Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in  $W$ . It is known that the complement $X(\mathscr{A}(W))$ of the reflection arrangement $\mathscr{A}(W)$ is a $K(\unicode[STIX]{x1D70B},1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr{A}(W)$ , let $X(\mathscr{A}(W)^{Y})$ be the complement in $Y$ of the hyperplanes in $\mathscr{A}(W)$ not containing  $Y$ . We hope that $X(\mathscr{A}(W)^{Y})$ is always a $K(\unicode[STIX]{x1D70B},1)$ . We prove it in case of the monomial groups $W=G(r,p,\ell )$ . Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.

Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If $G= {GL }_n$ , then there is a degeneration process for obtaining from H a completely reducible subgroup $H'$ of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a G-completely reducible subgroup $H'$ of G, unique up to $G(k)$ -conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL _n$ and with Serre’s ‘G-analogue’ of semisimplification for subgroups of $G(k)$ from [19]). We also show that under some extra hypotheses, one can pick $H'$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf, and Rousseau.

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is said to be spherical if it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer’s list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Krämer’s classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.

Let G be a connected reductive algebraic group defined over an algebraically closed field of characteristic 0. We consider the commuting variety of the nilradical of the Lie algebra of a Borel subgroup B of G. In case B acts on with only a finite number of orbits, we verify that is equidimensional and that the irreducible components are in correspondence with the distinguishedB-orbits in . We observe that in general is not equidimensional, and determine the irreducible components of in the minimal cases where there are infinitely many B-orbits in .

Let $G(q)$ be a finite Chevalley group, where $q$ is a power of a good prime $p$ , and let $U(q)$ be a Sylow $p$ -subgroup of $G(q)$ . Then a generalized version of a conjecture of Higman asserts that the number $k(U(q))$ of conjugacy classes in $U(q)$ is given by a polynomial in $q$ with integer coefficients. In [S. M. Goodwin and G. Röhrle, J. Algebra 321 (2009) 3321–3334], the first and the third authors of the present paper developed an algorithm to calculate the values of $k(U(q))$ . By implementing it into a computer program using $\mathsf{GAP}$ , they were able to calculate $k(U(q))$ for $G$ of rank at most five, thereby proving that for these cases $k(U(q))$ is given by a polynomial in $q$ . In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of $k(U(q))$ for finite Chevalley groups of rank six and seven, except $E_7$ . We observe that $k(U(q))$ is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write $k(U(q))$ as a polynomial in $q-1$ , then the coefficients are non-negative.

Under the assumption that $k(U(q))$ is a polynomial in $q-1$ , we also give an explicit formula for the coefficients of $k(U(q))$ of degrees zero, one and two.

Suppose that W is a finite, unitary, reflection group acting on the complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in W, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of W-invariant polynomial functions on V to the algebra of C-invariant functions on X. In this note we consider the special case when W is a Coxeter group, V is the complexified reflection representation of W, and X is in the lattice of the arrangement of W, and give a simple, combinatorial characterization of when the restriction mapping is surjective in terms of the exponents of W and C. As an application of our result, in the case when W is the Weyl group of a semisimple, complex Lie algebra, we complete a calculation begun by Richardson in 1987 and obtain a simple combinatorial characterization of regular decomposition classes whose closure is a normal variety.

We consider the finite W-algebra U(𝔤,e) associated to a nilpotent element e∈𝔤 in a simple complex Lie algebra 𝔤 of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem for U(𝔤,e), we verify a conjecture of Premet, that U(𝔤,e) always has a 1-dimensional representation when 𝔤 is of type G2, F4, E6 or E7. 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.

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we consider particular classes of connected reductive subgroups H of G and show that the cocharacters of H that are associated to a given nilpotent element e in the Lie algebra of H are precisely the cocharacters of G associated to e that take values in H. In particular, we show that this is the case provided H is a connected reductive subgroup of G of maximal rank; this answers a question posed by J. C. Jantzen.

The conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.

Let $G$ be a universal Chevalley group defined over an algebraically closed field $F$ of arbitrary characteristic. In this paper we investigate the restrictions to parabolic subgroups of $G$ of the irreducible $FG$-modules corresponding to minuscule highest weights via a combinatorial scheme of “chamber systems” indexed over certain cosets in the Weyl group.