Symmetry breaking operators for strongly spherical reductive pairs and the Gross-Prasad conjecture for complex orthogonal groups

A real reductive pair $(G,H)$ is called strongly spherical if the homogeneous space $(G\times H)/{\rm diag}(H)$ is real spherical. This geometric condition is equivalent to the representation theoretic property that ${\rm dim\,Hom}_H(\pi|_H,\tau)<\infty$ for all smooth admissible representations $\pi$ of $G$ and $\tau$ of $H$. In this paper we explicitly construct for all strongly spherical pairs $(G,H)$ intertwining operators in ${\rm Hom}_H(\pi|_H,\tau)$ for $\pi$ and $\tau$ spherical principal series representations of $G$ and $H$. These so-called \textit{symmetry breaking operators} depend holomorphically on the induction parameters and we further show that they generically span the space ${\rm Hom}_H(\pi|_H,\tau)$. In the special case of multiplicity one pairs we extend our construction to vector-valued principal series representations and obtain generic formulas for the multiplicities between arbitrary principal series. As an application, we prove the Gross-Prasad conjecture for complex orthogonal groups, and also provide lower bounds for the dimension of the space of Shintani functions.Comment: 57 page

Structural change in Europe's rural regions: Farm livelihoods between subsistence orientation, modernisation and non-farm diversification

The contributions in this edited volume constitute the mini-symposium on 'Structural change in Europe's rural regions - Farm livelihoods between subsistence orientation, modernization and non-farm diversification' at the international conference of the International Association of Agricultural Economists (IAAE) on 'The New Landscape of Global Agriculture' in Beijing, China, August 16-22, 2009. Table of contents: Can we really talk about structural change? The issue of small-scale farms in rural Poland; Tomasz Wolek (WUDES) ...1-22 The role of farm activities for overcoming rural poverty in Romania; Cosmin Salasan (USAMVB) & Jana Fritzsch (IAMO)...23-41 Comparative Analysis of the contribution of subsistence production to household incomes in five EU New Member States: Lessons learnt; Sophia Davidova (UNIKENT), Lena Fredriksson (UNIKENT), Matthew Gorton (UNEW), Plamen Mishev (UNWE) & Dan Petrovici (UNIKENT)...43-68 The flexibility of family farms in Poland; Swetlana Renner, Heinrich Hockmann, Agata Pieniadz, & Thomas Glauben (IAMO)...69-89 Agriculture and rural structural change: An analysis of the experience of past accessions in selected EU15 regions; Carmen Hubbard & Matthew Gorton (UNEW)...91-112 Expanding biogas production in Germany and Hungary: Good prospects for small scale farms?; Lioudmila Möller (IAMO)...113-133 Impact of topical policies on the future of small-scale farms in Poland - A multiobjective approach; Stefan Wegener (IAMO), Jana Fritzsch (IAMO), Gertrud Buchenrieder (IAMO), Jarmila Curtiss (IPTS), & Sergio Gomez y Paloma (IPTS)...135-160 --