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Santosh Nadimpalli

Publications and source records attributed to Santosh Nadimpalli.

17 recordsLinked to original sources
arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

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arXiv2026

Uniqueness of Branching through regular unipotent elements

Let \(\mathrm G\) be a complex simple algebraic group and let \(\mathrm G_0\subset \mathrm G\) be a closed connected subgroup containing a regular unipotent element of \(\mathrm G\), with semisimple rank at least \(2\). Using Dynkin's classification, we prove that the restriction of an irreducible finite-dimensional representation of \(\mathrm G\) to \(\mathrm G_0\) determines the representation up to an outer automorphism of \(\mathrm G\) preserving \(\mathrm G_0\). We extend this method to the diagonal embedding $\mathrm G_0\hookrightarrow \mathrm G\times \mathrm G$ for the specific pairs $(\mathrm{SO}_{2k}(\mathbb C) \times\mathrm{SO}_{2k}(\mathbb C),\,\mathrm{SO}_{2k-1}(\mathbb C))$, $(E_6\times E_6,\,F_4)$ and $(Spin_8(\mathbb C) \times Spin_8(\mathbb C), G_2)$ and show that uniqueness continues to hold. Finally, we give examples showing that, in the diagonal setting, restriction to the principal \(\mathrm{SL}_2(\mathbb C)\) alone is not sufficient to establish uniqueness.

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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.

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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 ${\rm Sp}_4(\mathbb{F}_{q^\ell})$.

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arXiv2025

Character theory at a torsion element

The paper relates character value of an irreducible representation of a compact connected Lie group at certain elements of finite order with the dimension of a representation on another group, up to some precise constants, which all have significance. An important input is to analyse torsion elements of order d in an adjoint group with minimal dimensional centraliser, and to prove that in most cases when d divides the Coxeter number of G, this gives rise to a unique conjugacy class.

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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.

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arXiv2024

Twisted Jacquet modules: a conjecture of D. Prasad

In this note, we study the twisted Jacquet modules of sub-quotients of principal series representations of ${\rm GL}_2(D)$ where $D$ is a division algebra over a non-archimedean local field $F$. We begin with a proof of a conjecture due to D. Prasad on twisted Jacquet modules of Speh representations of ${\rm GL}_2(D)$ when $D$ is the quaternionic division algebra. Later, when $D$ is an arbitrary division algebra over $F$, we focus on depth-zero principal series and compute the dimensions of twisted Jacquet modules of generalised Speh representations and investigate their structure explicitly.

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arXiv2024

On the integrality of locally algebraic representations of $\mathrm{GL}_{2}(D)$

Emerton's theory of Jacquet modules for locally analytic representations provides necessary conditions for the existence of integral structures in locally analytic representations. These conditions are also expected to be sufficient for the integrality of generic irreducible locally algebraic representations. In this article, we prove the sufficiency of Emerton's conditions for some tamely ramified locally algebraic representations of $\mathrm{GL}_{2}(D)$ where $D$ is a $p$-adic division algebra.

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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 $\widehat{H}^0({\rm Gal}(E/F), \mathbb{W}^0(π_E, ψ_E))$.

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arXiv2022

Quotients of commuting schemes associated to Symmetric Pairs

Let $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ be a $\mathbb Z_2$-grading of a classical Lie algebra such that $(\mathfrak{g}, \mathfrak{g}_0)$ is a classical symmetric pair. Let $G$ be a classical group with Lie algebra $\mathfrak{g}$ and let $G_0$ be the connected subgroup of $G$ with ${\rm Lie} (G_0)=\mathfrak g_0$. For $d \geq 2$, let $\mathfrak{C}^d(\mathfrak{g}_1)$ be the $d$-th commuting scheme associated with the symmetric pair $(\mathfrak g, \mathfrak g_0)$. In this article, we study the categorical quotient $\mathfrak{C}^d(\mathfrak{g}_1)//{G_0}$ via the Chevalley restriction map. As a consequence we show that the categorical quotient scheme $\mathfrak C^d(\mathfrak g_1)//G_0$ is normal and reduced. As a part of the proof, we describe a generating set for the algebra $k[\mathfrak{g}_1^d]^{G_0}$, which are of independent interest.

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arXiv2022

A note on branching of $V(ρ)$

Let $\mathfrak{g}$ be a complex simple Lie algebra and let $\mathfrak{g}_0$ be the sub-algebra fixed by a diagram automorphism of $\mathfrak{g}$. Let $G$ be the complex, simply-connected, simple algebraic group with Lie algebra $\mathfrak{g}$, and let $G_0$ be the connected subgroup of $G$ with Lie algebra $\mathfrak{g}_0$. Let $ρ$ be the half sum of positive roots of $\mathfrak{g}$. In this article, we give a necessary and sufficient condition for a highest weight $\mathfrak{g}_0$-representation $V_0(dμ)$ to occur in the representation ${\rm res}_{\mathfrak{g}_0}V(dρ)$, for any saturation factor $d$ of the pair $(G_0, G)$.

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arXiv2019

On typical representations for depth-zero components of split classical groups

Let ${\bf G}$ be a split classical group over a non-Archimedean local field $F$ with the cardinality of the residue field $q_F>5$. Let $M$ be the group of $F$-points of a Levi factor of a proper $F$-parabolic subgroup of ${\bf G}$. Let $[M, σ_M]_M$ be an inertial class such that $σ_M$ contains a depth-zero Moy--Prasad type of the form $(K_M, τ_M)$, where $K_M$ is a hyperspecial maximal compact subgroup of $M$. Let $K$ be a hyperspecial maximal compact subgroup of ${\bf G}(F)$ such that $K$ contains $K_M$. In this article, we classify $\mathfrak{s}$-typical representations of $K$. In particular, we show that the $\mathfrak{s}$-typical representations of $K$ are precisely the irreducible subrepresentations of $\ind_J^Kλ$, where $(J, λ)$ is a level-zero $G$-cover of $(K\cap M, τ_M)$.

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arXiv2019

Generic cuspidal representations of $U(2,1)$

Let $F$ be any non-Archimedean local field with a Galois involution $σ$ and $F_0$ be the fixed field for the action of $σ$. When the residue characteristic of $F_0$ is odd, using the explicit construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of $U(2,1)(F/F_0)$.

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arXiv2019

Typical representations for level zero Bernstein components of ${\rm GL}_n(F)$

Let $F$ be a non-discrete non-Archimedean locally compact field. In this article for a level zero Bernstein component $s$, we classify those irreducible smooth representations of ${\rm GL}_n{\integers{F}}$ (called typical representations) whose appearance in a smooth irreducible representation $π$ of ${\rm GL}_n{F}$ implies that the cuspidal support of $π$ is $s$. These results extend, for level zero representations, the results of Henniart and Paškūnas on cuspidal representations. The results are independent of the characteristic of the base field.

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arXiv2019

On classification of typical representations for ${\rm GL}_3(F)$

Let $F$ be any non-Archimedean local field with residue field of cardinality $q_F$. In this article, we obtain a classification of typical representations for the Bernstein components associated to the inertial classes of the form $[{\rm GL}_n(F)\times F^\times, σ\otimesχ]$ with $q_F>2$, and for the principal series components with $q_F>3$. With this we complete the classification of typical representations for ${\rm GL}_3(F)$, for $q_F>2$.

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arXiv2017

On extensions of characters of affine pro-$p$ Iwahori--Hecke algebra

Let $K$ be a non-discrete non-Archimedean local field with residue characteristic $p$. Let $G$ be the group of $K$ rational points of a algebraic connected reductive group defined over $K$. In this article we compute the extensions between characters of affine pro-$p$ Iwahori--Hecke algebra $\mathcal{H}^{\text{aff}}$ over an algebraically closed field $R$ of characteristic $p$. In rank one case we deduce the relation between the blocks and $L$-packets.

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