Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= finite union of hyperplanes) whose Levi-Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements defined by finite complex reflection groups. We determine a parameter interval for which the metric is locally of Fubini-Study type, flat, or complex-hyperbolic. We find a finite subset of this interval for...

Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $\>2$. Consider the dual pair $H={\mathrm{SO}}_{2m},G={\mathrm{Sp}}_{2n}$ over $X$ with $H$ split. Write ${\mathrm{Bun}}_G$ and ${\mathrm{Bun}}_H$ for the stacks of $G$-torsors and $H$-torsors on $X$. The theta-kernel ${\mathrm{Aut}}_{G,H}$ on ${\mathrm{Bun}}_G\times {\mathrm{Bun}}_H$ yields theta-lifting functors $F_G:\mathrm{D}\left({\mathrm{Bun}}_H\right)\to \mathrm{D}\left({\mathrm{Bun}}_G\right)$ and $F_H:\mathrm{D}\left({\mathrm{Bun}}_G\right)\to \mathrm{D}\left({\mathrm{Bun}}_H\right)$ between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non ramified case)....

À partir de l’étude de l’intégrabilité de la structure adjointe sur un groupe de Lie $𝒢$, on est amené à introduire l’algèbre de Lie $h_g$ des opérateurs symétriques du crochet de l’algèbre de Lie $g$ de $𝒢$. On fait apparaître une décomposition canonique de toute algèbre de Lie de centre nul en somme directe $\sigma \oplus b$ d’idéaux caractéristiques, où $\sigma$ est somme de deux sous-algèbres abéliennes et où $h_b$ est formée d’opérateurs nilpotents.Nous montrons que l’étude de la platitude à l’ordre 2 de la structure adjointe...

On expose une preuve détaillée de la classification par Thurston des huit géométries modèles de dimension trois.

For a locally symmetric space $M$, we define a compactification $M\cup M\left(\infty \right)$ which we call the “geodesic compactification”. It is constructed by adding limit points in $M\left(\infty \right)$ to certain geodesics in $M$. The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give $M\left(\infty \right)$ for locally symmetric spaces. Moreover, $M\left(\infty \right)$ has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in...

Let $G$ be a quasi-Hermitian Lie group with Lie algebra $𝔤$ and $K$ be a compactly embedded subgroup of $G$. Let ${\xi }_0$ be a regular element of ${𝔤}^{*}$ which is fixed by $K$. We give an explicit $G$-equivariant diffeomorphism from a complex domain onto the coadjoint orbit $𝒪\left({\xi }_0\right)$ of ${\xi }_0$. This generalizes a result of [B. Cahen, Berezin quantization and holomorphic representations, Rend. Sem. Mat. Univ. Padova, to appear] concerning the case where $𝒪\left({\xi }_0\right)$ is associated with a unitary irreducible representation of $G$ which is holomorphically...

We study the representation theory of the solution space of the one-dimensional Schrödinger equation with singular potential V λ(x) = λx −2 as a representation of $\widetilde{SL(2,ℝ)}$ . The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category. By studying the subspace of K-finite vectors in this space, a distinguished family of potentials, parametrized by the triangular numbers is shown to generate a global representation of $\widetilde{SL(2,ℝ)}$ ⋉ H 3, where H...

In the first section of this paper we give a characterization of those closed convex cones (wedges) $W$ in the Lie algebra $sl(2,\mathbf{R}{)}^n$ which are invariant under the maximal compact subgroup of the adjoint group and which are controllable in the associated simply connected Lie group $Sl(2,\mathbf{R}{)}^{n^{\sim }}$, i.e., for which the subsemigroup $S=(expW)$ generated by the exponential image of $W$ agrees with the whole group $G$ (Theorem 13). In Section 2 we develop some algebraic tools concerning real root decompositions with respect to compactly...