The paper under review defines the notion of stable conjugacy in the Brylinski-Deligne covering (BD-covering for short) of the symplectic group over a local field \(F\) (whose characteristic is not 2). To justify the definition, the author constructed the epipelagic L-packets and showed their compatibility with his notion of stable conjugacy. The paper contains a careful and detailed account of many subjects, such as a survey on the Brylinski-Deligne covering, stable conjugacy in Sp(2n), and L-group of BD-covering. Therefore it also could be served as a great reference for these subjects.
Brylinski and Deligne built a functorial framework to produce central extensions of a reductive algebraic group. The theory is reviewed in Section 1. In the following, let \(1\rightarrow \mu_m\rightarrow \tilde{G} \xrightarrow{\ \ p \ \ } G\rightarrow 1\) be the \(m\)-fold BD-covering of \(G=\mathrm{Sp}(2n,F)\) where \(\mu_m\) is the group of \(m\)-th roots of unity. A representation of \(\tilde{G}\) is called genuine if \(\mu_m\) acts by the scalar multiplication. This paper is fundamental for the character theory of genuine representations of \(\tilde{G}\).
The set of good elements is of particular importance since every genuine invariant function (for example, the character of an irreducible genuine representation of \(\tilde{G}\)) on the \(\tilde{G}_{reg}\) vanishes off the good locus. The set of “good elements” in \(\tilde{G}\) is classified in Proposition 4.3.1.
A related question in the endoscopy theory is the stable conjugacy, i.e., conjugacy over the separable closure. Section 3 discussed the parameterization of stable conjugacy classes of regular semisimple elements in \(\mathrm{Sp}(2n,F)\). One of the necessary conditions for two regular semisimple elements \(\tilde{\delta}\) and \(\tilde{\eta}\) in \(\tilde{G}\) are stably conjugate is that their images in \(G\) are stably conjugate. However, the precise definition (Definition 4.3.10) of stable conjugacy in BD-coverings must refer to some additional data: a tuple \(\sigma\) of \(\pm\)-signs and some elements \(\delta_0,\eta_0\) in an isogeny of \(Z_G(\delta)\) (see \((4.1)\)). When \(4|m\) the reference to the additional data could be removed, see Corollary 4.4.4. The author shows in Section 9 that his definition of stable conjugacy is compatible with Adams’ definition for metaplectic groups (i.e., \(m=2\)) and Hiraga-Ikeda’s definition for \(\mathrm{SL}(2)\) (i.e., \(n=1\)). With the definition of stable conjugacy, the notion of stable character is given by Definition 4.4.5 (the case of \(4|m\) and that of \(4\not| m\) are treated separately).
In sections 5-7, the author discuss the definition of L-group for \(\tilde{G}\). Following Reeder-Yu and Kaletha, epipelagic L-parameters (Definition 7.4.1) and the corresponding L-package (Definition 7.3.6 and Definition 7.4.4) are defined. Note that when \(4|m\), there may be no epipelagic representations attached to a certain parameter. Then the stability of the invariant distributions (Definition 7.6.1) arose from the epipelagic L-packet is verified using Adler-Spice’s character formula in Section 7.6. Furthermore, the compatibility of the epipelagic L-packet of SO(2n+1) with that of the metaplectic group via theta-lifting is established in Theorem 9.3.3.