Berezin kernels and analysis on Makarevich spaces.

Jacques Faraut (1) , Michael Pevzner (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 LMR - Laboratoire de Mathématiques de Reims

Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels. We consider the conformal group Conf(V ) of a simple real Jordan algebra V . The maximal degenerate representations πs (s ∈ C) we shall study are induced by a character of a maximal parabolic subgroup P¯ of Conf(V ). These representations πs can be realized on a space Is of smooth functions on V. There is an invariant bilinear form Bs on the space Is . The problem we consider is to diagonalize this bilinear form Bs , with respect to the action of a symmetric subgroup G of the conformal group Conf(V ). This bilinear form can be written as an integral involving the Berezin kernel Bν , an invariant kernel on the Riemannian symmetric space G/K, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of B_\nu . From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity: $D(\nu)B_\nu = b(\nu)B_{\nu+1}, where D(\nu) is an invariant differential operator on G/K and b(\nu) is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from I_{−s} to I_s . Furthermore, we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group U of the conformal group Conf(V ).

Mots clés: Berezin Transform; Makarevich symmetric spaces

Domaines: Théorie des représentations [math.RT]

Jacques Faraut, Michael Pevzner. Berezin kernels and analysis on Makarevich spaces.. Indagationes Mathematicae, 2005, 16, pp.461-486. ⟨hal-00163356⟩