
Journal of Lie Theory 23 (2013), No. 4, 10111022 Copyright Heldermann Verlag 2013 Projections of Orbital Measures, GelfandTsetlin Polytopes, and Splines Grigori Olshanski Institute for Information Transmission Problems, 19 Bolshoy Karetny, Moscow 127994, Russia and: Independent University of Moscow, 11 Bolshoy Vlasyevsky, Moscow 119002, Russia olsh2007@gmail.com [Abstractpdf] The unitary group $U(N)$ acts by conjugations on the space ${\cal H}(N)$ of $N\times N$ Hermitian matrices, and every orbit of this action carries a unique invariant probability measure called an orbital measure. Consider the projection of the space ${\cal H}(N)$ onto the real line assigning to an Hermitian matrix its $(1,1)$entry. Under this projection, the density of the pushforward of a generic orbital measure is a spline function with $N$ knots. This fact was pointed out by Andrei Okounkov in 1996, and the goal of the paper is to propose a multidimensional generalization. Namely, it turns out that if instead of the $(1,1)$entry we cut out the upper left matrix corner of arbitrary size $K\times K$, where $K=2,\dots,N1$, then the pushforward of a generic orbital measure is still computable: its density is given by a $K\times K$ determinant composed from onedimensional splines. The result can also be reformulated in terms of projections of the GelfandTsetlin polytopes. Keywords: Orbital measure, GelfandTsetlin polytope, Bspline, HarishChandraItzyksonZuber integral. MSC: 22E30 41A15 [ Fulltextpdf (296 KB)] for subscribers only. 