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Zbl 0919.22005
Aslaksen, Helmer; Tan, Eng-Chye; Zhu, Chen-bo
On $O_m \times GL_n$ highest weight vectors.
(English)
[A] Gruber, Bruno (ed.), Symmetries in science VIII. Proceedings of a symposium, Bregenz, Austria, August 1994. New York, NY: Plenum Press. 1-11 (1995). ISBN 0-306-45119-0

The article under review is a summary of the authors' paper [J. Algebra 174, No. 1, 159-186 (1995; Zbl 0828.22019)]. 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 $V = \Bbb C^{m,n}$. Let $B = U T$ 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 $P(V)= P(V)^{O_m} \cdot {\cal H}$, where $${\cal H} = \{ f : (\forall 1 \leq j \leq k \leq n) \Delta_{j,k}.f = 0\}$$ is the space of harmonics, and $\Delta_{jk} = \sum_{s=1}^m {\partial^2 \over \partial x_{sj} \partial x_{sk}}.$ For $m \geq 3$ this leads directly to the description of $R_{m,1}$ as $\Bbb C [z_1, r_{11}^2]$ with $z_1 = x_{11} - i x_{21}$ and $r_{jk}^2 = \sum_{s=1}^m x_{sj} x_{sk}.$ An important source of information is the work of {\it M. Kashiwara} and {\it M. Vergne} [Invent. Math. 44, 1-47 (1978; Zbl 0375.22009)]. In the article under review the authors describe the ring $R_{m,2}$ and some of its applications to branching rules. They also describe a set of generators for $R_{m,3}$.
[K.-H.Neeb (Darmstadt)]
MSC 2000:
*22E46 Semi-simple Lie groups and their representations
22E45 Analytic repres.of Lie and linear algebraic groups over real fields
13A50 Invariant theory
14L30 Group actions on varieties or schemes
20G05 Representation theory of linear algebraic groups
22E30 Analysis on real and complex Lie groups

Keywords: highest weight; polynomial algebra; branching rule; harmonic polynomial; representation; highest weight vectors

Citations: Zbl 0828.22019; Zbl 0375.22009

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