TY - JOUR
T1 - The absolute continuity of convolutions of orbital measures in symmetric spaces
AU - Gupta, Sanjiv Kumar
AU - Hare, Kathryn E.
N1 - Funding Information:
This research is supported in part by NSERC #44597. The first author thanks the University of Waterloo for their hospitality when some of this research was done.
Publisher Copyright:
© 2017 Elsevier Inc.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces G/K of Cartan type III, where G is a non-compact, connected Lie group and K is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, KzK, in G. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z. A consequence of our work is to show that the convolution product of any rank G/K, continuous, K-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type An or D3, when rank G/K +1 is needed.
AB - We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces G/K of Cartan type III, where G is a non-compact, connected Lie group and K is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, KzK, in G. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z. A consequence of our work is to show that the convolution product of any rank G/K, continuous, K-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type An or D3, when rank G/K +1 is needed.
KW - Absolute continuity
KW - Double coset
KW - Orbital measure
KW - Symmetric space
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U2 - 10.1016/j.jmaa.2017.01.027
DO - 10.1016/j.jmaa.2017.01.027
M3 - Article
AN - SCOPUS:85009754101
SN - 0022-247X
VL - 450
SP - 81
EP - 111
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
ER -