The Alesker–Bernig–Schuster theorem asserts that each irreducible representation of the special orthogonal group appears with multiplicity at most one as a subrepresentation of the space of continuous translation-invariant valuations with fixed degree of homogeneity. Moreover, the theorem describes in terms of highest weights which irreducible representations appear with multiplicity one. In this paper, we present a refinement of this result, namely the explicit construction of a highest weight vector in each irreducible subrepresentation. We then describe how important natural operations on valuations (pullback, pushforward, Fourier transform, Lefschetz operator, Alesker–Poincaré pairing) act on these highest weight vectors. We use this information to prove the Hodge–Riemann relations for valuations in the case of Euclidean balls as reference bodies. Since special cases of the Hodge–Riemann relations have recently been used to prove new geometric inequalities for convex bodies, our work immediately extends the scope of these inequalities.

Alesker–Bernig–Schuster 定理断言，特殊正交群的每个不可约表示最多出现一个，作为具有固定同质度的连续平移不变估值空间的子表示。此外，该定理根据最高权重描述了哪些不可约表示以重数 1 出现。在本文中，我们对这一结果进行了改进，即在每个不可约子表示中显式构建最高权重向量。然后，我们描述了估值的自然操作（回调、前推、傅立叶变换、Lefschetz 算子、Alesker–Poincaré 配对）如何作用于这些最高权重向量。我们使用这些信息来证明在欧几里德球作为参考体的情况下的霍奇-黎曼关系。由于霍奇-黎曼关系的特例最近被用来证明凸体的新几何不等式，我们的工作立即扩展了这些不等式的范围。