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1. CJM Online first

Elduque, Alberto
Order $3$ elements in $G_2$ and idempotents in symmetric composition algebras
Order three elements in the exceptional groups of type $G_2$ are classified up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras. Over an algebraically closed field, there are two conjugacy classes of order three elements in $G_2$ in characteristic not $3$ and four of them in characteristic $3$. The centralizers in characteristic $3$ fail to be smooth for one of these classes.

Keywords:symmetric composition algebra, Okubo algebra, automorphism group, centralizer, idempotent
Categories:17A75, 14L15, 17B25, 20G15

2. CJM 2016 (vol 68 pp. 395)

Garibaldi, Skip; Nakano, Daniel K.
Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the question of whether a given complex representation is symplectic or orthogonal has been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from 2 hold by using similar proofs. This paper considers analogues of these results for simple, induced and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

Keywords:orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups
Categories:20G05, 11E39, 11E88, 15A63, 20G15

3. CJM 2014 (vol 66 pp. 1201)

Adler, Jeffrey D.; Lansky, Joshua M.
Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes
Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\tilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $(\tilde{G},\Gamma)$, and consider any group $G$ satisfying the axioms. If both $\tilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\tilde{G}^*$ and $G^*$. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in $G^*(k)$ to the analogous set for $\tilde{G}^*(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\tilde{G}(k)$, one obtains a mapping of such packets.

Keywords:reductive group, lifting, conjugacy class, representation, Lusztig series
Categories:20G15, 20G40, 20C33, 22E35

4. CJM 2009 (vol 61 pp. 950)

Tange, Rudolf
Infinitesimal Invariants in a Function Algebra
Let $G$ be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let $\g$ be its Lie algebra. First we extend a well-known result about the Picard group of a semi-simple group to reductive groups. Then we prove that if the derived group is simply connected and $\g$ satisfies a mild condition, the algebra $K[G]^\g$ of regular functions on $G$ that are invariant under the action of $\g$ derived from the conjugation action is a unique factorisation domain.

Categories:20G15, 13F15

5. CJM 2004 (vol 56 pp. 945)

Helminck, Aloysius G.; Schwarz, Gerald W.
Smoothness of Quotients Associated \\With a Pair of Commuting Involutions
Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$ and $K:=G^\theta$ be the fixed point groups. We have an action $(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson \cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$.

Categories:20G15, 20G20, 22E15, 22E46

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