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Simon M. Goodwin

Publications and source records attributed to Simon M. Goodwin.

32 records · Page 2Linked to original sources
arXiv2011

On changing highest weight theories for finite W-algebras

A highest weight theory for a finite W-algebra U(g,e) was developed in [BGK]. This leads to a strategy for classifying the irreducible finite dimensional U(g,e)-modules. The highest weight theory depends on the choice of a parabolic subalgebra of g leading to different parameterizations of the finite dimensional irreducible U(g,e)-modules. We explain how to construct an isomorphism preserving bijection between the parameterizing sets for different choices of parabolic subalgebra when g is of type A, or when g is of types C or D and e is an even multiplicity nilpotent element

math.RT↗
arXiv2010

Finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent orbits in classical Lie algebras

We consider finite W-algebras U(g,e) associated to even multiplicity nilpotent elements in classical Lie algebras. We give a classification of finite dimensional irreducible U(g,e)-modules with integral central character in terms of the highest weight theory for finite W-algebras. As a corollary, we obtain a parametrization of primitive ideals of U(g) with associated variety the closure of the adjoint orbit of e and integral central character.

math.RT↗
arXiv2010

On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

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, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal in $U(\g)$ whose associated variety is the coadjoint orbit corresponding to $e$.

math.RT↗
arXiv2009

Translation for finite W-algebras

A finite $W$-algebra $U(\g,e)$ is a certain finitely generated algebra that can be viewed as the enveloping algebra of the Slodowy slice to the adjoint orbit of a nilpotent element $e$ of a complex reductive Lie algebra $\g$. It is possible to give the tensor product of a $U(\g,e)$-module with a finite dimensional $U(\g)$-module the structure of a $U(\g,e)$-module; we refer to such tensor products as translations. In this paper, we present a number of fundamental properties of these translations, which are expected to be of importance in understanding the representation theory of $U(\g,e)$.

math.RT↗
arXiv2008

Calculating conjugacy classes in Sylow $p$-subgroups of finite Chevalley groups

In earlier work, the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow $p$-subgroup $U(q)$ of a finite Chevalley group $G(q)$, valid when $q$ is a power of a good prime for $G(q)$. In this paper we develop this algorithm and discuss an implementation in the computer algebra language {\sf GAP}. Using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in $U(q)$, when $G(q)$ is of rank at most 6. In these cases, we observe that the number of conjugacy classes of $U(q)$ is given by a polynomial in $q$ with integer coefficients.

math.GR↗
arXiv2008

On conjugacy of unipotent elements in finite groups of Lie type

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.

math.GR↗
arXiv2008

Highest weight theory for finite W-algebras

We define analogues of Verma modules for finite W-algebras. By the usual ideas of highest weight theory, this is a first step towards the classification of finite dimensional irreducible modules. Motivated by known results in type A, we then formulate some precise conjectures in the case of nilpotent orbits of standard Levi type.

math.RT↗
arXiv2007

Orbits of parabolic subgroups on metabelian ideals

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.

math.RT↗
arXiv2007

Parabolic conjugacy in general linear groups

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.

math.GR↗
arXiv2007

The orbit structure of Dynkin curves

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.

math.GR↗
arXiv2006

Richardson elements for parabolic subgroups of classical groups in positive characteristic

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 $\p = ł\oplus \u$, where $\u$ is the Lie algebra of the unipotent radical of $P$ and $ł$ 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.

math.RT↗
arXiv2006

Prehomogeneous spaces for Borel subgroups of general linear groups

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 $Δ$-good $\cA_{t,1}$--module of certain fixed $Δ$-dimension vectors.

math.RT↗
arXiv2005

Good grading polytopes

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.

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