27 problems

Let ${\mathfrak q}$ be a reductive Lie algebra and let $\xi\in{\mathfrak q}^*$. The index ${\rm ind}\,{\mathfrak q}^\xi$ is the index of the stabiliser of $\xi$ under the coadjoint…

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

Orbit Conjecture. Let $K$ be a compact connected Lie group acting unitarily on $V$. Then $(K,H_V)$ is a Gelfand pair if and only if the action of $K$ on $V$ is multiplicity-free.

Highest-weight extension conjecture. The additional parameters, namely the highest weights, correspond to the above-mentioned hypothetical extension of the dual space…

Characteristic-cycle conjecture. The cycle $C_\lambda$ equals the characteristic cycle of the sheaf $\mathcal F_\lambda$. This identification would provide a geometric way to compu…

Let $A$ be a matrix solution of the reflection equation, and let its quantum trace be denoted by $\operatorname{Tr}_q(A)$. A solution is non-degenerate when it is non-degenerate as…

A bisymmetric orbit is an orbit described in the paper's quantization framework by the relevant bisymmetric matrix data. Bisymmetric-orbit quantization conjecture. All bisymmetric…

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…

A coadjoint orbit of a classical compact Lie group is a homogeneous symplectic manifold associated with a coadjoint action, and the construction developed here provides an oscillat…

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…

Let $K$ be a compact Lie group, let $\mathcal{O}_{\lambda}$ be the coadjoint orbit through a point $\lambda$ in a positive Weyl chamber, and equip it with its Kostant--Kirillov--So…

Orbital-parameter conjecture. A large and important part of the collection of irreducible, unitary representations of $G_{\mathbb{R}}$ can be constructed from orbital parameters th…

Admissible-polarization conjecture. The character formula

Let $K$ be a compact Lie group, let $\mathfrak k^*$ be the dual of its Lie algebra, and let $(\mathcal{O}_{\lambda},\omega^{KKS})$ be the coadjoint orbit through a point $\lambda$…

Let $G$ be a compact connected simple Lie group, let $\lambda\in\operatorname{Lie}(G)^*$, and let $\mathcal{O}_\lambda$ be the coadjoint orbit through $\lambda$, equipped with its…

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…

Conjecture. The representation $\pi$ is holomorphic or anti-holomorphic if and only if $\operatorname{p}_1(\mathcal{O}_{\pi})$ is an open coadjoint $A_1N_1$-orbit in…

Duflo's conjecture. [The supplied candidate statement ends immediately after “Alors:”; no mathematical assertion is present in the input.]

Overalgebra almost separation conjecture. For every $n$, $\mathfrak{sl}(n,\mathbb R)$ does not admit an overalgebra almost separating of degree $n-1$, but it admits one of degree…

Let $G$ be a compact Lie group acting naturally and effectively on a coadjoint orbit, with the corresponding Hamiltonian action. Reznikov's conjecture. The induced homomorphism fro…