L 2 -Singular Dichotomy for Orbital Measures on Complex Groups

S. K. Gupta; K. E. Hare

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 3, page 409-419
  • ISSN: 0392-4041

Abstract

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It is known that all continuous orbital measures, μ on a compact, connected, classical simple Lie group G or its Lie algebra satisfy a dichotomy: either μ k ∈ L 2 or μ k is purely singular to Haar measure. In this note we prove that the same dichotomy holds for the dual situation, continuous orbital measures on the complex group G C . We also determine the sharp exponent k such that any k -fold convolution product of continuous G -bi-invariant measures on G C is absolute continuous with respect to Haar measure.

How to cite

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Gupta, S. K., and Hare, K. E.. "$L^{2}$-Singular Dichotomy for Orbital Measures on Complex Groups." Bollettino dell'Unione Matematica Italiana 3.3 (2010): 409-419. <http://eudml.org/doc/290673>.

@article{Gupta2010,
abstract = {It is known that all continuous orbital measures, $\mu$ on a compact, connected, classical simple Lie group $G$ or its Lie algebra satisfy a dichotomy: either $\mu^\{k\} \in L^\{2\}$ or $\mu^\{k\}$ is purely singular to Haar measure. In this note we prove that the same dichotomy holds for the dual situation, continuous orbital measures on the complex group $G^C$. We also determine the sharp exponent $k$ such that any $k$-fold convolution product of continuous $G$-bi-invariant measures on $G^\{C\}$ is absolute continuous with respect to Haar measure.},
author = {Gupta, S. K., Hare, K. E.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {409-419},
publisher = {Unione Matematica Italiana},
title = {$L^\{2\}$-Singular Dichotomy for Orbital Measures on Complex Groups},
url = {http://eudml.org/doc/290673},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Gupta, S. K.
AU - Hare, K. E.
TI - $L^{2}$-Singular Dichotomy for Orbital Measures on Complex Groups
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/10//
PB - Unione Matematica Italiana
VL - 3
IS - 3
SP - 409
EP - 419
AB - It is known that all continuous orbital measures, $\mu$ on a compact, connected, classical simple Lie group $G$ or its Lie algebra satisfy a dichotomy: either $\mu^{k} \in L^{2}$ or $\mu^{k}$ is purely singular to Haar measure. In this note we prove that the same dichotomy holds for the dual situation, continuous orbital measures on the complex group $G^C$. We also determine the sharp exponent $k$ such that any $k$-fold convolution product of continuous $G$-bi-invariant measures on $G^{C}$ is absolute continuous with respect to Haar measure.
LA - eng
UR - http://eudml.org/doc/290673
ER -

References

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  2. DOOLEY, A. - RICCI, F., Characterisation of G -invariant Fourier algebras, Un. Boll. Mat. Ital., 9 (1995), 37-45. Zbl0836.43008MR1324602
  3. GUPTA, S. - HARE, K., Singularity of orbits in classical Lie algebras, Geom. Func. Anal., 13 (2003), 815-844. Zbl1031.22004MR2006558DOI10.1007/s00039-003-0431-x
  4. GUPTA, S. - HARE, K., L 2 -singular dichotomy for orbital measures of classical compact Lie groups, Adv. Math, 222 (2009), 1521-1573. Zbl1179.43006MR2555904DOI10.1016/j.aim.2009.06.008
  5. GUPTA, S., - HARE, K. - SEYFADDINI, S., L 2 -singular dichotomy for orbital measures of classical simple Lie algebras, Math. Zeit., 262 (2009), 91-124. Zbl1168.43003MR2491602DOI10.1007/s00209-008-0364-z
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  8. RAGOZIN, D., Zonal measure algebras on isotropy irreducible homogeneous spaces, J. Func. Anal., 17 (1974), 355-376. Zbl0297.43002MR365044
  9. RICCI, F. - STEIN, E., Harmonic analysis on nilpotent groups and singular integrals II. Singular kernels supported on submanifolds, J. Func. Anal., 78 (1988), 56-84. Zbl0645.42019MR937632DOI10.1016/0022-1236(88)90132-2
  10. VARADARAJAN, V. S., Lie groups and Lie algebras and their representations, Springer-Verlag, New York (1984). Zbl0955.22500MR746308DOI10.1007/978-1-4612-1126-6

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