Let $G$ be a simply connected exponential solvable Lie group, $H$ a closed connected subgroup, and let $\tau$ be a representation of $G$ induced from a unitary character $\chi_f$ of $H$. The spectrum of $\tau$ corresponds via the orbit method to the set $G\cdot A_\tau / G$ of coadjoint orbits that meet the spectral variety $A_\tau = f + {\frak h}^\perp$. We prove that the spectral measure of $\tau $ is absolutely continuous with respect to the Plancherel measure if and only if $H$ acts freely on some point of $A_\tau$. As a corollary we show that if $G$ is nonunimodular, then $\tau$ has admissible vectors if and only if the preceding orbital condition holds.
1
Dept. of Mathematics and Computer Science, 220 N. Grand Blvd., St. Louis, MO 63103, U.S.A.
Bradley Currey; Vignon Oussa. Admissibility for Monomial Representations of Exponential Lie Groups. Journal of Lie Theory, Tome 22 (2012) no. 2, pp. 481-487. http://geodesic.mathdoc.fr/item/JOLT_2012_22_2_a5/
@article{JOLT_2012_22_2_a5,
author = {Bradley Currey and Vignon Oussa},
title = {Admissibility for {Monomial} {Representations} of {Exponential} {Lie} {Groups}},
journal = {Journal of Lie Theory},
pages = {481--487},
year = {2012},
volume = {22},
number = {2},
url = {http://geodesic.mathdoc.fr/item/JOLT_2012_22_2_a5/}
}
TY - JOUR
AU - Bradley Currey
AU - Vignon Oussa
TI - Admissibility for Monomial Representations of Exponential Lie Groups
JO - Journal of Lie Theory
PY - 2012
SP - 481
EP - 487
VL - 22
IS - 2
UR - http://geodesic.mathdoc.fr/item/JOLT_2012_22_2_a5/
ID - JOLT_2012_22_2_a5
ER -
%0 Journal Article
%A Bradley Currey
%A Vignon Oussa
%T Admissibility for Monomial Representations of Exponential Lie Groups
%J Journal of Lie Theory
%D 2012
%P 481-487
%V 22
%N 2
%U http://geodesic.mathdoc.fr/item/JOLT_2012_22_2_a5/
%F JOLT_2012_22_2_a5
