1. CJM Online first
 Stavrova, Anastasia

Nonstable $K_1$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 twodimensional
split torus $\mathbf{G}_m^2$.
We show that the natural map of nonstable $K_1$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 YoshiiNeher)
and the subgroup generated by
exponential automorphisms.
Keywords:loop reductive group, nonstable $K_1$functor, Whitehead group, Laurent polynomials, Lie torus Categories: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 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 quasisemisimplicity 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
fixedpoint 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 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 sousgroupe 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 $\mathrm{V}^Äe$ et $\mathrm{W}^Äe$, 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 padic reductive groups, types, contragredient, intertwining Category: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 blowups 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 variation Categories: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 socalled 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 group Categories:14A15, 14L15 
