|
Isaac Newton Institute for Mathematical Sciences Programme 2009:
Algebraic
Lie Theory.
ALT Report
DFG project: Serre's notion of Complete Reducibility and Geometric Invariant Theory (within the
DFG Priority Programme in Representation Theory).
Past EPSRC-funded research
projects.
Publications on
MathSciNet.
Recent preprints on the ArXiv.
Latest preprints:
Closed Orbits and uniform S-instability in Geometric Invariant Theory
with M. Bate, B. Martin, and R. Tange
Abstract:
In this paper we consider various problems involving the action of a reductive group G on an affine variety V. We prove some general rationality results about the G-orbits in V. In addition, we extend fundamental results of Kempf and Hesselink regarding optimal destabilizing parabolic subgroups of G for such general G-actions.
We apply our general rationality results to answer a question of Serre concerning how his notion of G-complete reducibility behaves under separable field extensions. Applications of our new optimality results also include a construction which allows us to associate an optimal destabilizing parabolic subgroup of G to any subgroup of G. Finally, we use these new optimality techniques to provide an answer to Tits' Centre Conjecture in a special case.
math.RT/0904.4853
Complete Reducibility and Conjugacy classes of tuples in Algebraic Groups and Lie algebras
with M. Bate, B. Martin, and R. Tange
Abstract:
Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on G^n, the n-fold Cartesian product of G with itself, by simultaneous conjugation.
We give a purely algebraic characterization of the closed H-orbits in G^n, generalizing work of Richardson which treats the case H = G.
This characterization turns out to be a natural generalization of Serre's notion of G-complete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We also consider some rationality questions.
math.RT/0905.0065
On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type
with Simon M. Goodwin and Glenn Ubly
Abstract:
We consider the finite $W$-algebra $U(\g,e)$ associated to
a nilpotent element $e \in \g$ in a simple complex Lie algebra $\g$ of
exceptional type. Using presentations obtained through an algorithm
based on the PBW-theorem, we verify a conjecture of Premet, that
$U(\g,e)$ always has a 1-dimensional representation, when $\g$ is of
type $G_2$, $F_4$, $E_6$ or $E_7$. Thanks to a theorem of Premet, this
allows one to deduce the existence of minimal dimension representations
of reduced enveloping algebras of modular Lie algebras of the above
types. In addition, we deduce that there exists a completely prime
primitive ideal in $U(\g)$ whose associated variety is the coadjoint
orbit corresponding to $e$.
math.RT/0905.3714
Former and current PhD students:
Simon Goodwin
(University of Birmingham)
Michael Bate
(University of York)
Russell Fowler (Npower)
Daniel Gold
Glenn Ubly
Sebastian Herpel
Editorial Activity:
Tbilisi
Mathematical Journal.
Computer
Programs
| 
