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Cocharacter-closure and spherical buildings
Let k be a field, let G be a reductive k-group and V an affine k-variety on which G acts. In this note we continue our study of the notion of cocharacter-closed G(k)-orbits in V . In earlier work we used a rationality condition on the point stabilizer of a G-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding G(k)-orbit in V . In the present paper we employ building-theoretic techniques to derive analogous results.
Inductive freeness of Ziegler’s canonical multiderivations for reflection arrangements
Let A be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction A 00 of A to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. We initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of re ection arrangements. More precisely, let A = A (W) be the re ection arrangement of a complex re ection group W. By work of Terao, each such re ection arrangement is free. Thus so is Ziegler's canonical multiplicity on the restriction A 00 of A to a hyperplane. We show that the latter is inductively free as a multiarrangement if and only if A 00 itself is inductively free.
On unipotent radicals of pseudo-reductive groups
We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, let k′ be a purely inseparable field extension of k of degree pe and let G denote the Weil restriction of scalars Rk′/k(G′) of a reductive k′-group G′. We prove that the unipotent radical Ru(Gk¯) of the extension of scalars of G to the algebraic closure k¯ of k has exponent e. Our main theorem is to give bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases.
Freeness of multi-reflection arrangements via primitive vector fields
In 2002, Terao showed that every reection multi-arrangement of a real reection group with constant multiplicity is free by providing a basis of the module of derivations. We rst generalize Terao's result to multi-arrangements stemming from well-generated unitary reection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reection representation. We then extend our results further to all imprimitive irreducible unitary reection groups. In this case the exponents turn out to depend on the exponents of a certain Galois twist of the dual reection representation that comes from a Beynon-Lusztig type semi-palindromicity of the fake degrees.
Calculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups of rank six and seven
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 [12], the first and the third authors developed an algorithm to calculate the values of k(U(q)). By implementing it into a computer program using GAP, they were able to calculate k(U(q)) for G of rank at most 5, 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 E7. 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.
Cocharacter-Closure and the Rational Hilbert-Mumford Theorem
For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.
On reflection subgroups of finite Coxeter groups
Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.
On commuting varieties of nilradicals of Borel subalgebras of reductive Lie algebras
Let G be a connected reductive algebraic group defined over an algebraically closed field k of characteristic zero. We consider the commuting variety C(u) of the nilradical u of the Lie algebra b of a Borel subgroup B of G. In case B acts on u with only a finite number of orbits, we verify that C(u) is equidimensional and that the irreducible components are in correspondence with the distinguished B-orbits in u. We observe that in general C(u) is not equidimensional, and determine the irreducible components of C(u) in the minimal cases where there are infinitely many B-orbits in u.
G-complete reducibility in non-connected groups
In this paper we present an algorithm for determining whether a subgroup H of a non-connected reductive group G is G-completely reducible. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of G0 is G0-cr. This essentially reduces the problem of determining G-complete reducibility to the connected case.
An inductive approach to Coxeter arrangements and Solomon’s descent algebra
In our recent paper [3], we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik-Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for nite Coxeter groups of rank up to 2.