A unified approach on Springer fibers in the hook, two-row and two-column cases

We consider the Springer fiber over a nilpotent endomorphism. Fix a Jordan basis and consider the standard torus relative to this. We deal with the problem to describe the flags fixed by the torus which belong to a given component of the Springer fiber. We solve the problem in the hook, two-row and two-column cases. We provide two main characterizations which are common to the three cases, and which involve dominance relations between Young diagrams and combinatorial algorithms. Then, for these three cases, we deduce topological properties of the components and their intersections.Comment: 42 page

Cohomological descent theory for a morphism of stacks and for equivariant derived categories

In the paper we answer the following question: for a morphism of varieties (or, more generally, stacks), when the derived category of the base can be recovered from the derived category of the covering variety by means of descent theory? As a corollary, we show that for an action of a reductive group on a scheme, the derived category of equivariant sheaves is equivalent to the category of objects, equipped with an action of the group, in the ordinary derived category.Comment: 28 page

Deformation theory of objects in homotopy and derived categories III: abelian categories

This is the third paper in a series. In part I we developed a deformation theory of objects in homotopy and derived categories of DG categories. Here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories. Then we consider examples from (noncommutative) algebraic geometry. In particular, we study noncommutative Grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in projective spaces.Comment: Alexander Efimov is a new co-author of this paper. Besides some minor changes, a new part (part 3) about noncommutative Grassmanians was adde

The orbit structure of Dynkin curves

Let G be a simple algebraic group over an algebraically closed field k; assume that Char k is zero or good for G. Let \cB be the variety of Borel subgroups of G and let e in Lie G be nilpotent. There is a natural action of the centralizer C_G(e) of e in G on the Springer fibre \cB_e = {B' in \cB | e in Lie B'} associated to e. In this paper we consider the case, where e lies in the subregular nilpotent orbit; in this case \cB_e is a Dynkin curve. We give a complete description of the C_G(e)-orbits in \cB_e. In particular, we classify the irreducible components of \cB_e on which C_G(e) acts with finitely many orbits. In an application we obtain a classification of all subregular orbital varieties admitting a finite number of B-orbits for B a fixed Borel subgroup of G.Comment: 12 pages, to appear in Math

Quotients for sheets of conjugacy classes

We provide a description of the orbit space of a sheet S for the conjugation action of a complex simple simply connected algebraic group G. This is obtained by means of a bijection between S/G and the quotient of a shifted torus modulo the action of a subgroup of the Weyl group and it is the group analogue of a result due to Borho and Kraft. We also describe the normalisation of the categorical quotient \overline{S}//G for arbitrary simple G and give a necessary and sufficient condition for S//G to be normal in analogy to results of Borho, Kraft and Richardson. The example of G_2 is worked out in detail

Quotients for sheets of conjugacy classes

We provide a description of the orbit space of a sheet S for the conjugation action of a complex simple simply connected algebraic group G. This is obtained by means of a bijection between S 15G and the quotient of a shifted torus modulo the action of a subgroup of the Weyl group and it is the group analogue of a result due to Borho and Kraft. We also describe the normalisation of the categorical quotient // for arbitrary simple G and give a necessary and sufficient condition for //G to be normal in analogy to results of Borho, Kraft and Richardson. The example of G2 is worked out in detail

Extended Joseph polynomials, quantized conformal blocks, and a q-Selberg type integral

We consider the tensor power $V=(C^N)^{\otimes n}$ of the vector representation of $gl_N$ and its weight decomposition $V=\oplus_{\lambda=(\lambda_1,...,\lambda_N)}V[\lambda]$. For $\lambda = (\lambda_1 \geq ... \geq \lambda_N)$, the trivial bundle V[\lambda]\times \C^n\to\C^n has a subbundle of q-conformal blocks at level l, where $l = \lambda_1-\lambda_N$ if $\lambda_1-\lambda_N> 0$ and l=1 if $\lambda_1-\lambda_N=0$. We construct a polynomial section $I_\lambda(z_1,...,z_n,h)$ of the subbundle. The section is the main object of the paper. We identify the section with the generating function $J_\lambda(z_1,...,z_n,h)$ of the extended Joseph polynomials of orbital varieties, defined in [DFZJ05,KZJ09]. For l=1, we show that the subbundle of q-conformal blocks has rank 1 and $I_\lambda(z_1,...,z_n,h)$ is flat with respect to the quantum Knizhnik-Zamolodchikov discrete connection. For N=2 and l=1, we represent our polynomial as a multidimensional q-hypergeometric integral and obtain a q-Selberg type identity, which says that the integral is an explicit polynomial

International registry of congenital porto-systemic shunts: a multi-centre, retrospective and prospective registry of neonates, children and adults with congenital porto-systemic shunts.

BACKGROUND Congenital portosystemic shunts (CPSS) are rare vascular malformations associated with the risk of life-threatening systemic conditions, which remain underdiagnosed and often are identified after considerable diagnostic delay. CPSS are characterized by multiple signs and symptoms, often masquerading as other conditions, progressing over time if the shunt remains patent. Which patients will benefit from shunt closure remains to be clarified, as does the timing and method of closure. In addition, the etiology and pathophysiology of CPSS are both unknowns. This rare disorder needs the strength of numbers to answer these questions, which is the purpose of the international registry of CPSS (IRCPSS). METHOD A retrospective and prospective registry was designed using secuTrial® by the ISO certified Clinical Research Unit. Given that a significant number of cases entered in the registry are retrospective, participants have the opportunity to use a semi-structured minimal or complete data set to facilitate data entry. In addition, the design allows subjects to be entered into the IRCPSS according to clinically relevant events. Emphasis is on longitudinal follow-up of signs and symptoms, which is paramount to garner clinically relevant information to eventually orient patient management. The IRCPSS includes also three specific forms to capture essential radiological, surgical, and cardiopulmonary data as many times as relevant, which are completed by the specialists themselves. Finally, connecting the clinical data registry with a safe image repository, using state-of-the-art pseudonymization software, was another major focus of development. Data quality and stewardship is ensured by a steering committee. All centers participating in the IRCPSS have signed a memorandum of understanding and obtained their own ethical approval. CONCLUSION Through state-of-the-art management of data and imaging, we have developed a practical, user-friendly, international registry to study CPSS in neonates, children, and adults. Via this multicenter and international effort, we will be ready to answer meaningful and urgent questions regarding the management of patients with CPSS, a condition often ridden with significant diagnostic delay contributing to a severe clinical course

International registry of congenital porto-systemic shunts: a multi-centre, retrospective and prospective registry of neonates, children and adults with congenital porto-systemic shunts

BACKGROUND: Congenital portosystemic shunts (CPSS) are rare vascular malformations associated with the risk of life-threatening systemic conditions, which remain underdiagnosed and often are identified after considerable diagnostic delay. CPSS are characterized by multiple signs and symptoms, often masquerading as other conditions, progressing over time if the shunt remains patent. Which patients will benefit from shunt closure remains to be clarified, as does the timing and method of closure. In addition, the etiology and pathophysiology of CPSS are both unknowns. This rare disorder needs the strength of numbers to answer these questions, which is the purpose of the international registry of CPSS (IRCPSS). METHOD: A retrospective and prospective registry was designed using secuTrial® by the ISO certified Clinical Research Unit. Given that a significant number of cases entered in the registry are retrospective, participants have the opportunity to use a semi-structured minimal or complete data set to facilitate data entry. In addition, the design allows subjects to be entered into the IRCPSS according to clinically relevant events. Emphasis is on longitudinal follow-up of signs and symptoms, which is paramount to garner clinically relevant information to eventually orient patient management. The IRCPSS includes also three specific forms to capture essential radiological, surgical, and cardiopulmonary data as many times as relevant, which are completed by the specialists themselves. Finally, connecting the clinical data registry with a safe image repository, using state-of-the-art pseudonymization software, was another major focus of development. Data quality and stewardship is ensured by a steering committee. All centers participating in the IRCPSS have signed a memorandum of understanding and obtained their own ethical approval. CONCLUSION: Through state-of-the-art management of data and imaging, we have developed a practical, user-friendly, international registry to study CPSS in neonates, children, and adults. Via this multicenter and international effort, we will be ready to answer meaningful and urgent questions regarding the management of patients with CPSS, a condition often ridden with significant diagnostic delay contributing to a severe clinical course