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1. CJM Online first

Stavrova, Anastasia
 Non-stable $K_1$-functors of multiloop groups Let $k$ be a field of characteristic 0. Let $G$ be a reductive group over the ring of Laurent polynomials $R=k[x_1^{\pm 1},...,x_n^{\pm 1}]$. Assume that $G$ contains a maximal $R$-torus, and that every semisimple normal subgroup of $G$ contains a two-dimensional split torus $\mathbf{G}_m^2$. We show that the natural map of non-stable $K_1$-functors, also called Whitehead groups, $K_1^G(R)\to K_1^G\bigl( k((x_1))...((x_n)) \bigr)$ is injective, and an isomorphism if $G$ is semisimple. As an application, we provide a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii-Neher) and the subgroup generated by exponential automorphisms. Keywords:loop reductive group, non-stable $K_1$-functor, Whitehead group, Laurent polynomials, Lie torusCategories:20G35, 19B99, 17B67

2. CJM Online first

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 groupsCategories: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 seriesCategories:20G15, 20G40, 20C33, 22E35

4. CJM 2013 (vol 66 pp. 1287)

Henniart, Guy; Sécherre, Vincent
 Types et contragrÃ©dientes Soit $\mathrm{G}$ un groupe rÃ©ductif $p$-adique, et soit $\mathrm{R}$ un corps algÃ©briquement clos. Soit $\pi$ une reprÃ©sentation lisse de $\mathrm{G}$ dans un espace vectoriel $\mathrm{V}$ sur $\mathrm{R}$. Fixons un sous-groupe ouvert et compact $\mathrm{K}$ de $\mathrm{G}$ et une reprÃ©sentation lisse irrÃ©ductible $\tau$ de $\mathrm{K}$ dans un espace vectoriel $\mathrm{W}$ de dimension finie sur $\mathrm{R}$. Sur l'espace $\mathrm{Hom}_{\mathrm{K}(\mathrm{W},\mathrm{V})}$ agit l'algÃ¨bre d'entrelacement $\mathscr{H}(\mathrm{G},\mathrm{K},\mathrm{W})$. Nous examinons la compatibilitÃ© de ces constructions avec le passage aux reprÃ©sentations contragrÃ©dientes $\mathrm{V}^Äe$ et $\mathrm{W}^Äe$, et donnons en particulier des conditions sur $\mathrm{W}$ ou sur la caractÃ©ristique de $\mathrm{R}$ pour que le comportement soit semblable au cas des reprÃ©sentations complexes. Nous prenons un point de vue abstrait, n'utilisant que des propriÃ©tÃ©s gÃ©nÃ©rales de $\mathrm{G}$. Nous terminons par une application Ã  la thÃ©orie des types pour le groupe $\mathrm{GL}_n$ et ses formes intÃ©rieures sur un corps local non archimÃ©dien. Keywords:modular representations of p-adic reductive groups, types, contragredient, intertwiningCategory:22E50

5. CJM 2011 (vol 64 pp. 409)

Rainer, Armin
 Lifting Quasianalytic Mappings over Invariants Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb C[V]^G$. We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$ over the mapping of invariants $\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in $\mathcal C$, e.g., the real analytic class, then $f$ admits a lift of the same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation. If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions. Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variationCategories:14L24, 14L30, 20G20, 22E45

6. CJM 2006 (vol 58 pp. 93)

Gordon, Julia
 Motivic Haar Measure on Reductive Groups We define a motivic analogue of the Haar measure for groups of the form $G(k\llp t\rrp)$, where~$k$ is an algebraically closed field of characteristic zero, and $G$ is a reductive algebraic group defined over $k$. A classical Haar measure on such groups does not exist since they are not locally compact. We use the theory of motivic integration introduced by M.~Kontsevich to define an additive function on a certain natural Boolean algebra of subsets of $G(k\llp t\rrp)$. This function takes values in the so-called dimensional completion of the Grothendieck ring of the category of varieties over the base field. It is invariant under translations by all elements of $G(k\llp t\rrp)$, and therefore we call it a motivic analogue of Haar measure. We give an explicit construction of the motivic Haar measure, and then prove that the result is independent of all the choices that are made in the process. Keywords:motivic integration, reductive groupCategories:14A15, 14L15
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