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Let $J$ be a (complex) semi-simple Jordan algebra, and consider the
action of
the automorphism group on the $n$-fold product of $J$ via the diagonal
action.
In the talk, geometric properties of this action are studied. In
particular, a
characterization of the closed orbits is given.
In the case of a complex reductive linear algebraic group and the
adjoint action
on its Lie algebra, the closed orbits are precisely the orbits through
semi-simple
elements. More generally, a result of R.W. Richardson characterizes the
closed
orbits of the diagonal action on the $n$-fold product of the Lie
algebra.
A similar condition can be found in the case of Jordan algebras. It
turns out that
the orbit through an $n$-tuple $x=(x_1,\ldots, x_n)$ is closed if and
only if the
Jordan subalgebra generated by $x_1,\ldots, x_n$ is semi-simple.
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Ghislain Fourier (Glasgow)
PBW filtrations, posets and symmetric functions
I will recall the PBW filtration on cyclic modules for a complex Lie algebra and the results known for irreducible modules for simple complex Lie algebras. Then I will generalize the construction of the graded modules and apply it to cyclic modules for the truncated current algebra. This will link conjectures on fusion products to conjectures on posets of symmetric functions.
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Dynkin has given an algorithm to classify the regular semisimple
subalgebras of a complex semisimple Lie algebra, up to conjugacy
by the inner automorphism group. Here we show how this can be
extended to semisimple Lie algebras defined over the real numbers.
Vinberg has shown that a classification of a certain type of regular
subalgebras (called carrier algebras) in a graded semisimple Lie algebra,
yields a classification of the nilpotent orbits in a homogeneous component
of the graded Lie algebra. Our methods can also be used to classify the
carrier algebras in a real graded semisimple Lie algebra. At the end we
will discuss what needs to be done to obtain a classification of the
nilpotent orbits from that. Such classifications of nilpotent orbits have
applications in differential geometry and in theoretical physics.
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t.b.a
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The geometric side of the Arthur-Selberg trace formula expresses a certain distribution $J(f)$ on an adelic reductive group as a sum of integrals of the test function $f$ over conjugacy classes with respect to certain non-invariant measures. Those measures are known only in special cases. I will present an approach to express them in terms of prehomogeneous zeta integrals. This has been realised for groups of rank up to 2. The problem is that $J(f)$ is defined as a sum indexed by cosets in parabolic subgroups with respect to their unipotent radicals, which is incompatible with the decomposition into conjugacy classes. The rearrangement uses induction of conjugacy classes, Siegel's mean value formula and canonical parabolics.
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The set $\mathcal{SUB}(G)$ of all closed subgroups of
any locally compact group $G$ carries a
canonical compact Hausdorff topology (nowadays
called $\textit{Chabauty topology}$).
In order to sample recent interest in this
functorial parameter,
Let $\mu_G\colon G\to \mathcal{SUB}(G)$ denote the function
which attaches to an element $g$ of $G$ the closed subgroup $\% \langle g \rangle$
generated by it. It is shown that $G$ is totally disconnected if and only if
$\mu_G$ is continuous.
Other functions $G\to\mathcal{SUB}(G)$ which associate
with an element of $G$ in a natural way a closed subgroup of $G$
are
$$\matrix{%
{\rm lev}_G(g){=}&\left\{x\in G \mid\{g^kxg^{-k}\}_{k\in{\bf Z}}
\hbox{ is precompact}\right\},%
\hbox{ the $\textit{Levi}$ subgroup of }g;\hbox{ and }\cr
{\rm par}_G(g){=}&\left\{x\in G \mid \{g^kxg^{-k}\}_{k\in{\bf N}} \hbox{ is precompact}\right\},\hbox{ the $\textit{parabolic}$ subgroup of }g.\cr}$$
They are shown to be continuous for totally disconnected $G$.
Other functions $G\to\mathcal{SUB}(G)$ which are natural fail to be continuous
even if $G$ is totally disconnected.
(Joint work with George W. Willis)
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A pseudoreflection group is a finite linear group over the real
numbers generated by transformations with codimension two fixed point subspace.
Such groups naturally arise in the theory of orbifolds and are closely related
to reflection groups. We explain their classification and characterize them in
terms of quotient spaces.
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Let $B$ be a Borel subgroup of a semisimple algebraic group $G$, and let
$\mathfrak a$ be an abelian ideal of $\mathfrak b=Lie(B)$. The ideal
$\mathfrak a$ is determined by a certain subset $\Delta_{\mathfrak a}$
of positive roots, and using $\Delta_{\mathfrak a}$ we give an explicit
classification of the $B$-orbits in $\mathfrak a$ and $\mathfrak a^*$.
Our description visibly demonstrates that there are finitely many $B$-orbits
in both cases. Then we describe the Pyasetskii correspondence between
the $B$-orbits in $\mathfrak a$ and $\mathfrak a^*$ and the invariant
algebras $k[\mathfrak a]^U$ and $k[\mathfrak a^*]^U$, where $U=(B,B)$.
As an application, the number of $B$-orbits in the abelian nilradicals is
computed. We also discuss related results of A.~Melnikov and others for
classical groups and state a general conjecture on the closure and dimension
of the $B$-orbits in the abelian nilradicals, which exploits a relationship
between between $B$-orbits and involutions in the Weyl group.
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In his seminal doctoral dissertation Kirillov constructed a bijection between the unitary dual $\hat{N}$ of a simply connected nilpotent Lie group $N$ and the orbit space $\frak{n}^*/N$ (orbit method). Later, Bernat extended this result to exponential groups $G$.
Traditionally one constructs a map $\frak{g}^*/G\rightarrow \hat{G}$ using (Pukanszky) polarizations. It requires some work to see the independence of the chosen polarizations.
Here we construct a family of bijections $\kappa _G:\hat{G}\rightarrow \frak{g}^*/G$, $G$ an exponential Lie group, in the opposite direction. This construction depends on the choice of an abelian ideal, which in a way is a milder arbitrariness. But still the independence has to be established.
Furthermore, the family $(\kappa _G)$ is canonical in the sense that it can be (uniquely) characterized in terms of a short list of plausible properties. If one restricts to nilpotent groups one has, with some extra work, an even nicer characterization of this family $(\kappa _N)$.
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This is a joint work with Victor Gerasimov (University of Belo Horisonte, Brasil).
Let a discrete group $G$ possess two convergence actions by homeomorphisms on compacta $X$ and $Y$. Consider
the following question: does there exist a convergence action of $G$ on a compactum $Z$ and continuous equivariant
maps $X$ $\leftarrow$ $Z$ $\rightarrow$ $Y$ ? We call the space $Z$ (and action of $G$ on it) pullback space (action). In such general setting
a negative answer follows from a recent result of O. Baker and T. Riley.
Suppose, in addition, that the initial actions are relatively hyperbolic that is they are non-parabolic and the
induced action on the space of distinct pairs of points is cocompact. In the case when $G$ is finitely generated the
universal pullback space exists by a theorem of V. Gerasimov.
We show that the situation drastically changes already in the case of countable non-finitely generated groups.
We provide an example of two relatively hyperbolic actions of the free group $G$ of countable rank for which the
pullback action does not exist.
Our main result is that the pullback space exists for two relatively hyperbolic actions of any group $G$ if and
only if the maximal parabolic subgroups of one of the actions are dynamically quasiconvex for the other one.
We study an analog of the geodesic flow for a large subclass of convergence groups including the relatively
hyperbolic ones. The obtained results imply the main result and seem to have an independent interest.
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We establish a surprising link between two a priori completely unrelated
objects: The space of isometry classes of separation coordinates for the
Hamilton-Jacobi equation on an n-dimensional sphere one one hand and
the Deligne-Mumford moduli space $M_{0,n+2}$ of stable algebraic curves of
genus zero with $n+2$ marked points on the other hand. This relation is
proved by realising separation coordinates as maximal abelian subalgebras
in a representation of the Kohno-Drinfeld Lie algebra. We use the rich combinatorial
structure of $M_{0,n+2}$ and the closely related Stasheff polytopes
in order to classify the different canonical forms of separation coordinates.
Moreover, we infer an explicit construction for separation coordinates and
the corresponding quadratic integrals from the mosaic operad on $M_{0,n+2}$.
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We will study Kahlerian coadjoint orbits of Hermitian Lie groups (e.g. the real
symplectic group $Sp(n,R)$, $SU(p,q)$ and others) and their convex hulls. It is known
that the momentum images of these orbits in a compact Cartan subalgebra are
convex polyhedra. Furthermore we will see that the faces of the convex hull
of an orbit are exposed and are given by the face structure of the momentum image.
