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Whittaker vectors and the Goodman-Wallach operators. (English) Zbl 0723.22019

Let G be a connected real semisimple Lie group, \(G=KA_ mN_ m\) an Iwasawa decomposition of G and \(P_ m=M_ mA_ mN_ m\) a minimal parabolic subgroup of G. Let \(P=MAN\) be a parabolic subgroup of G whose complexified Lie algebra \({\mathfrak p}={\mathfrak man}\) contains the Lie algebra of \(P_ m\) and let \(\psi\) be a character of N. For a left U(\({\mathfrak g})\)-module V let \(Wh_{{\mathfrak n},\psi}(V)\) [resp. \(Wh^*_{{\mathfrak n},\psi}(V)]\) denote the space of [resp. dual] Whittaker vectors and \(Wh^ G_{{\mathfrak n},\psi}(V)\) the space of global Whittaker vectors, i.e., the image of \(\Gamma\) : Hom\({}_{U({\mathfrak g})}(V,A(G,{\mathfrak n},\psi))\to Wh^*_{{\mathfrak n},\psi}(V)\) defined by \(\Gamma (F)(v)=(F(v))(e)\) (for \(v\in V)\), where A(G,\({\mathfrak n},\psi)\) is the space of Whittaker functions on G. Then under some restrictions on G,\({\mathfrak n},\psi\) the problems of the nontriviality of \(Wh^ G_{{\mathfrak n},\psi}(V)\) and \(Wh^*_{{\mathfrak n},\psi}(V)\), the finiteness of their dimensions and the explicit form of these, the coincidence of the two spaces, the relation between \(Wh^*_{{\mathfrak n},\psi}(V)\) and the associated variety of the annihilator of V in U(\({\mathfrak g})\) have been studied by various people. The author gives generalizations of the answers to the problems by constructing sufficiently many global Whittaker vectors.
One of the main results is the following. Let V be an irreducible Harish- Chandra module and suppose that there is a nonsingular pairing between V and the highest weight left U(\({\mathfrak g})\)-module L(\({\mathfrak p},\lambda)\) with highest weight \(\lambda \in P_ S^{++}\). Then a nontrivial global Whittaker vector on V is constructed from \(Wh_{{\mathfrak n},-\psi}(\hat L({\mathfrak p},\lambda)):\) the author proves that \(Wh_{{\mathfrak n},- \psi}(\hat L({\mathfrak p},\lambda))\subset L^{\chi}({\mathfrak p},\lambda)\) \((1\leq \chi <2)\), where L(\({\mathfrak p},\lambda)\) and \(L^{\chi}({\mathfrak p},\lambda)\) are the formal and Gevrey completions of L(\({\mathfrak p},\lambda)\) respectively. Then, by the same method as Goodman and Wallach used in the case when \({\mathfrak n}\) is a nilradical of a Borel subalgebra, the author obtains that if \(\psi\) is admissible, \(Wh^ G_{{\mathfrak n}_{m,\psi}}(V)=Wh^*_{{\mathfrak n}_{m,\psi}}(V)\).

MSC:

22E46 Semisimple Lie groups and their representations
22E30 Analysis on real and complex Lie groups
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B35 Universal enveloping (super)algebras
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