Unipotent Classes Research Articles

Maximality Properties of Generalized Springer Representations of SO (N, ℂ)

Abstract Let $C$ be a unipotent class of $G=\textrm{SO}(N,\mathbb{C})$, $\mathcal{E}$ an irreducible $G$-equivariant local system on $C$. The generalized Springer representation $\rho (C,\mathcal{E})$ appears in the top cohomology of some variety. Let $\bar \rho (C,\mathcal{E})$ be the representation obtained by summing over all cohomology groups of this variety. It is well known that $\rho (C,\mathcal{E})$ appears in $\bar \rho (C,\mathcal{E})$ with multiplicity $1$ and that its Springer support $C$ is strictly minimal in the closure ordering among the Springer supports of the irreducbile subrepresentations of $\bar \rho (C,\mathcal{E})$. Suppose $C$ is parametrized by an orthogonal partition with only odd parts. We prove that $\bar \rho (C,\mathcal{E})$ (resp. $\textrm{sgn}\otimes \bar \rho (C,\mathcal{E})$) has a unique multiplicity 1 “maximal” subrepresentation $\rho ^{\textrm{max}}$ (resp. “minimal” subrepresentation $\textrm{sgn}\otimes \rho ^{\textrm{max}}$), where $\textrm{sgn}$ is the sign representation. These are analogues of results for $\textrm{Sp}(2n,\mathbb{C})$ by Waldspurger.

Distinguished strata in a reductive group

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 also introduced and studied.

Representatives for unipotent classes and nilpotent orbits

Let G be a simple algebraic group over an algebraically closed field k of characteristic p. The classification of the conjugacy classes of unipotent elements of G(k) and nilpotent orbits of G on is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (resp. subalgebra), or a natural generalization if G is of type Dn . From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups.

Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes

Let G be a reductive group over an algebraically closed field and let W be its Weyl group. In a series of papers, Lusztig introduced a map from the set [W] of conjugacy classes of W to the set [Gu] of unipotent classes of G. This map, when restricted to the set of elliptic conjugacy classes [We] of W, is injective. In this paper, we show that Lusztig's map [We]→[Gu] is order-reversing, with respect to the natural partial order on [We] arising from combinatorics and the natural partial order on [Gu] arising from geometry.

Geck's conjecture and the generalized Gelfand-Graev representations in bad characteristic

For a connected reductive algebraic group G defined over a finite field Fq, Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of the finite group G(Fq) 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 for G. Here, we will give a proof of Geck's conjecture. Combined with Geck's pioneer work, our proof verifies Geck's conjectural characterization of special unipotent classes, and completes his definition of GGGRs in bad characteristics.

Springer’s work on unipotent classes and Weyl group representations

We discuss some of the contributions of T.A. Springer (1926–2011) to the theory of algebraic groups, with emphasis on his work on unipotent classes and representations of Weyl groups.

GENERALISED GELFAND\u2013GRAEV REPRESENTATIONS IN BAD CHARACTERISTIC ?

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 Gleft({mathbbm{F}}_qright) , 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 weighted Dynkin diagrams.

Finite-Dimensional Pointed Hopf Algebras Over Finite Simple Groups of Lie Type IV. Unipotent Classes in Chevalley and Steinberg Groups

We show that all unipotent classes in finite simple Chevalley or Steinberg groups, different from PSLn(q) and PSp2n(q), collapse (i.e. are never the support of a finite-dimensional Nichols algebra), with a possible exception on one class of involutions in PSUn(2m).

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].

Modular irreducibility of cuspidal unipotent characters

We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after $$\ell $$ -reduction. To this end, we construct a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand–Graev representations. This is achieved by showing that cuspidal representations appear in the head of generalised Gelfand–Graev representations attached to cuspidal unipotent classes, as defined and studied in Geck and Malle (J Lond Math Soc 2(53):63–78, 1996).

Tempered representations of p-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.

Geometric approach to parabolic induction

In this paper we construct a “restriction” map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on the parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of Lusztig–Spaltenstein on induced unipotent classes to all infinite fields. We also prove a group version of a theorem of Harish-Chandra about the density of the span of regular semisimple orbital integrals.

Unipotent class representatives for finite classical groups

Abstract We describe explicitly representatives of the conjugacy classes of unipotent elements of the finite classical groups.

GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS

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$F:\mathbf{G}\rightarrow \mathbf{G}$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 (Adv. Math.94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math.94(2) (1992), 139–179) is only valid under the assumption that$p$is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that$p$is an acceptable prime for$\mathbf{G}$($p$very good is sufficient but not necessary). As an application we show that every irreducible character of$G$, respectively, character sheaf of$\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever$p$is good for $\mathbf{G}$.

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type II: Unipotent classes in symplectic groups

We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterion to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes in Chevalley and Steinberg groups.

On classification of σq-conjugacy classes of a loop group

Let k be an algebraically closed field and L=k((ϵ)) the field of Laurent series over k. Let G be a connected reductive group over k such that the characteristic of k does not divide the order of the Weyl group of G. For q∈k×, the automorphism of L, defined by ∑aiϵi↦∑ai(qϵ)i, induces an automorphism σq of the loop group G(L). We show that, when q is not a root of unity, the classification of σq-conjugacy classes of G(L) can be reduced to the classification of unipotent classes of (non-connected) reductive groups. As an application, we recover the classification (of σq-conjugacy classes) for G=GLn.

Evaluating characteristic functions of character sheaves at unipotent elements

AssumeG\mathbf {G}is a connected reductive algebraic group defined over an algebraic closureK=F¯p\mathbb {K} = \overline {\mathbb {F}}_pof the finite field of prime orderp>0p>0. Furthermore, assume thatF:G→GF : \mathbf {G} \to \mathbf {G}is a Frobenius endomorphism ofG\mathbf {G}. In this article we give a formula for the value of anyFF-stable character sheaf ofG\mathbf {G}at a unipotent element. This formula is expressed in terms of class functions ofGF\mathbf {G}^Fwhich are supported on a single unipotent class ofG\mathbf {G}. In general these functions are not determined, however, we give an expression for these functions under the assumption thatZ(G)Z(\mathbf {G})is connected,G/Z(G)\mathbf {G}/Z(\mathbf {G})is simple andppis a good prime forG\mathbf {G}. In this case our formula is completely explicit.

Outer unipotent classes in automorphism groups of simple algebraic groups

We study the unipotent elements of disconnected algebraic groups of the formG〈τ〉, whereG is a simple algebraic group in characterisctic p possessing a graph automorphism τ of order p. We classify the unipotent classes in the coset Gτ and determine the corresponding centralizers, showing that these bear a close relation to classes in a certain natural connected overgroup of G〈τ〉. We also obtain a formula for the total number of outer unipotent elements in the finite group Gγ〈τ〉, where γ is a Frobenius morphism, analogous to the well-known Steinberg formula for the number of inner unipotent elements.

CONSTRUCTIONS OF GLOBAL INTEGRALS IN THE EXCEPTIONAL GROUP F4

Motivated by known examples of global integrals which represent automorphic L-functions, this paper initiates the study of a certain two-dimensional array of global integrals attached to any reductive algebraic group, indexed by maximal parabolic subgroups in one direction and by unipotent conjugacy classes in the other. Fourier coefficients attached to unipotent classes, the Gelfand-Kirillov dimension of automorphic representations, and an identity which, empirically, appears to constrain the unfolding process are presented in detail with examples selected from the exceptional groups. Two new Eulerian integrals are included among these examples.

On the Kazhdan–Lusztig order on cells and families

We consider the set Irr( W ) of (complex) irreducible characters of a finite Coxeter group W . The Kazhdan–Lusztig theory of cells gives rise to a partition of Irr( W ) into “families” and to a natural partial order \leq_{\mathcal{LR}} on these families. Following an idea of Spaltenstein, we show that \leq_{\mathcal{LR}} can be characterised (and effectively computed) in terms of standard operations in the character ring of W . If, moreover, W is the Weyl group of an algebraic group G , then \leq_{\mathcal{LR}} can be interpreted, via the Springer correspondence, in terms of the closure relation among the “special” unipotent classes of G .