Related papers: Minimal representations, spherical vectors, and ex…

We study a conjectural formula for the maximal elements in the wavefront set associated with a theta representation of a covering group over $p$-adic fields. In particular, it is shown that the formula agrees with the existing work in the…

To a finite quadratic module, that is, a finite abelian group D together with a non-singular quadratic form Q:D --> Q/Z, it is possible to associate a representation of either the modular group, SL(2,Z), or its metaplectic cover, Mp(2,Z),…

Given F a locally compact, non-discrete, non-archimedean field of characteristic different from 2 and R an integral domain such that a non-trivial smooth F-character with values in the multiplicative group of R exists, we construct the…

We construct an explicit realization of a minimal representation of G, where G is the conformal group of a real Jordan algebra N. We characterize spherical vectors for these representation and prove that they are closely related to the…

The classical Cartan-Helgason theorem characterises finite-dimensional spherical representations of reductive Lie groups in terms of their highest weights. We generalise the theorem to the case of a reductive symmetric supergroup pair…

We study the structure of minimal parabolic subgroups of the classical infinite dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras. This depends on the recently developed structure of…

We propose new formulas for singular vectors in Verma modules over the affine Lie superalgebra $\hat{sl}(2|1)$. We analyze the coexistence of singular vectors of different types and identify the twisted modules $N_{h,k;\theta}$ arising as…

We give explicit computations of the $\Gamma$-Euler characteristic of several families of orbit space definable translation groupoids. These include the translation groupoids associated to finite-dimensional linear representations of the…

We classify the finite-dimensional rational representations $V$ of the exceptional algebraic groups $G$ with $\mathfrak g={\sf Lie}(G)$ such that the symmetric invariants of the semi-direct product $\mathfrak g\ltimes V$, where $V$ is an…

We construct almost holomorphic and holomorphic modular forms by considering theta series for quadratic forms of signature $(n-1,1)$. We include homogeneous and spherical polynomials in the definition of the theta series (generalizing a…

We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the…

Let $\mathfrak{o}$ be the ring of integers of a non-archimedean local field with the maximal ideal $\wp$ and the finite residue field of characteristic $p.$ Let $\mathbf{G}$ be the General Linear or Special Linear group with entries from…

We construct the positive principal series representations for $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by $\mathbb{R}_{\geq 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We…

We establish a connection between certain unique models, or equivalently unique functionals, for representations of p-adic groups and linear characters of their corresponding Hecke algebras. This allows us to give a uniform evaluation of…

Let $U_\epsilon(\mathfrak g)$ be the simply connected quantized enveloping algebra associated to a finite-dimensional complex simple Lie algebra $\mathfrak g$ at the roots of unity. The De Concini-Kac-Procesi conjecture on the dimension of…

Schur-Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog…

We describe algebraically, diagrammatically and in terms of weight vectors, the restriction of tensor powers of the standard representation of quantum $\mathfrak{sl}_2$ to a coideal subalgebra. We realise the category as module category…

We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R}) $ and $\mathrm{O}^*(2n)$. These cycles are (covered by) locally…

Let $\mathbf{G}$ be a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. We consider bivariate zeta functions of groups of the form $\mathbf{G}(\mathcal{O})$…

In this article, we introduce and study the concept of $\textit{spherical-vectors}$, which can be perceived as a natural extension of the arguments of complex numbers in the context of quaternions. We initially establish foundational…