Dual pair \(G_{\mathcal J}\times PGL_ 2 G_{\mathcal J}\) is the automorphism group of the Jordan algebra \({\mathcal J}\)
Let \(G\) be a simple, split, simply laced group over a nonarchimedean local field \(F\) of characteristic different from 2. Let \(J(\chi_{\text{sr}})\) be the spherical representation of \(G\) with Satake parameter \(\chi_{\text{sr}}\). The author computes the wavefront set of \(J(\chi_{\text{sr}})\) by computing the wavefront set of the representation \(\pi_0\) constructed by \textit{D. Kazhdan} and \textit{G. Savin} [Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his 60th birthday, Pt. I (Tel Aviv 1989), Israel Math. Conf. Proc. 2, 209-233 (1990; Zbl 0737.22008)] and shows that if \(G\) is of types \(A_{2n-1}\), \(D_n\) or \(E_n\) \((n=6,7,8)\), then \(J(\chi_{\text{sr}})\) is a minimal representation of \(G\). Let \(\mathcal G\) be a (split) Jordan algebra and \(G_{\mathcal G}\) be the automorphism group of the Jordan algebra \(\mathcal G\). Using methods of \textit{S. Sahi} [Invent. Math. 110, 409-418 (1992; Zbl 0779.22006)], the author obtains results for the dual pair \(G_{\mathcal G} \times \text{PGL}_2\). This generalizes in a natural way results of \textit{R. Howe} [J. Funct. Anal. 32, 297-303 (1979; Zbl 0408.22018)] and others in the classical situation, i.e., when \(\mathcal G\) is the algebra of symmetric \(n\times n\) matrices over \(F\). By this generalization it is shown that the dual pair is \(O(n)\times \text{SL}_2\) and the minimal representation is the Weil representation.
scientific article; zbMATH DE number 703880