arXiv subjects

Arindam Jana

Publications and source records attributed to Arindam Jana.

5 recordsLinked to original sources

arXiv2026

On the $I(1)$-invariants: Non-abelian Hecke algebra case

Let $F$ be a finite extension of $\mathbb{Q}_p$. The so-called supersingular representations are the basic building blocks in the theory of mod $p$ representations of ${\rm GL}_2(F)$. The space of pro-$p$-Iwahori invariants of a universal module played a crucial role in the construction of the supersingular representations of ${\rm GL}_2(\mathbb{Q}_p)$. In this paper, we give an explicit description of the pro-$p$-Iwahori invariants of the universal module $π_r$ for $r = 0, q - 1$ using the Iwahori-Hecke model. We also determine the action of the pro-$p$-Iwahori-Hecke algebra on these newly found invariants. As an application, we recover $π_r$ functorially from its space of $I(1)$-invariants and extend a theorem of Ollivier for any totally ramified extension of $\mathbb{Q}_p$ other than itself.

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arXiv2026

On the depth of the adjoint representations

Let $F$ be a non-archimedean local field of odd residual characteristic $p$. The depth of a smooth representation of ${\rm GL}_n(F)$ is an invariant of Local Langlands Correspondence (LLC). The analogous notion on the Galois side of LLC is known as the slope of a local Galois representation. The slope is well related to the Swan conductor for irreducible Galois representations, whereas its behavior is subtle for reducible Galois representations. In this article, we provide an explicit formula for the slope of the adjoint of a Carayol representation of the local Galois group.

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arXiv2024

Modular representations of $\mathrm{GL}_2({\mathbb F}_q)$ using calculus

We show that certain modular induced representations of $\mathrm{GL}_2({\mathbb F}_q)$ can be written as cokernels of operators acting on symmetric power representations of $\mathrm{GL}_2({\mathbb F}_q)$. When the induction is from the Borel subgroup, respectively the anisotropic torus, the operators involve multiplication by newly defined twisted Dickson polynomials, respectively, twisted Serre operators. Our isomorphisms are explicitly defined using differential operators. As a corollary, we improve some periodicity results for quotients in the theta filtration.

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arXiv2021

Orthogonality of invariant vectors

Let $G$ be a finite group with given subgroups $H$ and $K$. Let $π$ be an irreducible complex 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 the standard $G$-invariant inner product $\langle ~,~ \rangle_π$ on $π$. Our interest is in computing the square of the absolute value of $\langle v_H,v_K \rangle_π$. This is the correlation constant $c(π;H,K)$ defined by Gross. In this paper, we give a sufficient condition for $\langle v_H, v_K \rangle_π$ to be zero and a sufficient condition for it to be non-zero (i.e., $H$ and $K$ are correlated with respect to $π$), when $G={\rm GL}_2(\mathbb F_q)$, where $\mathbb F_q$ is the finite field of $q=p^f$ elements of odd characteristic $p$, $H$ is its split torus and $K$ is a non-split torus. The key idea in our proof is to analyse the mod $p$ reduction of $π$. We give an explicit formula for $|\langle v_H,v_K \rangle_π|^2$ modulo $p$. Finally, we study the behaviour of $\langle v_H,v_K \rangle_π$ under the Shintani base change and give a sufficient condition for $\langle v_H,v_K \rangle_π$ to vanish for an irreducible representation $π={\rm BC}(τ)$ of ${\rm PGL}_2(\mathbb E)$, in terms of the epsilon factor of the base changing representation $τ$ of ${\rm PGL}_2(\mathbb F)$, where $\mathbb E/\mathbb F$ is a finite extension of finite fields. This is reminiscent of the vanishing of $L(1/2, {\rm BC}(τ))$, in the theory of automorphic forms, when the global root number of $τ$ is $-1$.

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arXiv2021

Iwahori-Hecke model for mod p representations of GL(2,F)

For a $p$-adic field $F$, the space of pro-$p$-Iwahori invariants of a universal supersingular mod $p$ representation $τ$ of ${\rm GL}_2(F)$ is determined in the works of Breuil, Schein, and Hendel. The representation $τ$ is introduced by Barthel and Livné and this is defined in terms of the spherical Hecke operator. In earlier work of Anandavardhanan-Borisagar, an Iwahori-Hecke approach was introduced to study these universal supersingular representations in which they can be characterized via the Iwahori-Hecke operators. In this paper, we construct a certain quotient $π$ of $τ$, making use of the Iwahori-Hecke operators. When $F$ is not totally ramified over $\mathbb Q_p$, the representation $π$ is a non-trivial quotient of $τ$. We determine a basis for the space of invariants of $π$ under the pro-p Iwahori subgroup. A pleasant feature of this "new" representation $π$ is that its space of pro-$p$-Iwahori invariants admits a more uniform description vis-à-vis the description of the space of pro-$p$-Iwahori invariants of $τ$.

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