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1201  Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes Adler, Jeffrey D.; Lansky, Joshua M.
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.


1225  Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$ Cortadellas Benítez, Teresa; D'Andrea, Carlos
We exhibit a set of minimal generators of the defining ideal of the
Rees Algebra associated with the ideal of three bivariate homogeneous
polynomials parametrizing a proper rational curve in projective plane,
having a minimal syzygy of degree 2.


1250  Symplectic Degenerate Flag Varieties Feigin, Evgeny; Finkelberg, Michael; Littelmann, Peter
A simple finite dimensional module $V_\lambda$ of a simple complex
algebraic group $G$ is naturally endowed with a filtration induced by the PBWfiltration
of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module
for the group $G^a$, which can be roughly described as a semidirect product of a
Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy
to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$,
we call the closure
$\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$
of the $G^a$orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$.
In general this is a
singular variety, but we conjecture that it has many nice properties similar to
that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group.
The symplectic case is important for the conjecture
because it is the first known case where even for fundamental weights $\omega$ the varieties
$\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit
construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations,
similar to the BottSamelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$
are normal locally complete intersections with terminal and rational singularities.
We also show that these varieties are Frobenius split. Using the above mentioned results, we
prove an analogue of the BorelWeil theorem and obtain a $q$character formula
for the characters of irreducible $Sp_{2n}$modules via the AtiyahBottLefschetz fixed
points formula.


1287  Types et contragrédientes Henniart, Guy; Sécherre, Vincent
Soit $\mathrm{G}$ un groupe réductif $p$adique, et soit $\mathrm{R}$
un corps algébriquement clos.
Soit $\pi$ une représentation 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 représentation
lisse irréductible $\tau$ 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'algèbre
d'entrelacement $\mathscr{H}(\mathrm{G},\mathrm{K},\mathrm{W})$.
Nous examinons la compatibilité de ces constructions avec le passage aux
représentations contragrédientes $\mathrm{V}^ėe$ et $\mathrm{W}^ėe$, et donnons en
particulier des conditions sur $\mathrm{W}$ ou sur la caractéristique
de $\mathrm{R}$ pour que
le comportement soit semblable au cas des représentations complexes.
Nous prenons un point de vue abstrait, n'utilisant que des propriétés
générales de $\mathrm{G}$.
Nous terminons par une application à la théorie des types pour le groupe
$\mathrm{GL}_n$ et ses formes intérieures sur un corps local non archimédien.


1305  Congruence Relations for Shimura Varieties Associated with $GU(n1,1)$ Koskivirta, JeanStefan
We prove the congruence relation for the mod$p$ reduction of Shimura
varieties associated to a unitary similitude group $GU(n1,1)$ over
$\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$
is even was obtained by T. Wedhorn and O. B?ltel, as a special case
of a result of B. Moonen, when the $\mu$ordinary locus of the $p$isogeny
space is dense. This condition fails in our case. We show that every
supersingular irreducible component of the special fiber of $p\textrm{}\mathscr{I}sog$
is annihilated by a degree one polynomial in the Frobenius element
$F$, which implies the congruence relation.


1327  Obstructions of Connectivity Two for Embedding Graphs into the Torus Mohar, Bojan; Skoda, Petr
The complete set of minimal obstructions for embedding graphs
into the torus is still not determined.
In this paper, we present all obstructions for the torus of connectivity
2. Furthermore, we
describe the building blocks of obstructions of connectivity
2 for any orientable surface.


1358  Sharp Localized Inequalities for Fourier Multipliers Osėkowski, Adam
In the paper we study sharp localized $L^q\colon L^p$ estimates for
Fourier multipliers resulting from modulation of the jumps of
Lévy
processes.
The proofs of these estimates rest on probabilistic methods and
exploit related sharp bounds for differentially subordinated
martingales, which are of independent interest. The lower bounds
for
the constants involve the analysis of laminates, a family of
certain
special probability measures on $2\times 2$ matrices. As an
application, we obtain new sharp bounds for the real and imaginary
parts of the BeurlingAhlfors operator .


1382  Weighted Carleson Measure Spaces Associated with Different Homogeneities Wu, Xinfeng
In this paper, we introduce weighted Carleson measure spaces associated
with different homogeneities and prove that these spaces are the dual spaces
of weighted Hardy spaces studied in a forthcoming paper.
As an application, we establish
the boundedness of composition of two CalderónZygmund operators with
different homogeneities on the weighted Carleson measure spaces; this,
in particular, provides the weighted endpoint estimates for the operators
studied by PhongStein.


1413  Generalized KählerEinstein Metrics and Energy Functionals Zhang, Xi; Zhang, Xiangwen
In this paper, we consider a generalized
KählerEinstein equation on Kähler manifold $M$. Using the
twisted $\mathcal K$energy introduced by Song and Tian, we show
that the existence of generalized KählerEinstein metrics with
semipositive twisting $(1, 1)$form $\theta $ is also closely
related to the properness of the twisted $\mathcal K$energy
functional. Under the condition that the twisting form $\theta $ is
strictly positive at a point or $M$ admits no nontrivial Hamiltonian
holomorphic vector field, we prove that the existence of generalized
KählerEinstein metric implies a MoserTrudinger type inequality.

