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We extend Prasad’s results on the existence of trilinear forms on representations of $G L_{2}$ of a local field, by permitting one or more of the representations to be reducible principal series, with infinite-dimensional irreducible quotient. We apply this in a global setting to compute (unconditionally) the dimensions of the subspaces of motivic cohomology of the product of two modular curves constructed by Beilinson.

A relation between automorphic representations of $GL (2)$ and $GL (3)$

Stephen Gelbart, Hervé Jacquet (1978)

Annales scientifiques de l'École Normale Supérieure

A unitary criterion for p-adic groups.

Dan Barbasch, Allen Moy (1989)

Inventiones mathematicae

Admissible distributions on p-adic symmetric spaces.

Jeffrey Hakim (1994)

Journal für die reine und angewandte Mathematik

Admissible Representations of a Semi-Simple Group over a Local Field with Vector Fixed under an Iwahori Subgroup.

Armand Borel (1976)

Inventiones mathematicae

An application of the building to orbital integrals

Jonathan D. Rogawski (1980)

Compositio Mathematica

An explicit computation of $p$ -stabilized vectors

Michitaka MIYAUCHI, Takuya YAMAUCHI (2014)

Journal de Théorie des Nombres de Bordeaux

In this paper, we give a concrete method to compute $p$ -stabilized vectors in the space of parahori-fixed vectors for connected reductive groups over $p$ -adic fields. An application to the global setting is also discussed. In particular, we give an explicit $p$ -stabilized form of a Saito-Kurokawa lift.

An explicit construction of the metaplectic representation over a finite field.

Neuhauser, Markus (2002)

Journal of Lie Theory

An inequality for local unitary Theta correspondence

Z. Gong, L. Grenié (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

Given a representation $π$ of a local unitary group $G$ and another local unitary group $H$ , either the Theta correspondence provides a representation $θ_{H} (π)$ of $H$ or we set $θ_{H} (π) = 0$ . If $G$ is fixed and $H$ varies in a Witt tower, a natural question is: for which $H$ is $θ_{H} (π) \neq 0$ ? For given dimension $m$ there are exactly two isometry classes of unitary spaces that we denote $H_{m}^{\pm}$ . For $ε \in {0, 1}$ let us denote $m_{ε}^{\pm} (π)$ the minimal $m$ of the same parity of $ε$ such that $θ_{H_{m}^{\pm}} (π) \neq 0$ , then we prove that $m_{ε}^{+} (π) + m_{ε}^{-} (π) \geq 2 n + 2$ where $n$ is the dimension of $π$ .

Analyse harmonique des groupes d'automorphismes d'arbres de Bruhat-Tits

Francis M. Choucroun (1994)

Mémoires de la Société Mathématique de France

Arithmetic aspect of operator algebras

R. J. Plymen, C. W. Leung (1991)

Compositio Mathematica

Asymptotic behaviour of supercuspidal characters of $p$ -adic ${𝐆𝐒𝐩}_{(4)}$

Fiona Murnaghan (1991)

Compositio Mathematica

Automorphic forms on $PGSp (2)$ .

Flicker, Yuval Z. (2004)

Electronic Research Announcements of the American Mathematical Society [electronic only]

Banach-Hecke algebras and $p$ -adic Galois representations.

Schneider, P., Teitelbaum, J. (2007)

Documenta Mathematica

Base change for Bernstein centers of depth zero principal series blocks

Thomas J. Haines (2012)

Annales scientifiques de l'École Normale Supérieure

Let $G$ be an unramified group over a $p$ -adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for $G$ and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with $Γ_{1} (p)$ -level structure initiated by M. Rapoport and the author in [15].

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