Spherical Conjugacy Research Articles

Conjugacy languages in virtual graph products

In this paper we study the behaviour of conjugacy languages in virtual graph products, extending results by Ciobanu et al. [13]. We focus primarily on virtual graph products in the form of a semi-direct product. First, we study the behaviour of twisted conjugacy representatives in right-angled Artin and Coxeter groups. We prove regularity of the conjugacy geodesic language for virtual graph products in certain cases, and highlight properties of the spherical conjugacy language, depending on the automorphism and ordering on the generating set. Finally, we give a criterion for when the spherical conjugacy language is not unambiguous context-free for virtual graph products. We can extend this further in the case of virtual RAAGs, to show the spherical conjugacy language is not context-free.

On Lusztig's map for spherical unipotent conjugacy classes in disconnected groups

AbstractLet be a simple algebraic group over an algebraically closed field and let be a graph‐automorphism of . We classify the spherical unipotent conjugacy classes in the coset . As a by‐product, we show that J.‐H. Lu's characterization in characteristic zero of spherical conjugacy classes in by the dimension formula also holds for spherical unipotent conjugacy classes in in positive characteristic. If has order 2, we provide an alternative description of the restriction to spherical unipotent conjugacy classes in , of Lusztig's map from the set of unipotent conjugacy classes in to the set of twisted conjugacy classes of the Weyl group of . We also show that a twisted conjugacy class in the Weyl group has a unique maximal length element if and only if it has maximum in the Bruhat order (a result previously proved by X. He).

A classification of spherical conjugacy classes

Let G be a simple algebraic group over an algebraically closed field k. We classify the spherical conjugacy classes of G.

A Katsylo theorem for sheets of spherical conjugacy classes

We show that, for a sheet or a Lusztig stratum S S containing spherical conjugacy classes in a connected reductive algebraic group G G over an algebraically closed field in good characteristic, the orbit space S / G S/G is isomorphic to the quotient of an affine subvariety of G G modulo the action of a finite abelian 2 2 -group. The affine subvariety is a closed subset of a Bruhat double coset and the abelian group is a finite subgroup of a maximal torus of G G . We show that sheets of spherical conjugacy classes in a simple group are always smooth and we list which strata containing spherical classes are smooth.

Lusztig’s partition and sheets (with an Appendix by M. Bulois)

We show that, for a connected reductive algebraic group G over an algebraically closed field of zero or good characteristic, the parts, called strata, in the partition of G recently introduced by Lusztig are unions of sheets of conjugacy classes. For G simple and adjoint we refine the parametrization of such sheets obtained in previous work with F. Esposito. We give a simple combinatorial description of strata containing spherical conjugacy classes, showing that Lusztig's correspondence induces a bijection between unions of spherical conjugacy classes and unions of classes of involutions in the Weyl group. Using ideas from the Appendix by M. Bulois, we show that the closure of a stratum is not necessarily a union of strata.

Conjugacy growth series and languages in groups

In this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series, as well as on regularity of the geodesic conjugacy language and spherical conjugacy language. In particular, we show that regularity of the geodesic conjugacy language is preserved by the graph product construction, and rationality of the geodesic conjugacy growth series is preserved by both direct and free products. 2010 Mathematics Subject Classification: 20F65, 20E45.

A classification of unipotent spherical conjugacy classes in bad characteristic

Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of every involutorial automorphism (involution) of G is a spherical subgroup of G.

A classification of spherical conjugacy classes in good characteristic

We classify spherical conjugacy classes in a simple algebraic group over an algebraically closed field of good, odd characteristic.

On the coordinate ring of spherical conjugacy classes

Let G be a simple algebraic group over an algebraically closed field k of characteristic zero and $${\mathcal{O}}$$ be a spherical conjugacy class of G. We determine the decomposition of the coordinate ring $${k[\mathcal{O}]}$$ of $${\mathcal{O}}$$ into simple G-modules.

Spherical conjugacy classes and involutions in the Weyl group

Let G be a simple algebraic group over an algebraically closed field of characteristic zero or positive odd, good characteristic. Let B be a Borel subgroup of G. We show that the spherical conjugacy classes of G intersect only the double cosets of B in G corresponding to involutions in the Weyl group of G. This result is used in order to prove that for a spherical conjugacy class $${\mathcal{O}}$$ with dense B-orbit v 0 ⊂ BwB there holds $$\ell(w) + {\rm rk}(1-w) = \dim {\mathcal{O}}$$ extending to the case of groups over fields of odd, good characteristic a characterization of spherical conjugacy classes obtained by Cantarini, Costantini and the author. It is also shown that the weights occurring in the G-module decomposition of the ring of regular functions on $${\mathcal{O}}$$ are self-adjoint and they lie in the −1-eigenspace of the element w.

Spherical orbits and representations of Uε(g)

Let Ue(g) be the simply connected quantized enveloping algebra at roots of one associated to a finite dimensional complex simple Lie algebra g. The De Concini-Kac-Procesi conjecture on the dimension of the irreducible representations of Ue(g) is proved for the representations corresponding to the spherical conjugacy classes of the simply connected algebraic group G with Lie algebra g. We achieve this result by means of a new characterization of the spherical conjugacy classes of G in terms of elements of the Weyl group.