Bernstein degree of singular unitary highest weight representations of the metaplectic group
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.
scientific article; zbMATH DE number 1310584
scientific article