Unruffled extensions and flatness over central subalgebras

A condition on an affine central subalgebra $Z$ of a noetherian algebra $A$ of finite Gelfand-Kirillov dimension, which we call here \emph{unruffledness}, is shown to be equivalent in some circumstances to the flatness of $A$ as a $Z$-module. Unruffledness was studied by Borho and Joseph in work on enveloping algebras of complex semisimple Lie algebras, and we discuss applications of our result to enveloping algebras, as well as beginning the study of this condition for more general algebras

Springer theory via the Hitchin fibration

In this paper, we translate the Springer theory of Weyl group representations into the language of symplectic topology. Given a semisimple complex group G, we describe a Lagrangian brane in the cotangent bundle of the adjoint quotient g/G that produces the perverse sheaves of Springer theory. The main technical tool is an analysis of the Fourier transform for constructible sheaves from the perspective of the Fukaya category. Our results can be viewed as a toy model of the quantization of Hitchin fibers in the Geometric Langlands program.Comment: 37 pages; to appear in Compos. Mat

On sheets of conjugacy classes in good characteristic

We show that the sheets for a connected reductive algebraic group G over an algebraically closed field in good characteristic acting on itself by conjugation are in bijection with G-conjugacy classes of triples (M, Z(M)^\circ t, O) where M is the connected centralizer of a semisimple element in G, Z(M)^\circ t is a suitable coset in Z(M)/Z(M)^\circ and O is a rigid unipotent conjugacy class in M. Any semisimple element is contained in a unique sheet S and S corresponds to a triple with O={1}. The sheets in G containing a unipotent conjugacy class are precisely those corresponding to triples for which M is a Levi subgroup of a parabolic subgroup of G and such a class is unique.Comment: After acceptance and typesetting of the journal version of the paper, we discovered that the notions of Levi envelope, Jordan class and exceptional element in an algebraic group had already appeared in G. Lusztig's paper "Intersection cohomology complexes on a reductive group", Invent. Math. 75 no. 2, 205-272 (1984); online final version published online in International Mathematics Research Notices 201

Nilpotent pairs, dual pairs, and sheets

Recently, V.Ginzburg introduced the notion of a principal nilpotent pair (= pn-pair) in a semisimple Lie algebra {\frak g}. It is a double counterpart of the notion of a regular nilpotent element. A pair (e_1,e_2) of commuting nilpotent elements is called a pn-pair, if the dimension of their simultaneous centralizer is equal to the rank of {\frak g} and some bi-homogeneity condition is satisfied. Ginzburg proved that many familiar results of the `ordinary' theory have analogues for pn-pairs. The aim of this article is to develop the theory of nilpotent pairs a bit further and to present some applications of it to dual pairs and sheets. It is shown that a large portion of Ginzburg's theory can be extended to the pairs whose simultaneous centraliser is of dimension rk{\frak g}+1. Such pairs are called almost pn-pairs. It is worth noting that the very existence of almost pn-pairs is a purely "double" phenomenon, because the dimension of "ordinary" orbits is always even. We prove that to any principal or almost nilpotent pair one naturally associates a dual pair. Moreover, this dual pair is reductive if and only if e_1 and e_2 can be included in commuting sl_2-triples. We also study sheets containing members of pn-pairs. Some cases are described, where these sheets are smooth and admit a section.Comment: 27 pages, LaTeX 2.0

Young, unaccompanied refugees’ expectations of social workers and social worker roles

Background: Young people who have travelled to another country, unaccompanied and with refugee status, are a both resilient and vulnerable group with specific needs. Supporting them is often challenging for social workers, and providing this support is mediated by the expectations that these young people have of social workers and social worker roles. Aim: In this study, we explore how young unaccompanied refugees (YURs) perceive the roles of social workers in the national context of Norway, where concerns about the quality of social work for this group have been highlighted. Method: Using the theoretical lens of role theory, semi-structured in-depth interviews were conducted with 11 Afghan boys between 16 and 23 years of age, living under the protection of the Child Welfare Services (CWS) in two municipalities in Norway. The interviews explored the boys’ positive and negative experiences of the social worker. A thematic analysis was conducted, in which the coding framework was informed by the premise that actual experience informs our expectations of other individuals’ behaviour and roles. Findings: YURs’ expectations are more than instrumental, and more than a task they expect the social worker to perform. They also expect the task to be performed in a person-centred, therapeutic alliance (e.g. with humour and trust), and that the social worker exhibits particular personal characteristics or competences, besides being culturally competent and sensitive. Conclusion: We find that YURs’ descriptions of the social worker’s roles of being a caregiver and practical helper are similar to what other young people in contact with the CWS expect. However, YURs expect an additional role, which is specific to this field of social work, namely that of an integration helper. However, the expectations that each individual young person has of social workers are individual, in flux and contextual, and not consistent over time. Therefore, we recommend prioritizing learning more about the young person’s individual expectations of the social worker roles, as well as a useful weighting of these roles for each individual young refugee.publishedVersio

Asymptotic cone of semisimple orbits for symmetric pairs

Let G be a reductive algebraic group over the complex field and O_h be a closed adjoint orbit through a semisimple element h. By a result of Borho and Kraft (1979), it is known that the asymptotic cone of the orbit O_h is the closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup whose Levi component is the centralizer Z_G(h) in G. In this paper, we prove an analogue on a semisimple orbit for a symmetric pair (G, K).Comment: 14 page

Orbit closures in the enhanced nilpotent cone

We study the orbits of $G=\mathrm{GL}(V)$ in the enhanced nilpotent cone $V\times\mathcal{N}$, where $\mathcal{N}$ is the variety of nilpotent endomorphisms of $V$. These orbits are parametrized by bipartitions of $n=\dim V$, and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato's exotic nilpotent cone in type C, proving that the closure ordering is the same, and conjecturing that the intersection cohomology is the same but with degrees doubled.Comment: 32 pages. Update (August 2010): There is an error in the proof of Theorem 4.7, in this version and the almost-identical published version. See the corrigendum arXiv:1008.1117 for independent proofs of later results that depend on that statemen

Cohomology of the minimal nilpotent orbit

We compute the integral cohomology of the minimal non-trivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the sub-root system generated by the long simple roots. The modulo $\ell$ reduction of the Springer correspondent representation involves the sign representation exactly when $\ell$ divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes.Comment: 29 pages, v2 : Leray-Serre spectral sequence replaced by Gysin sequence only, corrected typo