Sur les caractères des groupes de Lie réductifs non connexes. (On the characters of non-connected reductive Lie groups)
The paper deals with the following class of Lie groups G: the Lie algebra \({\mathfrak g}\) of G is reductive; G contains an abelian normal subgroup \(\Gamma\) so that \(\Gamma G_ 0\) is of finite index in G. (An essential point of the paper is that G is not assumed to being connected, nor even of Harish-Chandra class.) For such a group Duflo has described certain representations \(\Pi (f_ 0,\tau_ 0)\), parametrized by pairs \((f_ 0,\tau_ 0)\), where \(f_ 0\) is an element of \({\mathfrak g}^*\) whose centralizer \({\mathfrak g}(f_ 0)\) is a Cartan subalgebra and \(\tau_ 0\) is a unitary irreducible representation of a canonical double cover of \(G(f_ 0)\) so that \(\tau_ 0(\exp X)=e^{if_ 0(X)}\) and \(\tau_ 0(\epsilon)=-1\), \(\epsilon\) the non-trivial element of the kernel of the covering. (If G is of Harish-Chandra class, these are the tempered representations with regular infinitesimal character.) The paper gives a formula for the character \(\Theta (f_ 0,\tau_ 0)\) of such a representation in a neighbourhood of 1 in the centralizer Z of each semisimple element s of G. The formula may be viewed as a formula of the Kirillov type adapted to Harish-Chandra's method of descent: it represents the character in exponential coordinates as an integral over \(G\cdot f_ 0\cap {\mathfrak z}^*\), which is a union of a finite number of Z-orbits. For \(s=1\) it reduces to the usual Kirillov-type formula; for s regular to Harish-Chandra's formula. (The proof in general uses these cases.) When G is connected the formula (for relative discrete series) is due to \textit{M. Duflo}, \textit{G. Heckman}, and \textit{M. Vergne} [Mém. Soc. Math. Fr., Nouv. Sér. 15, 65-128 (1984; Zbl 0575.22014)].
scientific article; zbMATH DE number 4008702
scientific article