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

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…

Lifting quasianalytic mappings over invariants

Let $ρ: G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $σ_1,σ_n$ be a system of generators of the algebra of invariant polynomials $\mathbb{C}[V]^G$. We study the problem of lifting mappings $f : \mathbb{R}^q \supseteq U \to σ(V) \subseteq \mathbb{C}^n$ over the mapping of invariants $σ=(σ_1,σ_n) : V \to σ(V)$. Note that $σ(V)$ 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 subcl…

Intrinsic construction of invariant functions on simple Lie algebras

An algorithm for constructing primitive adjoint-invariant functions on a complex simple Lie algebra is presented. The construction is intrinsic in the sense that it does not resort to any representation. A primitive invariant function on the whole Lie algebra is obtained by lifting a coordinate function on a Kostant slice of the Lie algebra. Such an intrinsic construction of invariant functions is most useful for the bigger exceptional Lie algebras such as the E's. The Maple implementation of this algorithm is outlined at the end and will be applied to these exceptional Lie algebras in a futur…

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…

Lifting automorphisms of quotients of adjoint representations

Let $\mathfrak g_i$ be a simple complex Lie algebra, $1\leq i \leq d$, and let $G=G_1\times...\times G_d$ be the corresponding adjoint group. Consider the $G$-module $V=\oplus r_i\mathfrak g_i$ where $r_i\geq 1$ for all $i$. We say that $V$ is \emph{large} if all $r_i\geq 2$ and $r_i\geq 3$ if $G_i$ has rank 1. In [Schwarz12] we showed that when $V$ is large any algebraic automorphism $ψ$ of the quotient $Z:= V//G$ lifts to an algebraic mapping $Ψ\colon V\to V$ which sends the fiber over $z$ to the fiber over $ψ(z)$, $z\in Z$. (Most cases were already handled in [Kuttler11]). We also showed th…

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。

Lie algebras of vector fields on smooth affine varieties

We reprove the results of Jordan [18] and Siebert [31] and show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not depend on papers mentioned above. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered.

On central extensions and simply laced Lie algebras

Let $Λ$ be a simply laced root lattice and $w$ an elliptic automorphism of $Λ$ of order $d$. This paper gives a construction that begins with a central extension of the group of coinvariants $Λ_w$ and produces a semisimple Lie algebra of Dykin type $Λ$ with an automorphism of order $d$ lifting $w$. The input for this construction naturally arises when considering certain families of algebraic curves. The construction generalizes one used by Thorne to study plane quartics and one used by the author and Thorne to study a family of genus-2 curves.

Tannaka duality for proper Lie groupoids

The main contribution of this thesis is a Tannaka duality theorem for proper Lie groupoids. This result is obtained by replacing the category of smooth vector bundles over the base manifold of a Lie groupoid with a larger category, the category of smooth Euclidean fields, and by considering smooth actions of Lie groupoids on smooth Euclidean fields. The notion of smooth Euclidean field that is introduced here is the smooth, finite dimensional analogue of the familiar notion of continuous Hilbert field. In the second part of the thesis, ordinary smooth representations of Lie groupoids on smooth…

光滑向量的指数可解李群表示的强因子分解定理

Strong factorization theorem for smooth vectors of exponential solvable Lie group representations

本文改进了Dixmier-Malliavin定理，为指数可解李群在弗雷歇空间上的光滑向量建立了新的强因子分解性质。

Smooth vectors and Weyl-Pedersen calculus for representations of nilpotent Lie groups

We present some recent results on smooth vectors for unitary irreducible representations of nilpotent Lie groups. Applications to the Weyl-Pedersen calculus of pseudo-differential operators with symbols on the coadjoint orbits are also discussed.

A non-smooth continuous unitary representation of a Banach-Lie group

In this note we show that the representation of the additive group of the Hilbert space $L^2([0,1],\R)$ on $L^2([0,1],\C)$ given by the multiplication operators $π(f) := e^{if}$ is continuous but its space of smooth vectors is trivial. This example shows that a continuous unitary representation of an infinite dimensional Lie group need not be smooth.

A Tannaka Theorem for Proper Lie Groupoids

By replacing the category of smooth vector bundles over a manifold with the category of what we call smooth Euclidean fields, which is a proper enlargement of the former, and by considering smooth actions of Lie groupoids on smooth Euclidean fields, we are able to prove a Tannaka duality theorem for proper Lie groupoids. The notion of smooth Euclidean field we introduce here is the smooth, finite dimensional analogue of the usual notion of continuous Hilbert field.

Projective unitary representations of infinite dimensional Lie groups

For an infinite dimensional Lie group $G$ modelled on a locally convex Lie algebra $\mathfrak{g}$, we prove that every smooth projective unitary representation of $G$ corresponds to a smooth linear unitary representation of a Lie group extension $G^{\sharp}$ of $G$. (The main point is the smooth structure on $G^{\sharp}$.) For infinite dimensional Lie groups $G$ which are 1-connected, regular, and modelled on a barrelled Lie algebra $\mathfrak{g}$, we characterize the unitary $\mathfrak{g}$-representations which integrate to $G$. Combining these results, we give a precise formulation of the co…

Norm continuous unitary representations of Lie algebras of smooth sections

We give a complete description of the bounded (i.e. norm continuous) unitary representations of the Fréchet-Lie algebra of all smooth sections, as well as of the LF-Lie algebra of compactly supported smooth sections, of a smooth Lie algebra bundle whose typical fiber is a compact Lie algebra. For the Lie algebra of all sections, bounded unitary irreducible representations are finite tensor products of so-called evaluation representations, hence in particular finite-dimensional. For the Lie algebra of compactly supported sections, bounded unitary irreducible (factor) representations are possibl…