17 research outputs found
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 of the building of a
connected reductive algebraic group , where is a subgroup of .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