We study the Newton stratification of the adjoint quotient of a connected split reductive group G with simply connected derived group over the field F of formal Laurent series in one variable over the field of complex numbers. Our main result describes the centralizer of a regular semisimple element in G(F) whose image in the adjoint quotient lies in a certain generic subset of a given Newton stratum. Other noteworthy results include analogues of some results of Springer on regular elements of finite reflection groups, as well as a geometric construction of a well known homomorphism from the fundamental group of a reduced and irreducible root system to the Weyl group of the system.
The set of strata of a reductive group can be viewed as an enlargement of the set of unipotent classes. In this paper the notion of distinguished unipotent class is extended to this larger set. The strata of a Weyl group are introduced and studied.
For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. In this paper we give an almost general proof of that conjecture.
Let $D$ be a connected component of a possibly disconnected reductive group $G$ over an algebraic closed field. We define a partition of $D$ into finitely many Strata each of which is a union of $G^0$-conjugacy classes of fixed dimension. In the case where $D=G^0$ this recovers a known partition.
Let $G$ be a non-connected reductive algebraic group over an algebraically closed field $\mathbb{K}$ and let $D$ be a connected component of $G$. We investigate Jordan classes of $D$ and we obtain a description of the regular part of the closure of a Jordan class in terms of induction of $G^{\circ}$-orbits. We use this result to show that Lusztig strata in a non-connected reductive algebraic group are locally closed.
The Newton strata of a reductive $p$-adic group are introduced in \cite{Newton} and play some role in the representation theory of $p$-adic groups. In this paper, we give a geometric interpretation of the Newton strata.
Let $G$ be a connected reductive group over an algebraically closed field $\Bbbk$. Under mild restrictions on the characteristic of $\Bbbk$, we show that any $G$-module with a good filtration also has a good filtration as a module for the reductive part of the centralizer of a nilpotent element $x$ in its Lie algebra.
Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata such that each stratum is a union of conjugacy classes of fixed dimension; the strata are indexed by a set defined in terms of the Weyl group which is independent of the characteristic. In the case where $G$ is replaced by the corresponding loop group we define an analogous decomposition of the set of regular semisimple compact elements into countably many strata.
Let LG be an algebraic loop group associated to a reductive group G. A fundamental stratum is a triple consisting of a point x in the Bruhat-Tits building of LG, a nonnegative real number r, and a character of the corresponding depth r Moy-Prasad subgroup that satisfies a non-degeneracy condition. The authors have shown in previous work how to associate a fundamental stratum to a formal flat G-bundle and used this theory to define its slope. In this paper, the authors study fundamental strata that satisfy an additional regular semisimplicity condition. Flat G-bundles that contain regular strat…
This paper defines and studies a stratification of the adjoint quotient of the Lie algebra of a reductive group over a Laurent power series field. The stratification arises naturally in the context of affine Springer fibers.
We prove analogues of fundamental results of Kostant on the universal centralizer of a connected reductive algebraic group for algebraically closed fields of positive characteristic (with mild assumptions), and for integral coefficients. As an application, we use these results to obtain a "mixed modular" analogue of the derived Satake equivalence of Bezrukavnikov-Finkelberg.
In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group G defined over a finite field with corresponding Frobenius map F and derive the number of F-stable semisimple classes of G with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters to their inertia group in the full automorphism group. As a consequence, we prove that "twisted" and "untwisted" simple groups of type E_6 are "good" in defining characteristic, which is a contribution to …
Let G be a connected reductive algebraic group over an algebraically closed field k. We consider the strata in G defined by Lusztig as fibers of a map given in terms truncated induction of Springer representations. We extend to arbitrary characteristic the following two results: Lusztig's strata are locally closed and the irreducible components of a stratum X are those sheets for the G-action on itself that are contained in X.
We show that any two homomorphisms of the alternating group A_5 into E_8 (over the complex numbers) whose images has finite centralizers are conjugate under E_8. This, in conjunction with earlier work of Frey, completes the classification up to G-conjugacy of homomorphisms of A_5 into a simple adjoint algebraic group over the complex numbers.
For a connected reductive group G and a finite-dimensional G-module V, we study the invariant Hilbert scheme that parameterizes closed G-stable subschemes of V affording a fixed, multiplicity-finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine G-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of G. We show that V contains only finitely many multiplicity-free G-subva…
We prove homological mirror symmetry for the universal centralizer $J_G$ (a.k.a the Toda space), associated to any complex reductive Lie group $G$. The A-side is a partially wrapped Fukaya category on $J_G$, and the B-side is the category of coherent sheaves on the categorical quotient of a dual maximal torus by the Weyl group action (with some modification if the center of $G$ is not connected).
Let \mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}} be a basic classical Lie superalgebra over an algebraically closed field \mathbb{K} whose characteristic p>0 is a good prime for \mathfrak{g}. Let G_{\bar{0}} be the reductive algebraic group over \mathbb{K} such that \mathrm{Lie}(G_{\bar{0}})=\mathfrak{g}_{\bar{0}}. Suppose e\in\mathfrak{g}_{\bar{0}} is nilpotent. Write \mathfrak{g}^{e} for the centralizer of e in \mathfrak{g} and \mathfrak{z}(\mathfrak{g}^{e}) for the centre of \mathfrak{g}^{e}. We calculate a basis for \mathfrak{g}^{e} and \mathfrak{z}(\mathfrak{g}^{e}) by …
Let V be a G-module where G is a complex reductive group. Let Z:=V//G denote the categorical quotient. One can ask if the Luna stratification of Z is intrinsic. That is, if phi : Z\to Z is any automorphism, does phi send strata to strata? In a paper of Kuttler and Reichstein the answer was shown to be yes for V a direct sum of sufficiently many copies of a G-module W. We show that the answer is yes for almost all V. The key is to consider the vector fields on Z. Our methods also show that complex analytic automorphisms preserve the stratification.
Let $(G,G')$ be a reductive dual pair of a symplectic group and an orthogonal group over a finite field of odd characteristic. The Howe correspondence establishes a correspondence between a subset of irreducible characters of $G$ and a subset of irreducible characters of $G'$. The Lusztig correspondence is a bijection between the Lusztig series indexed by the conjugacy class of a semisimple element $s$ in the connected component $(G^*)^0$ of the dual group of $G$ and the set of irreducible unipotent characters of the centralizer of $s$ in $G^*$. In this paper, we prove the commutativity (up to…
The theory of character sheaves on a reductive group is extended to a class of varieties which includes the strata of the De Concini-Procesi completion of an adjoint group.
Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u in G unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the second author done for simple, adjoint G over the complex numbers. When G is simple and adjoint, the previous work of the second author makes our description combinatorial and explicit; moreover, it turns out that knowledge of the conjugacy classes suffices to determine the group structure of A(u). Thus we obtain the result, previously known through case-checki…
Let G be a reductive algebraic group over the complex field and O_h be a closed adjoint orbit through a semisimple element h. By a result of Borho and Kraft (1979), it is known that the asymptotic cone of the orbit O_h is the closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup whose Levi component is the centralizer Z_G(h) in G. In this paper, we prove an analogue on a semisimple orbit for a symmetric pair (G, K).
For arbitrary reductive groups $G$ defined over a finite field, we decompose Newton strata in the special fiber of moduli spaces of global $G$-shtukas into a product of Rapoport-Zink spaces and Igusa varieties. This allows us to compare the $\ell$-adic cohomology of these spaces together with the various actions on them.
We show that, for a connected reductive algebraic group G over an algebraically closed field of zero or good characteristic, the parts, called strata, in the partition of G recently introduced by Lusztig are unions of sheets of conjugacy classes. For G simple and adjoint we refine the parametrization of such sheets obtained in previous work with F. Esposito. We give a simple combinatorial description of strata containing spherical conjugacy classes, showing that Lusztig's correspondence induces a bijection between unions of spherical conjugacy classes and unions of classes of involutions in th…
Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F) has finitely many nilpotent orbits in Lie(G)(F). When the residue field is finite, we obtain a proof that nilpotent orbital integrals …
In adjoint reductive groups $H$ of type $\mathsf{D}$ we show that for every semisimple element $s$, its centralizer splits over its connected component, i.e., $C_H(s) = C_H(s)^\circ \rtimes \check A$ for some complement $\check A$ with strong stability properties. We derive several consequences about the action of automorphisms on semisimple conjugacy classes. This helps to parametrize characters of the finite groups $\mathsf{D}_{l,\text{sc}}(q)$ and $^2\mathsf{D}_{l,\text{sc}}(q)$ and describe the action of automorphisms on them. It is also a contribution to the final proof of the McKay conje…
Let F be a non-archimedean local field and let G be a connected reductive group defined over F. We assume that G splits over a tame extension of F and that the residual characteristic p does not divide the order of the Weyl group. To each discrete Langlands parameter of the Weil group of F into the complex L-group of G we associate explicitly a finite set of irreducible supercuspidal representations of G(F), and relate its internal structure to the centralizer of the parameter. We give evidence that this assignment is an explicit realization of the local Langlands correspondence.
We generalise to the equivariant case a result of J. Denef and F. Loeser about trigonometric sums on tori; on the other hand, we study the Thom-Boardman stratification associated to the multiplication of global sections of line bundles on a curve. We prove a subtle inequaliity about the dimensions of these strata. Our motivation comes from the geometric Langlands program. Based on works of W. T. Gan, N. Gurevich, D. Jiang and S. Lysenko, we propose, for the reductive group $G$ of type $\mathrm G_2$, a conjectural construction of the automorphic sheaf whose Arthur parameter is unipotent and sub…
Recently, Lusztig constructed for each reductive group a partition by unions of sheets of conjugacy classes, which is indexed by a subset of the set of conjugacy classes in the associated Weyl group. Sevostyanov subsequently used certain elements in each of these Weyl group conjugacy classes to construct strictly transverse slices to the conjugacy classes in these strata, generalising the classical Steinberg slice, and similar cross sections were built out of different Weyl group elements by He-Lusztig. In this paper we observe that He-Lusztig's and Sevostyanov's Weyl group elements share a ce…
We show that the sheets for a connected reductive algebraic group G over an algebraically closed field in good characteristic acting on itself by conjugation are in bijection with G-conjugacy classes of triples (M, Z(M)^\circ t, O) where M is the connected centralizer of a semisimple element in G, Z(M)^\circ t is a suitable coset in Z(M)/Z(M)^\circ and O is a rigid unipotent conjugacy class in M. Any semisimple element is contained in a unique sheet S and S corresponds to a triple with O={1}. The sheets in G containing a unipotent conjugacy class are precisely those corresponding to triples fo…
