Springer Correspondence Research Articles

Kirillov Polynomials for the Exceptional Lie Algebra g2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak g_{2}$$\\end{document}

As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of q elements and fixed Jordan type. One obtains polynomials with respect to q with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra g2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak g_2$$\\end{document}. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.

On the labelling of characters of Weyl groups of type $$F_4$$

AbstractIn the literature on finite groups of Lie type, there exist two different conventions about the labelling of the irreducible characters of Weyl groups of type $$F_4$$ F 4 . We point out some issues concerning these two conventions and their effect on tables about unipotent characters or the Springer correspondence. Using experiments related to these issues with the computer algebra system , we spotted an error in Spaltenstein’s tables for the generalised Springer correspondence in type $$E_7$$ E 7 .

Unipotent character sheaves and strata of a reductive group

Let H H be a connected reductive group over an algebraically closed field. We define a surjective map from the set C S ( H ) CS(H) of unipotent character sheaves on H H (up to isomorphism) to the set of strata of H H . To do this we use the generalized Springer correspondence. We also give a new parametrization of C S ( H ) CS(H) in terms of data coming from bad characteristic.

On the generalised Springer correspondence for groups of type 𝐸₈

We complete the determination of the generalised Springer correspondence for connected reductive algebraic groups, by proving a conjecture of Lusztig on the last open cases which occur for groups of type E 8 E_8 .

A new approach to the generalized Springer correspondence

The Springer resolution of the nilpotent cone is used to give a geometric construction of the irreducible representations of Weyl groups. Borho and MacPherson obtain the Springer correspondence by applying the decomposition theorem to the Springer resolution, establishing an injective map from the set of irreducible Weyl group representations to simple equivariant perverse sheaves on the nilpotent cone. In this manuscript, we consider a generalization of the Springer resolution using a variety defined by the first author. Our main result shows that in the type A case, applying the decomposition theorem to this map yields all simple perverse sheaves on the nilpotent cone with multiplicity as predicted by Lusztig’s generalized Springer correspondence.

Perverse Sheaves and the Cohomology of Regular Hessenberg Varieties

We use the Springer correspondence to give a partial characterization of the irreducible representations which appear in the Tymoczko dot action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Lie type. We use this extension to prove that regular Hessenberg varieties, though not necessarily smooth, always have the “Kähler package.”

Symmetric Spaces Associated to Classical Groups with Even Characteristic

Let G=GL(V) for an N-dimensional vector space V over an algebraically closed field k, and Gθ the fixed point subgroup of G under an involution θ on G. In the case where Gθ=O(V), the generalized Springer correspondence for the unipotent variety of the symmetric space G/Gθ was described in [SY], assuming that ch k≠2. The definition of θ given there, and of the symmetric space arising from θ, make sense even if ch k=2. In this paper, we discuss the Springer correspondence for those symmetric spaces with even characteristic. We show, if N is even, that the Springer correspondence is reduced to that of symplectic Lie algebras in ch k=2, which was determined by Xue. While if N is odd, the number of Gθ-orbits in the unipotent variety is infinite, and a very similar phenomenon occurs as in the case of exotic symmetric space of higher level, namely of level r=3.

Springer basic sets and modular Springer correspondence for classical types

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic p, and with coefficients in a field of odd characteristic ℓ≠p): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We provide a quick proof, by sorting characters according to the dimension of the corresponding Springer fibre, an invariant which is directly computable from symbols.

Sign signatures and characters of Weyl Groups

Let GR be the set of real points of a complex linear reductive group and Gˆλ, its classes of irreducible admissible representations with infinitesimal integral regular character λ. In this case, each cell of representations is associated to a special nilpotent orbit. This helps organize the corresponding set of irreducible Harish-Chandra modules. The goal of this paper is to bypass the need for the character table of the Weyl group associated with Gˆλ in the Springer correspondence, by describing a new and less computationally intensive parametrization of irreducible representations of simple complex Weyl groups.

Affine Hecke algebras and their representations

This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible representations.Only at the end we establish a new result: a natural bijection between the set of irreducible representations of an affine Hecke algebra with parameters in R≥1, and the set of irreducible representations of the affine Weyl group underlying the algebra. This can be regarded as a generalized Springer correspondence with affine Hecke algebras.

Generalized Springer Correspondence for Symmetric Spaces Associated to Orthogonal Groups

Let $G = GL_N(\mathbf{k})$, where $\mathbf{k}$ is an algebraically closed field of $\operatorname{ch} \mathbf{k} \ne 2$, and $\theta$ an involutive automorphism of $G$ such that $H = (G^{\theta})^0$ is isomorphic to $SO_N(\mathbf{k})$. Then $G^{\iota\theta} = \{ g \in G \mid \theta(g) = g^{-1}\}$ is regarded as a symmetric space $G/G^{\theta}$. Let $G^{\iota\theta}_{\operatorname{uni}}$ be the set of unipotent elements in $G^{\iota\theta}$. $H$ acts on $G^{\iota\theta}_{\operatorname{uni}}$ by the conjugation. As an analogue of the generalized Springer correspondence in the case of reductive groups, we establish in this paper the generalized Springer correspondence between $H$-orbits in $G^{\iota\theta}_{\operatorname{uni}}$ and irreducible representations of various symmetric groups.

Formality and Lusztig’s Generalized Springer Correspondence

We prove a derived equivalence between each block of the derived category of sheaves on the nilpotent cone and the category of differential graded modules over a degeneration of Lusztig’s graded Hecke algebra. Along the way, we construct and study a mixed version of the geometric category. This work can be viewed as giving a derived version of the generalized Springer correspondence.

Hessenberg varieties, intersections of quadrics, and the Springer correspondence

In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair ( S L ( N ) , S O ( N ) ) (SL(N),SO(N)) in [Compos. Math. 154 (2018), pp. 2403–2425].

Springer correspondence, hyperelliptic curves, and cohomology of Fano varieties

In [CVX3] (T. H. Chen, K. Vilonen, and T. Xue, “Springer correspondence for the split symmetric pair in type A”, Compos. Math. 154 (2018), no. 11, 2403–2425), we have established a Springer theory for the symmetric pair (SL(N),SO(N)). In this setting we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These representations arise from cohomology of families of certain (Hessenberg) varieties. In this paper we determine the Springer correspondence explicitly for IC sheaves supported on order 2 nilpotent orbits. In this process we encounter universal families of hyperelliptic curves. As an application we calculate the cohomolgy of Fano varieties of k-planes in the smooth intersection of two quadrics in an even dimensional projective space.

Dirac index and associated cycles of Harish-Chandra modules

Let GR be a simple real linear Lie group with maximal compact subgroup KR and assume that rank(GR)=rank(KR). For any representation X of Gelfand-Kirillov dimension 12dim⁡(GR/KR), we consider the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing X. Under a technical condition involving the Springer correspondence, we establish an explicit relationship between this polynomial and the multiplicities of the irreducible components occurring in the associated cycle of X. This relationship was conjectured in [12].

Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups

We consider the derived category D^b_G(V) of coherent sheaves on a complex vector space V equivariant with respect to an action of a finite reflection group G . In some cases, including Weyl groups of type A , B , G_2 , F_4 , as well as the groups G(m,1,n)=(\mu_m)^n\rtimes S_n , we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of G . The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces V^g/C(g) , where C(g) is the centralizer subgroup of g\in G . In the case of the Weyl groups the construction uses some key results about the Springer correspondence, due to Lusztig, along with some formality statement generalizing a result of Deligne [23]. We also construct global analogs of some of these semiorthogonal decompositions involving derived categories of equivariant coherent sheaves on C^n , where C is a smooth curve.

Springer correspondence for the split symmetric pair in type

In this paper we establish Springer correspondence for the symmetric pair $(\text{SL}(N),\text{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at $q=-1$. Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at $q=-1$ arise in geometry.

Generalized Springer theory for D-modules on a reductive Lie algebra

Given a reductive group $G$, we give a description of the abelian category of $G$-equivariant $D$-modules on $\mathfrak{g}=\mathrm{Lie}(G)$, 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 $(L,\mathcal{E})$, consisting of a Levi subgroup $L$, and a cuspidal local system $\mathcal{E}$ on a nilpotent $L$-orbit. Each block is equivalent to the category of $D$-modules on the center $\mathfrak{z}(\mathfrak{l})$ of $\mathfrak{l}$ which are equivariant for the action of the relative Weyl group $N_G(L)/L$. 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.

A partial order on bipartitions from the generalized Springer correspondence

In [1], Lusztig gives an explicit formula for the bijection between the set of bipartitions and the set \mathcal{N} of unipotent classes in a spin group which carry irreducible local systems equivariant for the spin group but not equivariant for the special orthogonal group. The set \mathcal{N} has a natural partial order and therefore induces a partial order on bipartitions. We use the explicit formula given in [1] to prove that this partial order on bipartitions is the same as the dominance order appeared in Dipper–James–Murphy's work [2].

Generalizations of the Springer correspondence and cuspidal Langlands parameters

Let $${\mathcal H}$$ be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for $${\mathcal H}$$ , which conjecturally puts supercuspidal $${\mathcal H}$$ -representations in bijection with such L-parameters. We also define a cuspidal support map and Bernstein components for enhanced L-parameters, in analogy with Bernstein’s theory of representations of p-adic groups. We check that for several well-known reductive groups these analogies are actually precise. Furthermore we reveal a new structure in the space of enhanced L-parameters for $${\mathcal H}$$ , that of a disjoint union of twisted extended quotients. This is an analogue of the ABPS conjecture (about irreducible $${\mathcal H}$$ -representations) on the Galois side of the local Langlands correspondence. Only, on the Galois side it is no longer conjectural. These results will be useful to reduce the problem of finding a local Langlands correspondence for $${\mathcal H}$$ -representations to the corresponding problem for supercuspidal representations of Levi subgroups of $${\mathcal H}$$ . The main machinery behind this comes from perverse sheaves on algebraic groups. We extend Lusztig’s generalized Springer correspondence to disconnected complex reductive groups G. It provides a bijection between, on the one hand, pairs consisting of a unipotent element u in G and an irreducible representation of the component group of the centralizer of u in G, and, on the other hand, irreducible representations of a set of twisted group algebras of certain finite groups. Each of these twisted group algebras contains the group algebra of a Weyl group, which comes from the neutral component of G.