No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
A Topological Approach to Springer's Representations
... De nombreuses autres constructions ontété proposées par la suite. Par exemple, Kazhdan et Lusztig ont proposé une approche topologique [KL80b], et Slodowy a construit ces représentations par monodromie [Slo80a]. Au début des années 1980, l'essor de la cohomologie d'inter-section a permis de réinterpréter la correspondance de Springer en termes de faisceaux pervers [Lus81,BM81]. ...
... A la fin de leur article original [KL80b], Kazhdan et Lusztig mentionnent le cas de Sp 4 . On a deuxéléments de longueur trois dans le groupe de Weyl. ...
... Many other constructions were subsequently proposed by other mathematicians. For example, Kazhdan and Lusztig proposed a topological approach [KL80b], and Slodowy constructed these representations by monodromy [Slo80a]. At the beginning of the 1980's, the blossoming of intersection cohomology permitted to reinterpret Springer correspondence in terms of perverse sheaves [Lus81,BM81]. ...
In 1976, Springer defined a correspondence making a link between the irreducible ordinary (characteristic zero) representations of a Weyl group and the geometry of the associated nilpotent variety. In this thesis, we define a modular Springer correspondence (in positive characteristic), and we show that the decomposition numbers of a Weyl group (for example the symmetric group) are particular cases of decomposition numbers for equivariant perverse sheaves on the nilpotent variety. We calculate explicitly the decomposition numbers associated to the regular and subregular classes, and to the minimal and trivial classes. We determine the correspondence explicitly in the case of the symmetric group, and show that James's row and column removal rule is a consequence of a smooth equivalence of nilpotent singularities obtained by Kraft and Procesi. The first chapter contains generalities about perverse sheaves with Z_l and F_l coefficients.
... Then the equalities e w,v = P w,v (1) , ∀v ∈ W P , (1.2) hold if and only if the characteristic cycle of the intersection homology (IH) sheaf of the Schubert variety X P w is irreducible; see Section 8.1. In general, the problem of finding the decomposition of the characteristic cycle of the IH sheaves into irreducible components is open, although some particular cases are known; see for example [KL80,KT84,BFL90,BF97,EM99,Bra02,Wil15] and also Section 8 below for more details. We note that since CSM classes of Schubert cells can be explicitly calculated [AM09, AM16, RV18], equation (1.1) shows that giving an algorithm to calculate Mather classes is equivalent to giving one for the local Euler obstructions. ...
... Kazhdan and Lusztig [KL80] conjectured the irreducibility of characteristic cycles of the IH sheaf in type A. However, Kashiwara and Tanisaki [KT84], and then Kashiwara and Saito [KS97], found counterexamples for the full flag manifolds of Lie type B and type A, respectively; see also [Bra02,Wil15]. Boe and Fu [BF97] found the decompositions of the characteristic cycles of the Schubert varieties in cominuscule spaces of Lie types B, C. Next, we use the methods of this paper to recover an example of Kashiwara and Tanisaki of a reducible IH characteristic cycle. ...
... Kazhdan, Lusztig, Ginzburg, and Vasserot considered the Steinberg varieties on the Weyl and Schur sides St and St d n respectively (see [5,9,10,17]). Those are G = GL(n)-invariant Lagrangian subvarieties in T * (B) 2 and T * (F ) 2 respectively. ...
... i.e. the kth row of the matrix K is obtained from the matrix N by adding up rows corresponding to the non-zero entries of the kth column of the matrix M. (3,9). ...
We use Drinfeld style generators and relations to define an algebra which is a "q=0" version of the affine quantum group of We then use the convolution product on the equivariant K-theory of spaces of pairs of partial flags in a d-dimensional vector space V to define affine zero-Schur algebras and to prove that for every d there exists a surjective homomorphism from to ${\mathbb S}_0^{\operatorname{aff}}(n,d).
... The fact that covexillary Schubert varieties have the same singularities as Grassmannian Schubert varieties yields some immediate consequences. The characteristic cycle of the IC sheaf of a Schubert variety F l w is irreducible if and only if for each T -fixed point v, the local Euler obstruction of F l w at v equals the evaluation of the Kazhdan-Lusztig polynomial [KL80a,KL80b] Finally, we turn our attention to the conormal variety N * F l w of F l w in F l, which we relate to the conormal variety N * Gr v of Gr v in Gr(n, 2n) via the commutative diagram in Equation (1). Here f # : T * M → T * N denotes the open immersion induced on cotangent bundles from an open immersion f : M → N of smooth varieties. ...
... Euler obstruction computations are useful in the study of the irreducibility of the characteristic cycle of the IC sheaf of a Schubert variety. Precisely, the characteristic cycle is irreducible if and only if the Euler obstruction of F l w at a point v equals the value of the Kazhdan-Lusztig polynomial [KL80a,KL80b] Since both the Kazhdan-Lusztig polynomials and the Euler obstructions are local invariants, the following is an immediate consequence of Theorem 1.6. Theorem 1.10. ...
A permutation is called covexillary if it avoids the pattern 3412. We construct an open embedding of a covexillary matrix Schubert variety into a Grassmannian Schubert variety. As applications of this embedding, we show that the characteristic cycles of covexillary Schubert varieties are irreducible, and provide a new proof of Lascoux's model computing Kazhdan-Lusztig polynomials of vexillary permutations. Combining the above embedding with earlier work of the author on the conormal varieties of Grassmannian Schubert varieties, we develop an algebraic criterion identifying the conormal varieties of covexillary Schubert and matrix Schubert varieties as subvarieties of the respective cotangent bundles.
... In general, the problem of finding the decomposition of the characteristic cycle of the IH sheaves into irreducible components is open, although some particular cases are known; see e.g. [KL80,KT84,BFL90,BF97,EM99,Bra02,Wil15], and also Section 10 below for more details. We note that since CSM classes of Schubert cells can be explicitly calculated [AM09,AM16,RV18], equation (1) shows that giving an algorithm to calculate Mather classes is equivalent to one for the local Euler obstructions. ...
... The problem of finding the multiplicities of the characteristic cycle seems to be very difficult. Kazhdan and Lusztig [KL80] conjectured the irreducibility of characteristic cycles of the IH sheaf in type A. However, Kashiwara and Tanisaki [KT84], then Kashiwara and Saito [KS97] found counterexamples for the full flag manifolds of Lie type B and type A respectively. See also [Bra02,Wil15] for more about this. ...
Let G/P be a complex cominuscule flag manifold. We prove a type independent formula for the torus equivariant Mather class of a Schubert variety in G/P, and for a Schubert variety pulled back via the natural projection . We apply this to find formulae for the local Euler obstructions of Schubert varieties, and for the torus equivariant localizations of the conormal spaces of these Schubert varieties. We conjecture positivity properties for the local Euler obstructions and for the Schubert expansion of Mather classes. We check the conjectures in many cases, by utilizing results of Boe and Fu about the characteristic cycles of the intersection homology sheaves of Schubert varieties. We also conjecture that certain `Mather polynomials' are unimodal in general Lie type, and log concave in type A.
... More precisely, the Springer fiber S X associated to X is the collection of flags S X = {gB ∈ G/B : g −1 Xg ∈ b} Springer used them to construct an action of the Weyl group on the cohomology H * (S X ) that has many remarkable properties, described further in Section 4.2. For this reason, and to understand more deeply the geometry underlying the construction, many have constructed Springer's representation (or its dual) [22,4,20,28,15]. These constructions often relied on geometric properties of Springer fibers like purity [39,40], closed formulas for dimensions [38,10], or identification of the components [39]. ...
... Originally proven by Springer [41], the theorem has many different proofs using different approaches and perspectives, some recovering only parts of the theorem as we have stated it and others much stronger. For instance there are proofs due to Kazhdan and Lusztig [22], Borho and MacPherson [4], Lusztig [28], Garsia and Procesi [15], and many others. Of course there is a second irreducible S n -representation of the same dimension as the irreducible representation associated to λ, namely its dual (obtained by tensoring with the sign representation). ...
This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.
... (See also [GKM98] where a similar result is proven in the case of affine Springer fibers.) The Springer representation is somewhat mysterious because W does not act (in an algebraic way) on X, but rather it acts via deformations of X [L81], [KL80]. However, it turns out that the action of W on the equivariant cohomology is easy to describe and that, in fact, the Weyl group can be naturally identified as the group of automorphisms, ...
... In [Spr76], T. A. Springer constructed a surprising representation of the Weyl group W of G on the cohomology H * (X a ) of each Springer fiber. This representation has since been constructed by other means ( [L81], [KL80]) and it has been thoroughly analyzed. Each Springer fiber X a admits an action of a certain algebraic torus T . ...
If an algebraic torus T acts on a complex projective algebraic variety X, then the affine scheme Spec H*T (X; C) associated with the equivariant cohomology is often an arrangement of linear subspaces of the vector space HT2(X; C). In many situations the ordinary cohomology ring of X can be described in terms of this arrangement.
... the variety S to study the varieties ðG=PÞ v ¼ fgP j g À1 vgAPg for an arbitrary unipotent element v in G: Subsequently, Kazhdan and Lusztig [9] have defined a subvariety Z of G u  B  B; where G u is the variety of unipotent elements in G and B is the variety of Borel subgroups of G; by Z ¼ fðu; B; B 0 Þ j uAG u -B-B 0 g: Thus, Z may be identified with the union in G  B  B of Steinberg's varieties SðC; B; BÞ as C varies over the unipotent conjugacy classes in G: We call the variety Z; or any variety canonically isomorphic to it, the Steinberg variety of G. When G is a complex, semisimple, algebraic group, Kazhdan and Lusztig used the Steinberg variety of G to study the representations of the Weyl group of G defined by Springer on the cohomology of the varieties B u ¼ fBAB j uABg; where u is a unipotent element in G: Since then, the Steinberg variety of G has turned out to be important in the classification of primitive ideals in the universal enveloping algebra of LieðGÞ [2] [3] [8] and in the proof of the partial solution to the Deligne–Langlands–Lusztig conjecture given by Kazhdan and Lusztig [10]. ...
... Tanisaki [19] and also Chriss and Ginzburg [7] have strengthened the connection between the geometry of Steinberg varieties and the representation theory of reductive algebraic groups by defining a convolution multiplication that gives ring structures on both the graded, rational Borel–Moore homology group of Z and the G Â C Ã equivariant K-group of Z: With these ring structures, they show that the top Borel–Moore homology group of Z is isomorphic to the group algebra of the Weyl group of G and that the equivariant K-group is isomorphic to the generic, extended, affine Hecke algebra of G: A crucial ingredient in their convolution construction is the fact that Z can be expressed as a fiber product of smooth varieties that map properly onto a base space. Set g ¼ LieðGÞ and let N denote the variety of nilpotent elements in g: In Section 2 we will define generalizations of the Steinberg variety, X P;Q c;d ; depending on four parameters and study some of their basic properties using ideas introduced by Steinberg [18], Kazhdan and Lusztig [9], and Lusztig [11]. Here P and Q are two conjugacy classes of parabolic subgroups of G; c is a G-equivariant function from P to the set of closed subvarieties of N with the property that for P in P; the nilradical of LieðPÞ is contained in cðPÞ and cðPÞ is contained in LieðPÞ; and d is a map from Q to N with similar properties. ...
For a reductive, algebraic group, G, the Steinberg variety of G is the set of all triples consisting of a unipotent element, u, in G and two Borel subgroups of G that contain u. We define generalized Steinberg varieties that depend on four parameters and analyze in detail two special cases that turn out to be related to distinguished double coset representatives in the Weyl group. Using one of the two special cases, we define a parabolic version of a map from the Weyl group to a set of nilpotent orbits of G in Lie(G) defined by Joseph and study some of its properties.
... 6]. More recently Kazhdan-Lusztig and others compared the Springer basis to the Kazhdan-Lusztig basis, and in fact the Kazhdan-Lusztig polynomials encode the transition matrix between those bases [10,16,17]. Garsia-McLarnan relate the Kazhdan-Lusztig and Specht bases, showing the change-of-basis matrix with respect to a certain ordering of basis elements is upper-triangular with ones along the diagonal [11,Theorem 5.3]. ...
We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representations which arises in this context through the geometry of varieties called Springer fibers. We describe a graph that encapsulates combinatorial relations between each of these bases, prove that there is a unique way (up to scaling) to map the Specht basis into the web representation, and use this to recover a result of Garsia-McLarnan that the transition matrix between the Specht and web bases is upper-triangular with ones along the diagonal. We then strengthen their result to prove vanishing of certain additional entries unless a nesting condition on webs is satisfied. In fact we conjecture that the entries of the transition matrix are nonnegative and are nonzero precisely when certain directed paths exist in the web graph.
... In the setting of highest weight modules and Schubert varieties, Kazhdan-Lusztig [KL,§7] first raised the question of whether characteristic cycles in Type A were always irreducible. (Borho-Brylinski's results [BB] implied that this is equivalent to the question of whether characteristic cycles of Harish-Chandra bimodules for GL(n, C) were always irreducible.) ...
We give examples of reducible characteristic cycles for irreducible Harish-Chandra modules for by analyzing a four-dimensional singular subvariety of . We relate this singularity to the Kashiwara-Saito singularity arising for Schubert varieties for with reducible characteristic cycles, as well as recent related examples of Williamson. In particular, this gives another explanation of the simple highest weight modules for with reducible associated varieties that Williamson discovered.
... One remarkable aspect of Springer's representation is that W does not act on X itself, except when X = G/B. Springer's original construction uses ℓ-adic cohomology, but, later, more direct constructions for ordinary cohomology became available (cf. [BBM,Car2,KL,Slod,Spr4]). In [CK] (also see [Car2]), the author and K. Kaveh showed that Springer's action when G = SL(n, C) is a consequence of a general result on torus actions (S, X), where X is a projective variety with no odd cohomology. ...
This note is motivated by the problem of understanding Springer's remarkable action of the Weyl group of a semi-simple complex linear algebraic group G, with maximal torus T, on the cohomology algebra of an arbitrary Springer variety in the flag variety of G from the viewpoint of torus actions. Continuing the work [CK] which gave a sufficient condition for a group acting on the fixed point set of an algebraic torus action (S,X) on a complex projective variety X to lift to a representation of on the cohomology algebra (over ), we describe when the representation on is equivalent to the representation of on the cohomology of the fixed point set. As a consequence of this theorem, we give a simple proof in type A of the Alvis-Lusztig-Treumann Theorem, which describes Springer's representation of W for Springer varieties corresponding to nilpotents in a Levi subalgebra of Lie(G). In the final two sections, we describe the local structure of the moment graph of a special torus action (S,X), and we also show that if a finite group acts on the moment graph of X, then induces pair of actions on , namely the left and right or dot and star actions of Knutson [Knu] and Tymoczko [Tym] respectively. In particular, W acts on the moment (or Bruhat) graph of (T,G/P) for any parabolic P in G containing T, and the right action of W on is an induced representation. Furthermore, we show the left action of W on is trivial.
... It would be interesting to study this basis, and determine whether it has properties that resemble the canonical bases introduced by Kazhdan and Lusztig in various contexts. In type A, a version of this question was first posed in [KL80] and subsequently restated in terms of the irreducibility of certain characteristic cycles by ). Subsequent work by Kashiwara-Saito ( [KS97]) and Williamson ([Wil15]) found counterexamples to the irreducibility conjectures. ...
Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson-Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of the exotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces.
... For the purposes of this paper, this is an element IH(Y ) ∈ H 2 dim X C * (T * (X)), the middle C * -equivariant cohomology of T * (X), where C * acts by fibrewise dilation. The group H 2 dim X A fundamental problem in geometric representation theory is to calculate the multiplicities m w,v := m v Xw , in the special case when Y = X w is a Schubert variety [KL80,KT84,BFL90,BF97,EM99,Bra02,Wil15]. Besides their intrinsic interest, these multiplicities are related to problems in representation theory in characteristic p; see e.g. ...
Let G/P be a complex cominuscule flag manifold of type . We prove that each characteristic cycle of the intersection homology (IH) complex of a Schubert variety in G/P is irreducible. The proof utilizes an earlier algorithm by the same authors which calculates local Euler obstructions, then proceeds by direct computer calculation using Sage. This completes to the exceptional Lie types the characterization of irreducibility of IH sheaves of Schubert varieties in cominuscule G/P obtained by Boe and Fu. As a by-product, we also obtain that the Mather classes, and the Chern-Schwartz-MacPherson classes of Schubert cells in cominuscule G/P of type , are strongly positive.
... In Lie type A n−1 the top-degree cohomology of H(X, b) for nilpotent X is an irreducible representation, and varying over all nilpotent conjugacy classes of X recovers every irreducible representation of the Weyl group S n exactly once in that case. The Springer representation has since been constructed in many ways, using tools across geometry, topology, algebra, and combinatorics, among many others [10,21,37,42,43]. ...
Hessenberg varieties form a class of subvarieties of the flag variety G/B, parameterized by an operator X and certain subspaces H of the Lie algebra of G. We identify several families of Hessenberg varieties in type that are T-stable subvarieties of G/B, as well as families that are invariant under a subtorus K of T. In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Among the T-stable Hessenberg varieties, we identify several that are {\it GKM spaces}, meaning T acts with isolated fixed points and a finite number of one-dimensional orbits, though we also show that not all Hessenberg varieties with torus actions and finitely many fixed points are GKM. We conclude with a series of open questions about Hessenberg varieties, both in type and in general Lie type.
... Harish-Chandra showed that any invariant eigendistribution on (a real form of) G satisfies a certain system of differential equations [36]. The Harish-Chandra systems were reinterpreted by Hotta and Kashiwara [39] using D-module theory, and explained the connection to Springer theory (the geometric construction of representations of the Weyl group), as developed by Springer [62], Kazhdan-Lusztig [43], and Borho-MacPherson [17]. In his seminal paper [50] Lusztig defined and classified cuspidal local systems and proved the generalized Springer Correspondence, which classifies equivariant perverse sheaves on the unipotent cone of G. ...
Given a reductive group G, we give a description of the abelian category of G-equivariant D-modules on , which specializes to Lusztig’s generalized Springer correspondence upon restriction to the nilpotent cone. More precisely, the category has an orthogonal decomposition in to blocks indexed by cuspidal data , consisting of a Levi subgroup L, and a cuspidal local system on a nilpotent L-orbit. Each block is equivalent to the category of D-modules on the center of which are equivariant for the action of the relative Weyl group . The proof involves developing a theory of parabolic induction and restriction functors, and studying the corresponding monads acting on categories of cuspidal objects. It is hoped that the same techniques will be fruitful in understanding similar questions in the group, elliptic, mirabolic, quantum, and modular settings.
... 6]. More recently Kazhdan-Lusztig and others compared the Springer basis to the Kazhdan-Lusztig basis, and in fact the Kazhdan-Lusztig polynomials encode the transition matrix between those bases [10,16,17]. Garsia-McLarnan relate the Kazhdan-Lusztig and Specht bases, showing the change-of-basis matrix with respect to a certain ordering of basis elements is upper-triangular with ones along the diagonal [11,Theorem 5.3]. ...
We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representations which arises in this context through the geometry of varieties called Springer fibers. We describe a graph that encapsulates combinatorial relations between each of these bases, prove that there is a unique way (up to scaling) to map the Specht basis into the web representation, and use this to recover a result of Garsia-McLarnan that the transition matrix between the Specht and web bases is upper-triangular with ones along the diagonal. We then strengthen their result to prove vanishing of certain additional entries unless a nesting condition on webs is satisfied. In fact we conjecture that the entries of the transition matrix are nonnegative and are nonzero precisely when certain directed paths exist in the web graph.
... This construction was conjectured to give the same action of W on H * (B e ) as the one in Theorem 1.5.1. A similar construction by Rossmann appeared in [38, §2], in which the author identified his action with that constructed by Kazhdan and Lusztig in [20], and the latter was known to be the same as Springer's action. Thus all these constructions give the same W-action as in Theorem 1.5.1. ...
These are the expanded lecture notes from the author's mini-course during the graduate summer school of the Park City Math Institute in 2015. The main topics covered are: geometry of Springer fibers, affine Springer fibers and Hitchin fibers; representations of (affine) Weyl groups arising from these objects; relation between affine Springer fibers and orbital integrals.
... Then many other approaches to Springer correspondence were discovered. For example, Kazhdan and Lusztig found a topological approach [KL80], and Slodowy constructed Springer representations by monodromy [Slo80a]. Links between different constructions were established, as in Hotta's work [Hot81]. ...
The Springer correspondence makes a link between the characters of a Weyl
group and the geometry of the nilpotent cone of the corresponding semisimple
Lie algebra. In this article, we consider a modular version of the theory, and
show that the decomposition numbers of a Weyl group are particular cases of
decomposition numbers for equivariant perverse sheaves on the nilpotent cone.
We give some decomposition numbers which can be obtained geometrically. In the
case of the symmetric group, we show that James' row and column removal rule
for the symmetric group can be derived from a smooth equivalence between
nilpotent singularities proved by Kraft and Procesi. We give the complete
structure of the Springer and Grothendieck sheaves in the case of .
Finally, we determine explicitly the modular Springer correspondence for
exceptional types.
... In this way, simple perverse sheaves naturally give rise to distinguished bases in geometric representation theory. The Kazhdan-Lusztig basis of a Hecke algebra associated to a Weyl group W arise in this way [30]. Likewise, simple integrable representations and their tensor products of quantum Kac-Moody algebra possess canonical bases [42,43,44,45,27,28]. ...
Categorification is a process of lifting structures to a higher categorical
level. The original structure can then be recovered by means of the so-called
"decategorification" functor. Algebras are typically categorified to additive
categories with additional structure and decategorification is usually given by
the (split) Grothendieck group. In this expository article we study an
alternative decategorification functor given by the trace or the zeroth
Hochschild--Mitchell homology. We show that this form of decategorification
endows any 2-representation of the categorified quantum sl(n) with an action of
the current algebra U(sl(n)[t]) on its center.
... Kazhdan and Lusztig conjectured (still for G = SL n (C)) that the characteristic varieties of all L y are irreducible [KL80a] (that is, that m x,y = 0 if x = y). Of course this would imply an affirmative answer to the above question. ...
We show that simple highest weight modules for sl_12 may have reducible
characteristic variety. This answers a question of Borho-Brylinski and Joseph
from 1984. The relevant singularity under Beilinson-Bernstein localization is
the (in)famous Kashiwara-Saito singularity. We sketch the rather indirect route
via the p-canonical basis, W-graphs and decomposition numbers for perverse
sheaves that led us to examine this singularity.
... The characteristic cycle is topologically important for it is related to the local Euler obstruction (see [4]), the absolute polar varieties and local Morse inequalities (see Theorem 7.5 and Corollary 5.5 of [17]), and index formulas for the vanishing cycles (see [8], [23], [15], and below). In addition, the characteristic cycle of the intersection cohomology complex is of great importance in representation theory (see [12] and [2]). The Morse modules, or coefficients of the characteristic cycle, of a complex A @BULLET can be described in terms of vanishing cycles A @BULLET along functions with complex non-degenerate critical points. ...
If A
• is a bounded, constructible complex of sheaves on a complex analytic space X, and
and
are complex analytic functions, then the iterated vanishing cycles φ
g
[−1](φ
f
[−1]A
•) are important for a number of reasons. We give a formula for the stalk cohomology H*(φ
g
[−1]φ
f
[−1]A
•)x
in terms of relative polar curves, algebra, and Morse modules of A
•.
... We remark that there are several alternative constructions of the group operation of W on the Borel-Moore homology/ singular cohomology of the Springer fibres. In [Ara01], section 5.5 you find an understandable treatment of Lusztig's approach to this operation using intermediate extensions for perverse sheaves and Arabia provides a list of other authors and approaches to this (first Springer [Spr76], [Spr78], then Kazhdan-Lusztig [KL80], Slodowy [Slo80a], Lusztig [Lus81], Rossmann [Ros91]) and these operations differ between each other by at most by multiplication with a sign character (see [Hot81]). Also, Springer proves with taking (co)homology of Springer fibres with rational coefficients that the simple W -representations are all even defined over Q, a result which our approach does not give because the simple C(x)-modules are not necessarily all defined over Q (cp. ...
This is not standard in the sense that we understand a Springer map to be a
collapsing of homogeneous bundles. Apart from that we use mostly techniques
from Chriss and Ginzbergs book but we work in the equivariant derived category
of Bernstein and Lunts. We define Steinberg algebras as (equivariant)
Borel-Moore homology algebras of the associated Steinberg varieties. The data
of the BBD-decomposition theorem applied to the Springer map give a
parametrization of projective graded and simple graded modules over the
Steinberg algebra. Also, the projective graded modules are equivalent to a
category of shifts of perverse sheaves. This has as a consequence for example
the Springer correspondence. We call classical Springer Theory what is usually
considered as Springer Theory. The main results are parametrizations of simple
modules of different types of Hecke algebras. Our second main example is
quiver-graded Springer theory (due to Lusztig), here the Steinberg algebras are
the quiver Hecke algebras. We also explain Lusztig's and Khovanov-Lauda's
monoidal categorification of the negative half of the quantum group using the
categories of shifts of perverse sheaves and projective graded modules over the
quiver Hecke algebra respectively.
The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules Sλ. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases—essentially, only the cases λ=(d,d) and λ=(d,d,d). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with λ of the form (d,d,1ℓ). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.
Let x ∈ Mn(𝕂) satisfy x2 = 0. Let Fx be the Springer fiber over x. The components of Fx are labeled by standard Young tableaux with two columns. For a Young tableau T with two columns, one can define a numerical invariant ρ(T). The component FT is singular if and only if ρ(T) ≥ 2. In this paper, we construct the stratification of FT for T with ρ(T) = 2, which reflects also the inner structure of the singular locus, and show that such components have equivalent stratifications if and only if they are isomorphic as algebraic varieties and provide the combinatorial procedure which partitions the set of such components into the classes of isomorphic components. We explain why the stratification of FT for T with ρ(T) ≥ 3 is very complex and does not reflect the inner structure of the singular locus.
We compute characteristic cycles of the IC-sheaves associated to the K-orbits on Grassmannians.
The purpose of this note is to describe the action of a (say finite) group on the cohomology algebra of a projective variety X over in the following setting: there exists an equivariantly formal action on X by an algebraic torus S such that acts on the equivariant cohomology algebra as a graded algebra and -module and on as a graded -algebra and, in addition, the cohomology restriction map is -equivariant. A sufficient condition for an action of a group on to define an action on which admits a lift to satisfying the above conditions was given in \cite{CK}. In this note we show that under the above conditions, the actions of on and are equivalent. In particular, if acts on the fixed point set , then the action of on is a sum of induced representations, one for each component of . This gives a simple proof of the Alvis-Lusztig-Treumann Theorem which describes Springer's Weyl group action on the cohomology of a Springer variety in a flag variety of type A.
We give examples of reducible characteristic cycles for irreducible Harish-Chandra modules for by analyzing a four-dimensional singular subvariety of . We relate this singularity to the Kashiwara-Saito singularity arising for Schubert varieties for with reducible characteristic cycles, as well as recent related examples of Williamson. In particular, this gives another explanation of the simple highest weight modules for with reducible associated varieties that Williamson discovered.
Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection will be described explicitly in a sequel. Note that this is in contrast with ordinary type C Springer fibres, where the parametrization of irreducible components, and the resulting geometric Robinson-Schensted bijection, are more complicated.
The purpose of this article is to study in detail the actions of a semisimple Lie or algebraic group on its Lie algebra by the adjoint representation and on itself by the adjoint action. We will focus primarily on orbits through nilpotent elements in the Lie algebra; these are called nilpotent orbits for short. Many deep results about such orbits have been obtained in the last thirty-five years; we will collect some of the most significant of these that have found wide application to representation theory. We will primarily work in the setting of a semisimple Lie algebra and its adjoint group over an algebraically closed field of characteristic zero, but we will extend much of what we do to semisimple Lie algebras over the reals or an algebraically closed field of prime characteristic, and to conjugacy classes in semisimple algebraic groups. We will give detailed proofs of many results, including some which are difficult to ferret out of the literature. Other results will be summarized with reasonably complete references. The treatment is a more comprehensive version of that in [CM93]; there is also some overlap with Humphreys’s book [Hu95]. In the last chapter we summarize some of the most recent work being done in this topic and indicate some directions of current research.
This chapter briefly explains orbital varieties, Goldie rank polynomials, and unitary highest weight modules. The Goldie rank polynomials are intimately related to the geometry of the nilpotent orbits via what was at first just a strange coincidence with the Springer theory. However, one can now directly attach to a nilpotent orbit polynomials that are not quite the same, but that span the same space. These are the characteristic polynomials of the orbital varieties attached to a given orbit. It turns out that these are given by exactly the same formulae, but involving geometric analogs of the Kazhdan-Lusztig polynomials that can be and are different. This gives an entirely new twist to the orbit method and insight into geometry from representation theory rather than vice versa. The relationship between Goldie rank and characteristic polynomials allows one to conclude that the latter (and hence the former) can be expressed as a positive linear combination of products of (distinct) positive roots.
Let G
R
be a connected Lie group, gR
its Lie algebra, and gR
* the vector space dual of gR
. Each coadjoint orbit, i.e., GR
-orbit in gR
*, has an intrinsically defined G
R
-invariant symplectic structure, and thus carries a distinguished G
R
-invariant measure. Kirillov’s character formula — in those cases when it applies — expresses the irreducible unitary characters of G
R
as Fourier transforms of the distinguished measures on coadjoint orbits, which are then lifted from the Lie algebra to the group via the exponential map. Kirillov [Ki] originally established the formula for nilpotent groups, and Auslander-Kostant [AK] for a large class of solvable groups.
LetG be a semi-simple complex algebraic group with Lie algebra and flag varietyX=G/B. For each primitive idealJ with trivial central character in the enveloping algebraU() we define a characteristic variety in the cotangent bundle ofX, which projects under the Springer resolution mapT
*
X onto the closure of a nilpotent orbit. We prove that this characteristic variety is theG-saturation of the characteristic variety of a highest weight module with annihilatorJ. We conjecture that it is irreducible forG=SL
n
. Our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of Weyl group representations, as achieved by A. Joseph. The presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space, and to study more generally characteristic varieties of Harish-Chandra modules.
We consider a real reductive dual pair (G′, G) of type I, with rank
. Given a nilpotent coadjoint orbit
, let
denote the complex orbit containing
. Under some condition on the partition λ′ parametrizing
, we prove that, if λ is the partition obtained from λ by adding a column on the very left, and
is the nilpotent coadjoint orbit parametrized by λ, then
, where
are the moment maps. Moreover, if
, where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to
with corresponds to
via the Springer correspondence.
The special basis in spaces of finite dimensional representation ofS
N and GL(n) is constructed and its properties are studied.
We generalize Springer representations to the context of groups over a global function field. The global counterpart of the Grothendieck simultaneous resolution is the parabolic Hitchin fibration. We construct an action of the graded double affine Hecke algebra (DAHA) on the direct image complex of the parabolic Hitchin fibration. In particular, we get representations of the degenerate graded DAHA on the cohomology of parabolic Hitchin fibers, providing the first step towards a global Springer theory.
We examine the accelerating 'obesity epidemic' in the US from the perspective of recently developed theory relating a cognitive hypothalamic- pituitary-adrenal axis to an embedding 'language' of structured psy- chosocial stress. Using a Rate Distortion argument, the obesity epi- demic is found to represent the literal writing of an image of a ratch- eting pathological social hierarchy onto the bodies of American adults and children. This process, while stratified by the usual divisions of class and ethnicity, is nonetheless relentlessly engulfing even auent majority populations. Our perspective places the common explana- tion that 'obesity occurs when people eat too much and get too little exercise' in the same category as the remark by US President Calvin
We introduce an elliptic version of the Grothendieck-Springer sheaf and
establish elliptic analogues of the basic results of Springer theory. From a
geometric perspective, our constructions specialize geometric Eisenstein series
to the resolution of degree zero, semistable G-bundles by degree zero B-bundles
over an elliptic curve E. From a representation theory perspective, they
produce a full embedding of representations of the elliptic or double affine
Weyl group into perverse sheaves with nilpotent characteristic variety on the
moduli of G-bundles over E. The resulting objects are principal series examples
of elliptic character sheaves, objects expected to play the role of character
sheaves for loop groups.
We prove a conjecture of Tanisaki in the classical case, which says roughly that the matrix of orbital varieties occurring in the associated variety of a simple highest weight module is upper triangular.
We observe that the characteristic cycle of a D-module gives bounds for
decomposition numbers of intersection cohomology complexes.