相关论文: Bounds for asymptotic characters of simple Lie gro…

Let $G$ be a real linear semisimple algebraic group without compact factors and $\Gamma$ a Zariski dense subgroup of $G$. In this paper, we use a probabilistic counting in order to study the asymptotic properties of $\Gamma$ acting on the…

For a fixed finite solvable group $G$ and number field $K$, we prove an upper bound for the number of $G$-extensions $L/K$ with restricted local behavior (at infinitely many places) and ${\rm inv}(L/K)<X$ for a general invariant $"{\rm…

In this article, we study symmetric $(v, k, \lambda)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle $X$ is a finite simple exceptional group of Lie type. We prove a reduction theorem, severely…

For a field $F$ of characteristic zero and an additive subgroup $G$ of $F$, a Lie algebra $B(G)$ of lock type is defined with basis $\{L_{a,i},c|a \in G, i>-2\}$ and relations…

The following short note provides an alternative proof of a result of Coornaert: namely, that given a non-elementary word-hyperbolic group $G$ with a finite generating set $X$, there exist constants $\lambda,D > 1$ such that \[…

In this paper the authors consider four questions of primary interest for the representation theory of reductive algebraic groups: (i) Donkin's Tilting Module Conjecture, (ii) the Humphreys-Verma Question, (iii) whether $\operatorname{St}_r…

The asymptotics of characters $\chi_{k\lambda}(\exp(h/k))$ of irreducible representations of a compact Lie group $G$ for large values of the scaling factor $k$ are given by Duistermaat-Heckman (DH) integrals over coadjoint orbits of $G$.…

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the Generalized Riemann Hypothesis, we give an asymptotic expansion of this quantity in…

Let $\mathbf{G}$ be a connected reductive algebraic group over a $p$-adic local field $F$. In this paper we study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular element $\gamma $ of $\mathbf{G}(F)$…

Let $K$ be an algebraically closed field of characteristic $2$, $G$ be the algebraic group $\mathrm{SL}_2$ over $K$, and $V$ be the natural representation of $G$. Let $b_k^{G,V}$ denote the number of $G$-indecomposable factors of…

Let $G$ be a simple, simply-connected algebraic group defined over $\mathbb{F}_p$. Given a power $q = p^r$ of $p$, let $G(\mathbb{F}_q) \subset G$ be the subgroup of $\mathbb{F}_q$-rational points. Let $L(\lambda)$ be the simple rational…

Let $H_k$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq 2$ for the full modular group $SL(2, \mathbb{Z})$, and let $j\geq 3$ be any fixed integer. For $f\in H_k$, we write $\lambda_{{\rm{sym}^j…

Let H be a subgroup of some locally compact group G. Assume H is approximable by discrete subgroups and G admits neighborhood bases which are "almost-invariant" under conjugation by finite subsets of H. Let $m: G \to \mathbb{C}$ be a…

We use recently obtained bounds for sums of Kloosterman sums to bound the sum $\sum_{-D\leq d\leq D} \int_{-D}^D |\zeta(1/2+it,\lambda^d)|^4| \sum_{0<|\mu|^2\leq M} A(\mu)\lambda^d((\mu)) |\mu|^{-2it}|^2 {\rm d}t$, where $\lambda^d$ is the…

We prove an asymptotically tight lower bound on $|A+\lambda A|$ for $A\subset \mathbb{C}$ and algebraic integer $\lambda$. The proof combines strong version of Freiman's theorem, structural theorem on dense subsets of a hypercubic lattice…

In this paper, we study the asymptotic behavior of the sum of twisted traces of self-dual or conjugate self-dual discrete automorphic representations of $\mathrm{GL}_n$ for the level aspect of principal congruence subgroups under some…

The characters of irreducible finite dimensional representations of compact simple Lie group G are invariant with respect to the action of the Weyl group W(G) of G. The defining property of the new character-like functions ("hybrid…

We prove that for any transitive subshift $X$ with word complexity function $c_n(X)$, if $\liminf \frac{\log (c_n(X)/n)}{\log \log \log n} = 0$, then the quotient group $\textrm{Aut}(X,\sigma) / \langle \sigma\rangle$ of the automorphism…

Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\{\lambda_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $\lambda_f(n)$, we…

Let $\lambda$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_{\alpha} \left | \sum_{x < n \leq x + H} \lambda(n) e(-\alpha n) \right | dx = o ( X H) $$ for all $H \geq X^{\theta}$ with $\theta >…