An inequality for local unitary Theta correspondence

Z. Gong; L. Grenié

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Displaying similar documents to “An inequality for local unitary Theta correspondence”

Congruences modulo $ℓ$ between $ϵ$ factors for cuspidal representations of $G L (2)$

Marie-France Vignéras (2000)

Journal de théorie des nombres de Bordeaux

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Let $ℓ \neq p$ be two different prime numbers, let $F$ be a local non archimedean field of residual characteristic $p$ , and let ${\bar{𝐐}}_{ℓ}, {\bar{𝐙}}_{ℓ}, {\bar{𝐅}}_{ℓ}$ be an algebraic closure of the field of $ℓ$ -adic numbers $𝐐_{ℓ}$ , the ring of integers of ${\bar{𝐐}}_{ℓ}$ , the residual field of ${\bar{𝐙}}_{ℓ}$ . We proved the existence and the unicity of a Langlands local correspondence over ${\bar{𝐅}}_{ℓ}$ for all $n \geq 2$ , compatible with the reduction modulo $ℓ$ in [V5], without using $L$ and $ϵ$ factors of pairs. We conjecture that the Langlands local correspondence over ${\bar{𝐐}}_{ℓ}$ respects congruences...

Davenport-Hasse relations and an explicit Langlands correspondence, II : twisting conjectures

Colin J. Bushnell, Guy Henniart (2000)

Journal de théorie des nombres de Bordeaux

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Let $F / ℚ_{p}$ be a finite field extension. The Langlands correspondence gives a canonical bijection between the set $𝒢_{F}^{0} (n)$ of equivalence classes of irreducible $n$ -dimensional representations of the Weil group $𝒲_{F}$ of $F$ and the set $𝒜_{F}^{0} (n)$ of equivalence classes of irreducible supercuspidal representations of GL $_{n} (F)$ . This paper is concerned with the case where $n = p^{m}$ . In earlier work, the authors constructed an explicit bijection $π : 𝒢_{F}^{0} (p^{m}) \to 𝒜_{F}^{0} (p^{m})$ using a non-Galois tame base change map. If this tame base change satisfies a certain...

Towards the Jacquet conjecture on the Local Converse Problem for $p$ -adic ${GL}_{n}$

Dihua Jiang, Chufeng Nien, Shaun Stevens (2015)

Journal of the European Mathematical Society

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The Local Converse Problem is to determine how the family of the local gamma factors $γ (s, π \times τ, ψ)$ characterizes the isomorphism class of an irreducible admissible generic representation $π$ of ${GL}_{n} (F)$ , with $F$ a non-archimedean local field, where $τ$ runs through all irreducible supercuspidal representations of ${GL}_{r} (F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r = 1, 2, ..., [\frac{n}{2}]$ . Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet...

On a variant of Kazhdan's property (T) for subgroups of semisimple groups

Mohammed Bachir Bekka, Nicolas Louvet (1997)

Annales de l'institut Fourier

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Let $Γ$ be an irreducible lattice in a product $G$ of simple groups. Assume that $G$ has a factor with property (T). We give a description of the topology in a neighbourhood of the trivial one dimensional representation of $Γ$ in terms of the topology of the dual space $\hat{G}$ of $G$ . We use this result to give a new proof for the triviality of the first cohomology group of $Γ$ with coefficients in a finite dimensional unitary representation.

Complete Pick positivity and unitary invariance

Angshuman Bhattacharya, Tirthankar Bhattacharyya (2010)

Studia Mathematica

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The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foiaş. Just as a contraction is related to the Szegö kernel $k_{S} (z, w) = {(1 - z w ̅)}^{- 1}$ for |z|,|w| < 1, by means of $(1 / k_{S}) (T, T *) \geq 0$ , we consider an arbitrary open connected domain Ω in ℂⁿ, a complete Pick kernel k on Ω and a tuple T = (T₁, ..., Tₙ) of commuting bounded operators on a complex separable Hilbert space ℋ such that (1/k)(T,T*) ≥ 0. For a complete Pick kernel the 1/k functional calculus makes sense...

A descriptive view of unitary group representations

Simon Thomas (2015)

Journal of the European Mathematical Society

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In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if $G$ and $H$ are countable amenable non-type I groups, then the unitary duals of $G$ and $H$ are Borel isomorphic.

On the angles between certain arithmetically defined subspaces of $𝐂^{n}$

Robert Brooks (1987)

Annales de l'institut Fourier

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If ${v_{i}}$ and ${w_{j}}$ are two families of unitary bases for $C^{n}$ , and $θ$ is a fixed number, let $V^{n}$ and $W^{n}$ be subspaces of $C^{n}$ spanned by $[θ \cdot n]$ vectors in ${v_{i}}$ and ${w_{j}}$ respectively. We study the angle between $V^{n}$ and $W^{n}$ as $n$ goes to infinity. We show that when ${v_{i}}$ and ${w_{j}}$ arise in certain arithmetically defined families, the angles between $V^{n}$ and $W^{n}$ may either tend to $0$ or be bounded away from zero, depending on the behavior of an associated eigenvalue problem.

Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group

Benjamin Cahen (2013)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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Let $G$ be a quasi-Hermitian Lie group with Lie algebra $𝔤$ and $K$ be a compactly embedded subgroup of $G$ . Let $ξ_{0}$ be a regular element of $𝔤^{*}$ which is fixed by $K$ . We give an explicit $G$ -equivariant diffeomorphism from a complex domain onto the coadjoint orbit $𝒪 (ξ_{0})$ of $ξ_{0}$ . This generalizes a result of [B. Cahen, Berezin quantization and holomorphic representations, Rend. Sem. Mat. Univ. Padova, to appear] concerning the case where $𝒪 (ξ_{0})$ is associated with a unitary irreducible representation of $G$ which...

Spherical unitary dual of general linear group over non-Archimidean local field

Marko Tadic (1986)

Annales de l'institut Fourier

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Let $F$ be a local non-archimedean field. The set of all equivalence classes of irreducible spherical representations of $G L (n, F)$ is described in the first part of the paper. In particular, it is shown that each irreducible spherical representation is parabolically induced by an unramified character. Bernstein’s result on the irreducibility of the parabolically induced representations of $G L (n, F)$ by irreducible unitary ones, and Ol’shanskij’s construction of complementary series give directly a description...

Representation numbers of five sextenary quadratic forms

Ernest X. W. Xia, Olivia X. M. Yao, A. F. Y. Zhao (2015)

Colloquium Mathematicae

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For nonnegative integers a, b, c and positive integer n, let N(a,b,c;n) denote the number of representations of n by the form $\sum_{i = 1}^{a} (x ²_{i} + x_{i} y_{i} + y ²_{i}) + 2 \sum_{j = 1}^{b} (u ²_{j} + u_{j} v_{j} + v ²_{j}) + 4 \sum_{k = 1}^{c} (r ²_{k} + r_{k} s_{k} + s ²_{k})$ . Explicit formulas for N(a,b,c;n) for some small values were determined by Alaca, Alaca and Williams, by Chan and Cooper, by Köklüce, and by Lomadze. We establish formulas for N(2,1,0;n), N(2,0,1;n), N(1,2,0;n), N(1,0,2;n) and N(1,1,1;n) by employing the (p, k)-parametrization of three 2-dimensional theta functions due to Alaca, Alaca and Williams.