TY - JOUR
T1 - Convolution of orbital measures in symmetric spaces
AU - Anchouche, Boudjemǎa
AU - Gupta, Sanjiv Kumar
PY - 2011/6
Y1 - 2011/6
N2 - Let G/K be a noncompact symmetric space, Gc/K its compact dual, g = t ⊕ p the Cartan decomposition of the Lie algebra g of G, a a maximal abelian subspace of a, H be an element of a, a=exp (H) , and ac =exp (iH). In this paper, we prove that if for some positive integer r, v acr is absolutely continuous with respect to the Haar measure on Gc, then vra is absolutely continuous with respect to the left Haar measure on G, where ac (respectively a) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare ['Singular dichotomy for orbital measures on complex groups', Boll.Unione Mat.Ital.(9)III(2010), 409-419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.
AB - Let G/K be a noncompact symmetric space, Gc/K its compact dual, g = t ⊕ p the Cartan decomposition of the Lie algebra g of G, a a maximal abelian subspace of a, H be an element of a, a=exp (H) , and ac =exp (iH). In this paper, we prove that if for some positive integer r, v acr is absolutely continuous with respect to the Haar measure on Gc, then vra is absolutely continuous with respect to the left Haar measure on G, where ac (respectively a) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare ['Singular dichotomy for orbital measures on complex groups', Boll.Unione Mat.Ital.(9)III(2010), 409-419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.
KW - convolution
KW - double coset
KW - orbital measures
KW - symmetric spaces
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U2 - 10.1017/S0004972710002017
DO - 10.1017/S0004972710002017
M3 - Article
AN - SCOPUS:79956151490
SN - 0004-9727
VL - 83
SP - 470
EP - 485
JO - Bulletin of the Australian Mathematical Society
JF - Bulletin of the Australian Mathematical Society
IS - 3
ER -