Derivations of the trigonometric BC(n) Sutherland model by quantum Hamiltonian reduction

The BC(n) Sutherland Hamiltonian with coupling constants parametrized by three arbitrary integers is derived by reductions of the Laplace operator of the group U(N). The reductions are obtained by applying the Laplace operator on spaces of certain vector valued functions equivariant under suitable symmetric subgroups of U(N)\times U(N). Three different reduction schemes are considered, the simplest one being the compact real form of the reduction of the Laplacian of GL(2n,C) to the complex BC(n) Sutherland Hamiltonian previously studied by Oblomkov.

💡 Research Summary

The paper presents a systematic derivation of the trigonometric BC(n) Sutherland model by means of quantum Hamiltonian reduction applied to the Laplace operator on the compact unitary group U(N). The authors start from the well‑known Laplacian Δ_U on U(N), which can be written as a left‑right invariant differential operator on the product group U(N)×U(N). By restricting Δ_U to spaces of vector‑valued functions that are equivariant under a pair of symmetric subgroups G₁ and G₂ of U(N), one obtains a reduced Hamiltonian acting on a much smaller configuration space.

Three distinct reduction schemes are examined. In the first (and simplest) scheme the subgroups are taken as G₁ = G₂ = U(n)×U(n) with N = 2n. Functions in the equivariant space satisfy
f(g₁ g g₂⁻¹) = ρ₁(g₁) f(g) ρ₂(g₂)⁻¹,
where ρ₁ and ρ₂ are finite‑dimensional representations labeled by two integers k₁ and k₂. After imposing this equivariance, the Laplacian collapses to a differential operator in n real variables x₁,…,xₙ. The resulting effective Hamiltonian is precisely the BC(n) Sutherland Hamiltonian:

H = –∑{i=1}^n ∂{x_i}² + ∑_{i<j}