The size of characters of compact Lie groups

Kathryn Hare

Studia Mathematica (1998)

  • Volume: 129, Issue: 1, page 1-18
  • ISSN: 0039-3223

Abstract

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Pointwise upper bounds for characters of compact, connected, simple Lie groups are obtained which enable one to prove that if μ is any central, continuous measure and n exceeds half the dimension of the Lie group, then μ n ∈ L 1 . When μ is a continuous, orbital measure then μ n is seen to belong to L 2 . Lower bounds on the p-norms of characters are also obtained, and are used to show that, as in the abelian case, m-fold products of Sidon sets are not p-Sidon if p < 2m/(m+1).

How to cite

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Hare, Kathryn. "The size of characters of compact Lie groups." Studia Mathematica 129.1 (1998): 1-18. <http://eudml.org/doc/216489>.

@article{Hare1998,
abstract = {Pointwise upper bounds for characters of compact, connected, simple Lie groups are obtained which enable one to prove that if μ is any central, continuous measure and n exceeds half the dimension of the Lie group, then $μ^n ∈ L^1$. When μ is a continuous, orbital measure then $μ^n$ is seen to belong to $L^2$. Lower bounds on the p-norms of characters are also obtained, and are used to show that, as in the abelian case, m-fold products of Sidon sets are not p-Sidon if p < 2m/(m+1).},
author = {Hare, Kathryn},
journal = {Studia Mathematica},
keywords = {compact Lie group; central measures; Cartan-Weyl theory; irreducible representations; irreducible characters; simple Lie groups; Sidon sets},
language = {eng},
number = {1},
pages = {1-18},
title = {The size of characters of compact Lie groups},
url = {http://eudml.org/doc/216489},
volume = {129},
year = {1998},
}

TY - JOUR
AU - Hare, Kathryn
TI - The size of characters of compact Lie groups
JO - Studia Mathematica
PY - 1998
VL - 129
IS - 1
SP - 1
EP - 18
AB - Pointwise upper bounds for characters of compact, connected, simple Lie groups are obtained which enable one to prove that if μ is any central, continuous measure and n exceeds half the dimension of the Lie group, then $μ^n ∈ L^1$. When μ is a continuous, orbital measure then $μ^n$ is seen to belong to $L^2$. Lower bounds on the p-norms of characters are also obtained, and are used to show that, as in the abelian case, m-fold products of Sidon sets are not p-Sidon if p < 2m/(m+1).
LA - eng
KW - compact Lie group; central measures; Cartan-Weyl theory; irreducible representations; irreducible characters; simple Lie groups; Sidon sets
UR - http://eudml.org/doc/216489
ER -

References

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  1. [1] T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer, New York, 1985. Zbl0581.22009
  2. [2] D. Cartwright and J. McMullen, A structural criterion for the existence of infinite Sidon sets, Pacific J. Math. 96 (1981), 301-317. Zbl0445.43006
  3. [3] A. Dooley, Norms of characters and lacunarity for compact Lie groups, J. Funct. Anal. 32 (1979), 254-267. Zbl0404.43007
  4. [4] S. Giulini, P. Soardi and G. Travaglini, Norms of characters and Fourier series on compact Lie groups, ibid. 46 (1982), 88-101. Zbl0494.22009
  5. [5] K. Hare, Properties and examples of ( L p , L q ) multipliers, Indiana Univ. Math. J. 38 (1989), 211-227. Zbl0655.43003
  6. [6] K. Hare, Central Sidonicity for compact Lie groups, Ann. Inst. Fourier (Grenoble) 45 (1995), 547-564. Zbl0820.43003
  7. [7] K. Hare and D. Wilson, A structural criterion for the existence of infinite central Λ (p) sets, Trans. Amer. Math. Soc. 337 (1993), 907-925. Zbl0796.43003
  8. [8] K. Hare and D. Wilson, Weighted p-Sidon sets, J. Austral. Math. Soc. 61 (1996), 73-95. Zbl0874.43005
  9. [9] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, New York, 1972. Zbl0254.17004
  10. [10] J. Lopez and K. Ross, Sidon Sets, Lecture Notes in Pure and Appl. Math. 13, Marcel Dekker, New York, 1975. 
  11. [11] M. Marcus and G. Pisier, Random Fourier Series with Applications to Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., 1981. Zbl0474.43004
  12. [12] M. Mimura and H. Toda, Topology of Lie Groups, Transl. Math. Monographs 91, Amer. Math. Soc., Providence, R.I., 1991. Zbl0757.57001
  13. [13] D. Ragozin, Central measures on compact simple Lie groups, J. Funct. Anal. 10 (1972), 212-229. Zbl0286.43002
  14. [14] F. Ricci and G. Travaglini, L p - L q estimates for orbital measures and Radon transforms on compact Lie groups and Lie algebras, ibid. 129 (1995), 132-147. Zbl0843.43011
  15. [15] D. Rider, Central lacunary sets, Monatsh. Math. 76 (1972), 328-338. Zbl0258.43008
  16. [16] V. Varadarajan, Lie Groups, Lie Algebras and their Representations, Springer, New York, 1984. Zbl0955.22500
  17. [17] R. Vrem, L p -improving measures on hypergroups, in: Probability Measures on Groups, IX (Oberwolfach, 1988), Lecture Notes in Math. 1379, Springer, Berlin, 1989, 389-397. 

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