A Theorem About Maximal Cohen–Macaulay Modules

It is shown that for any local strongly $F$-regular ring there exists natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen–Macaulay module, then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite.

UN SCINDAGE DU MORPHISME DE FROBENIUS SUR L’ALGÈBRE DES DISTRIBUTIONS D’UN GROUPE RÉDUCTIF

Pour un groupe algébrique semi-simple simplement connexe sur un corps algébriquement clos de caractéristique positive, nous avons précédemment construit un scindage de l’endomorphisme de Frobenius sur son algèbre des distributions. Nous généralisons la construction au cas de des groupes réductifs connexes et en dégageons les corollaires correspondants. For a simply connected semisimple algebraic group over an algebraically closed field of positive characteristic we have already constructed a splitting of the Frobenius endomorphism on its algebra of distributions. We generalize the construction to the case of general connected reductive groups and derive the corresponding corollaries.

The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups

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 parametrization 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 In an unpublished manuscript, Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped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. Using our investigations into root data we give new proofs of Asai's results and generalize 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.

On the Mackey formula for connected centre groups

Let {\mathbf{G}} be a connected reductive algebraic group over {\overline{\mathbb{F}}_{p}} and let {F:\mathbf{G}\to\mathbf{G}} be a Frobenius endomorphism endowing {\mathbf{G}} with an {\mathbb{F}_{q}} -rational structure. Bonnafé–Michel have shown that the Mackey formula for Deligne–Lusztig induction and restriction holds for the pair {(\mathbf{G},F)} except in the case where {q=2} and {\mathbf{G}} has a quasi-simple component of type {\mathsf{E}_{6}} , {\mathsf{E}_{7}} , or {\mathsf{E}_{8}} . Using their techniques, we show that if {q=2} and {Z(\mathbf{G})} is connected then the Mackey formula holds unless {\mathbf{G}} has a quasi-simple component of type {\mathsf{E}_{8}} . This establishes the Mackey formula, for instance, in the case where {(\mathbf{G},F)} is of type {\mathsf{E}_{7}(2)} . Using this, together with work of Bonnafé–Michel, we can conclude that the Mackey formula holds on the space of unipotently supported class functions if {Z(\mathbf{G})} is connected.

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 $\mathbf{G}$.