1
vote
0answers
169 views

quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement. Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...
2
votes
1answer
452 views

Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $

My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature of the coadjoint representation is the fact that all coadjoint orbits possess a ...
5
votes
2answers
318 views

Information from Moment Polytopes

Let $T$ be a compact real torus, and $X$ a Hamiltonian $T$-manifold (which you may take to be a smooth complex projective variety) with moment map $\mu:X\rightarrow\frak{t}^*$. If ...
5
votes
4answers
319 views

Prequantization and Hilbert space

In the definition of pre-quantization on a symplectic manifold $(M,\omega)$, we represent a function $f\in C^{\infty}(M)$(with Lie algebra structure) to $\hat{f}$ in the Hilbert space $L^2(M,L,\mu)$ ...
1
vote
0answers
134 views

Level sets of momentum map for diagonal action on two coadjoint orbits.

Hi, I'm trying to get a better understanding of multiplicities in geometric quantization, and so I've been concentrating on a specific simple case: let $\mathcal{O}\subset\mathfrak{g}^*$ be an ...
5
votes
1answer
284 views

The Fukaya category of a simple singularity (reference request)

I have heard that for an ADE singularity $f$, $ D^b\mathrm{Fuk}(f) \simeq D^b(\mathrm{Rep}\\ Q)$ where $Q$ is the corresponding Dynkin quiver. (As one would hope, if $\mathrm{Fuk}$ is some kind of ...
5
votes
3answers
313 views

Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator

Consider the metaplectic representation of $Mp(n)$ on $L^2(\mathbb R^n)$. We can view $U(n)$ as a subgroup of $Sp(n)$ and so inside $Mp(n)$ is a double cover $\tilde U(n)$ of $U(n)$. The restriction ...
12
votes
3answers
807 views

When is a coadjoint orbit an integrable system (in a weak sense explained below)?

Let $X$ be an affine holomorphic symplectic variety of dimension $2n$, with the associated Poisson bracket { , }. Let's say it's an integrable system when there are $n$ algebraically independent ...
5
votes
1answer
301 views

Splitting principle in equivariant cohomology

The following is a weaker version of what is called splitting principle in Appendix C, page 12, see also for a lighter version Brions Eq cohom and eq intersection theory, page 6: Let $G$ be a compact ...
1
vote
0answers
190 views

(Non-)Surjectivity of the Maslov index

Let $V$ be a symplectic space over a field $k$ (for simplicity, the characteristic of $k$ is not $2$). The Maslov index sends a collection of $n$ lagrangian subspaces of $V$ to a quadratic space over ...
2
votes
1answer
288 views

Non-Abelian Duistermaat-Heckman Measure (not just a reference request)

Let $V$ be a representation of some torus $T$. It is then well-known that the Duistermaat-Heckman measure for $P(V)$ is the weak limit of the properly rescaled distribution of multiplicities of ...
20
votes
0answers
540 views

Why are there so few quaternionic representations of simple groups?

Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...
4
votes
2answers
680 views

Which tensor fields on a symplectic manifold are invariant under all Hamiltonian vector fields?

Consider a connected symplectic manifold $(M, \omega)$ of dimension $m=2n$. A few preliminary reminders (mostly to fix the notation): A vector field $X$ is symplectic if its flow preserves the ...