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CONVOLUTION OF ORBITAL MEASURES ON SYMMETRIC SPACES OF TYPE $C_{p}$ AND $D_{p}$

  • P. GRACZYK (a1) and P. SAWYER (a2)

Abstract

We study the absolute continuity of the convolution ${\it\delta}_{e^{X}}^{\natural }\star {\it\delta}_{e^{Y}}^{\natural }$ of two orbital measures on the symmetric spaces $\mathbf{SO}_{0}(p,p)/\mathbf{SO}(p)\times \mathbf{SO}(p)$ , $\mathbf{SU}(p,p)/\mathbf{S}(\mathbf{U}(p)\times \mathbf{U}(p))$ and $\mathbf{Sp}(p,p)/\mathbf{Sp}(p)\times \mathbf{Sp}(p)$ . We prove sharp conditions on $X$ , $Y\in \mathfrak{a}$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions.

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References

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[1]Flensted-Jensen, M. and Koornwinder, T., ‘The convolution structure for Jacobi function expansions’, Ark. Mat. 11 (1973), 245262.
[2]Friedberg, S. H., Insel, A. J. and Spence, L. E., Linear Algebra (Prentice-Hall, Upper Saddle River, NJ, 1997).
[3]Graczyk, P., Loeb, J. J. and Zak, T., ‘Strong central limit theorem for isotropic random walks in Rd’, Probab. Theory Related Fields 151(1–2) (2011), 153172.
[4]Graczyk, P. and Sawyer, P., ‘The product formula for the spherical functions on symmetric spaces of noncompact type’, J. Lie Theory 13 (2003), 247261.
[5]Graczyk, P. and Sawyer, P., ‘Absolute continuity of convolutions of orbital measures on Riemannian symmetric spaces’, J. Funct. Anal. 259 (2010), 17591770.
[6]Graczyk, P. and Sawyer, P., ‘A sharp criterion for the existence of the density in the product formula on symmetric spaces of type A n’, J. Lie Theory 20 (2010), 751766.
[7]Graczyk, P. and Sawyer, P., ‘On the product formula on noncompact Grassmannians’, Colloq. Math. 133 (2013), 145167.
[8]Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press, New York, 1978).
[9]Helgason, S., Groups and Geometric Analysis (Academic Press, Orlando, FL, 1984).
[10]Helgason, S., Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs, 83 (American Mathematical Society, Providence, RI, 1994).
[11]Jaworski, W., ‘Strong approximate transitivity, polynomial growth, and spread out random walks on locally compact groups’, Pacific J. Math. 170(2) (1995), 517533.
[12]Jaworski, W. and Raja, C. R. E., ‘The Choquet–Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth’, New York J. Math. 13 (2007), 159174.
[13]Koornwinder, T., ‘Jacobi polynomials. II. An analytic proof of the product formula’, SIAM J. Math. Anal. 5 (1974), 125137.
[14]Lin, M. and Wittmann, R., ‘Convolution powers of spread-out probabilities’, Ann. Inst. Henri Poincaré Probab. Stat. 32(5) (1996), 661667.
[15]Ostrovskii, I. V., ‘Description of the I 0class in a special semigroup of probability measures’, Sov. Math. Dokl. 14 (1973), 525529 (transl. Dokl. Akad. Nauk SSSR 209 (1973), 788–791).
[16]Rösler, M., ‘Positive convolution structure for a class of Heckman–Opdam hypergeometric functions of type BC’, J. Funct. Anal. 258(8) (2010), 27792800.
[17]Sawyer, P., ‘Spherical functions on SO0(p, q)∕SO(p) ×SO(q)’, Canad. Math. Bull. 42(4) (1999), 486498.
[18]Trukhina, I. P., ‘Arithmetic of spherically symmetric measures in Lobachevskij space’, Teor. Funkts. Funkts. Anal. Prilozh. 34 (1980), 136146.
[19]Voit, M., ‘Factorization of probability measures on symmetric hypergroups’, J. Aust. Math. Soc. 50(3) (1991), 417467.
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CONVOLUTION OF ORBITAL MEASURES ON SYMMETRIC SPACES OF TYPE $C_{p}$ AND $D_{p}$

  • P. GRACZYK (a1) and P. SAWYER (a2)

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