Estimations at infinity of Bessel functions associated to the representations of a Jordan algebra (Q2730484)

Let \(V\) be a simple euclidean Jordan algebra with identity element \(e\) and let \(\Phi:V\to {\mathcal H}(E)\) be a regular representation of \(V\) on the euclidean space \(E\). Let \(Q: E\to V\) be the corresponding quadratic mapping. The Stiefel variety \(\Sigma=\{\sigma\in E|\;Q(\sigma)=e\}\) is then a compact analytic subvariety of \(E\). The Bessel function \(j\) associated to the representation \(\Phi\) is the Fourier transform of the normalized measure of \(\Sigma: j(\xi)=\int_\Sigma \exp(-i\langle \xi,\sigma\rangle_E) d\sigma\). This function \(j\) is radial. The author studies the asymptotic behaviour along a singular ray of the function \(J(x^2)=j(\Phi(x)^*\sigma_0)\), where \(x\) is an element of the closure of the symmetric cone associated to \(V\) (and \(\sigma_0\) a fixed element of \(\Sigma\)). Using the method of stationary phase [see \textit{J. J. Duistermaat, J. A. C. Kolk} and \textit{V. S. Varadarajan}, Compos. Math. 49, 309-398 (1983; Zbl 0524.43008)] he shows that for \(\tau\to\infty\) NEWLINE\[NEWLINEJ(\tau^2 x^2)\sim \sum_{m\in M(x)}\exp(i\tau g_x(\sigma)+ \tfrac{1}{4} i\pi g_x''(\sigma)) \sum_{l=0}\infty\tau^{-1/2n_m-l}p_{ml}NEWLINE\]NEWLINE where \(g_x(\sigma)=\langle\Phi(x)\sigma,\sigma_0\rangle_E\) and where \(M(x)\) is a set of multi-indices which depends only on the multiplicity of the eigenvalues of \(x\).