Component groups of unipotent centralizers in good characteristic
George J. McNinch, Eric Sommers
备注：
13 pages; AMS LaTeX. This is the final version; it will appear in the Steinberg birthday volume of the Journal of Algebra. This version corrects an oversight pointed out by the referee; see Prop 23
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-checking, that the structure of the component group A(u) is independent of good characteristic.
Let X and Y be commuting nilpotent K-endomorphisms of a vector space V, where K is a field of characteristic p >= 0. If F=K(t) is the field of rational functions on the projective line, consider the K(t)-endomorphism A=X+tY of V. If p=0, or if the (p-1)-st power of A is 0, we show here that X and Y are tangent to the unipotent radical of the centralizer of A in GL(V). For all geometric points (a:b) of a suitable open subset of the projective line, it follows that X and Y are tangent to the unipotent radical of the centralizer of aX+bY. This answers a question of J. Pevtsova.
Let G be a semisimple algebraic group over a field K whose characteristic is very good for G, and let sigma be any G-equivariant isomorphism from the nilpotent variety to the unipotent variety; the map sigma is known as a Springer isomorphism. Let y in G(K), let Y in Lie(G)(K), and write C_y = C_G(y) and C_Y= C_G(Y) for the centralizers. We show that the center of C_y and the center of C_Y are smooth group schemes over K. The existence of a Springer isomorphism is used to treat the crucial cases where y is unipotent and where Y is nilpotent. Now suppose G to be quasisplit, and write C for the …
Let $G$ be a classical group with natural module $V$ over an algebraically closed field of good characteristic. For every unipotent element $u$ of $G$, we describe the Jordan block sizes of $u$ on the irreducible $G$-modules which occur as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of $u$, for which recursive formulae are known.
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 simple algebraic group of adjoint type over an algebraically closed field of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs (M,O) where M is the identity component of the centralizer of a semisimple element in G and O is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.
Let G be a connected unipotent group over a finite field F_q with q elements. In this article we propose a definition of L-packets of complex irreducible representations of the finite group G(F_q) and give an explicit description of L-packets in terms of the so-called "admissible pairs" for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension of every complex irreducible representation of G(F_q) is a power of q, confirming a conjecture of V. Drinfeld. This paper is the first in a series of three papers exploring the relatio…
We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a conseque…
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 $\bfG$ be a connected reductive algebraic group defined over $\F_q$, where $q$ is a power of a prime $p$ that is good for $\bfG$. Let $F$ be the Frobenius morphism associated with the $\FF_q$-structure on $\bfG$ and set $G = \bfG^F$, the fixed point subgroup of $F$. Let $\bfP$ be an $F$-stable parabolic subgroup of $\bfG$ and let $\bfU$ be the unipotent radical of $\bfP$; set $P = \bfP^F$ and $U = \bfU^F$. Let $G_\uni$ be the set of unipotent elements in $G$. In this note we show that the number of conjugacy classes of $U$ in $G_\uni$ is given by a polynomial in $q$ with integer coefficien…
For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies some simple condition.
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…
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $\ell$ over an algebraically closed field $\Bbbk$ of characteristic zero, and let $(e,h,f)$ be an $\mathfrak{sl}_2$-triple of g. Denote by $\mathfrak{g}^{e}$ the centralizer of $e$ in $\mathfrak{g}$ and by ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ the algebra of symmetric invariants of $\mathfrak{g}^{e}$. We say that $e$ is good if the nullvariety of some $\ell$ homogenous elements of ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ in $(\mathfrak{g}^{e})^{*}$ has codimension $\ell$. If $e$ is good then ${\rm S}(\mathfrak…
Let g be a finite-dimensional simple Lie algebra of rank r over an algebraically closed field of characteristic zero, and let e be a nilpotent element of g. Denote by g^e the centralizer of e in g and by S(g^e)^{g^e} the algebra of symmetric invariants of g^e. We say that e is good if the nullvariety of some r homogeneous elements of S(g^e)^{g^e} in the dual of g^{e} has codimension r. If e is good then S(g^e)^{g^e} is polynomial. The main result of this paper stipulates that if for some homogeneous generators of S(g^e)^{g^e}, the initial homogeneous component of their restrictions to e+g^f ar…
We show that the decomposition matrix of unipotent $\ell$-blocks of a finite reductive group $\mathbf{G}(\mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell$ is very good for $\mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka…
In this article we extend independent results of Lusztig and Hézard concerning the existence of irreducible characters of finite reductive groups, (defined in good characteristic and arising from simple algebraic groups), satisfying a strong numerical relationship with their unipotent support. Along the way we obtain some results concerning quasi-isolated semisimple elements.
In previous work of Gow, Ohmori, Lusztig and the author, the Schur indices of all unipotent characters of finite groups of Lie type have been explicitly determined except for six cases in groups of type $F_4$, $E_7$ and $E_8$. In this paper, we show that the Schur indices of all cuspidal unipotent characters for type $F_4$ and $E_8$ are~1, assuming that the group is defined over a field of ``good'' characteristic. This settles four out of the six open cases. For type $E_7$, we show that the Schur indices are at most~2.
Let G be a reductive algebraic group over the algebraic closure of a finite field F_q of good characteristic. In this paper, we demonstrate a remarkable compatibility between the Springer correspondence for G and the parametrization of unipotent characters of G(F_q). In particular, we show that in a suitable sense, "large" portions of these two assignments in fact coincide. This extends earlier work of Lusztig on Springer representations within special pieces of the unipotent variety.
Let F be an algebraically closed field and let G be a semisimple F-algebraic group for which the characteristic of F is *very good*. If X in Lie(G) = Lie(G)(F) is a nilpotent element in the Lie algebra of G, and if C is the centralizer in G of X, we show that (i) the root datum of a Levi factor of C, and (ii) the component group C/C^o both depend only on the Bala-Carter label of X; i.e. both are independent of very good characteristic. The result in case (ii) depends on the known case when G is (simple and) of adjoint type. The proofs are achieved by studying the centralizer C of a nilpotent s…
Let $G$ be a simple and simply connected algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p>0$. Assume that $p$ is good for the root system of $G$ and that the covering map $G_{sc} \rightarrow G$ is separable. In previous work we proved the existence of a (not necessarily unique) Springer isomorphism for $G$ that behaved like the exponential map on the resticted nullcone of $G$. In the present paper we give a formal definition of these maps, which we call `generalized exponential maps.' We provide an explicit and uniform construction of such maps for all root syste…
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 simple algebraic group defined over a finite field of good characteristic, with associated Frobenius endomorphism $F$. In this article we extend an observation of Lusztig, (which gives a numerical relationship between an ordinary character of $G^F$ and its unipotent support), to the case where $Z(G)$ is disconnected. We then use this observation in some applications to the ordinary character theory of $G^F$.
We classify good Z-gradings of basic Lie superalgebras over an algebraically closed field of characteristic zero. Good Z-gradings are used in quantum Hamiltonian reduction for affine Lie superalgebras, where they play a role in the construction of super W-algebras. We also describe the centralizer of a nilpotent even element and of an sl(2)-triple in gl(m|n) and osp(m|2n).
Assume $\mathbf{G}$ is a connected reductive algebraic group defined over an algebraic closure $\mathbb{K} = \overline{\mathbb{F}}_p$ of the finite field of prime order $p>0$. Furthermore, assume that $F : \mathbf{G} \to \mathbf{G}$ is a Frobenius endomorphism of $\mathbf{G}$. In this article we give a formula for the value of any $F$-stable character sheaf of $\mathbf{G}$ at a unipotent element. This formula is expressed in terms of class functions of $\mathbf{G}^F$ which are supported on a single unipotent class of $\mathbf{G}$. In general these functions are not determined, however we give …
Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman for unipotent elements in G; moreover, we show that the same formula determines the p-nilpotence degree of the corresponding nilpotent elements in the Lie algebra of G. Second, if G is semisimple and p is sufficiently large, we show that G always has a faithful representation (r,V) with the property that the exponential of dr(X) lies in r(G) for each p-nilpote…
Let $k$ be an algebraically closed field of positive characteristic, $G$ a reductive group over $k$, and $V$ a finite dimensional $G$-module. Let $B$ be a Borel subgroup of $G$, and $U$ its unipotent radical. We prove that if $S=\Sym V$ has a good filtration, then $S^U$ is $F$-pure.
Let $G$ be a classical group with natural module $V$ and Lie algebra $\mathfrak{g}$ over an algebraically closed field $K$ of good characteristic. For rational irreducible representations $f: G \rightarrow \operatorname{GL}(W)$ occurring as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, we describe the Jordan normal form of $\mathrm{d} f(e)$ for all nilpotent elements $e \in \mathfrak{g}$. The description is given in terms of the Jordan block sizes of the action of $e$ on $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, for which recursive formulae are known. Our results are…
Recently, a strong exponential character bound has been established in [3] for all elements $g \in \mathbf{G}^F$ of a finite reductive group $\mathbf{G}^F$ which satisfy the condition that the centraliser $C_{\mathbf{G}}(g)$ is contained in a $(\mathbf{G},F)$-split Levi subgroup $\mathbf{M}$ of $\mathbf{G}$ and that $\mathbf{G}$ is defined over a field of good characteristic. In this paper, assuming a weak version of Lusztig's conjecture relating irreducible characters and characteristic functions of character sheaves holds, we considerably generalize this result by removing the condition that…
Let G be a quasisimple algebraic group over an algebraically closed field of characteristic p>0. We suppose that p is very good for G; since p is good, there is a bijection between the nilpotent orbits in the Lie algebra and the unipotent classes in G. If the nilpotent X in Lie(G) and the unipotent u in G correspond under this bijection, and if u has order p, we show that the partitions of ad(X) and Ad(u) are the same. When G is classical or of type G_2, we prove this result with no assumption on the order of u. In the cases where u has order p, the result is achieved through an application of…
Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ of the finite field of prime order $p$ and let $F : \mathbf{G} \to \mathbf{G}$ be a Frobenius endomorphism with $G = \mathbf{G}^F$ the corresponding $\mathbb{F}_q$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig, which decomposes Kawanaka's Generalised Gelfand-Graev Representations (GGGRs) in terms of characteristic functions of intersection c…
Let $k$ be an algebraically closed field of positive characteristic, $G$ a reductive group over $k$, and $V$ a finite dimensional $G$-module. Let $P$ be a parabolic subgroup of $G$, and $U_P$ its unipotent radical. We prove that if $S$=\textyen $Sym V$ has a good filtration, then the ring of invariants $S^{U_P}$ is strongly $F$-regular.
We determine approximations to the decomposition matrices for unipotent $\ell$-blocks of several series of finite reductive groups of classical and exceptional type over $\FF_q$ of low rank in non-defining good characteristic~$\ell$.
A good knowledge of the Jordan canonical form (JCF) for a tensor product of `Jordan blocks' is key to understanding the actions of $p$-groups of matrices in characteristic $p$. The JCF corresponds to a certain partition which depends on the characteristic $p$, and the study of these partitions dates back to Aitken's work in 1934. Equivalently each JCF corresponds to a certain permutation $π$ introduced by Norman in 1995. These permutations $π= π(r,s,p)$ depend on the dimensions $r$, $s$ of the Jordan blocks, and on $p$. We give necessary and sufficient conditions for $π(r,s,p)$ to be trivial, …
We determine the decomposition matrices of unipotent $\ell$-blocks of defect $Φ_4^2$ for exceptional groups of Lie type up to a few unknowns. For this we employ the new cohomological methods of the first author, together with properties of generalized Gelfand-Graev characters which were recently shown to hold whenever the underlying characteristic is good.
Let G be a reductive group over an algebraically closed field of characteristic p, and let u in G be a unipotent element of order p. Suppose that p is a good prime for G. We show in this paper that there is a homomorphism phi:SL_2/k --> G whose image contains u. This result was first obtained by D. Testerman (J. Algebra, 1995) using case considerations for each type of simple group (and using, in some cases, computer calculations with explicit representatives for the unipotent orbits). The proof we give is free of case considerations (except in its dependence on the Bala-Carter theorem). Our c…
For a connected reductive algebraic group $G$ defined over a finite field $\mathbb F_q$, Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of the finite group $G(\mathbb F_q)$ in the case where $q$ is a power of a good prime for $G$. This representation has been widely studied and used in various contexts. Recently, Geck proposed a conjecture, characterizing Lusztig's special unipotent classes in terms of weighted Dynkin diagrams. Based on this conjecture, he gave a guideline for extending the definition of GGGRs to the case where $q$ is a power of a bad prime…
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 …
This article is part of a project which aims to describe as explicitly as possible the Arthur packets of classical real groups and to prove a multiplicity one result for them. Let $G$ be a symplectic or special orthogonal real group, and $ψ: W_{\mathbb R}\times \mathbf{SL}_2(\mathbb C)\rightarrow {}^LG$ be an Arthur parameter for $G$. Let $A(ψ)$ the component group of the centralizer of $ψ$ in $\hat G$. Attached to $ψ$ is a finite length unitary representation $π^A(ψ)$ of $G\times A(ψ)$, which is characterized by the endoscopic identities (ordinary and twisted) it satisfies. In [arXiv:1703.072…
Let $G$ be a connected reductive algebraic group defined over a finite field with $q$ elements. In the 1980's, Kawanaka introduced generalised Gelfand-Graev representations of the finite group $G(F_q)$, assuming that $q$ is a power of a good prime for $G$. These representations have turned out to be extremely useful in various contexts. Here we investigate to what extent Kawanaka's construction can be carried out when we drop the assumptions on~$q$. As a curious by-product, we obtain a new, conjectural characterisation of Lusztig's concept of special unipotent classes of $G$ in terms of weight…
We prove the existence of certain rationally rigid triples E8 in good characteristic and thereby show that these groups over the prime field occur as Galois groups over the field of rational numbers. We show that these triples give rise to rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic 0. As a byproduct we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a classification of possible overgroups in exceptional groups containing regular unipotent elements.
Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p$, and assume that $p$ is a very good prime for $G$. Let $P$ be a parabolic subgroup whose unipotent radical $U_P$ has nilpotence class less than $p$. We show that there exists a particularly nice Springer isomorphism for $G$ which restricts to a certain canonical isomorphism $\text{Lie}(U_P) \xrightarrow{\sim} U_P$ defined by J.-P. Serre. This answers a question raised both by G. McNinch in \cite{M2}, and by J. Carlson \textit{et. al} in \cite{CLN}. For the groups $SL_n, SO_n$, and $Sp_{2n}$, viewed in …
Let $R$ be an algebraically closed field and $\ell$ be its characteristic. Let $G$ be a locally profinite group having a compact open subgroup of invertible pro-order in $R$. Take $N$ a closed subgroup of $G$ exhausted by compact subgroups of invertible pro-orders in $R$ and fix a smooth character $θ$ of $N$. For $π$ an irreducible smooth $R$-representation of $G$ whose matrix coefficients are compactly supported modulo the center (we call it $Z$-compact), we show that the dimensions $\mathrm{Hom}_{N}(π,θ)$ and $\mathrm{Hom}_{N}(π^\vee,θ^{-1})$ are equal provided one of the two is finite. We d…
