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Published online by Cambridge University Press: 29 May 2025
We review some recent results on connections between Brownian motion,
Whittaker functions, random matrices and representation theory.
1. Harish-Chandra formula, Duistermaat–Heckman measure and Gelfand–Tsetlin patterns
DefineJλ(x) = h(λ)-1 det(eλi xj) For each x, Jλ(x) is an analytic function of λ; in particular,
The functions Jλ(x) play a central role in random matrix theory. For example, if
Λ and X are Hermitian matrices with eigenvalues given by λ and x, respectively, Then
where the integral is with respect to normalised Haar measure on the unitary
group. This is known as the Harish-Chandra, or Itzykson–Zuber, formula.
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