On certain rings of highest weight vectors
In this paper the authors consider the representation of the group \(G = O_m \times GL_n\) on the polynomial algebra \(P(V)\) of \(m \times n\)- matrices induced by the natural action by left, resp., right multiplication on \(V = \mathbb{C}^{m,n}\). Let \(B = UT\) be a Borel subgroup of \(G\), where \(T\) is a maximal torus and \(U\) a maximal unipotent subgroup. Then \(T\) acts semisimply on the space \(P(V)^U\) of \(U\)- invariants in \(P(V)\). The eigenvectors for this representation of \(T\) are called highest weight vectors and \(R_{m,n} := P(V)^U\) the ring of highest weight vectors. The main objective of the authors is to study the structure of this ring. One major tool to do this is the decomposition \(P(V) = P(V)^{O_m} \cdot {\mathcal H}\), where \({\mathcal H} = \{f : (\forall 1 \leq j \leq k \leq n) \Delta_{j,k} \cdot f = 0\}\) is the space of harmonics, and \(\Delta_{jk} = \sum^m_{s = 1} {\partial^2 \over \partial x_{sj} \partial x_{sk}}\). For \(m \geq 3\) this leads directly to the description of \(R_{m,1}\) as \(\mathbb{C} [z_1, r^2_{11}]\) with \(z_1 = x_{11} - ix_{21}\) and \(r^2_{jk} = \sum^m_{s = 1} x_{sj} x_{sk}\). Another important source of information is the work of \textit{M. Kashiwara} and \textit{M. Vergne} [Invent. Math. 44, 1-47 (1978; Zbl 0375.22009)]. In the article under review the authors determine the ring \(R_{m,2}\) and use it to compute for the holomorphic representations of \(\text{Sp}_4\) the Poincaré series, the Gelfand- Kirillov dimension and the Berenstein degree. They also describe a set of generators for \(R_{m,3}\).
scientific article; zbMATH DE number 783859