# zbMATH — the first resource for mathematics

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.

##### 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.)
Full Text: