Abstract
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 ac r is absolutely continuous with respect to the Haar measure on Gc, then vr a 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.
| Original language | English |
|---|---|
| Pages (from-to) | 470-485 |
| Number of pages | 16 |
| Journal | Bulletin of the Australian Mathematical Society |
| Volume | 83 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - Jun 2011 |
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Keywords
- convolution
- double coset
- orbital measures
- symmetric spaces
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Convolution of orbital measures in symmetric spaces. / Anchouche, Boudjemǎa; Gupta, Sanjiv Kumar.
In: Bulletin of the Australian Mathematical Society, Vol. 83, No. 3, 06.2011, p. 470-485.Research output: Contribution to journal › Article
}
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 ac r is absolutely continuous with respect to the Haar measure on Gc, then vr a 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 ac r is absolutely continuous with respect to the Haar measure on Gc, then vr a 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
UR - http://www.scopus.com/inward/record.url?scp=79956151490&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=79956151490&partnerID=8YFLogxK
U2 - 10.1017/S0004972710002017
DO - 10.1017/S0004972710002017
M3 - Article
AN - SCOPUS:79956151490
VL - 83
SP - 470
EP - 485
JO - Bulletin of the Australian Mathematical Society
JF - Bulletin of the Australian Mathematical Society
SN - 0004-9727
IS - 3
ER -