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Dipendra Prasad

Publications and source records attributed to Dipendra Prasad.

At least 19 recordsLinked to original sources
arXiv2026

Dimension statistics of representations of finite groups

The first part of this paper deals with unipotent and reductive groups over finite fields with $q$ elements in which either $q$ goes to infinity or $G=GL_n(q)$ and $n$ goes to infinity. The second part of the paper deals with the symmetric group $S_n$. The main conclusion that we want to bring out in the case of reductive groups $G(q)$, $q$ varying, is that the dimension data, resp. the size of conjugacy classes, is in a statistical sense, ``roughly'' constant and the same (up to taking the squares). We introduce the notion of {\it asympototically constant}, and {\it asympototically log constant} to make precise these notions, which we apply to various groups discussed in this paper including the symmetric groups $S_n$.

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arXiv2025

Some questions in Diophantine approximation: real and p-adics

The Weak approximation theorem describes the closure of $G(Q)$ inside $G(Q_p)$ as well as inside $G(R)$ for $G$ an algebraic group over $Q$; the closure is always an open normal subgroup with finite abelian quotient, and is well understood in a certain sense even if precise results are not always available (such as for tori!). In this paper, for a finitely generated subgroup $ L \subset G(Q)$ we consider the topological closure of $ L$ inside $G(Q_p)$ and $G(R)$. The paper is written mostly for $G$ a torus or an abelian variety, but eventually considers a variant of the question for $G$ a semisimple group. The paper is written with the wishful thinking that when dealing with questions on topological closure of algebraic points in an algebraic group defined over a number field, the simplest answers hold, a well-known principle known as ``Occum's razor''.

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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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arXiv2024

Character of Irreducible Representations Restricted to Finite Order Elements -- An Asymptotic Formula

Let $G$ be a connected reductive group over the complex numbers and let $T\subset G$ be a maximal torus. For any $t\in T$ of finite order and any irreducible representation $V(λ)$ of $G$ of highest weight $λ$, we determine the character $ch(t, V(λ))$ by using the Lefschetz Trace Formula due to Atiyah-Singer and explicitly determining the connected components and their normal bundles of the fixed point subvariety $(G/P)^t\subset G/P$ (for any parabolic subgroup $P$). This together with Wirtinger's theorem gives an asymptotic formula for $ch(t, V(nλ))$ when $n$ goes to infinity.

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arXiv2024

Homological aspects of branching laws

In this mostly expository article, we consider certain homological aspects of branching laws for representations of a group restricted to its subgroups in the context of $p$-adic groups. We follow our earlier paper, ICM 2018 proceedings, updating it with some more recent works. In particular, following Chan and Chan-Savin, see many of their papers listed in the bibliography, we have emphasized in this work that the restriction of a (generic) representation $π$ of a group $G$ to a closed subgroup $H$ (most of the paper is written in the context of GGP) turns out to be a projective representation on most Bernstein blocks of the category of smooth representations of $H$. Further, once $π|_H$ is a projective module in a particular Bernstein block, it has a simple structure.

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arXiv2023

Twisted GGP Problems and Conjectures

In an earlier work, we considered a family of restriction problems for classical groups (over local and global fields) and proposed precise answers to these problems using the local and global Langlands correspondence. These restriction problems were formulated in terms of a pair $W \subset V$ of orthogonal, Hermitian, symplectic, or skew-Hermitian spaces. In this paper, we consider a twisted variant of these conjectures in one particular case -- that of a pair of skew-Hermitian spaces $W = V$.

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arXiv2022

Relations between cusp forms sharing Hecke eigenvalues

In this paper we consider the question of when the set of Hecke eigenvalues of a cusp form on $GL_n(A_F)$ is contained in the set of Hecke eigenvalues of a cusp form on $GL_m(A_F)$ for $n \leq m$.This question is closely related to a question about finite dimensional representations of an abstract group, which also we consider in this work.

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arXiv2022

Homological duality for covering groups of reductive $p$-adic groups

In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre duality with respect to the Bernstein center and provide a proof of the folklore result that on admissible modules this functor is nothing but the contragredient duality. We single out a necessary and sufficient condition for when these three dualities agree on finite length modules in a given block. In particular, we show this is the case for all cuspidal blocks as well as, due to a result of Roche, on all blocks with trivial stabilizer in the relative Weyl group.

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arXiv2021

Cohomological representations for real reductive groups

For a connected reductive group $G$ over ${\mathbb R}$, we study cohomological $A$-parameters, which are Arthur parameters with the infinitesimal character of a finite-dimensional representation of $G({\mathbb C})$. We prove a structure theorem for such $A$-parameters, and deduce from it that a morphism of $L$-groups which takes a regular unipotent element to a regular unipotent element respects cohomological $A$-parameters. This is used to give complete understanding of cohomological $A$-parameters for all classical groups. We review the parametrization of Adams-Johnson packets of cohomological representations of $G({\mathbb R})$ by cohomological $A$-parameters and discuss various examples. We prove that the sum of the ranks of cohomology groups in a packet on any real group (and with any infinitesimal character) is independent of the packet under consideration, and can be explicitly calculated. This result has a particularly nice form when summed over all pure inner forms.

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arXiv2020

Branching laws for Classical Groups: the non-tempered case

This paper generalizes the GGP conjectures which were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the nongeneric L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of A-parameters which governs the branching laws for $GL_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in our earlier work including Bessel models and Fourier-Jacobi models.

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arXiv2020

Multiplicities for tensor products on Special linear versus Classical groups

In this paper, using computations done through the LiE software, we compare the tensor product of irreducible selfdual representations of the special linear group with those of classical groups to formulate some conjectures relating the two. In the process a few other phenomenon present themselves which we record as questions. More precisely, under the natural correspondence of irreducible finite dimensional selfdual representations of ${\rm SL}_{2n}({\mathbb C})$ with those of ${\rm Spin}_{2n+1}({\mathbb C})$, it is easy to see that if the tensor product of three irreducible representations of ${\rm Spin}_{2n+1}({\rm C})$ contains the trivial representation, then so does the tensor product of the corresponding representations of ${\rm SL}_{2n}({\rm C})$. The paper formulates a conjecture in the reverse direction. We also deal with the pair $({\rm SL}_{2n+1}({\rm C}), {\rm Sp}_{2n}({\rm C}))$.

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arXiv2020

Multiplicities under basechange: finite field case

A general proposition is proved relating multiplicities (of restriction of a representation of a group to a subgroup) under basechange, and used to calculate some multiplicities for cuspidal representations which become principal series representations under basechange for which multiplicities can be calculated by geometric methods.

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arXiv2020

A character relationship between symmetric group and hyperoctahedral group

We relate character theory of the symmetric groups $S_{2n}$ and $S_{2n+1}$ with that of the hyperoctahedral group $B_n = ({\mathbb Z}/2)^n \rtimes S_n$, as part of the expectation that the character theory of reductive groups with diagram automorphism and their Weyl groups, is related to the character theory of the fixed subgroup of the diagram automorphism.

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arXiv2019

Relating Tate-Shafarevich group of an elliptic curve with class group

The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb Z}/p)$ operates by its standard 2 dimensional representation over ${\mathbb Z}/p$. We establish such a relationship in most cases.

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arXiv2019

Multiplicity upon restriction to the derived subgroup

We present a conjecture on multiplicity of irreducible representations of a subgroup $H$ contained in the irreducible representations of a group $G$, with $G$ and $H$ having the same derived groups. We point out some consequences of the conjecture, and verification of some of the consequences. We give an explicit example of multiplicity $2$ upon restriction, as well as certain theorems in the context of classical groups where the multiplicity is $1$.

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arXiv2019

Generic representations for symmetric spaces

For a connected quasi-split reductive algebraic group $G$ over a field $k$, which is either a finite field or a non-archimedean local field, $θ$ an involutive automorphism of $G$ over $k$, let $K =G^θ$. Let $K^1=[K^0,K^0]$, the commutator subgroup of $K^0$, the connected component of identity of $K$. In this paper, we provide a simple condition on $(G,θ)$ for there to be an irreducible admissible generic representations $π$ of $G$ with ${\rm Hom}_{K^1}[π,{\mathbb C}] \not = 0$. The condition is most easily stated in terms of a real reductive group $G_θ({\mathbb R})$ associated to the pair $(G,θ)$ being quasi-split.

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arXiv2018

A mod-p Artin-Tate conjecture, and generalized Herbrand-Ribet

Following the natural instinct that when a group operates on a number field then every term in the class number formula should factorize `compatibly' according to the representation theory (both complex and modular) of the group, we are led to some natural questions about the $p$-part of the classgroup of any CM Galois extension of $\Q$ as a module for $\Gal(K/Q)$, in the spirit of Herbrand-Ribet's theorem on the $p$-component of the class number of $Q(ζ_p)$. In trying to formulate these questions, we are naturally led to consider $L(0,ρ)$, for $ρ$ an Artin representation, in situations where this is known to be nonzero and algebraic, and it is important for us to understand if this is $p$-integral for a prime $\p$ of the ring of algebraic integers $\bar{Z}$ in $C$, that we call {\it mod-$p$ Artin-Tate conjecture}. The most minor term in the class number formula, the number of roots of unity, plays an important role for us --- it being the only term in the denominator, is responsible for all poles!

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