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 --

Minimal representations via Bessel operators

We construct an L^2-model of "very small" irreducible unitary representations of simple Lie groups G which, up to finite covering, occur as conformal groups Co(V) of simple Jordan algebras V. If $V$ is split and G is not of type A_n, then the representations are minimal in the sense that the annihilators are the Joseph ideals. Our construction allows the case where G does not admit minimal representations. In particular, applying to Jordan algebras of split rank one we obtain the entire complementary series representations of SO(n,1)_0. A distinguished feature of these representations in all cases is that they attain the minimum of the Gelfand--Kirillov dimensions among irreducible unitary representations. Our construction provides a unified way to realize the irreducible unitary representations of the Lie groups in question as Schroedinger models in L^2-spaces on Lagrangian submanifolds of the minimal real nilpotent coadjoint orbits. In this realization the Lie algebra representations are given explicitly by differential operators of order at most two, and the key new ingredient is a systematic use of specific second-order differential operators (Bessel operators) which are naturally defined in terms of the Jordan structure