Bernstein degree of singular unitary highest weight representations of the metaplectic group
type: Item
instance of: scholarly article
title: Bernstein degree of singular unitary highest weight representations of the metaplectic group
zbMATH Open document ID: 0947.11020
author: Kyo Nishiyama
author: Hiroyuki Ochiai
published in: Proceedings of the Japan Academy. Series A
publication date: 2000-11-12T00:00:00Z
review text: Let \(\omega\) denote the Weil representation of the metaplectic double cover \(G=\text{Mp}(2n,{\mathbb R})\) of the symplectic group \(\text{Sp}(2n,{\mathbb R})\) of rank \(n\). Decompose the \(m\)-fold tensor product \(\omega^{\otimes m}\) as a \(G\times O(m)\) module: \(\omega^{\otimes m}=\oplus L(\sigma)\otimes \sigma\), where the sum is over \(\sigma\in\hat O(m)\). In this article, the authors give a formula for the Bernstein degree of \(L({\mathbf 1}_m)\), \(1\leq m\leq n\), where \({\mathbf 1}_m\) denotes the trivial representation. The representation \(L({\mathbf 1}_m)\) is an irreducible singular unitary highest weight representation of \(G\) and it has one-dimensional minimal \(K\)-type; if \(m\) is even then it factors through and gives an irreducible representation of \(\text{Sp}(2n,{\mathbb R})\). The authors' formula is obtained by using the Gindikin gamma function on a positive Hermitian cone. By a result from invariant theory, this formula is equivalent to Giambelli's formula for the degree of the determinantal subvariety of symmetric complex \(n\times n\) matrices of rank \(\leq m\) contained in the variety of symmetric matrices modulo scalars. Thus the authors' approach gives a new proof of this formula.
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 1310584
zbMATH Keywords: Bernstein degree
zbMATH Keywords: metaplectic group
zbMATH Keywords: determinantal variety
zbMATH Keywords: Giambelli's formula
reviewed by: Solomon Friedberg
MaRDI profile type: Publication
full work available at URL: pjaa.75.9
OpenAlex ID: W2015377618
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DOI: 10.3792/PJAA.75.9
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description: scientific article; zbMATH DE number 1310584