Berezin Kernels and Analysis on Makarevich Spaces
| dc.creator | Faraut, J. | |
| dc.creator | Pevzner, M. | |
| dc.date | 2004-11-12 | |
| dc.date.accessioned | 2026-07-07T05:14:17Z | |
| dc.date.available | 2026-07-07T05:14:17Z | |
| dc.description | 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 ${\rm Conf}(V)$ of a simple real Jordan algebra $V$. The maximal degenerate representations $π_s$ ($s\in {\mathbb C}$) we shall study are induced by a character of a maximal parabolic subgroup $\bar P$ of ${\rm Conf}(V)$. These representations $π_s$ can be realized on a space $I_s$ of smooth functions on $V$. There is an invariant bilinear form ${\mathfrak B}_s$ on the space $I_s$. The problem we consider is to diagonalize this bilinear form ${\mathfrak B}_s$, with respect to the action of a symmetric subgroup $G$ of the conformal group ${\rm Conf}(V)$. This bilinear form can be written as an integral involving the Berezin kernel $B_ν$, 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 $B_ν$. From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity : $$D(ν)B_ν =b(ν)B_{ν+1},$$ where $D(ν)$ is an invariant differential operator on $G/K$ and $b(ν)$ 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 ${\rm Conf}(V)$. | |
| dc.identifier | https://arxiv.org/abs/math/0411294 | |
| dc.identifier | http://arxiv.org/abs/math/0411294 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73219 | |
| dc.subject | Representation Theory | |
| dc.subject | 43A35, 22E46 | |
| dc.title | Berezin Kernels and Analysis on Makarevich Spaces | |
| dc.type | text |