Consider the action of a connected complex reductive group on a
finite-dimensional vector space. A fundamental result in invariant theory
states that the orbit closure of a vector v is separated from the origin if and
only if some homogeneous invariant polynomial is nonzero on v, i.e. v is not in
the null cone. Thus, efficiently finding the minimum distance between the orbit
closure and the origin can lead to deterministic algorithms for null cone
membership, an important polynomial identity testing problem including the
non-commutative Edmonds problem. This connection to optimization has recently
led to efficient algorithms for many problems in invariant theory.
Here we explore a refinement of the famous duality between orbit closures and
invariant polynomials, which holds that the following two quantities coincide:
(1) the logarithm of the Euclidean distance between the orbit closure and the
origin and (2) the rate of exponential growth of the ‘invariant part’ of
$v^{\otimes k}$ in the semiclassical limit as k tends to infinity. This result
can be deduced from work of S. Zhang (Geometric reductivity at Archimedean
places, 1994), which uses sophisticated tools in arithmetic geometry. We
provide a new and independent elementary proof inspired by the Fourier-analytic
proof of the local central limit theorem. We generalize the result to
projections onto highest weight vectors and isotypical components, and explore
connections between such semiclassical limits and the asymptotic behavior of
multiplicities in representation theory, large deviations theory in classical
and quantum statistics, and the Jacobian conjecture as reformulated by Mathieu.
Our formulas imply that they can be computed, in many cases efficiently, to
arbitrary precision.
