On Tits' Centre Conjecture for Fixed Point Subcomplexes

We give a short and uniform proof of a special case of Tits' Centre Conjecture using a theorem of J-P. Serre and a result from our earlier work. We consider fixed point subcomplexes $X^H$ of the building $X = X(G)$ of a connected reductive algebraic group $G$, where $H$ is a subgroup of $G$.Comment: 4 pages; minor changes, to appear in C. R. Acad. Sci. Paris Ser. I Mat

Complete Reducibility and Commuting Subgroups

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p. We study J.-P. Serre's notion of G-complete reducibility for subgroups of G. In particular, for a subgroup H and a normal subgroup N of H, we look at the relationship between G-complete reducibility of N and of H, and show that these properties are equivalent if H/N is linearly reductive, generalizing a result of Serre. We also study the case when H = MN with M a G-completely reducible subgroup of G which normalizes N. We show that if G is connected, N and M are connected commuting G-completely reducible subgroups of G, and p is good for G, then H = MN is also G-completely reducible.Comment: 21 pages; to appear in J. Reine Angew. Math. final for

SINFONI - Integral Field Spectroscopy at 50 milli-arcsecond resolution with the ESO VLT

SINFONI is an adaptive optics assisted near-infrared integral field spectrometer for the ESO VLT. The Adaptive Optics Module (built by the ESO Adaptive Optics Group) is a 60-elements curvature-sensor based system, designed for operations with natural or sodium laser guide stars. The near-infrared integral field spectrometer SPIFFI (built by the Infrared Group of MPE) provides simultaneous spectroscopy of 32 x 32 spatial pixels, and a spectral resolving power of up to 3300. The adaptive optics module is in the phase of integration; the spectrometer is presently tested in the laboratory. We provide an overview of the project, with particular emphasis on the problems encountered in designing and building an adaptive optics assisted spectrometer.Comment: This paper was published in Proc. SPIE, 4841, pp. 1548-1561 (2003), and is made available as an electronic reprint with permission of SPIE. Copyright notice added to first page of articl

Complete reducibility and separability

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module.Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and G is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1

A Geometric approach to complete reducibility

Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert–Mumford–Kempf from geometric invariant theory. We deduce that a normal subgroup of a G-completely reducible subgroup of G is again G-completely reducible, thereby providing an affirmative answer to a question posed by J.-P. Serre, and conversely we prove that the normalizer of a G-completely reducible subgroup of G is again G-completely reducible. Some rationality questions and applications to the spherical building of G are considered. Many of our results extend to the case of non-connected G

Community Psychology in higher education in Europe: Results of a survey and discussion of the basic competency approach

This report from the Standing Committee on Community Psychology of the European Federation of Psychological Association provides an overview of higher education in Community Psychology (CP) in 14 European countries. Our findings show that 10 countries have some kind of CP teaching in their educational system. Twenty European universities offer a CP-oriented Master degree, two universities at the Bachelor level and 16 universities also have CP-oriented Ph.D. programmes. The profiles of the universities focus on two areas: Community psychology in a pure form and a combination of social psychology and community psychology. The other universities vary between clinical, organisational psychology, and a pedagogical focus. Within a certain European emphasis, these universities are analysing and changing the social conditions of community life and mental health. The responding universities failed to report adequately on comprehensive core competences and key elements in CP. To compensate for this deficit, the Standing Committee on Community Psychology proposes to develop a primer of basic CP competences for inclusion in programmes like EuroPsy