Spectral decomposition of \(L^ 2(N\backslash GL(2),\eta)\) (Q1078341)

Let G (resp. \(G^ 1)\) denote GL(2,F) (resp. SL(2,F)), where F is \({\mathbb{R}}\) or \({\mathbb{C}}\) or a p-adic field. Let \(\eta\) be a non-trivial character of the unipotent upper triangular group N in G. In the first section, the author constructs a spectral decomposition of \(L^ 2\)(N\(\setminus G^ 1,\eta)\). (By doing this one obtains a spectral decomposition of \(L^ 2(N\setminus G,\eta)\), too.) First a special subspace \(S_ 0(F^ 2)\) of \(S(F^ 2)\) (space of Schwartz-Bruhat functions) is defined and mapped into \(L^ 2(N\setminus G,\eta)\). If \({}^ 0V\) (resp. \({}^ 0V_ K)\) denotes the orthocomplement of the image of \(S_ 0(F^ 2)\) (resp. the \(K^ 1\)-finite part of this orthocomplement), then \({}^ 0V_ K\) is admissible and can be identified with the discrete part \(V_{dis}\) of \(L^ 2(N\setminus G^ 1,\eta)\). As a consequence, one obtains a decomposition \(V_{dis}=\oplus_{\pi \in \Sigma (\eta)}\pi\), where \(\Sigma\) (\(\eta)\) denotes the set of all irreducible, inequivalent, square-integrable \(G^ 1\)-modules having a Whittaker model relative to \(\eta\). In the last two sections, L-functions are defined for some special functions in \(L^ 2(N\setminus G)\) coming from functions of \(S_ 0(F^ 2)\) resp. of \(S(F^ 2)\). This supplies some relations between L-functions and \(\epsilon\)-factors. Especially, one obtains certain scalar product formulas.