On functional equations of complex powers
Let X be a prehomogeneous vector space (in the sense of Sato-Shintani) over a local field K of characteristic 0, i.e. an affine space together with an irreducible polynomial f over K on X and a connected reductive algebraic K-subgroup \(G\subset GL(X)\) acting transitively on \(Y:=X\setminus f^{-1}(0)\). There exist finitely many \(G_ K\)-orbits \(Y_ 1,...,Y_{\ell}\) making up Y(K); for each \(Y_ i\) and any \(\omega\) in the space \(\Omega\) of all quasi-characters on \(K^{\times}=K\setminus \{0\}\), tempered distributions \(Z_ i(\omega)\) are defined by \[ Z_ i(\omega)(\Phi)=\int_{Y_ i}(\omega \omega_{\kappa})^{-1}(f(x)) \Phi (x) dx \] for Schwartz-Bruhat functions \(\Phi\) on \(X_ K\), where \(\omega_ s(t):=| t|^ s_ K\), \(| \quad |_ K\) denotes valuation in K and \(\kappa:=\dim X/\deg f\). These \(Z_ i\) depend holomorphically on \(\omega\) when restricted to an open subset of \(\Omega\) and admit meromorphic continuations to all of \(\Omega\) with functional equations \[ Z_ i(\omega)^*=\sum^{\ell}_{j=1}\gamma_{ij}(\omega) Z_ j(\omega_{\kappa}\omega^{-1}) \] where * denotes Fourier transform and \(\gamma_{ij}\) are meromorphic on \(\Omega\). The \(\Gamma\)-matrix \((\gamma_{ij}(\omega))\) apparently under the influence of the (polynomial) b-function of f, has been explicitly computed (up to sign) for \(\omega =\omega_ s\), \(K={\mathbb{C}}\) (and reductive G) by earlier authors (e.g. Sato). Now, by introducing a ''renormalized'' \(\Gamma\)-matrix \(a_ K(G,\omega)\) involving \(\gamma\) (\(\omega)\) and the b-function, the author establishes (under the additional restrictions for p-adic fields K that G is irreducible and K-split with the b-function having all its roots in \({\mathbb{Z}})\) that \(a_ K(G,\omega)\) depends only on the K- equivalence class of G and on \(\omega\) (once the ordering of \(Y_ i\) and of similar 'dual orbits' is fixed) and is ''intrinsic'' for \(\ell =1\), in particular. It is explicitly determined and shown to be 1, for \(K={\mathbb{C}}\) and similarly for \(K={\mathbb{R}}\) with \(\ell =1\) or p-adic fields K, with the stated restrictions). One may refer in this connection to a preprint of \textit{F. Sato} on ''Remarks on functional equations of zeta distributions''.
connected reductive algebraic K-subgroup
scientific article; zbMATH DE number 3966232
scientific article