Related papers: On Classical groups detected by the tensor third r…

This paper performs the following steps toward the proof of GLC in the de Rham setting: (i) We deduce GLC for G=GL_n; (ii) We prove that the Langlands functor L_G constructed in [GLC1], when restricted to the cuspidal category, is…

All possible permutations in the discrete $S_4$ group are classified by three rotation angles associated with the orthogonal group $O(3)$. We construct a spinor representation ${\bf 2}_D$ of $O(3)$, which is transformed by three 4$\times$4…

Selfdual representations of any group fall into two classes when they are irreducible: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the…

Based on Furusawa's theory, we present an integral representation for the L-function L(s,\pi \times \tau), where \pi is a cuspidal automorphic representation on GSp(4) related to a holomorphic Siegel modular form, and where \tau is an…

Let $\pi$ be a unitary automorphic cuspidal representation of $GL_2(\mathbb{Q}_\mathbb{A})$ with Fourier coefficients $\lambda_\pi(n)$. Asymptotic expansions of certain sums of $\lambda_\pi(n)$ are proved using known functorial liftings…

The Langlands functoriality conjecture envisaged in the bisemialgebra framework is proved to correspond to the nonorthogonal completely reducible cuspidal representations of the bilinear algebraic semigroups.

We continue our study of the poles of local Langlands L-functions through the theory of induced from supercuspidal representations of quasi-split groups. Here we study the odd special orthogonal groups, and hence determine poles of Rankin…

For a generic $L$-parameter of $U(n)\times U(n)$, it is conjectured that there is a unique representation in their associated relevant Vogan $L$-packet which produces the unique Fourier-Jacobi model. We investigated this conjecture for some…

Suppose G is a real reductive Lie group in Harish-Chandra's class. We propose here a structure for the set \Pi_u(G) of equivalence classes of irreducible unitary representations of G. (The subscript u will be used throughout to indicate…

We consider a new integral representation for $L(s_1, \Pi \times \tau_1) L(s_2, \Pi \times \tau_2),$ where $\Pi$ is a globally generic cuspidal representation of $GSp_4,$ and $\tau_1$ and $\tau_2$ are two cuspidal representations of $GL_2$…

In this paper, we explicitly determine the local $2$-adic component of a non-selfdual automorphic representation $\Pi$ of $\mathrm{GL}_3$ constructed by van Geemen and Top. We prove that $\Pi_2$ is a parabolically induced representation of…

In this paper, we begin the study of poles of partial L-functions L^S(sigma tensor tau,s), where sigma tensor tau is an irreducible, automorphic, cuspidal, generic (i.e. with nontrivial Whittaker coefficient) representation of G_A x…

We introduce and motivate -- based on ongoing joint work with Germ\'an Stefanich -- the notion of potent categorical representations of a complex reductive group $G$, specifically a conjectural Langlands correspondence identifying potent…

Let $\boldsymbol\Lambda_3(\mathbb C)\,(=\mathbb C^{27})$ be the space of structure vectors of $3$-dimensional algebras over $\mathbb C$ considered as a $G$-module via the action of $G={\rm GL}(3,\mathbb C)$ on $\boldsymbol\Lambda_3(\mathbb…

In the second part of our work on observables we have shown that quantum observables in the sense of von Neumann, i.e.bounded selfadjoint operators in some von Neumann subalgebra $R$ of $L(H)$, can be represented as bounded continuous…

Let $G$ be a $p$-adic Lie group with reductive Lie algebra $\mathfrak{g}$. In analogy to the translation functors introduced by Bernstein and Gelfand on categories of $U(\mathfrak{g})$-modules we consider similarly defined functors on the…

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times…

In this paper, we study extra-twists for automorphic representations of $\mathrm{GL}_n$ and use them to give a precise description of the image of the Galois representations associated with regular algebraic cuspidal automorphic…

Let $S$ be the sphere of dimension $n-1, n\geq 4$. Let $(\pi_{\lambda})_{\lambda\in \mathbb C}$ be the scalar principle series of representations of the conformal group $SO_0(1,n)$, realized on $\mathcal C^\infty(S)$. For $\boldsymbol…

In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is…