arXiv subjects

U. K. Anandavardhanan

Publications and source records attributed to U. K. Anandavardhanan.

16 recordsLinked to original sources

arXiv2025

Simultaneous nonvanishing of the correlation constant

For $q=p^m$, where $p$ is an odd prime number, we study the correlation coefficient $c(π;H,K)$ of an irreducible (complex) representation $π$ of $G={\rm GL}_2(\mathbb F_q)$ with respect to a split torus $H$ and a non-split torus $K$. We consider a family of non-split tori $K_{α,u}$ indexed by $u \in \mathbb F_q$ and $α\in \mathbb F_q^\times \setminus \mathbb F_q^{\times 2}$. We show that under any identification of $\mathbb C$ with $\overline{\mathbb Q}_p$, and writing $π= π_r$ where $0 \leq r \leq (q-1)/2$ depending on this identification, we have \[c(π_r;H,K_{α,u}) \equiv [P_r(u/\sqrtα)]^2 \mod p, \] where $P_r(X) \in \mathbb Z[\frac{1}{2}][X]$ is the $r$-th Legendre polynomial. As a corollary, when $m \geq 2$, we prove that there exists $u \in \mathbb F_q^\times$ such that $c(π;H,K_{α,u}) \neq 0$ for all irreducible representations $π$ of $G$ admitting fixed vectors for both $H$ and $K$.

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arXiv2024

The sign of linear periods

Let $G$ be a group with subgroup $H$, and let $(π,V)$ be a complex representation of $G$. The natural action of the normalizer $N$ of $H$ in $G$ on the space $\mathrm{Hom}_H(π,\mathbb{C})$ of $H$-invariant linear forms on $V$, provides a representation $χ_π$ of $N$ trivial on $H$, which is a character when $\mathrm{Hom}_H(π,\mathbb{C})$ is one dimensional. If moreover $G$ is a reductive group over a local field, and $π$ is smooth irreducible, it is an interesting problem to express $χ_π$ in terms of the possibly conjectural Langlands parameter $ϕ_π$ of $π$. In this paper we consider the following situation: $G=\mathrm{GL}_m(D)$ for $D$ a central division algebra of dimension $d^2$ over a local field $F$ of characteristic zero, $H$ is the centralizer of a non central element $δ\in G$ such that $δ^2$ is in the center of $G$, and $π$ has generic Jacquet-Langlands transfer to $\mathrm{GL}_{md}(F)$. In this setting the space $\mathrm{Hom}_H(π,\mathbb{C})$ is at most one dimensional. When $\mathrm{Hom}_H(π,\mathbb{C})\simeq \mathbb{C}$ and $H\neq N$, we prove that the value of the $χ_π$ on the non trivial class of $\frac{N}{H}$ is $(-1)^mε(ϕ_π)$ where $ε(ϕ_π)$ is the root number of $ϕ_π$. Along the way we extend many useful multiplicity one results for linear and Shalika models to the case of non split $G$. When $F$ is $p$-adic we also classify standard modules with linear periods and Shalika models, which are new results even when $D=F$.

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arXiv2023

Toric periods for a $p$-adic quaternion algebra

Let $G$ be a compact group with two given subgroups $H$ and $K$. Let $π$ 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 $\langle ~,~ \rangle_π$ on $π$. We are interested in calculating the correlation coefficient \[c(π;H,K) = |\langle v_H,v_K \rangle_π|^2.\] 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 $π$ is such an irreducible representation of odd minimal conductor with non-trivial invariant vectors for two tori $H$ and $K$, then its root number $\varepsilon(π)$ is $\pm 1$ and $c(π; H, K)$ is non-vanishing precisely when $\varepsilon(π) = 1$.

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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

Distinction and Base Change

An irreducible smooth representation of a $p$-adic group $G$ is said to be distinguished with respect to a subgroup $H$ if it admits a non-trivial $H$-invariant linear form. When $H$ is the fixed group of an involution on $G$ it is suggested by the works of Hervé Jacquet from the nineties that distinction can be characterized in terms of the principle of functoriality. If the involution is the Galois involution then a recent conjecture of Dipendra Prasad predicts a formula for the dimension of the space of invariant linear forms which once again involves base change. We will describe the proof of this conjecture (in the generic case) for $SL(n)$ which is joint work with Dipendra Prasad. Then we describe one more newly discovered connection between distinction and base change which is that base change information appears in the constant of proportionality between two natural invariant linear forms on a distinguished representation. This latter result is for discrete series for $GL(n)$ and is joint with Nadir Matringe. This paper is a report on the author's talk in the International Colloquium on Arithmetic Geometry held in January 2020 at TIFR Mumbai.

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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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arXiv2020

Some questions on global distinction for $\mathrm{SL}(n)$

Let $E/F$ be a quadratic extension of number fields and let $π$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished cuspidal automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Using an unfolding argument, we prove that an element of the $\mathrm{L}$-packet of $π$ is distinguished if and only if it is $ψ$-generic for a non-degenerate character $ψ$ of $N_n(\mathbb{A}_E)$ trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $\mathrm{SL}_n$. We then use this result to analyze the non-vanishing of the period integral on different realizations of a distinguished cuspidal automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$ with multiplicity $> 1$, and show that in general some canonical copies of a distinguished representation inside different $\mathrm{L}$-packets can have vanishing period. We also construct examples of everywhere locally distinguished representations of $\mathrm{SL}_n(\mathbb{A}_E)$ the $\mathrm{L}$-packets of which do not contain any distinguished representation.

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arXiv2020

Distinction inside L-packets of SL(n)

If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $\mathrm{SL}_n(F)$-distinguished representations inside a distinguished unitary L-packet of $\mathrm{SL}_n(E)$ are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of $N(E)/N(F)$. Then we establish a global analogue of this result. For this, let $E/F$ be a quadratic extension of number fields and let $π$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished square integrable automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Let $(σ,d)$ be the unique pair associated to $π$, where $σ$ is a cuspidal representation of $\mathrm{GL}_r(\mathbb{A}_E)$ with $n=dr$. Using an unfolding argument, we prove that an element of the L-packet of $π$ is distinguished with respect to $\mathrm{SL}_n(\mathbb{A}_F)$ if and only if it has a degenerate Whittaker model for a degenerate character $ψ$ of type $r^d:=(r,\dots,r)$ of $N_n(\mathbb{A}_E)$ which is trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $\mathrm{SL}_n$. As a first application, under the assumptions that $E/F$ splits at infinity and $r$ is odd, we establish a local-global principle for $\mathrm{SL}_n(\mathbb{A}_F)$-distinction inside the L-packet of $π$. As a second application we construct examples of distinguished cuspidal automorphic representations $π$ of $\mathrm{SL}_n(\mathbb{A}_E)$ such that the period integral vanishes on some canonical copy of $π$, and of everywhere locally distinguished representations of $\mathrm{SL}_n(\mathbb{A}_E)$ such that their L-packets do not contain any distinguished representation.

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arXiv2019

Test vectors for finite periods and base change

Let $E/F$ be a quadratic extension of finite fields. By a result of Gow, an irreducible representation $π$ of $G = {\rm GL}_n(E)$ has at most one non-zero $H$-invariant vector, up to multiplication by scalars, when $H$ is ${\rm GL}_n(F)$ or ${\rm U}(n,E/F)$. If $π$ does have an $H$-invariant vector it is said to be $H$-distinguished. It is known, from the work of Gow, that $H$-distinction is characterized by base change from ${\rm U}(n,E/F)$, due to Kawanaka, when $H$ is ${\rm GL}_n(F)$ (resp. from ${\rm GL}_n(F)$, due to Shintani, when $H$ is ${\rm U}(n,E/F)$). Assuming $π$ is generic and $H$-distinguished, we give an explicit description of the $H$-invariant vector in terms of the Bessel function of $π$. Let $ψ$ be a non-degenerate character of $N_G/N_H$ and let $B_{π,ψ}$ be the (normalized) Bessel function of $π$ on the $ψ$-Whittaker model. For the $H$-average \[W_{π,ψ} = \frac{1}{|H|} \sum_{h\in H} π(h) B_{π,ψ}\] of the Bessel function, we prove that \[W_{π,ψ}(I_n) = \frac{{\rm dim}ρ}{{\rm dim} π} \cdot \frac{|{\rm GL}_n(E)|}{|{\rm GL}_n(F)| |{\rm U}(n,E/F)|},\] where $ρ$ is the representation of ${\rm U}(n,E/F)$ (resp. ${\rm GL}_n(F)$) that base changes to $π$ when $H$ is ${\rm GL}_n(F)$ (resp. ${\rm U}(n,E/F)$). As an application we classify the members of a generic $L$-packet of ${\rm SL}_n(E)$ that admit invariant vectors for ${\rm SL}_n(F)$. Finally we prove a $p$-adic analogue of our result for square-integrable representations in terms of formal degrees by employing the formal degree conjecture of Hiraga-Ichino-Ikeda \cite{hii08}.

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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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arXiv2016

Test vectors for local periods

Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic zero. An irreducible admissible representation $π$ of $GL(n,E)$ is said to be distinguished with respect to $GL(n,F)$ if it admits a non-trivial linear form that is invariant under the action of $GL(n,F)$. It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the $F$-points of the mirabolic subgroup when $π$ is unitary and generic. In this paper, we prove that the essential vector of [JPSS81] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local $L$-value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.

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arXiv2016

Distinguished representations for SL(n)

For $E/F$ a quadratic extension of local fields, and $π$ an irreducible admissible generic representation of $SL_n(E)$, we calculate the dimension of $Hom_{SL_n(F)}[π,C]$ and relate it to fibers of the base change map corresponding to base change of representations of $SU_n(F)$ to $SL_n(E)$ as suggested in a recent work of the second author. We also deal with finite fields.

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arXiv2011

A local global question in automorphic forms

In this paper, we consider the $\SL(2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\GL(2)$: one due to Harder-Langlands-Rapoport, and the other due to Waldspurger.

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arXiv2005

On the SL(2) period integral

Let E/F be a quadratic extension of number fields. For a cuspidal representation $π$ of SL(2,A_E), we study the non-vanishing of the period integral on SL(2,F)\SL(2,A_F). We characterise the non-vanishing of the period integral of $π$ in terms of $π$ being generic with respect to characters of E\A_E which are trivial on A_F. We show that the period integral in general is not a product of local invariant functionals, and find a necessary and sufficient condition when it is. We exhibit cuspidal representations of SL(2,A_E) whose period integral vanishes identically while each local constituent admits an SL(2)-invariant linear functional. Finally, we construct an automorphic representation $π$ on SL(2,A_E) which is abstractly SL(2,A_F) distinguished but none of the elements in the global L-packet determined by $π$ is distinguished by SL(2,A_F).

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arXiv2004

Distinguished non-Archimedean representations

For a symmetric space (G,H), one is interested in understanding the vector space of H-invariant linear forms on a representation πof G. In particular an important question is whether or not the dimension of this space is bounded by one. We cover the known results for the pair (G=R_{E/F}GL(n),H=GL(n)), and then discuss the corresponding SL(n) case. In this paper, we show that (G=R_{E/F}SL(n),H=SL(n)) is a Gelfand pair when n is odd. When $n$ is even, the space of H-invariant forms on πcan have dimension more than one even when πis supercuspidal. The latter work is joint with Dipendra Prasad.

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arXiv2004

Distinguished representations, base change, and reducibility for unitary groups

We show the equality of the local Asai L-functions defined via the Rankin-Selberg method and the Langlands-Shahidi method for a square integrable representation of GL(n,E). As a consequence we characterise reducibility of certain induced representations of U(n,n), and the image of the base change map from U(n) to GL(n,E) in terms of GL(n,F)-distinguishedness.

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