Related papers: Representing some II$_1$ factors in $L^2(\Lambda \…

We give combinatorial models for complex, smooth, non-spherical, generic, irreducible representations of the group G=PGL(2,F), where F is a non-archimedean locally compact field. They use the graphs X_k lying above the tree of G, introduced…

Let $F$ be a non-Archimedean local field. Let $\mathcal{A}_n(F)$ be the set of equivalence classes of irreducible admissible representations of $\textrm{GL}_n(F)$, and $\mathcal{G}_n(F)$ be the set of equivalence classes of n-dimensional…

We provide a necessary and sufficient condition for the representability of a function as the classical multidimensional Laplace transform, when the support of the representing measure is contained in some generalized semi-algebraic set.…

We determine the coefficients of the terms multiplying the gauge fields, gravitational field and cosmological term in a scheme whereby properties are characterized by $N$ anticommuting scalar Grassmann variables. We do this for general $N$,…

Let $\mathcal{B} = (B_1,\ldots, B_h)$ be an $h$-tuple of sets of positive integers. Let $g_{\mathcal{B} }(n)$ count the number of representations of $n$ in the form $n = b_1\cdots b_h$, where $b_i \in B_i$ for all $i \in \{1,\ldots, h\}$.…

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $\Gamma$ of a closed hyperbolic surface $\Sigma$ in ${\rm PSL}(2,\mathbb R)^n$. We identify the boundary…

Piatetski-Shapiro--Rallis discovered an integral representation construction, known as the doubling method, for the tensor product $L$-function of a cuspidal automorphic representation of $G \times \mathrm{GL}_1$, where $G$ is a classical…

To every $n$-dimensional lens space $L$, we associate a congruence lattice $\mathcal L$ in $\mathbb Z^m$, with $n=2m-1$ and we prove a formula relating the multiplicities of Hodge-Laplace eigenvalues on $L$ with the number of lattice…

We prove the density hypothesis for congruence subgroups of an irreducible uniform lattice in $\mathrm{PSL}_2(\mathbb{R})^d$, extending previous results on the spherical density hypothesis to bound multiplicities of non-tempered…

We prove logarithm laws and shrinking target properties for unipotent flows on the homogenous space $\Gamma\bs G$ with $G=\SL_2(\bbR)^{r_1}\times\SL_2(\bbC)^{r_2}$ and $\Gamma\subseteq G$ an irreducible non-uniform lattice. Our method…

We prove that for any k greater or equal to 2, given a smooth compact k-dimensional manifold and a multiplicative k-1-gerbe on a Lie group, together with an integrable connection, there is a line bundle on the corresponding…

Let $G$ be an $N \times N$ real matrix whose entries are independent identically distributed standard normal random variables $G_{ij} \sim \mathcal{N}(0,1)$. The eigenvalues of such matrices are known to form a two-component system…

Let G be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if G has the congruence subgroup property, then the number of n-dimensional irreducible representations of G grows like n^a, where a is a…

We use the relations between the base change representations, theta lifts and Whittaker model, to give a new proof to the period problems of $GL(2)$ over a quadratic local field extension $E/F.$ And we classify both local and global…

We present a class of lattices in R^d (d >= 2) which we call GL-lattices and conjecture that any lattice is such. This conjecture is referred to as GLC. Littlewood's conjecture amounts to saying that Z^2 is GL. We then prove existence of GL…

Let $\Gamma$ be a finitely generated nilpotent group and let G be a complex reductive algebraic group. The representation variety $\mathrm{Hom}(\Gamma,G)$ and the character variety $\mathrm{Hom}(\Gamma,G)//G$ each carry a natural topology,…

Let F be a totally real number field, n a prime integer, and G a unitary group of rank n defined over F that is compact at every infinite place. We prove an asymptotic formula for the number of cuspidal automorphic representations of G…

Let $L$ be a positive definite (non-classic) ternary $\z$-lattice and let $p$ be a prime such that a $\frac 12\z_p$-modular component of $L_p$ is nonzero isotropic and $4\cdot dL$ is not divisible by $p$. For a nonnegative integer $m$, let…

We construct a $GL$-invariant measure on a semi-infinite Grassmannian over a finite field, describe the natural group of symmetries of this measure, and decompose the space $L^2$ over the Grassmannian on irreducible representations. The…

For $G=GL(n,\mathbb{C})$ and a parabolic subgroup $P=LN$ with a two-block Levi subgroup $L=GL(n_1)\times GL(n_2)$, the space $G\cdot (\mathcal{\mathcal{O}}+\mathfrak{n})$, where $\mathcal{O}$ is a nilpotent orbit of $\mathfrak{l}$, is a…