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We consider the action of a parabolic subgroup of the General Linear Group on a metabelian ideal. For those actions, we classify actions with finitely many orbits using methods from representation theory., Comment: 10 pages, 6 eps figures

Let $\bfG$ be a connected reductive algebraic group defined over $\F_q$, where $q$ is a power of a prime $p$ that is good for $\bfG$. Let $F$ be the Frobenius morphism associated with the $\FF_q$-structure on $\bfG$ and set $G = \bfG^F$, the fixed point subgroup of $F$. Let $\bfP$ be an $F$-stable parabolic subgroup of $\bfG$ and let $\bfU$ be the unipotent radical of $\bfP$; set $P = \bfP^F$ and $U = \bfU^F$. Let $G_\uni$ be the set of unipotent elements in $G$. In this note we show that the number of conjugacy classes of $U$ in $G_\uni$ is given by a polynomial in $q$ with integer coefficients., Comment: 9 pages, Minor changes and corrections

Let q be a power of a prime and n a positive integer. Let P(q) be a parabolic subgroup of the finite general linear group GL(n,q). We show that the number of P(q)-conjugacy classes in GL(n,q) is, as a function of q, a polynomial in q with integer coefficients. This answers a question of J. Alperin., Comment: 12 pages; to appear in Journal of Algebraic Combinatorics

Let $G$ be a simple algebraic group of classical type over an algebraically closed field $k$. Let $P$ be a parabolic subgroup of $G$ and let $\p = \Lie P$ be the Lie algebra of $P$ with Levi decomposition $\p = {\l}\oplus \u$, where $\u$ is the Lie algebra of the unipotent radical of $P$ and $\l$ is a Levi complement. Thanks to a fundamental theorem of R. W. Richardson, $P$ acts on $\u$ with an open dense orbit; this orbit is called the {\em Richardson orbit} and its elements are called {\em Richardson elements}. Recently, the first author gave constructions of Richardson elements in the case $k = \C$ for many parabolic subgroups $P$ of $G$. In this note, we observe that these constructions remain valid for any algebraically closed field $k$ of characteristic not equal to 2 and we give constructions of Richardson elements for the remaining parabolic subgroups., Comment: 20 pages

Let G be a simple algebraic group over an algebraically closed field k; assume that Char k is zero or good for G. Let \cB be the variety of Borel subgroups of G and let e in Lie G be nilpotent. There is a natural action of the centralizer C_G(e) of e in G on the Springer fibre \cB_e = {B' in \cB | e in Lie B'} associated to e. In this paper we consider the case, where e lies in the subregular nilpotent orbit; in this case \cB_e is a Dynkin curve. We give a complete description of the C_G(e)-orbits in \cB_e. In particular, we classify the irreducible components of \cB_e on which C_G(e) acts with finitely many orbits. In an application we obtain a classification of all subregular orbital varieties admitting a finite number of B-orbits for B a fixed Borel subgroup of G., Comment: 12 pages, to appear in Math Z

Let $k$ be an algebraically closed field. Let $B$ be the Borel subgroup of $\mGL_n(k)$ consisting of nonsingular upper triangular matrices. Let $\frb = \mLie B$ be the Lie algebra of upper triangular $n \times n$ matrices and $\fru$ the Lie subalgebra of $\frb$ consisting of strictly upper triangular matrices. We classify all Lie ideals $\frn$ of $\frb$, satisfying $\fru' \subseteq \frn \subseteq \fru$, such that $B$ acts (by conjugation) on $\frn$ with a dense orbit. Further, in case $B$ does not act with a dense orbit, we give the minimal codimension of a $B$--orbit in $\frn$. This can be viewed as a first step towards the difficult open problem of classifying of all ideals $\frn \subseteq \fru$ such that $B$ acts on $\frn$ with a dense orbit. The proofs of our main results require a translation into the representation theory of a certain quasi-hereditary algebra $\cA_{t,1}$. In this setting we find the minimal dimension of $\mExt^1_{\cA_{t,1}}(M,M)$ for a $\Delta$-good $\cA_{t,1}$--module of certain fixed $\Delta$-dimension vectors., Comment: 27 pages, 6 figures, uses epsfig, latexsym, amsfonts, amsmath, amsthm, xy

Let $G$ be a connected reductive algebraic group defined over the finite field $\FF_q$, where $q$ is a power of a good prime for $G$. We write $F$ for the Frobenius morphism of $G$ corresponding to the $\FF_q$-structure, so that $G^F$ is a finite group of Lie type. Let $P$ be an $F$-stable parabolic subgroup of $G$ and $U$ the unipotent radical of $P$. In this paper, we prove that the number of $U^F$-conjugacy classes in $G^F$ is given by a polynomial in $q$, under the assumption that the centre of $G$ is connected. This answers a question of J. Alperin in \cite{alperin}. In order to prove the result mentioned above, we consider, for unipotent $u \in G^F$, the variety $\CP^0_u$ of $G$-conjugates of $P$ whose unipotent radical contains $u$. We prove that the number of $\FF_q$-rational points of $\CP^0_u$ is given by a polynomial in $q$ with integer coefficients. Moreover, in case $G$ is split over $\FF_q$ and $u$ is split (in the sense of \cite[\S5]{shoji}), the coefficients of this polynomial are given by the Betti numbers of $\CP^0_u$. We also prove the analogous results for the variety $\CP_u$ consisting of conjugates of $P$ that contain $u$., Comment: minor changes; to appear in Trans. Amer. Math. Soc

Let g be a finite dimensional semisimple Lie algebra over C and e be a nilpotent element. Elashvili and Kac have recently classified all good Z-gradings for e. We instead consider good R-gradings, which are naturally parameterized by an open convex polytope in a Euclidean space arising from the reductive part of the centralizer of e in g. As an application, we prove that the isomorphism type of the finite W-algebra attached to a good R-grading for e is independent of the particular choice of good grading., Comment: 27 pages