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1. CJM 2003 (vol 55 pp. 1155)
The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$ |
The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$ The main problem that is solved in this paper has the following simple
formulation (which is not used in its solution). The group $K =
\mathrm{O}_p ({\bf C}) \times \mathrm{O}_q ({\bf C})$ acts on the
space $M_{p,q}$ of $p\times q$ complex matrices by $(a,b) \cdot x =
axb^{-1}$, and so does its identity component $K^0 = \SO_p ({\bf C})
\times \SO_q ({\bf C})$. A $K$-orbit (or $K^0$-orbit) in $M_{p,q}$ is said
to be nilpotent if its closure contains the zero matrix. The closure,
$\overline{\mathcal{O}}$, of a nilpotent $K$-orbit (resp.\ $K^0$-orbit)
${\mathcal{O}}$ in $M_{p,q}$ is a union of ${\mathcal{O}}$ and some
nilpotent $K$-orbits (resp.\ $K^0$-orbits) of smaller dimensions. The
description of the closure of nilpotent $K$-orbits has been known for
some time, but not so for the nilpotent $K^0$-orbits. A conjecture
describing the closure of nilpotent $K^0$-orbits was proposed in
\cite{DLS} and verified when $\min(p,q) \le 7$. In this paper we
prove the conjecture. The proof is based on a study of two
prehomogeneous vector spaces attached to $\mathcal{O}$ and
determination of the basic relative invariants of these spaces.
The above problem is equivalent to the problem of describing the
closure of nilpotent orbits in the real Lie algebra $\mathfrak{so}
(p,q)$ under the adjoint action of the identity component of the real
orthogonal group $\mathrm{O}(p,q)$.
Keywords:orthogonal $ab$-diagrams, prehomogeneous vector spaces, relative invariants Categories:17B20, 17B45, 22E47 |
2. CJM 2002 (vol 54 pp. 595)
Lie Algebras of Pro-Affine Algebraic Groups We extend the basic theory of Lie algebras of affine algebraic groups
to the case of pro-affine algebraic groups over an algebraically
closed field $K$ of characteristic 0. However, some modifications
are needed in some extensions. So we introduce the pro-discrete
topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine
algebraic group $G$ over $K$, which is discrete in the
finite-dimensional case and linearly compact in general. As an
example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show
that the closure of $[L,L]$ in $\mathcal{L}(G)$ is algebraic in
$\mathcal{L}(G)$.
We also discuss the Hopf algebra of representative functions $H(L)$ of
a residually finite dimensional Lie algebra $L$. As an example, we
show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$
is connected, then the canonical Hopf algebra morphism from $K[G]$
into $H(L)$ is injective if and only if $L$ is algebraically dense
in $\mathcal{L}(G)$.
Categories:14L, 16W, 17B45 |
3. CJM 1998 (vol 50 pp. 929)
Decomposition varieties in semisimple Lie algebras The notion of decompositon class in a semisimple Lie algebra is a
common generalization of nilpotent orbits and the set of
regular semisimple elements. We prove that the closure of a
decomposition class has many properties in common with nilpotent
varieties, \eg, its normalization has rational singularities.
The famous Grothendieck simultaneous resolution is related to the
decomposition class of regular semisimple elements. We study the
properties of the analogous commutative diagrams associated to
an arbitrary decomposition class.
Categories:14L30, 14M17, 15A30, 17B45 |