\[S_\infty(\mathcal{H}) = \{M: M=M^*, \mathrm{tr}(M)=0\}\]
with eigenvalues \(\lambda_1 >\dots> \lambda_r\), eigenspaces \(P_1,\dots,P_r\), let \(Q_i=\sum_{j\le i}P_j\) be the associated flag in \(\mathbb{C}^n\). We can recover \(M\) from the flag and weights \(\lambda_i\).
Theorem 3. For \(X\) as above, suppose \(F\) is a convex function
The key thing is co-adjoint orbits. Let \(G\) be any Lie group, we have co-adjoint action on the dual of the lie algebra \(\mathfrak{g}^*\). We have \(0\neq \theta\in \mathfrak{g}^*\), then this gets some stablizer \(\Gamma_\theta\subset G\). The co-adjoint orbit \(G/\Gamma_\theta\).
\[\theta:\mathfrak{g}\to \mathbb{R}\]
we can restrict it to \(\theta|_{\mathrm{Lie}(\Gamma_\theta)} : \mathrm{Lie}(\Gamma_\theta)\to \mathbb{R}\),
Exercise: \(\theta\) defines a lie algebra morphism \(\mathrm{Lie}(\Gamma_\theta)\to \mathbb{R}\).
we say the orbit of \(\theta\) is integral (integrable?) if this lifts to a group homomorphism \(\Gamma_\theta\to S^1\).
\(G\) is compact, then the irreducible unitary representations of \(G\) is equivalent to integral co-adjoint orbits.
\[ \{\text{unitary irreps of } G\} \leftrightarrow \{\text{integral co-adjoint orbits in } \mathfrak{g}^*\}\]
\(G=\mathrm{SU}(2)\), \(\mathfrak{g}^*=\mathbb{R}^*\), non-zero orbits \(S^2_\mathbb{R}\subset \mathbb{R}^3\). Integral iff \(\mathbb{R}\) positive integer.
Representations \(S^k\) is the \(k\)-th symmetric power of \(\mathbb{C}^2\) of dimension \(k+1\).
\(\sigma:\Gamma_\theta\to S^1\), \(G\to G/\Gamma_\theta=M\), associated complex line bundle \(G\times_\sigma \mathbb{C}\).
\(M=G/\Gamma_\theta\subset \mathfrak{g}^*\) co-adjoint orbit, has \(G\)-invariant symplectic structure. Take a point \(\theta\in M\), the tangent space \(\mathfrak{g}\to TM_\theta\), for vectors \(v_1, v_2\in TM_\theta\) we can choose lifts \(\xi_1,\xi_2\in \mathfrak{g}\)
\[\omega(v_1,v_2) := \theta([\xi_1,\xi_2])\]
which is obviously skew-symmetric, we can prove that this is independent of the choice of lifts.
Poisson structure always have foliations by symplectic leaves.
If \(G\) has a transitive Hamiltonian action on a symplectic manifold \(N\), then \(N\cong\) co-adjoint orbit.
General Principle: Symplectic geometry, quantization.
classical mechanics
quantum mechanics
There should be some correspondences.
\(G\) action, \(\{G\mathrel{\circlearrowright}(N,\omega)\} \Leftrightarrow \{\text{unitary rep of } G\}\).