Multiplicity one theorems: the Archimedean case

Let $G$ be one of the classical Lie groups \GL_{n+1}(\R), \GL_{n+1}(\C), \oU(p,q+1), \oO(p,q+1), \oO_{n+1}(\C), \SO(p,q+1), \SO_{n+1}(\C), and let $G'$ be respectively the subgroup \GL_{n}(\R), \GL_{n}(\C), \oU(p,q), \oO(p,q), \oO_n(\C), \SO(p,q), \SO_n(\C), embedded in $G$ in the standard way. We show that every irreducible Casselman-Wallach representation of $G'$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $G$. Similar results are proved for the Jacobi groups \GL_{n}(\R)\ltimes \oH_{2n+1}(\R), \GL_{n}(\C)\ltimes \oH_{2n+1}(\C), \oU(p,q)\ltimes \oH_{2p+2q+1}(\R), \Sp_{2n}(\R)\ltimes \oH_{2n+1}(\R), \Sp_{2n}(\C)\ltimes \oH_{2n+1}(\C), with their respective subgroups \GL_{n}(\R), \GL_{n}(\C), \oU(p,q), \Sp_{2n}(\R), \Sp_{2n}(\C).Comment: To appear in Annals of Mathematic

A New Technique for the Design of Multi-Phase Voltage Controlled Oscillators

© 2017 World Scientific Publishing Company.In this work, a novel circuit structure for second-harmonic multi-phase voltage controlled oscillator (MVCO) is presented. The proposed MVCO is composed of (Formula presented.) ((Formula presented.) being an integer number and (Formula presented.)2) identical inductor–capacitor ((Formula presented.)) tank VCOs. In theory, this MVCO can provide 2(Formula presented.) different phase sinusoidal signals. A six-phase VCO based on the proposed structure is designed in a TSMC 0.18(Formula presented.)um CMOS process. Simulation results show that at the supply voltage of 0.8(Formula presented.)V, the total power consumption of the six-phase VCO circuit is about 1(Formula presented.)mW, the oscillation frequency is tunable from 2.3(Formula presented.)GHz to 2.5(Formula presented.)GHz when the control voltage varies from 0(Formula presented.)V to 0.8(Formula presented.)V, and the phase noise is lower than (Formula presented.)128(Formula presented.)dBc/Hz at 1(Formula presented.)MHz offset frequency. The proposed MVCO has lower phase noise, lower power consumption and more outputs than other related works in the literature.Peer reviewedFinal Accepted Versio