Geometric Satake, Springer correspondence, and small representations II

For a split reductive group scheme $G$ over a commutative ring $k$ with Weyl group $W$, there is an important functor $Rep(G,k) \to Rep(W,k)$ defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dual group to $G$. The translation from representation theory to geometry is via the Satake equivalence and the Springer correspondence. This generalizes the result for the $k=\mathbb{C}$ case proved by the first two authors, and also provides a better explanation than in that earlier paper, since the current proof is uniform across all types.Comment: Version 4: minor revisions; 73 page

Modular generalized Springer correspondence: an overview

This is an overview of our series of papers on the modular generalized Springer correspondence. It is an expansion of a lecture given by the second author in the Fifth Conference of the Tsinghua Sanya International Mathematics Forum, Sanya, December 2014, as part of the Master Lecture `Algebraic Groups and their Representations' Workshop honouring G. Lusztig. The material that has not appeared in print before includes some discussion of the motivating idea of modular character sheaves, and heuristic remarks about geometric functors of parabolic induction and restriction.Comment: 19 pages. Version 2 includes more examples and tables in Section

Modular generalized Springer correspondence II: classical groups

We construct a modular generalized Springer correspondence for any classical group, by generalizing to the modular setting various results of Lusztig in the case of characteristic-$0$ coefficients. We determine the cuspidal pairs in all classical types, and compute the correspondence explicitly for $\mathrm{SL}(n)$ with coefficients of arbitrary characteristic and for $\mathrm{SO}(n)$ and $\mathrm{Sp}(2n)$ with characteristic-$2$ coefficients.Comment: 52 pages. Version 2 corrects a minor mistake in the combinatorics of the type D case; no numbered statements are affected. Version 3 has minor additions, mostly in Section 2; final version, to appear in J. Eur. Math. So

Modular generalized Springer correspondence III: exceptional groups

We complete the construction of the modular generalized Springer correspondence for an arbitrary connected reductive group, with a uniform proof of the disjointness of induction series that avoids the case-by-case arguments for classical groups used in previous papers in the series. We show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the Sylow subgroups of the Weyl group. Under some assumptions, we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character. We also provide tables and other information on the modular generalized Springer correspondence for quasi-simple groups of exceptional type, including a complete classification of cuspidal pairs in the case of good characteristic, and a full determination of the correspondence in type $G_2$.Comment: 40 pages. Version 2: added section 7.5, modified Table 5.2 to match current conventions of GAP3. Version 3 has minor edits suggested by the referee, including a slight strengthening of Proposition 3.2; final version, to appear in Math. Annale

Adaptive Simulation Using Perfect Control Variates

Adaptive Simulation Using Perfect Control Variate

Navigating the ‘meaningless’ of social innovation: perspectives of social care practitioners in Scotland

Social innovation is an umbrella concept that allows space for a diverse range of perspectives to co-exist. In this paper, we explore how practitioners negotiate this complexity. Conducting 19 interviews with stakeholders involved in social enterprise and social care in Scotland, we show that almost anything can be conceived of as a social innovation as defined by the European Union. The EU definition can be a useful tool for organisations to demonstrate to funders how and why they are socially innovative. However, in failing to interrogate the power dimension of social innovation, the EU definition neglects any transformative potential

Cycling stability of a hybrid activated carbon//poly(3- methylthiophene) supercapacitor with N-butyl-Nmethylpyrrolidinium bis(trifluoromethanesulfonyl)imide ionic liquid as electrolyte

A long cycle-life, high-voltage supercapacitor featuring an activated carbon//poly(3-methylthiophene) hybrid configuration with N-butyl-N-methylpyrrolidinium bis(trifluoromethanesulfonyl)imide ionic liquid, a solvent-free green electrolyte, was developed. The cyclability of a laboratory scale cell with electrode mass loading sized for practical uses was tested at 60 °C over 16,000 galvanostatic charge–discharge cycles at 10 mA cm−2 in the 1.5 and 3.6 V voltage range. The reported average and maximum specific energy and power, specific capacitance and capacity, equivalent series resistance and coulombic efficiency over cycling demonstrate the long-term viability of this ionic liquid as green electrolyte for high-voltage hybrid supercapacitors

Constructible sheaves on nilpotent cones in rather good characteristic

International audienceWe study some aspects of modular generalized Springer theory for a complex reductive group G with coefficients in a field k under the assumption that the characteristic l of k is rather good for G, i.e., l is good and does not divide the order of the component group of the centre of G. We prove a comparison theorem relating the characteristic-l generalized Springer correspondence to the characteristic-0 version. We also consider Mautner's characteristic-l 'cleanness conjecture'; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone

Modular generalized Springer correspondence: an overview

International audienceThis is an overview of our series of papers on the modular generalized Springer correspondence. It is an expansion of a lecture given by the second author in the Fifth Conference of the Tsinghua Sanya International Mathematics Forum, Sanya, December 2014, as part of the Master Lecture 'Algebraic Groups and their Representations' Workshop honouring G. Lusztig. The material that has not appeared in print before includes some discussion of the motivating idea of modular character sheaves, and heuristic remarks about geometric functors of parabolic induction and restriction