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Dipendra Prasad

Publications and source records attributed to Dipendra Prasad.

At least 37 records · Page 2Linked to original sources
arXiv2018

Symplectic models for Unitary groups

In analogy with the study of representations of $GL_{2n}(F)$ distinguished by $Sp_{2n}(F)$, where $F$ is a local field, in this paper we study representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$. (Only quasi-split unitary groups are considered in this paper since they are the only ones which contain $Sp_{2n}(F)$.) We prove that there are no cuspidal representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$ for $F$ a non-archimedean local field. We also prove the corresponding global theorem that there are no cuspidal representations of $U_{2n}({\mathbb A}_k)$ with nonzero period integral on $Sp_{2n}(k) \backslash Sp_{2n}({\mathbb A}_k)$ for $k$ any number field or a function field. We completely classify representations of quasi-split unitary group in four variables over local and global fields with nontrivial symplectic periods using methods of theta correspondence. We propose a conjectural answer for the classification of all representations of a quasi-split unitary group distinguished by $Sp_{2n}(F)$.

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arXiv2018

Reducible principal series representations, and Langlands parameters for real groups

The work of Bernstein-Zelevinsky and Zelevinsky gives a good understanding of irreducible subquotients of a reducible principal series representation of $GL_n(F)$, $F$ a $p$-adic field (without specifying their multiplicities which is done by a Kazhdan-Lusztig type conjecture). In this paper we make a proposal of a similar kind for principal series representations of $GL_n({\mathbb R})$. Our investigation on principal series representations naturally led us to consider the Steinberg representation for real groups, which has curiuosly not been paid much attention to in the subject (unlike the $p$-adic case). Our proposal for Steinberg is the simplest possible: for a real reductive group $G$, the Steinberg of $G({\mathbb R})$ is a discrete series representation if and only if $G({\mathbb R})$ has a discrete series, and makes up a full $L$-packet of representations of $G({\mathbb R})$ (of size $W_G/W_K$), so is typically not irreducible.

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arXiv2018

Generalizing the MVW involution, and the contragredient

For certain quasi-split reductive groups $G$ over a general field $F$, we construct an automorphism $ι_G$ of $G$ over $F$, well-defined as an element of ${\rm Aut}(G)(F)/jG(F)$ where $j:G(F) \rightarrow {\rm Aut}(G)(F)$ is the inner-conjugation action of $G(F)$ on $G$. The automorphism $ι_G$ generalizes (although only for quasi-split groups) an involution due to Moeglin-Vigneras-Waldspurger in [MVW] for classical groups which takes any irreducible admissible representation $π$ of $G(F)$ for $G$ a classical group and $F$ a local field, to its contragredient $π^\vee$. The paper also formulates a conjecture on the contragredient of an irreducible admissible representation of $G(F)$ for $G$ a reductive algebraic group over a local field $F$ in terms of the (enhanced) Langlands parameter of the representation.

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arXiv2017

Ext-analogues of Branching laws

We consider the Ext-analogues of branching laws of representations for a group to its subgroups in the context of $p$-adic groups.

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arXiv2017

On a duality theorem of Schneider-Stuhler

We extend a duality theorem of Schneider-Stuhler about $Ext^i[π_1,π_2]$ proved there for smooth representations of a $p$-adic group $G$ with central characters to all smooth representations assuming their result for only irreducible representations by generalities in homological algebra.

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arXiv2017

Restriction of representations of metaplectic $GL_{2}(F)$ to tori

Let $F$ be a non-Archimedean local field. We study the restriction of an irreducible admissible genuine representations of the two fold metaplectic cover $\widetilde{GL}_{2}(F)$ of $GL_{2}(F)$ to the inverse image in $\widetilde{GL}_{2}(F)$ of a maximal torus in $GL_{2}(F)$.

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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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arXiv2016

Euler Poincare Characteristic for the Oscillator Representation

Suppose $(G,G')$ is a dual pair of subgroups of a metaplectic group. The dual pair correspondence is a bijection between (subsets of the) irreducible representations of $G$ and $G'$, defined by the non-vanishing of Hom$(ω,π\timesπ')$, where $ω$ is the oscillator representation. Alternatively one considers Hom$_G(ω,π)$ as a $G'$-module. It is fruitful to replace Hom with Ext$^i$, and general considerations suggest that the Euler-Poincare characteristic EP$(ω,π)$, the alternating sum of Ext$^i(ω,π)$, will be a more elementary object. We restrict to the case of $p$-adic groups, and prove that EP$(ω,π)$ is a well defined element of the Grothendieck group of finite length representations of $G'$, and show that it is indeed more elementary than Hom$(ω,π)$. We expect that computation of EP, together with vanishing results for higher Ext groups, will be a useful tool in computing the dual pair correspondence, and will help to elucidate the structure of Hom$(ω,π)$.

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arXiv2015

A `relative' local Langlands correspondence

For $E/F$ quadratic extension of local fields and $G$ a reductive algebraic group over $F$, the paper formulates a conjecture classifying irreducible admissible representations of $G(E)$ which carry a $G(F)$ invariant linear form, and the dimension of the space of these invariant forms, in terms of the Langlands parameter of the representation. The paper studies parameter spaces of Langlands parameters, and morphisms between them associated to morphisms of $L$-groups. The conjectural answer to the question on the space of $G(F)$-invariant linear forms is in terms of fibers of a particular finite map between parameter spaces.

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arXiv2015

Notes on Central Extensions

These are the notes for some lectures given by this author at Harish-Chandra Research Institute, Allahabad in March 2014 for a workshop on Schur multipliers. The lectures aimed at giving an overview of the subject with emphasis on groups of Lie type over finite, real and $p$-adic fields.

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arXiv2014

A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians

We construct examples of number fields which are not isomorphic but for which their idele class groups are isomorphic. We also construct examples of projective algebraic curves which are not isomorphic but for which their Jacobian varieties are isomorphic. Both are constructed using an example in group theory provided by Leonard Scott of a finite group $G$ and subgroups $H_1$ and $H_2$ which are not conjugate in $G$ but for which the $G$-module ${\mathbb Z}[G/H_1]$ is isomorphic to ${\mathbb Z}[G/H_2]$.

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arXiv2014

Multiplicity formula for restriction of representations of $\widetilde{\rm GL}_{2}(E)$ to $\widetilde{\rm SL}_{2}(E)$

In this note we prove a certain multiplicity formula regarding the restriction of an irreducible admissible genuine representation of a 2-fold cover $\widetilde{\rm GL}_{2}(E)$ of ${\rm GL}_{2}(E)$ to the 2-fold cover $\widetilde{\rm SL}_{2}(E)$ of ${\rm SL}_{2}(E)$, and find in particular that this multiplicity may not be one, a result that seems to have been noticed before. The proofs follow the standard path via Waldspurger's analysis of theta correspondence between $\widetilde{\rm SL}_{2}(E)$ and ${\rm PGL}_{2}(E)$.

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arXiv2014

Notes on modular representations of $p$-adic groups, and the Langlands correspondence

These are expanded notes of some lectures given by the author for a workshop held at the Indian Statistical Institute, Bangalore in June, 2010, giving an exposition on the modular representations of finite groups of Lie type and $p$-adic groups, and the modular Langlands correspondence. These notes give an overview of the subject with several examples.

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arXiv2014

Half the sum of positive roots, the Coxeter element, and a theorem of Kostant

Interchanging character and co-character groups of a torus $T$ over a field $k$ introduces a contravariant functor $T \rightarrow \widehat{T}$. Interpreting $ρ:T\rightarrow {\mathbb C}^\times$, half the sum of positive roots for $T$ a maximal torus in a simply connected semi-simple group $G$ (over ${\mathbb C}$) using this duality, we get a co-character $\widehatρ: {\mathbb C}^\times \rightarrow \widehat{T}$ whose value at $e^{\frac{2 πi}{h}}$ ($h$ the Coxeter number) is the Coxeter conjugacy class of the dual group $\widehat{G}$. This point of view gives a rather transparent proof of a theorem of Kostant on the character values of irreducible finite dimensional representations of $G$ at the Coxeter element: the proof amounting to the fact that in $\widehat{G}_{sc}$, the simply connected cover of $\widehat{G}$, there is a unique regular conjugacy class whose image in $\widehat{G}$ has order $h$ (which is the Coxeter conjugacy class).

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arXiv2014

A character relationship on $GL_n$

In this paper we consider the character of a finite dimensional algebraic representation of $GL_{mn}({\mathbb C})$ restricted to a particular disconnected component of the normalizer of the Levi subgroup $GL_m({\mathbb C})^n$ of $GL_{mn}(\mathbb C)$, generalizing a theorem of Kostant on the character values at the Coxeter element.

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arXiv2013

Dimension of zero weight space: An algebro-geometric approach

Let G be a connected, adjoint, simple algebraic group over the complex numbers with a maximal torus T and a Borel subgroup B containing T. The study of zero weight spaces in irreducible representations of G has been a topic of considerable interest; there are many works which study the zero weight space as a representation space for the Weyl group. In this paper, we study the variation on the dimension of the zero weight space as the irreducible representation varies over the set of dominant integral weights for T which are lattice points in a certain polyhedral cone. The theorem proved here asserts that the zero weight spaces have dimensions which are piecewise polynomial functions on the polyhedral cone of dominant integral weights.

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