Let G be a free group with r generators, 1 < r < ∞. All the eigenfunctions of an operator on G which plays the same role of the Laplace Beltrami operator on semisimple Lie groups are characterized. Furthermore, an analytic family of representations πz of G on functions on the boundary Ω is considered, defined by π_z(x)ƒ(ω) = p^z(x, ω)ƒ(x^{−1}ω), where p(x, ω) is the Poisson kernel relative to the action of G on Ω. It is proved that, for 0 < s = Re z < 1, π_z is uniformly bounded on an appropriate Hilbert space H_s(Ω). Finally the uniform boundedness of other special representations of G, obtained by considering the free group either as a subgroup of the group of all isometries of a tree or as a subgroup of GL(2, Q_p) is proved.
The Poisson transform and representations of a free group
MANTERO, ANNA MARIA;ZAPPA, ANNA
1983-01-01
Abstract
Let G be a free group with r generators, 1 < r < ∞. All the eigenfunctions of an operator on G which plays the same role of the Laplace Beltrami operator on semisimple Lie groups are characterized. Furthermore, an analytic family of representations πz of G on functions on the boundary Ω is considered, defined by π_z(x)ƒ(ω) = p^z(x, ω)ƒ(x^{−1}ω), where p(x, ω) is the Poisson kernel relative to the action of G on Ω. It is proved that, for 0 < s = Re z < 1, π_z is uniformly bounded on an appropriate Hilbert space H_s(Ω). Finally the uniform boundedness of other special representations of G, obtained by considering the free group either as a subgroup of the group of all isometries of a tree or as a subgroup of GL(2, Q_p) is proved.
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