Suppose that $\widetilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\widetilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\widetilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $\left( \tilde{G},\Gamma \right)$, and consider any group $G$ satisfying the axioms. If both $\widetilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\widetilde{{{G}^{*}}}$ 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 $\widetilde{{{G}^{*}}}\,(k)$. 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 $\widetilde{G}\,(k)$, one obtains a mapping of such packets.

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.

A simple finite dimensional module ${{V}_{\lambda }}$ of a simple complex algebraic group $G$ is naturally endowed with a filtration induced by the PBW-filtration of $U\,(\text{Lie}\,G)$. The associated graded space $\text{V}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ is a module for the group ${{G}^{a}}$, which can be roughly described as a semi-direct 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 $\overline{{{G}^{a}}\,.\,[{{v}_{\text{ }\!\!\lambda\!\!\text{ }}}]}\,\,\subset \,\,\mathbb{P}\,(V_{\text{ }\!\!\lambda\!\!\text{ }}^{a})$ of the ${{G}^{a}}$-orbit through the highest weight line the degenerate flag variety $\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$. 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 $\mathcal{F}_{\text{ }\!\!\omega\!\!\text{ }}^{a}$ differ from ${{\mathcal{F}}_{\text{ }\!\!\omega\!\!\text{ }}}$. We give an explicit construction of the varieties $\text{Sp}\,\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ and construct desingularizations, similar to the Bott–Samelson resolutions in the classical case. We prove that $\text{Sp}\,\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ 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 Borel–Weil theorem and obtain a $q$-character formula for the characters of irreducible $\text{S}{{\text{p}}_{2\pi }}$-modules via the Atiyah–Bott–Lefschetz fixed points formula.

Soit $\text{G}$ un groupe réductif $p$-adique, et soit $\text{R}$ un corps algébriquement clos. Soit $\text{ }\!\!\pi\!\!\text{ }$ une représentation lisse de $\text{G}$ dans un espace vectoriel $\text{V}$ sur $\text{R}$. Fixons un sous-groupe ouvert et compact $\text{K}$ de $\text{G}$ et une représentation lisse irréductible $\varrho$ de $\text{K}$ dans un espace vectoriel $\text{W}$ de dimension finie sur $\text{R}$. Sur l'espace $\text{Ho}{{\text{m}}_{\text{K}}}(\text{W,}\,\text{V)}$ agit l'algèbre d'entrelacement $\mathcal{H}(\text{G,}\,\text{K,}\,\text{W)}$. Nous examinons la compatibilité de ces constructions avec le passage aux représentations contragrédientes ${{\text{V}}^{\vee }}$ et ${{\text{W}}^{\vee }}$, et donnons en particulier des conditions sur $\text{W}$ ou sur la caractéristique de $\text{R}$ 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 $\text{G}$. Nous terminons par une application á la théorie des types pour le groupe $\text{G}{{\text{L}}_{n}}$ et ses formes intérieures sur un corps local non archimédien.

We prove the congruence relation for the $\bmod -p$ reduction of Shimura varieties associated with a unitary similitude group $GU(n\,-\,1,\,1)$ over $\mathbb{Q}$ when $p$ is inert and $n$ odd. The case when $n$ is even was obtained by T. Wedhorn and O. Bültel, 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 $p-I\text{sog}$ is annihilated by a degree one polynomial in the Frobenius element $F$, which implies the congruence relation.

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.

In this paper we study sharp localized ${{L}^{q}}\,\to \,{{L}^{p}}$ 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 2×2 matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling–Ahlfors operator.

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 Calderón–Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong–Stein.

In this paper, we consider a generalized Kähler–Einstein equation on a Kähler manifold $M$. Using the twisted $\mathcal{K}$–energy introduced by Song and Tian, we show that the existence of generalized Kähler–Einstein metrics with semi–positive 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 Kähler–Einstein metric implies a Moser–Trudinger type inequality.