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.