# Tagged Questions

0answers
186 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 $\mu_0$ ...
1answer
532 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 ...
2answers
333 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 ...
4answers
334 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 $L^2(M,L,\mu)$ ...
0answers
140 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 ...
1answer
292 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 ...
3answers
319 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 ...
3answers
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### 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 ...
1answer
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### 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 ...
0answers
194 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 ...
1answer
294 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 ...
0answers
557 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 ...
2answers
700 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 ...