Differential operators and highest weight representations.

*(English)*Zbl 0759.22015
Mem. Am. Math. Soc. 455, 102 p. (1991).

Let \((G,K)\) be a Hermitian symmetric pair. We denote by \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) the corresponding Cartan decomposition of the complexified Lie algebra \({\mathfrak g}\) of \(G\) and \({\mathfrak p}={\mathfrak p}_ ++{\mathfrak p}_ -\) the decomposition of \(\mathfrak p\) into two irreducible \(K\)-submodules. From the algebraic point of view, the classification of unitarizable highest weight representations \((\pi,L)\) was given by T. Enright, R. Howe, and N. Wallach [Prog. Math. 40, 97-143 (1983; Zbl 0535.22012)] and by H. P. Jakobsen [J. Funct. Anal. 52, 385-412 (1983; Zbl 0517.22014)].

One of the themes of this memoir, from the analytic point of view, is to realize \((\pi,L)\) on a certain space of vector valued polynomials. When we regard \(L\) as the unique irreducible quotient \(L(\lambda)\) of the associated generalized Verma module \(N=N(\lambda+\rho)\), a nondegenerate pairing between \(N\) and \(N^*\) exists through differentiation and gives a correspondence between \(L\) and a finite system \({\mathcal D}_ \lambda\) of polynomial differential operators. Here \({\mathcal D}_ \lambda\) is defined by any set of generators for the maximal submodule of \(N\) as an \(S({\mathfrak p}_ -)\)-module and \(L\) is recovered infinitesimally as the kernel of the system \(D_ \lambda\).

The second theme concerns the so-called cone decomposition of the set of highest weights. Let \(\Lambda_ r\) denote the set of highest weights \(\lambda\) of the unitarizable highest weight module \(L\) such that \(N\) is reducible. Then \(\Lambda_ r\) is a disjoint union of a finite set of cones \(\Lambda_ a\), and the factorization theorem asserts that \({\mathcal D}_ \lambda\) (\(\lambda\in\Lambda_ a\)) is given as a shift of \({\mathcal D}_{\lambda_ a}\), where \(\lambda_ a\) is the vertex of the cone \(\Lambda_ a\).

These results are investigated more explicitly in the setting of harmonic polynomials and oscillator representations. The above shift is given by a multiplication of polynomials, a finite system \({\mathcal F}_ a\) of differential operators corresponding to the vertex \(\lambda_ a\) determines all of the relevant modules in \(\Lambda_ a\), and \(L\) is realized in the coordinate ring of an affine variety. As illustrative examples, characterizations of both the ladder and Wallach representations are treated in the last two sections.

One of the themes of this memoir, from the analytic point of view, is to realize \((\pi,L)\) on a certain space of vector valued polynomials. When we regard \(L\) as the unique irreducible quotient \(L(\lambda)\) of the associated generalized Verma module \(N=N(\lambda+\rho)\), a nondegenerate pairing between \(N\) and \(N^*\) exists through differentiation and gives a correspondence between \(L\) and a finite system \({\mathcal D}_ \lambda\) of polynomial differential operators. Here \({\mathcal D}_ \lambda\) is defined by any set of generators for the maximal submodule of \(N\) as an \(S({\mathfrak p}_ -)\)-module and \(L\) is recovered infinitesimally as the kernel of the system \(D_ \lambda\).

The second theme concerns the so-called cone decomposition of the set of highest weights. Let \(\Lambda_ r\) denote the set of highest weights \(\lambda\) of the unitarizable highest weight module \(L\) such that \(N\) is reducible. Then \(\Lambda_ r\) is a disjoint union of a finite set of cones \(\Lambda_ a\), and the factorization theorem asserts that \({\mathcal D}_ \lambda\) (\(\lambda\in\Lambda_ a\)) is given as a shift of \({\mathcal D}_{\lambda_ a}\), where \(\lambda_ a\) is the vertex of the cone \(\Lambda_ a\).

These results are investigated more explicitly in the setting of harmonic polynomials and oscillator representations. The above shift is given by a multiplication of polynomials, a finite system \({\mathcal F}_ a\) of differential operators corresponding to the vertex \(\lambda_ a\) determines all of the relevant modules in \(\Lambda_ a\), and \(L\) is realized in the coordinate ring of an affine variety. As illustrative examples, characterizations of both the ladder and Wallach representations are treated in the last two sections.

Reviewer: T.Kawazoe (Yokohama)

##### MSC:

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |

22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |