12 problems

Let $G/H$ be a homogeneous space in the reductive setting, let $\mathfrak g$ and $\mathfrak h$ be the Lie algebras of $G$ and $H$, and let $\mathfrak h^{\perp}\subset\mathfrak g^*$…

Let $G/H$ be a homogeneous space in the reductive setting, let $\mathfrak g$ and $\mathfrak h$ be the Lie algebras of $G$ and $H$, and let $\mathfrak h^{\perp}\subset\mathfrak g^*$…

Let $G$ be a compact Lie group with Lie algebra $\mathfrak g$, let $G^c$ be its complexification, and let $\Omega$ be the data defining the examples associated to $G$. Denote the r…

Let $G$ be an algebraic group with Lie algebra ${\mathfrak g}$, let ${\mathcal N}$ be the nilpotent cone, and let $e\in{\mathcal N}$. Write ${\mathfrak g}^e$ for the centraliser of…

Let ${\mathfrak g}$ be a simple Lie algebra. A Lie subalgebra is Frobenius when its index is zero; let ${\EuScript M}_{\sf Fr}$ denote the corresponding maximal dimension invariant…

Let $X$ be a vector field in the setting of three-dimensional volume-preserving flows, and let its corresponding helicity level set be the set of vector fields with the same helici…

Unique representatives conjecture. For every coadjoint orbit, there exists a unique linear form $\lambda\in S$ lying on that orbit.

Coadjoint-orbit conjecture. The moduli space of weighted prequantizable Lagrangian embeddings is a symplectic manifold whose connected components are related to the coadjoint orbit…

Let $F$ be a non-Archimedean local field, let $\bf G$ be an algebraic reductive group over $F$, and let $G=\bf G(F)$. For $\pi\in\operatorname{Irr}(G)$, let…

Admissible-polarization conjecture. The character formula

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $G$ act on $\mathfrak{g}^{\ast}$ by the coadjoint action. Assume that every coadjoint orbit is locally closed in…

Let $G$ be a reductive Lie group, let $\mathfrak{g}$ be its Lie algebra, and let $\mathcal{O}\subset \mathfrak{g}^*$ be a nilpotent coadjoint orbit. Let $H\subset G$ be a Cartan su…