Some spectral results on $L^2\left(H_n\right)$ related to the action of U(p,q)

Godoy, T., and Saal, L.. "Some spectral results on $L^{2}(H_{n})$ related to the action of U(p,q)." Colloquium Mathematicae 86.2 (2000): 177-187. <http://eudml.org/doc/210848>.

@article{Godoy2000,
abstract = {Let $H_\{n\}$ be the (2n+1)-dimensional Heisenberg group, let p,q be two non-negative integers satisfying p+q=n and let G be the semidirect product of U(p,q) and $H_\{n\}$. So $L^\{2\}(H_\{n\})$ has a natural structure of G-module. We obtain a decomposition of $L^\{2\}(H_\{n\})$ as a direct integral of irreducible representations of G. On the other hand, we give an explicit description of the joint spectrum σ(L,iT) in $L^\{2\}(H_\{n\})$ where $L=\sum _\{j=1\}^\{p\} (X_\{j\}^\{2\}+Y_\{j\}^\{2\}) - \sum _\{j=p+1\}^\{n\} (X_\{j\}^\{2\}+Y_\{j\}^\{2\})$, and where $\{X_\{1\},Y_\{1\},...,X_\{n\},Y_\{n\},T\}$ denotes the standard basis of the Lie algebra of $H_\{n\}$. Finally, we obtain a spectral characterization of the bounded operators on $L^\{2\}(H_\{n\})$ that commute with the action of G.},
author = {Godoy, T., Saal, L.},
journal = {Colloquium Mathematicae},
keywords = {spectral decomposition; joint spectrum; irreducible representation; Heisenberg group},
language = {eng},
number = {2},
pages = {177-187},
title = {Some spectral results on $L^\{2\}(H_\{n\})$ related to the action of U(p,q)},
url = {http://eudml.org/doc/210848},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Godoy, T.
AU - Saal, L.
TI - Some spectral results on $L^{2}(H_{n})$ related to the action of U(p,q)
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 2
SP - 177
EP - 187
AB - Let $H_{n}$ be the (2n+1)-dimensional Heisenberg group, let p,q be two non-negative integers satisfying p+q=n and let G be the semidirect product of U(p,q) and $H_{n}$. So $L^{2}(H_{n})$ has a natural structure of G-module. We obtain a decomposition of $L^{2}(H_{n})$ as a direct integral of irreducible representations of G. On the other hand, we give an explicit description of the joint spectrum σ(L,iT) in $L^{2}(H_{n})$ where $L=\sum _{j=1}^{p} (X_{j}^{2}+Y_{j}^{2}) - \sum _{j=p+1}^{n} (X_{j}^{2}+Y_{j}^{2})$, and where ${X_{1},Y_{1},...,X_{n},Y_{n},T}$ denotes the standard basis of the Lie algebra of $H_{n}$. Finally, we obtain a spectral characterization of the bounded operators on $L^{2}(H_{n})$ that commute with the action of G.
LA - eng
KW - spectral decomposition; joint spectrum; irreducible representation; Heisenberg group
UR - http://eudml.org/doc/210848
ER -