Generalized Springer Correspondence Research Articles

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.

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.

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.

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.

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.

A vanishing theorem for Dirac cohomology of standard modules

This paper studies the Dirac cohomology of standard modules in the setting of graded Hecke algebras with geometric parameters. We prove that the Dirac cohomology of a standard module vanishes if and only if the module is not twisted-elliptic tempered. The proof makes use of two deep results. One is some structural information from the generalized Springer correspondence obtained by S. Kato and Lusztig. Another one is a computation of the Dirac cohomology of tempered modules by Barbasch–Ciubotaru–Trapa and Ciubotaru.We apply our result to compute the Dirac cohomology of ladder representations for type An. For each of such representations with non-zero Dirac cohomology, we associate to a canonical Weyl group representation. We use the Dirac cohomology to conclude that such representations appear with multiplicity one.

𝐙/𝐦-graded Lie algebras and perverse sheaves, I

We give a block decomposition of the equivariant derived category arising from a cyclically graded Lie algebra. This generalizes certain aspects of the generalized Springer correspondence to the graded setting.

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 $G_2$.

Generalized Springer theory and weight functions

We show that each Weyl group which enters in the generalized Springer correspondence carries a natural weight function.

Constructible sheaves on nilpotent cones in rather good characteristic

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.

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 SL( n ) with coefficients of arbitrary characteristic and for SO( n ) and Sp(2 n ) with characteristic-2 coefficients.

A homological study of Green polynomials*

We interpret the orthogonality relation of Kostka polynomials arising from complex reection groups ([Shoji, Invent. Math. 74 (1983), J. Algebra 245 (2001)] and [Lusztig, Adv. Math. 61 (1986)]) in terms of homological algebra. This leads us to the notion of Kostka system, which can be seen as a categorical counterpart of Kostka polynomials. Then, we show that every generalized Springer correspondence ([Lusztig, Invent. Math. 75 (1984)]) in a good characteristic gives rise to a Kostka system. This enables us to see the top-term generation property of the (twisted) homology of generalized Springer bers, and the transition formula of Kostka polynomials between two generalized Springer correspondences of type BC. The latter provides an inductive algorithm to compute Kostka polynomials by upgrading [Ciubotaru-Kato-K, Invent. Math. 178 (2012)] x3 to its graded version. In the appendices, we present purely algebraic proofs that Kostka systems exist for type A and asymptotic type BC cases, and therefore one can skip geometric sections x3{5 to see the key ideas and basic examples/techniques.

Modular generalized Springer correspondence I: the general linear group

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.

Le groupe ${\bf GL}_{N}$ tordu, sur un corps $p$-adique, 2ème partie

Let $F$ be a nonarchimedean local field of characteristic zero, and let $N$ be an even integer at least $2$. Consider the following algebraic groups, both defined and split over $F:$ ${\bf G}={\bf GL}_{N}$ and ${\bf H}={\bf SO}(N+1)$. Let ${\bf G}^+={\bf G}\rtimes \{1,\theta\}$, where $\theta^2=1$ and $\theta$ act on ${\bf G}$ as the nontrivial outer automorphism. It is a nonconnected group. Let $\tilde{{\bf G}}={\bf G}\theta$ be the connected component that contains $\theta$. We consider $L$-packets of admissible irreducible discrete series representations of ${\bf H}(F)$ of unipotent reduction. In a previous article with C. Moeglin [MW], we have described these $L$-packets. Let $\Pi^H$ be such an $L$-packet. We associate to $\Pi^H$ an admissible irreducible representation $\pi^+$ of ${\bf G}^+(F)$. We prove that the restriction ${\rm trace}_{\tilde{{\bf G}}}\,\pi^+$ to $\tilde{{\bf G}}(F)$ of the character of $\pi^+$ is a stable distribution. Granting a fundamental lemma for the pair (${\bf G}^+$, ${\bf H}$) to be true, we prove then that ${\rm trace}_{\tilde{{\bf G}}}\,\pi^+$ is an endoscopic transfer of the sum of the characters of the representations belonging to $\Pi^H$. The proof uses the Schneider-Stuhler theory to express ${\rm trace}_{\tilde{{\bf G}}}\,\pi^+$ as an orbital integral of a pseudocoefficient of $\pi^{+}$. We use the generalized Springer correspondence to compute this pseudocoefficient.

Produit scalaire elliptique

Suppose that W is a Weyl group, let C ( W ) be a space of functions on W, with complex values, invariant under conjugation. We can define an “elliptic scalar product” on C ( W ) . It is a natural ingredient to the representation theory of p-adic reductive groups. Let G be a reductive group over the algebraic closure of a finite field. The generalized Springer correspondence gives a bijection between two sets: – the set of pairs ( U , E ) , where U is an unipotent orbit of G and E is a G-equivariant irreducible local system on U; – the disjoint union of the sets of irreducible representations of certain Weyl groups related to G. Using Kazhdan–Lusztig polynomials, we modify the generalized Springer correspondence. By the modified correspondence, a pair ( U , E ) as above maps to a representation of a certain Weyl group, and this representation is, in general, reducible. There is no simple formula that relates the elliptic scalar product and the generalized Springer correspondence. But a simple formula does exist, and we prove it, that relates the elliptic scalar product and the modified generalized Springer correspondence. Our result is, in fact, a corollary of a theorem of Lusztig on the restriction of character-sheaves to the unipotent variety.

Generalized Springer correspondence and Green functions for type [formula omitted] graded Hecke algebras

This paper presents a combinatorial approach to the tempered representation theory of a graded Hecke algebra H of classical type B or C, with arbitrary parameters. We present various combinatorial results which together give a uniform combinatorial description of what becomes the Springer correspondence in the classical situation of equal parameters. More precisely, by using a general version of Lusztig's symbols which describe the classical Springer correspondence, we associate to a discrete series representation of H with central character W 0 c , a set Σ ( W 0 c ) of W 0 -characters (where W 0 is the Weyl group). This set Σ ( W 0 c ) is shown to parametrize the central characters of the generic algebra which specialize into W 0 c . Using the parabolic classification of the central characters of H ^ t ( R ) on the one hand, and a truncated induction of Weyl group characters on the other hand, we define a set Σ ( W 0 c ) for any central character W 0 c of H ^ t ( R ) , and show that this property is preserved. We show that in the equal parameter situation we retrieve the classical Springer correspondence, by considering a set U of partitions which replaces the unipotent classes of SO 2 n + 1 ( C ) and Sp 2 n ( C ) , and a bijection between U and the central characters of H ^ t ( R ) . We end with a conjecture, which basically states that our generalized Springer correspondence determines H ^ t ( R ) exactly as the classical Springer correspondence does in the equal label case. In particular, we conjecture that Σ ( W 0 c ) indexes the modules in H ^ t ( R ) with central character W 0 c , in the following way. A module M in H ^ t ( R ) has a natural grading for the action of W 0 , and the W 0 -representation χ ( M ) in its top degree is irreducible. When M runs through the modules in H ^ t ( R ) with central character W 0 c , χ ( M ) runs through Σ ( W 0 c ) . Moreover, still in analogy with the equal parameter case, we conjecture that the W 0 -structure of the modules in H ^ t ( R ) can be computed using (generalized) Green functions.