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Let $W_0$ be a reflection subgroup of a finite complex reflection group $W$, and let $B_0$ and $B$ be their respective braid groups. In order to construct a Hecke algebra $\widetilde{H}_0$ for the normalizer $N_W(W_0)$, one first considers a natural subquotient $\widetilde{B}_0$ of $B$ which is an extension of $N_W(W_0)/W_0$ by $B_0$. We prove that this extension is split when $W$ is a Coxeter group, and deduce a standard basis for the Hecke algebra $\widetilde{H}_0$. We also give classes of both split and non-split examples in the non-Coxeter case., 22 pages. To appear in Annales de l'Institut Fourier

Computer vision (CV) is used throughout life sciences for tissue and cell imaging; this paper presents the first demonstration of CV as a biomolecular sensor. We present an integrated sensing and fluid-actuation platform that combines CV with bead-based chemistries to perform both biomolecular and cell-based assays. By rapidly detecting the size and shape of objects at 5-50 micron length scales, the CV system can accommodate a variety of assays that traditionally require separate analytical tools and workflows, providing a greater degree of data integration and automation. This paper demonstrates 3 initial use cases: 1) Semi-quantitation of a biomarker (IgG) using a clustering assay; 2) Enumeration of white blood cells expressing CD45 surface markers using functionalized nanoparticles; and 3) Cell viability detection using dyes. All assays are performed using a single CV platform.

We define a generalized Springer correspondence for the group GL(n) over any field. We also determine the cuspidal pairs, and compute the correspondence explicitly. Finally we define a stratification of the category of equivariant perverse sheaves on the nilpotent cone of GL(n) satisfying the `recollement' properties, and with subquotients equivalent to categories of representations of a product of symmetric groups., v1: 27 pages; v2: 29 pages, minor corrections; v3: 29 pages, final version, to appear in JEMS

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 pages

Muscle sympathetic nerve activity (MSNA) is greatly elevated in patients with obstructive sleep apnea (OSA) during normoxic daytime wakefulness. Increased MSNA is a precursor to hypertension and elevated cardiovascular morbidity and mortality. However, the mechanisms underlying the high MSNA in OSA are not well understood. In this study we used concurrent microneurography and magnetic resonance imaging to explore MSNA-related brainstem activity changes and anatomical changes in 15 control and 15 OSA subjects before and after 6 and 12 months of continuous positive airway pressure (CPAP) treatment. We found that following 6 and 12 months of CPAP treatment, resting MSNA levels were significantly reduced in individuals with OSA. Furthermore, this MSNA reduction was associated with restoration of MSNA-related brainstem activity and structural changes in the medullary raphe, rostral ventrolateral medulla, dorsolateral pons, and ventral midbrain. This restoration occurred after 6 months of CPAP treatment and was maintained following 12 months CPAP. These findings show that continual CPAP treatment is an effective long-term treatment for elevated MSNA likely due to its effects on restoring brainstem structure and function.

In this note, we point out an error in the proof of Theorem 4.7 of [P.N. Achar, A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. Math. 219 (1) (2008) 27–62], a statement about the existence of affine pavings for fibres of a certain resolution of singularities of an enhanced nilpotent orbit closure. We also give independent proofs of later results that depend on that statement, so all other results of that paper remain valid.

We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$, and Kato's exotic nilpotent cone. We prove that the number of $\F_q$-points in each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent cone. This is a finer version of Lusztig's result that corresponding special pieces in types $B_n$ and $C_n$ have the same number of $\F_q$-points. The proof requires studying the case of characteristic 2, where more direct connections between the three nilpotent cones can be established. We also prove that the type-$B$ and type-$C$ pieces of the exotic nilpotent cone are smooth in any characteristic., Comment: 32 pages

Let $H^*(\calB_e)$ be the cohomology of the Springer fibre for the nilpotent element $e$ in a simple Lie algebra $\g$, on which the Weyl group $W$ acts by the Springer representation. Let $\Lambda^i V$ denote the $i$th exterior power of the reflection representation of $W$. We determine the degrees in which $\Lambda^i V$ occurs in the graded representation $H^*(\calB_e)$, under the assumption that $e$ is regular in a Levi subalgebra and satisfies a certain extra condition which holds automatically if $\g$ is of type A, B, or C. This partially verifies a conjecture of Lehrer--Shoji, and extends the results of Solomon in the $e=0$ case and Lehrer--Shoji in the $i=1$ case. The proof proceeds by showing that $(H^*(\calB_e) \otimes \Lambda^* V)^W$ is a free exterior algebra on its subspace $(H^*(\calB_e)\otimes V)^W$., Comment: 8 pages

Let $W$ be a Weyl group, and let $\CT_W$ be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of $W$, and its weight lattice. The real locus $\CT_W(\R)$ is a smooth, connected, compact manifold with a $W$-action. We give a formula for the equivariant Euler characteristic of $\CT_W(\R)$ as a generalised character of $W$. In type $A_{n-1}$ for $n$ odd, one obtains a generalised character of $\Sym_n$ whose degree is (up to sign) the $n^{\text{th}}$ Euler number., Comment: 10 pages. Version 2 incorporates slight revisions made at the suggestion of the referee

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 statement

We give a formula for the character of the representation of the symmetric group $S_n$ on each isotypic component of the cohomology of the set of regular elements of a maximal torus of $SL_n$, with respect to the action of the centre., 14 pages; slightly revised in the light of math.RT/0508162

The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively, the affine symmetric group). We explain how to simplify this description using a combinatorial cancellation procedure, and derive some consequences for representation theory., 34 pages

We generalize Joyal's theory of species to the case of functors from the groupoid of finite sets to the category of varieties over Fq. These have cycle index series defined by counting fixed points of twisted Frobenius maps. We give an application to configuration spaces.

Using a general result of Lusztig, we give explicit formulas for the dimensions of K F -invariants in irreducible representations of G F , when G = GL n , F :G→G is a Frobenius map, and K is an F -stable subgroup of finite index in G θ for some involution θ :G→G commuting with F . The proofs use some combinatorial facts about characters of symmetric 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. Annalen

We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $\mathbb k$ under the assumption that the characteristic $\ell$ of $\mathbb k$ is rather good for $G$, i.e., $\ell$ 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-$\ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautner's characteristic-$\ell$ `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., Comment: 26 pages

Alexander Anievas, Nivi Manchanda and Robbie Shilliam - Confronting the Global Colour Line: an Introduction, PART 1: CONCEPTUALISING THE INTERNATIONAL RELATIONS OF RACE AND RACISM, 2. Errol Henderson - Hidden in Plain Sight: Racism in International Relations Theory, 3. Debra Thompson - Through, Against, and Beyond the Racial State: The Transnational Stratum of Race, 4. Branwen Gruffydd-Jones - "Good Governance' and 'State Failure': the Pseudo-Science of Statesmen in Our Times, 5. John M. Hobson - Re-Embedding the Global Colour Line within Post-1945 International Theory, 6. Srdjan Vucetic - Against Race Taboos: The Global Colour Line in Philosophical Discourse, PART 2: INTERNATIONAL PRACTICES OF RACE AND RACISM, 7. Randolph B. Persaud - Colonial Violence: Race and Gender on the Sugar Plantations of British Guiana, 8. Sankaran Krishna - A Postcolonial Racial/Spatial Order: Gandhi, Ambedkar and the Construction of the International, 9. Richard Seymour - The Cold War, American Anticommunism and the Global 'Colour Line', 10. Robert Knox - Race, Racialisation and Rivalry in the International Legal Order, PART 3: REFLECTIONS ON THE GLOBAL COLOUR LINE, 11. David Roediger - What Would It Mean to Transform International Relations?, 12. Charles W. Mills - Unwriting and Unwhitening the World

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., 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. Soc

We prove that in the symmetric space setting the functors of Fourier transform (in the sense of Deligne) and parabolic induction (in the sense of Lusztig) commute. We derive two consequences: the first is a new proof of Lusztig's description of the intersection cohomology of nilpotent orbit closures for GL n , and the second is an analogous description for GL2n /Sp2n .

We apply Lusztig’s theory of character sheaves to the problem of calculating the spherical functions of G ( F q 2 ) / G ( F q ) G(\mathbb {F}_{q^{2}})/G(\mathbb {F}_q) , where G G is a connected reductive algebraic group. We obtain the solution for generic spherical functions for any G G , and for all spherical functions when G = G L n G=GL_n . The proof includes a result about convolution of character sheaves and its interaction with the associated two-sided cells.

We show that two Weyl group actions on the Springer sheaf with arbitrary coefficients, one defined by Fourier transform and one by restriction, agree up to a twist by the sign character. This generalizes a familiar result from the setting of l-adic cohomology, making it applicable to modular representation theory. We use the Weyl group actions to define a Springer correspondence in this generality, and identify the zero weight spaces of small representations in terms of this Springer correspondence., Comment: 27 pages; version 2: accepted version, with minor changes recommended by referee

This is an expository article on the singularities of nilpotent orbit closures in simple Lie algebras over the complex numbers. It is slanted towards aspects that are relevant for representation theory, including Maffei's theorem relating Slodowy slices to Nakajima quiver varieties in type A. There is one new observation: the results of Juteau and Mautner, combined with Maffei's theorem, give a geometric proof of a result on decomposition numbers of Schur algebras due to Fang, Henke and Koenig., Comment: 30 pages. Version 2 has very minor modifications; final version, to appear in proceedings of the 5th Japanese-Australian Workshop on Real and Complex Singularities

We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the `split-quotient quiver' introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the quiver varieties of type A correspond to small self-dual representations, we show that the diagram involutions coincide with classical involutions of two-row Slodowy varieties. It follows that certain quiver varieties of type D are isomorphic to Slodowy varieties for orthogonal or symplectic Lie algebras., 43 pages. In version 2, at the referee's suggestion, we slightly expand some statements (Theorem 1.2 and Proposition 3.19) to include the relevant affine varieties. This version is to appear in Advances in Mathematics

This bold and refreshing approach to Lie algebras assumes only modest prerequisites (linear algebra up to the Jordan canonical form and a basic familiarity with groups and rings), yet it reaches a major result in representation theory: the highest-weight classification of irreducible modules of the general linear Lie algebra. The author's exposition is focused on this goal rather than aiming at the widest generality and emphasis is placed on explicit calculations with bases and matrices. The book begins with a motivating chapter explaining the context and relevance of Lie algebras and their representations and concludes with a guide to further reading. Numerous examples and exercises with full solutions are included. Based on the author's own introductory course on Lie algebras, this book has been thoroughly road-tested by advanced undergraduate and beginning graduate students and it is also suited to individual readers wanting an introduction to this important area of mathematics.

The toric variety corresponding to the Coxeter fan of type A can also be described as a De Concini-Procesi wonderful model. Using a general result of Rains which relates cohomology of real De Concini-Procesi models to poset homology, we give formulas for the Betti numbers of the real toric variety, and the symmetric group representations on the rational cohomologies. We also show that the rational cohomology ring is not generated in degree 1.

We study the action of the symplectic group on pairs of a vector and a flag. Considering the irreducible components of the conormal variety, we obtain an exotic analogue of the Robinson-Schensted correspondence. Conjecturally, the resulting cells are related to exotic character sheaves., Comment: 14 pages

For a simply-connected simple algebraic group $G$ over $\C$, we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of $G$, generalizing a well-known fact about $GL_n$. Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group., Comment: Version 2: minor revisions, 33 pages

We study the rational cohomology groups of the real De Concini-Procesi model corresponding to a finite Coxeter group, generalizing the type-A case of the moduli space of stable genus 0 curves with marked points. We compute the Betti numbers in the exceptional types, and give formulae for them in types B and D. We give a generating-function formula for the characters of the representations of a Coxeter group of type B on the rational cohomology groups of the corresponding real De Concini-Procesi model, and deduce the multiplicities of one-dimensional characters in the representations, and a formula for the Euler character. We also give a moduli space interpretation of this type-B variety, and hence show that the action of the Coxeter group extends to a slightly larger group., Comment: 27 pages. The main change in Version 2 is a type-independent proof of Cohen-Macaulayness

Using a general result of Lusztig, we find the decomposition into irreducibles of certain induced characters of the projective general linear group over a finite field of odd characteristic., 17 pages

We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we write the cohomology groups as direct sums of inductions of one-dimensional characters of subgroups. This relies on an equivariant description of the Orlik-Solomon algebras of full monomial reflection groups (wreath products of the symmetric group with a cyclic group). The combinatorial models involved are certain representations of these wreath products which possess bases indexed by labelled trees., 30 pages; in third version, more references are added, making further connections with general theory of cohomology of local systems on hyperplane complements; also the consequences of the main result for cohomology dimensions and module generators are stated more explicitly

We prove that the local intersection cohomology of nilpotent orbit closures of cyclic quivers is trivial when the two orbits involved correspond to partitions with at most two rows. This gives a geometric proof of a result of Graham and Lehrer, which states that standard modules of the affine Hecke algebra of $GL_d$ corresponding to nilpotents with at most two Jordan blocks are multiplicity-free., Revised version, clarified in various places. 17 pages, LaTeX and Pstricks

The wreath product W(r,n) of the cyclic group of order r and the symmetric group S n acts on the corresponding projective hyperplane complement and on its wonderful compactification as defined by De Concini and Procesi. We give a formula for the characters of the representations of W(r,n) on the cohomology groups of this compactification, extending the result of Ginzburg and Kapranov in the r=1 case. As a corollary, we get a formula for the Betti numbers, which generalizes the result of Yuzvinsky in the r=2 case. Our method involves applying to the nested-set stratification a generalization of Joyal's theory of tensor species, which includes a link between polynomial functors and plethysm for general r. We also give a new proof of Lehrer's formula for the representations of W(r,n) on the cohomology groups of the hyperplane complement.