Abstract. Let G be a connected reductive algebraic group over an algebraic closure Fp of the finite field of prime order p and let F : G → G be a Frobenius endomorphism with G = G F the corresponding Fq-rational structure. One of the strongest links we have between the representation theory of G and the geometry of the unipotent conjugacy classes of 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 G (p very good is sufficient but not necessary). As an application we show that every irreducible character of G, respectively, character sheaf of G, has a unique wave front set, respectively, unipotent support, whenever p is good for G.
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 .
Assume G is a finite symplectic group Sp 2n (q) over a finite field F q of odd characteristic. We describe the action of the automorphism group Aut(G) on the set Irr(G) of ordinary irreducible characters of G. This description relies on the equivariance of Deligne-Lusztig induction with respect to automorphisms. We state a version of this equivariance which gives a precise way to compute the automorphism on the corresponding Levi subgroup; this may be of independent interest. As an application we prove that the global condition in Späth's criterion for the inductive McKay condition holds for the irreducible characters of Sp 2n (q).
We investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition we obtain a parameterisation of the isomorphism classes of all root data. By working at the level of root data we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms such embeddings were constructed by Benjamin Martin.In an unpublished manuscript Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped with a Frobenius endomorphism, to those whose derived subgroup is simple and simply connected. By using our investigations into root data we give new proofs of Asai's results and generalise them so that they are compatible with Steinberg endomorphisms. As an illustration of these ideas, we answer a question posed to us by Olivier Dudas concerning unipotent supports.1. Introduction 1.1. Let K be an algebraically closed field of characteristic p 0 then we say G is a K-group if it is an algebraic group defined over K. We will denote by G K the set of isomorphism classes of connected reductive K-groups. In this article we will be concerned with the following natural question. Problem 1.2. Give a nice parameterisation of the set of isomorphism classes G K .
1.3.Here the term nice is subjective. However, for each element of G K we would like to give it a computable label which distinguishes it uniquely. If p > 0 then it is difficult to approach Problem 1.2 in the language of algebraic groups. This is primarily because a bijective morphism of algebraic groups need not be an isomorphism in positive characteristic. For instance, if p > 0, then the product map SL p (K) × G m → GL p (K), where G m denotes the multiplicative group of the field, is a bijective morphism of algebraic groups but the K-groups SL p (K) × G m and GL p (K) are not isomorphic.
1.4.To get around these subtleties we will provide an answer to Problem 1.2 using the language of root data. If V is a set then we denote by QV = Q ⊗ Z ZV the Q-vector space obtained from the free Z-module ZV by extending scalars. With this we recall that, roughly, a root datum is a quadruple (X, Φ, q X, q Φ) where X and q X are a pair of finite rank free Z-modules in duality and Φ ⊆ X ⊆ QX and q Φ ⊆ q X ⊆ Q q X are root systems. Let R denote the set of isomorphism classes of root data. By a classical theorem of Chevalley we have a bijective map G K → R defined by G → R(G), where R(G) is the root datum of G defined with respect to some (any) maximal torus of G. Hence, Problem 1.2 is equivalent to the following.
In previous work Regev used part of the representation theory of Lie superalgebras to compute the values of a character of the symmetric group whose decomposition into irreducible constituents is described by semistandard (k, ℓ)-tableaux. In this short note we give a new proof of Regev's result using skew characters.
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