Session 18. Harmonic analysis, orthogonal expansions and Dunkl theory

Convolution of orbital measures on symmetric spaces

Piotr Graczyk, LAREMA Université d'Angers, 
The presented results are based on a series of recent articles common with P. Sawyer(Laurentian University, Sudbury, Canada). We consider a Riemannian symmetric space of non-compact type \(G/K\) and we study the existence of the density of the convolution \[\delta_{e^X}^\natural \star\delta_{e^Y}^\natural\] of two orbital measures, when \(X\) and \(Y\) are singular elements of the Cartan space. This density intervenes in the product formula for spherical functions. Studying of its existence is also motivated by applications in probability theory on \(G/K\). We will survey our earlier results in this area: the existence of the density when \(X\) is regular and \(Y\not=0\) and the characterization of singular \(X,Y\) such that the density exists for the spaces \({\bf SL}(n,{\bf F})/ {\bf SU}(n,{\bf F})\), as well as their complex and quaternionic versions. Our recent results concern the symmetric spaces of type \(B_p\), \(C_p\) et \(D_p\), i.e. the non-compact Grassmanians \({\bf SO}_0(p,q)/{\bf SO}(p)\times{\bf SO}(q)\) and the symmetric spaces \({\bf SO}_0(p,p)/{\bf SO}(p)\times{\bf SO}(p)\), \({\bf SU}(p,p)/{\bf S}({\bf U}(p)\times{\bf U}(p))\) and \({\bf Sp}(p,p)/{\bf Sp}(p)\times{\bf Sp}(p)\). We will finish by discussing analogous problems on symmetric spaces of Euclidean type and of compact type.
References
  1. P. Graczyk, P. Sawyer, Convolution of orbital measures on symmetric spaces of type \(C_p\) and \(D_p\), submitted 2014, http://hal.archives-ouvertes.fr/hal-00965263, arXiv : 1403.6098
  2. P. Graczyk, P. Sawyer, On the product formula on noncompact Grassmannians, Coll.Math. 133 (2013), 145-167.
  3. P. Graczyk, P. Sawyer, A sharp criterion for the existence of the product formula on symmetric spaces of type \(A_n\), J. Lie Theory 20 (2010), 751-766.
  4. P. Graczyk, P. Sawyer, Absolute continuity of convolutions of orbital measures on Riemannian symmetric spaces, Journal of Functional Analysis 259 (2010), 1759-1770.
  5. P. Graczyk and P. Sawyer,The product formula for the spherical functions on symmetric spaces of noncompact type, J. of Lie Theory, 13(2003), p. 247-261.
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