Branching rules obtained from explicit correspondences of automorphic representations
The paper under review is concerned with applications of the classical Casselman--Shalika formula [\textit{W. Casselman} and \textit{J. Shalika}, Compos. Math. 41, 207--231 (1980; Zbl 0472.22005), Theorem 5.4] to ``branching rules'' for finite-dimensional representations of complex groups. More specifically, note that, via the Weyl character formula, the cited formula expresses the unramified Whittaker function (of a generic, unramified representation) on a \(p\)-adic group \(G(F_v)\) in terms of characters of its complex Langlands dual group \({^LG}(\mathbb C)\). Historically, one has combined this formula with an understanding of the finite-dimensional, irreducible representations of the complex \(L\)-groups to deduce instances of Langlands functoriality, i.e., lifting of automorphic representations between adèlic groups \(H(\mathbb A)\) and \(G(\mathbb A)\). In the paper under review, Ginzburg demonstrates that the current state of understanding of explicit liftings is sufficiently advanced to allow information to flow the other way; that is, that one may use known examples of Langlands functoriality to deduce information about the decomposition into irreducible representations of \({^LH}(\mathbb C)\) of the restriction of an irreducible representation of \({^LG}(\mathbb C)\). The paper starts by recovering the classical branching rule for the restriction to \(\text{SO}_{2n - 1}(\mathbb C) = {^L\text{Sp}_{2(n - 1)}}(\mathbb C)\) of a representation of \(\text{SO}_{2n}(\mathbb C) = {^L\text{SO}_{2n}}(\mathbb C)\). We describe this in detail, since it is the model for what follows. The groups \(\text{SO}_{2n}\) and \(\text{Sp}_{2(n - 1)}\) (almost) form a dual pair in a large symplectic group, and one may use the Weil representation on the corresponding metaplectic group to obtain a lifting. (As observed on p.~31, \textit{R. Howe, E.-C. Tan} and \textit{J. F. Willenbring} [Trans. Am. Math. Soc. 357, No. 4, 1601--1626 (2005; Zbl 1069.22006)] have previously considered the application of the theory of dual pairs to branching rules.) \textit{D. Ginzburg, S. Rallis} and \textit{D. Soudry} [J. Reine Angew. Math. 487, 85--114 (1997; Zbl 0928.11025)] gave an integral formula relating the unramified Whittaker functions of a suitable representation and its lift, and most of the paper under review is dedicated to the explicit evaluation of this and similar integrals. The main tools are the Iwasawa decomposition; explicit commutation relations for Weyl elements and root subgroups; and the translation-invariance (under suitable subgroups) of the integrand. Finally, one applies the Casselman--Shalika formula to obtain branching rules. The other situations considered are: {\parindent5mm \begin{itemize}\item[3] the branching rule from \({^LG} = \text G_2\) to \({^LH} = \text{SL}_3\); \item[4] the branching rule from \({^LG} = \text{Spin}_7\) to \({^LH} = \text G_2\); and \item[5] the computation of certain functionals on the orthogonal groups. \end{itemize}} The approach of this paper is intriguing. It seems in a sense to proceed from the more complicated (local-field Whittaker functions) to the simpler (finite-dimensional representations); but the results that are recovered, while usually not completely new, are nonetheless more explicit than those previously available.
scientific article; zbMATH DE number 5610595
scientific article