Lifting smooth curves over invariants for representations of compact Lie groups, III

Addendum to: "Lifting smooth curves over invariants for representations of compact Lie groups, III" [J. Lie Theory 16 (2006), No. 3, 579-600.]

We improve the main results in the paper from the title using a recent refinement of Bronshtein's theorem due to Colombini, Orrú, and Pernazza. They are then in general best possible both in the hypothesis and in the outcome. As a consequence we obtain a result on lifting smooth mappings in several variables.

Lifting smooth curves over invariants for representations of compact Lie groups. II

Any sufficiently often differentiable curve in the orbit space of a compact Lie group representation can be lifted to a once differentiable curve into the representation space.

On unitary deformations of smooth modular representations

Let G be a locally analytic group. We use deformation theory and Emerton's 'augmented' Iwasawa modules to study the possible liftings of a given absolutely irreducible smooth modular representation of G to a unitary Banach space representation. Final version. Some minor items changed. Appears in Israel J. Math.

A generalization of Puiseux's theorem and lifting curves over invariants

Let $ρ: G \to \operatorname{GL}(V)$ be a rational representation of a reductive linear algebraic group $G$ defined over $\mathbb C$ on a finite dimensional complex vector space $V$. We show that, for any generic smooth (resp. $C^M$) curve $c : \mathbb R \to V // G$ in the categorical quotient $V // G$ (viewed as affine variety in some $\mathbb C^n$) and for any $t_0 \in \mathbb R$, there exists a positive integer $N$ such that $t \mapsto c(t_0 \pm (t-t_0)^N)$ allows a smooth (resp. $mathbb C^M$) lift to the representation space near $t_0$. ($C^M$ denotes the Denjoy--Carleman class associated w…

Geometric theta-lifting for the dual pair GSp_{2n}, GSO_{2m}

Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Consider the dual pair H=GSO_{2m}, G=GSp_{2n} over X, where H splits over an etale two-sheeted covering of X. Write Bun_G and Bun_H for the stacks of G-torsors and H-torsors on X. We show that for m\le n (respectively, for m>n) the theta-lifting functor from D(Bun_H) to D(Bun_G) (respectively, from D(Bun_G) to D(Bun_H)) commutes with Hecke functors with respect to a morphism of the corresponding L-groups involving the SL_2 of Arthur. So, they realize the geometric Langlands functoriality for the corresp…

提升经典群的平凡表示

Lifting banal representations of classical groups

本文研究了局部非阿基米德域上辛群或分裂正交群的平凡素数表示，证明了任何光滑不可约表示均可提升至Q_lbar，同时探讨了更一般的经典群的类似结果，并应用此证明了强平凡情况下辛-正交或单元ary对的Howe对偶性。