Related papers: Invariant triple functionals over $U_q\frak{sl}_2$

Foundations of the formal series $*$ -- calculus in deformation quantisation are discussed. Several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced. The notion of nonnegativity…

This paper introduces the $u$-deformed homogeneous functions $\mathrm{R}_{\alpha}(x,y;u|q)$, for all $\alpha\in\mathbb{C}$. Basic properties of the functions $\mathrm{R}_{\alpha}(x,y;u|q)$ are given, along with recurrence relations, their…

Goldman defined a symplectic form on the smooth locus of the $G$-character variety of a closed, oriented surface $S$ for a Lie group $G$ satisfying very general hypotheses. He then studied the Hamiltonian flows associated to $G$-invariant…

In this paper we establish a connection between the associated variety of a representation and the existence of certain degenerate Whittaker functionals, for both smooth and K-finite vectors, for all quasi-split real reductive groups,…

Let $K$ be one of the complex classical groups ${\rm O}_k$, ${\rm GL}_k$, or ${\rm Sp}_{2k}$. Let $M \subseteq K$ be the block diagonal embedding ${\rm O}_{k_1} \times \cdots \times {\rm O}_{k_r}$ or ${\rm GL}_{k_1} \times \cdots \times…

We describe a new method to estimate the trilinear period on automorphic representations of PGL(2,R). Such a period gives rise to a special value of the triple L-function. We prove a bound for the triple period which amounts to a…

In a previous paper a multi-species version of the q-Boson stochastic particle system is introduced and the eigenfunctions of its backward generator are constructed by using a representation of the Hecke algebra. In this article we prove a…

When $G$ is a real semisimple group, there is a surprising interplay between its representation theory and that of its motion group $G_0$, known as the Mackey analogy. The present paper extends this analogy to the framework of…

A new type of sl_3 basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key-ingredient in the sl_3 basic hypergeometric series is a…

A representation of a specialization of a q-deformed class one lattice gl(\ell+1}-Whittaker function in terms of cohomology groups of line bundles on the space QM_d(P^{\ell}) of quasi-maps P^1 to P^{\ell} of degree d is proposed. For…

Let $k$ be a local field of characteristic zero. Rankin-Selberg's local zeta integrals produce linear functionals on generic irreducible admissible smooth representations of $GL_n(k)\times GL_r(k)$, with certain invariance properties. We…

The rational invariants of the SL_2(q)-invariant quadratic forms on the real irreducible representations are determined. There is still one open question (see Remark 6.5) if q is an even square.

The concept of $q$-deformation, or ``$q$-analogue'' arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; $q$-deformations are important for knot invariants, combinatorial…

In this paper, we generalize the Quillen-Lichtenbaum Conjecture relating special values of Dedekind zeta functions to algebraic $\mathrm{K}$-groups. The former has been settled by Rost-Voevodsky up to the Iwasawa Main Conjecture. Our…

In \cite{lafforgue2012chtoucas}, Vicent Lafforgue attaches a semisimple Langlands parameter (or, what amounts to the same thing, a $\hat{G}$-pseudocharacter) to every cuspidal automorphic representation of a reductive group $G$ over the…

For the nonstandard $q$-deformed algebras $U_q(so_n)$, defined recently in terms of trilinear relations for generating elements, most general finite dimensional irreducible representations directly corresponding to those of nondeformed…

We determine a substantial part of the unitary representation theory of the Drinfeld double of a $q$-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every…

We consider a natural $q$-deformation of the classical Markov numbers. This $q$-deformation is closely related to $q$-deformed rational numbers recently introduced by two of us. Both notions, those of $q$-rationals and $q$-Markov numbers,…

As a sequel to our proof of the analog of Serre's conjecture for function fields in Part I of this work, we study in this paper the deformation rings of $n$-dimensional mod $\ell$ representations $\rho$ of the arithmetic fundamental group…

We construct the positive principal series representations for $U_q(g_R)$ where $g$ is of type $B_n$, $C_n$, $F_4$ or $G_2$, parametrized by $R^r$ where $r$ is the rank of $g$. We show that under the representations, the generators of the…