On the uniqueness of conical vectors
| Autor: | J. Lepowsky |
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| Rok vydání: | 1976 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 57:217-220 |
| ISSN: | 1088-6826 0002-9939 |
| Popis: | Certain conical vectors in induced modules for semisimple symmetric Lie algebras have been determined in a previous paper. Here the uniqueness aspects of those results are proved in a simpler way, and a refinement of one of the earlier results is given. 1. Introduction. In this paper, we give a new proof of the uniqueness aspects of the main results (Theorems 10.1 and 10.2) of (3), concerning conical vectors and embeddings of induced modules (see Theorem 3.6 and Corollary 3.7). The proof, which we announced in the Introduction of (3), uses the general results on uniqueness of embeddings in (4), as well as an observation of B. Kostant on the usefulness of the action of the center of the enveloping algebra in limiting the possibilities for conical vectors in induced modules (cf. Theorem 3.2 below). We can thus avoid the Kostant-Mostow double transitivity theorem, fundamental commutation relation and transfer principles needed in (3) for proving the uniqueness. (But note that the last two of these were also used in (3) for the existence of conical vectors.) We also obtain an interesting refinement of Theorem 10.2 of (3) (see Theorem 3.8). The terminology and setting of (2, §2) and (3, §2) will be used here. We thank Bertram Kostant for a helpful discussion. |
| Databáze: | OpenAIRE |
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