Quotients, automorphisms and differential operators

Let $V$ be a $G$-module where $G$ is a complex reductive group. Let Z:=\quot VG denote the categorical quotient and let $\pi\colon V\to Z$ be the morphism dual to the inclusion \O(V)^G\subset\O(V). Let $\phi\colon Z\to Z$ be an algebraic automorphism. Then one can ask if there is an algebraic map $\Phi\colon V\to V$ which lifts $\phi$, i.e., $\pi(\Phi(v))=\phi(\pi(v))$ for all $v\in V$. In \cite{Kuttler} the case is treated where V=r\lieg is a multiple of the adjoint representation of $G$. It is shown that, for $r$ sufficiently large (often $r\geq 2$ will do), any $\phi$ has a lift. We consider the case of general representations (satisfying some mild assumptions). It turns out that it is natural to consider holomorphic lifting of holomorphic automorphisms of $Z$, and we show that if a holomorphic $\phi$ and its inverse lift holomorphically, then $\phi$ has a lift $\Phi$ which is an automorphism such that $\Phi(gv)=\sigma(g)\Phi(v)$, $v\in V$, $g\in G$ where $\sigma$ is an automorphism of $G$. We reduce the lifting problem to the group of automorphisms of $Z$ which preserve the natural grading of \O(Z)\simeq\O(V)^G. Lifting does not always hold, but we show that it always does for representations of tori in which case algebraic automorphisms lift to algebraic automorphisms. We extend Kuttler's methods to show lifting in case $V$ contains a copy of \lieg.Comment: 23 pages, minor revisions. To appear in J. London Math. Societ

A Policy Impact Evaluation Model For Scotland: Decoupling Single Farm Payments

The purpose of this paper is to assess the impacts of decoupling single farm payments in Scotland. It focuses on aggregate impacts on the agricultural products in domestic and external markets and the spill-over effect of this on the non-agricultural sector as well as an aggregate impact on the Scottish GDP. In order to capture system-wide impacts of the policy reform, a CGE model was formulated and implemented using a social accounting matrix constructed for Scotland. The simulation results suggest that the Scottish agricultural sector may encounter declines in output and factor us as a result of the policy reform. However, this critically depends on two factors: (a) the price effect of the policy reform on Scottish agricultural products relative to the EU average as well as the conditions of changes in world agricultural market prices; and (b) the extent to which customers would be sensitive to price effects of the policy reform. As far as the spill-over effect to the non-agricultural sector is concerned, decoupling of direct payments seems to have a positive spill-over effect. Similarly, the aggregate GDP effect is positive under all simulation scenarios. Critically, the simulation experiments indicate that policy shock may have a symmetrical outcome across the two sectors, with contractions in agriculture being accompanied by expansions in the non-agricultural sector, mainly because of factor market interactions between the two sectors.Cap reform; single farm payments; spill-over effects; Scotland

THE EFFECTS OF SINGLE FARM PAYMENTS ON SCOTTISH AGRICULTURE: A CGE MODELING APPROACH

Using a CGE model calibrated on Scottish data, this paper examines two important issues related to evaluating impacts of the Single Farm Payment. These are specification of product transformation functions and investigation into supply elasticity parameter. Simulation results from a standard CGE were compared with those from an alternative optimisation framework proposed in this study. The latter yielded a policy effect that is likely to represent behaviour of a profit maximising farmer. The parameter sensitivity analysis showed the important role differences in supply conditions can play; which implied a need for further econometric studies to estimate supply parameters.Single farm payments, decoupling, multi-output farming, farm types, CET function, CGE modelling, Agricultural Finance, Demand and Price Analysis, Farm Management, Research Methods/ Statistical Methods,