Induced conjugacy classes and induced U_e(G)-modules

By work of De Concini, Kac and Procesi the irreducible representations of the non-restricted specialization of the quantized enveloping algebra of the Lie algebra g at the roots of unity are parametrized by the conjugacy classes of a group G with Lie(G)=g. We show that there is a natural dimension preserving bijection between the sets of irreducible representations associated with conjugacy classes lying in the same Jordan class (decomposition class). We conjecture a relation for representations associated with classes lying in the same sheet of G, providing two alternative formulations. We underline some evidence and illustrate potential consequences.Comment: This paper is an expanded version of a lecture given at the conference "Hopf algebras and tensor categories", Almeri'a, July 2011. It will appear in the volume of Contemporary Math. containing the conference Proceeding

The Brauer group of modified supergroup algebras

The computation of the Brauer group BM of modified supergroup algebras is perfomed, yielding, in particular, the computation of the Brauer group of all finite-dimensional triangular Hopf algebras when the base field is algebraically closed and of characteristic zero. The results are compared with the computation of lazy cohomology and with Yinhuo Zhang's exact sequence. As an example, we compute explicitely the Brauer group and lazy cohomology for modified supergroup algebras with (extensions of) Weyl groups of irreducible root systems as a group datum and their standard representation as a representation datum.Comment: 47 pages, submitte

Spherical conjugacy classes and Bruhat decomposition

Let G be a connected, reductive algebraic group over an algebraically closed field of characteristic zero or good and odd. We characterize the spherical conjugacy classes of G as those intersecting only Bruhat cells corresponding to involutions in the Weyl group of G.Comment: Final version, to appear in Ann. Inst. Fourier, Grenobl

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 to prove that for a spherical conjugacy class O with dense B-orbit v_0 contained in BwB there holds l(w)+rk(1-w)=dim(O) extending a characterization of spherical conjugacy classes obtained by N. Cantarini, M. Costantini and the author to the case of groups over fields of odd, good characteristic.Comment: Lemma 3.8 had an incorrect proof and it is removed without effecting the main results of the paper. One reference is adde

A Katsylo theorem for sheets of spherical conjugacy classes

We show that, for a sheet or a Lusztig stratum S containing spherical conjugacy classes in a connected reductive algebraic group G over an algebraically closed field in good characteristic, the orbit space S/G is isomorphic to the quotient of an affine subvariety of G modulo the action of a finite abelian 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. 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

On Lusztig's map for spherical unipotent conjugacy classes

We provide an alternative description of the restriction to spherical unipotent conjugacy classes, of Lusztig's map Psi from the set of unipotent conjugacy classes in a connected reductive algebraic group to the set of conjugacy classes of its Weyl group. For irreducible root systems, we analyze the image of this restricted map and we prove that a conjugacy class in a finite Weyl group has a unique maximal length element if and only if it has a maximum.Comment: Refereed version, one reference added. The final version will appear in the Bulletin of the London Mathematical Societ

Some isomorphisms for the Brauer groups of a Hopf algebra

Using equivalences of categories we provide isomorphisms between the Brauer groups of different Hopf algebras. As an example, we show that when k is a field of characteristic different from 2 the Brauer groups BC(k,H_4,r_t) for every dual quasitriangular structure r_t on Sweedler's Hopf algebra H_4 are all isomorphic to the direct sum of (k,+) and the Brauer-Wall group of k. We provide an isomorphism between the Brauer group of a Hopf algebra H and the Brauer group of the dual Hopf algebra H^* generalizing a result of Tilborghs. Finally we relate the Brauer groups of H and of its opposite and co-opposite Hopf algebras.Comment: 23 pages, minor changes, one reference adde