A functional equation and degenerate principal series. (English) Zbl 1373.22021

Let \(G\) be a group for which there is a parabolic subgroup \(P\) such that \(P\) and its opposite parabolic are \(G\)-conjugate, \(P=LN\), \(L\) is the Levi subgroup, \(N\) is abelian, and the symmetric space corresponding to \(G\) is not of tube type. Such groups \(G\) arise from simple non-Euclidean Jordan algebras \(\mathfrak{n}\). The Levi subgroup \(L\) has a finite number of orbits on \(\mathfrak{n}\), and there is an open dense orbit defined by a relative invariant \(\nabla\neq 0\) of the prehomogeneous vector space \((L,\text{Ad},\mathfrak{n})\). The contragradient \((L,\text{Ad}, \overline{\mathfrak{n}})\) with a relative invariant \(\overline{\nabla}\) has the same properties.
The paper deals with an extension of the domain of the functional equation of some prehomogeneous vector spaces from the Schwartz space to the space of functions of certain degenerate principal series representations \(\text{Ind}_P^G(s)\), realized as functions on \(\overline{\mathfrak{n}}\) for all \(s\in\mathbb{C}\).


22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
20G05 Representation theory for linear algebraic groups
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