Spherical varieties and integral representations of L-functions
A very important method in studying automorphic \(L\)-functions is the construction of integral representations of such \(L\)-functions. Previously many important works were done in the construction of such integral representations, however there was not a general theory and the constructions are often considered as ``accidents''.NEWLINENEWLINEIn this paper, the author gives a uniform interpretation of a large number of the previous constructions. The theory of spherical varieties play an essential role in this interpretation.NEWLINENEWLINEGiven a spherical variety \(X\) for a split reductive group \(G\), the paper defines a space of Schwartz functions on \(X\), and constructs the space of theta functions and \(X\)-Eisenstein series associated to the Schwartz functions. It is conjectured that these \(X\)-Eisenstein series admit meromorphic continuations.NEWLINENEWLINEThe \(X\)-Eisenstein series will be generating functions for certain automorphic \(L\)-functions, at least when \(X\) has a structure of a ``preflag bundle''. The conjectured meromorphic continuation of \(X\)-Eisenstein series will give the continuation of the \(L\)-functions immediately.NEWLINENEWLINEUnder this setting, the problem of constructing integral representations of automorphic \(L\)-functions becomes a problem of classifying spherical varieties with the preflag bundle structure. In the case when \(X\) is a smooth affine spherical variety, the preflag bundle structure is there automatically. Knop--Van Steirteghem classified the smooth affine spherical varieties up to coverings, central tori and \(G_m\)-fibrations. The paper goes through the list of smooth affine varieties and recovers most known constructions of \(L-\)functions, as well as some new ones (for \(L(\pi_1\otimes\pi_2,s_2) L(\pi_2,s_1-\frac12(n-1))\) when \(\pi_1,\pi_2\) are cuspidal representations of \(\text{GL}_n\), and \(L(\pi_1\otimes\pi_2,s_2+\frac12)L(\pi_2,s_1-\frac12 n)\) where \(\pi_1\) is a cuspidal representation of \(\text{GL}_{n-1}\) and \(\pi_2\) of \(\text{GL}_n\)).NEWLINENEWLINEIn the last section of the paper, the author also gives an interpretation of the Jacquet-Rallis-Lai relative trace formula in terms of pairings of functions on spherical varieties. This suggests that the theory of relative trace formula is also closely related to the theory of spherical varieties.
scientific article; zbMATH DE number 6064724