Let $k$ be a $p$-adic field. Let $G$ be the group of $k$-rational points of a connected reductive group $𝖦$ defined over $k$, and let $𝔤$ be its Lie algebra. Under certain hypotheses on $𝖦$ and $k$, wethe tempered dual $\widehat{G}$ of $G$ via the Plancherel formula on $𝔤$, using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on $𝔤$ and $G$. As a consequence, we prove that any tempered representation contains a good minimal $𝖪$-type; we extend this result to irreducible...

Let $E$ be a CM number field, $p$ an odd prime totally split in $E$, and let $X$ be the $p$-adic analytic space parameterizing the isomorphism classes of $3$-dimensional semisimple $p$-adic representations of $\mathrm{Gal}(\overline{E}/E)$ satisfying a selfduality condition “of type $\mathrm{U}\left(3\right)$”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3[E:\mathbb{Q}]$. As important steps, and in any rank, we prove that any...

Let $V$ be a unitary space. For an arbitrary subgroup $G$ of the full symmetric group $S_m$ and an arbitrary irreducible unitary representation $\Lambda$ of $G$, we study the generalized symmetry class of tensors over $V$ associated with $G$ and $\Lambda$. Some important properties of this vector space are investigated.

A nonincreasing sequence $\pi =(d_1,...,d_n)$ of nonnegative integers is a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. In this case, $G$ is referred to as a realization of $\pi$. Given two graphs $G_1$ and $G_2$, A. Busch et al. (2014) introduced the potential-Ramsey number of $G_1$ and $G_2$, denoted by $r_{\mathrm{pot}}(G_1,G_2)$, as the smallest nonnegative integer $m$ such that for every $m$-term graphic sequence $\pi$, there is a realization $G$ of $\pi$ with $G_1\subseteq G$ or with $G_2\subseteq \overline{G}$, where $\overline{G}$ is the complement of $G$. For $t\ge 2$ and $0\le k\le \lfloor \frac{t}{2}\rfloor$, let $K_t^{-k}$ be the graph...

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...

For a complex character $\chi$ of a finite group $G$, it is known that the product $\mathrm{sh}\left(\chi \right)={\prod }_{l\in L\left(\chi \right)}(\chi \left(1\right)-l)$ is a multiple of $\left|G\right|$, where $L\left(\chi \right)$ is the image of $\chi$ on $G-\left\{1\right\}$. The character $\chi$ is said to be a sharp character of type $L$ if $L=L\left(\chi \right)$ and $\mathrm{sh}\left(\chi \right)=\left|G\right|$. If the principal character of $G$ is not an irreducible constituent of $\chi$, then the character $\chi$ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups $G$ with normalized sharp characters of type $\{-1,0,2\}$. Here we prove that such a group with nontrivial...

Let $\Omega \subset {ℂ}^n$ be a bounded, simply connected $ℂ$-convex domain. Let $\alpha \in {\mathbb{Z}}_{+}^n$ and let $f$ be a function on $\Omega$ which is separately $C^{2{\alpha }_j-1}$-smooth with respect to $z_j$ (by which we mean jointly $C^{2{\alpha }_j-1}$-smooth with respect to $\mathrm{Re}{z}_j$, $\mathrm{Im}{z}_j$). If $f$ is $\alpha$-analytic on $\Omega \setminus f^{-1}\left(0\right)$, then $f$ is $\alpha$-analytic on $\Omega$. The result is well-known for the case ${\alpha }_i=1$, $1\le i\le n$, even when $f$ a priori is only known to be continuous.

CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p,q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p,q}\left(B\right)$........ 235. Examples in $L_{p,q}$ theory...................................................................................

Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_1,...,x_m$. Suppose $G$ is a subgroup of $S_m$, and $\chi$ is an irreducible character of $G$. Let $H_d(G,\chi )$ be the symmetry class of polynomials of degree $d$ with respect to $G$ and $\chi$. For any linear operator $T$ acting on $V$, there is a (unique) induced operator $K_{\chi }\left(T\right)\in \mathrm{End}\left(H_d(G,\chi )\right)$ acting on symmetrized decomposable polynomials by $$K_{\chi }\left(T\right)(f_1*f_2*...*f_d)=T{f}_1*T{f}_2*...*T{f}_d.$$ In this paper, we show that the representation $T\mapsto K_{\chi }\left(T\right)$ of the general linear group $GL\left(V\right)$ is equivalent to the direct sum of $\chi \left(1\right)$ copies...