arXiv subjects

Basudev Pattanayak

Publications and source records attributed to Basudev Pattanayak.

7 recordsLinked to original sources

arXiv2025

Quotient branching laws and unitary Gan-Gross-Prasad relevance for general linear groups

This article addresses the quotient branching problem for the pair $(\mathrm{GL}_{n+1}(F), \mathrm{GL}_n(F))$ over a non-archimedean local field $F$. We present two primary contributions. First, we explicate Chan's recent general solution by providing a practical, step-by-step combinatorial algorithm to determine whether the space $\mathrm{Hom}_{\mathrm{GL}_n(F)}(\pi, \pi')$ is non-zero for any irreducible smooth representations $\pi$ and $\pi'$. This algorithm is designed to be verifiable by hand, given the representations' Langlands or Zelevinsky parameters. Second, we specialize to the unitary case, where we establish a unifying equivalence: a pair of irreducible unitary representations satisfies Chan's generalized GGP relevance criterion if and only if it satisfies a natural extension of the original Gan--Gross--Prasad relevance condition. This result resolves a previous inconsistency in the literature, corrects and extends prior work by Gurevich, and provides a complete, explicit criterion for unitary branching laws.

math.RT↗

arXiv2025

Algorithms for parabolic inductions and Jacquet modules in $\mathrm{GL}_n$

In this article, we present algorithms for computing parabolic inductions and Jacquet modules for the general linear group $G$ over a non-Archimedean local field. Given the Zelevinsky data or Langlands data of an irreducible smooth representation $\pi$ of $G$ and an essentially square-integrable representation $\sigma$, we explicitly determine the Jacquet module of $\pi$ with respect to $\sigma$ and the socle of the normalized parabolic induction $\pi \times \sigma$. Our result builds on and extends some previous work of M\oe glin-Waldspurger, Jantzen, M\'inguez, and Lapid-M\'inguez, and also uses other methods such as sequences of derivatives and an exotic duality. As an application, we give a simple algorithm for computing the highest derivative multisegment and an algorithm for computing the Langlands parameter of the highest Bernstein-Zelevinsky derivatives.

math.RT↗

arXiv2025

Classification of $\mathrm{GL}_{n}(\mathbb{C})$-Representations Distinguished by $\mathrm{GL}_n(\mathbb{R})$

This paper provides a complete classification of $\mathrm{GL}_n(\mathbb{R})$-distinguished irreducible representations of $\mathrm{GL}_n(\mathbb{C})$ when the representations are either generic or unitary. Additionally, for each such $\mathrm{GL}_n(\mathbb{R})$-distinguished representation, we explicitly construct the associated period and prove its non-vanishing on the distinguished minimal $K$-type. Furthermore, we offer some applications to the branching problem using theta correspondence.

math.RT↗

arXiv2023

Smooth representations and Hecke algebras of $p$-adic $\mathrm{GL}_n(\mathcal{D})$

The main question we are going to address in this paper is: How much does the representation theory of the $p$-adic group $\mathrm{GL}_n(\mathcal{D})$ depend on the $p$-adic division algebra $\mathcal{D}$? Let $\mathcal{D}$ be a central division algebra defined over some locally compact non-archimedean local field. Using Bushnell-Kutzko theory of types and S\'echerre-Stevens decomposition of spherical Hecke algebras associated to types, we obtain that the cuspidal blocks in the Bernstein decomposition of the category $\mathcal{R} \left( \mathrm{GL}_n(\mathcal{D}) \right)$ of smooth complex representations of $\mathrm{GL}_n(\mathcal{D})$ do not depend on the $p$-adic division algebra $\mathcal{D}$. In particular, when $n=1$ or $2$, the category $\mathcal{R} \left( \mathrm{GL}_n(\mathcal{D}) \right)$ does not depend on the $p$-adic division algebra $\mathcal{D}$.

math.RT↗

arXiv2023

Toric periods for a $p$-adic quaternion algebra

Let $G$ be a compact group with two given subgroups $H$ and $K$. Let $\pi$ be an irreducible representation of $G$ such that its space of $H$-invariant vectors as well as the space of $K$-invariant vectors are both one dimensional. Let $v_H$ (resp. $v_K$) denote an $H$-invariant (resp. $K$-invariant) vector of unit norm in a given $G$-invariant inner product $\langle ~,~ \rangle_\pi$ on $\pi$. We are interested in calculating the correlation coefficient \[c(\pi;H,K) = |\langle v_H,v_K \rangle_\pi|^2.\] In this paper, we compute the correlation coefficient of an irreducible representation of the multiplicative group of the $p$-adic quaternion algebra with respect to any two tori. In particular, if $\pi$ is such an irreducible representation of odd minimal conductor with non-trivial invariant vectors for two tori $H$ and $K$, then its root number $\varepsilon(\pi)$ is $\pm 1$ and $c(\pi; H, K)$ is non-vanishing precisely when $\varepsilon(\pi) = 1$.

math.NT↗

arXiv2021

Principal series component of Gelfand-Graev representation

Let $G$ be a connected reductive group defined over a non-archimedean local field $F$. Let $B$ be a minimal $F$-parabolic subgroup with Levi factor $T$ and unipotent radical $U$. Let $ψ$ be a non-degenerate character of $U(F)$ and $λ$ a character of $T(F)$. Let $(K,ρ)$ be a Bushnell-Kutzko type associated to the Bernstein block of $G(F)$ determined by the pair $(T,λ)$. We study the $ρ$-isotypical component $(c\text{-ind}_{U(F)}^{G(F)}ψ)^ρ$ of the induced space $c\text{-ind}_{U(F)}^{G(F)}ψ$ of functions compactly supported mod $U(F)$. We show that $(c\text{-ind}_{U(F)}^{G(F)}ψ)^ρ$ is cyclic module for the Hecke algebra $\mathcal{H}(G,ρ)$ associated to the pair $(K,ρ)$. When $T$ is split, we describe it more explicitly in terms of $\mathcal{H}(G,ρ)$. We make assumptions on the residue characteristic of $F$ and later also on the characteristic of $F$ and the center of $G$ depending on the pair $(T,λ)$. Our results generalize the main result of Chan and Savin in \cite{CS18} who treated the case of $λ=1$ for $T$ split.

math.RT↗

arXiv2019

A note on depth preservation

We show that for a wildly ramified torus, depth is not preserved in general under local Langlands correspondence for tori.

math.RT↗