Szegö kernel equivariant asymptotics under Hamiltonian Lie group actions

Asymptotic isometric minimal immersions of conic reductions into spheres for Hamiltonian actions

“…Let us assume that: Φ is transverse to the cone C(O ν ), 0 / ∈ Φ(M) and M G Oν = ∅. Theorem 1.1 below concerns asymptotic expansion for near-diagonal re-scaled displacements at a fixed diagonal point (x, x) ∈ X G Oν × X G Oν and the proof is very similar to the one of [Pa5,Theorem 1.3]. Suppose x ∈ X, and let x + (v, θ) be a Heisenberg local chart for X centered at x, as defined in [SZ]; in particular, if m := π(x) this unitarily identifies (T m M, h m ) and C n with its standard Hermitian structure.…”

“…In Section 2 we explain motivations and preliminaries; in particular in Section 2.1 we investigate the geometry of N G Oν . In Section 3 we prove the main theorem, which is based on computations made in [Pa5] and eventually we prove the aforementioned corollary.…”

“…In [Pa5] scaling asymptotics were investigated by making use of the Kirillov character formula. In fact, Theorem 1.1 is a simple generalization of the main Theorem in [Pa5].…”

“…where Π is the Szegő kernel, dV G is the Haar measure and χ kν and d kν are respectively the character and the dimension of the representation kν. In [Pa5], it is studied the dimensions of isotypes H(X) kν and the behavior of Π kν (x, x) as k → +∞. Now, we need to generalize the results contained in [Pa5] but let us pause to introduce some notation.…”

“…In [Pa5], it is studied the dimensions of isotypes H(X) kν and the behavior of Π kν (x, x) as k → +∞. Now, we need to generalize the results contained in [Pa5] but let us pause to introduce some notation. Denote by C(O ν ) the cone through the co-adjoint orbit O ν passing through ν in g * .…”