Let \(F\) be a local field of characteristic zero and \(G\) a connected reductive \(F\)-group, or a finite cover of such a group. The article under review proposes a generalization of the definition of formal degrees to all unitary irreducible representations of \(G\). Let \((\pi, V_\pi)\) be a unitary irreducible representation. One considers a pairing \(I(s; v_1, v_2, v_3, v_4) := \int_{G/Z_G} (\pi(g)v_1, v_2) \overline{(\pi(g)v_3, v_4)} \Delta(g)^s dg\), where \(v_1, \dots, v_4 \in V_\pi\) and \(\Delta\) is a certain “height function” so that \(I(s)\) defines a holomorphic function in \(s\) for \(\mathrm{Re}(s) \gg 0\). By imposing some conditions on \(\Delta\), the author’s program is to show (i) the meromorphic continuation of \(I(s)\) to all \(s \in \mathbb{C}\); (ii) the \(G \times G\)-invariance of the leading coefficient of \(I(s)\) at \(s=0\); (iii) a relation between the vanishing order of \(I(s)\) at \(s=0\) and that of the adjoint \(\gamma\)-factor of \(\pi\), at least when \(G\) is linear. Granting these conjectures, the leading coefficient of \(I(s)\) at \(s=0\) would give the generalized formal degree of \(\pi\).
When \(F\) is nonarchimedean, (i) is established by admitting the “volume hypothesis” (2.2.1) in harmonic analysis, which should hold for all \(G\). This can explicitly be verified in the case of split classical groups. For \(G=\mathrm{GL}(n)\), (ii) holds and (iii) is verified for generic \(\pi\) for \(F\) non-archimedean, and for every \(\pi\) for \(F=\mathbb{R}\).
One particularly interesting case is the twofold cover \(\mathrm{Mp}(2)\) of \(\mathrm{SL}(2)\), which is closely related to the Howe correspondence. The author verifies (i)-(iii) for all genuine unitary representations of \(\mathrm{Mp}(2)\). Moreover, he shows that the generalized formal degrees are preserved by the Howe correspondence for the dual pair \((\mathrm{O}(V,q), \mathrm{SL}(2))\) up to a factor \(|2|_F\), where \((V,q)\) is any \(3\)-dimensional quadratic space over \(F\).
The author gives detailed calculations as well as a comprehensive bibliography.