F)$ respectively. In this article, we study the compatibility between these two maps in the local base change setting. Further, an application of this compatibility is given in the context of linkage--which is the representation theoretic version of Brauer homomorphism.

math.RT↗

arXiv2024

Tate cohomology of Whittaker lattices and base change of generic representations of ${\rm GL}_n$

Let $p$ and $l$ be distinct odd primes and let $n\geq 2$ be a positive integer. Let $E$ be a finite Galois extension of degree $l$ of a $p$-adic field $F$. Let $q$ be the cardinality of the residue field of $F$. Let $\overlineπ_F$ be a generic mod-$l$ representation of ${\rm GL}_n(F)$ and let $π_F$ be an $l$-adic lift of $\overlineπ_F$. Let $\mathbb{W}^0(π_E, ψ_E)$ be the integral Whittaker model of $π_E$, i.e., the lattice of $\overline{\mathbb{Z}}_l$-valued functions in the Whittaker model of $π_E$. Assuming that $l$ does not divide $|{\rm GL}_{n-1}(\mathbb{F}_q)|$, we prove that the Frobenius twist of $\overlineπ_F$ is a $G_n(F)$ sub-quotient of the Tate cohomology group $\widehat{H}^0({\rm Gal}(E/F), \mathbb{W}^0(π_E, ψ_E))$.

math.NT↗