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The Alesker-Bernig-Schuster theorem asserts that each irreduciblerepresentation of the special orthogonal group appears with multiplicity atmost one as a subrepresentation of the space of continuoustranslation-invariant valuations with fixed degree of homogeneity. Moreover,the theorem describes in terms of highest weights which irreduciblerepresentations appear with multiplicity one. In this paper, we present arefinement of this result, namely the explicit construction of a highest weightvector in each irreducible subrepresentation. We then describe how importantnatural operations on valuations (pullback, pushforward, Fourier transform,Lefschetz operator, Alesker-Poincar\'e pairing) act on these highest weightvectors. We use this information to prove the Hodge-Riemann relations forvaluations in the case of Euclidean balls as reference bodies. Since specialcases of the Hodge-Riemann relations have recently been used to prove newgeometric inequalities for convex bodies, our work immediately extends thescope of these inequalities.

From harmonic analysis of translation-invariant valuations to geometric inequalities for convex bodies