On recursive properties of certain \(p\)-adic Whittaker functions (Q415252)

This paper establishes certain recursive properties of \(p\)-adic Whittaker functions which can be used to compute them explicitly in arbitrary dimensions. This is relevant for the Kudla program, a general framework which predicts close relations between heights of special cycles on certain Shimura varieties and special derivatives of Fourier coefficients of Eisenstein series.NEWLINENEWLINEThe main technical result deals with \(p\)-adic (or real, i.e.~archimedean) representation densities for three quadratic spaces \(L\), \(M\), \(N\), and establishes a natural compatibility between representation densities \(I(L, M)\), \(I(L, N)\), and the representation densities for certain subspaces of \(M\) and \(N\). This allows the author to recover a classical recursive property due to Kitaoka, as well as to prove a form of the desired orbit equation for representation densities. By interpolation, the author goes on to show that the corresponding facts also hold true for \(p\)-adic Whittaker functions.NEWLINENEWLINEIn the final sections, the application to Kudla's program is discussed for several Shimura varieties with references to the author's thesis, cf. [\textit{F. Hörmann}, The arithmetic and geometric volume of Shimura varieties of orthogonal type, (2011), \url{arXiv:1105.5357}], and a specific example is considered.