We study harmonic analysis on the symmetric space $\text{GL}_n \times \text{GL}_n \backslash \text{GL}_{2n}$ . We prove several standard results, e.g. Shalika germ expansion of orbital integrals, representability of the Fourier transform of orbital integrals and representability of spherical characters. These properties are not expected to hold for symmetric spaces in general.

Let $K/F$ be an unramified quadratic extension of a non-Archimedean local field. In a previous work [1], we proved a formula for the intersection number on Lubin–Tate spaces. The main result of this article is an algorithm for computation of this formula in certain special cases. As an application, we prove the linear Arithmetic Fundamental Lemma for $ \operatorname {{\mathrm {GL}}}_4$ with the unit element in the spherical Hecke Algebra.

We set up a formalism of endoscopy for metaplectic groups. By defining a suitable transfer factor, we prove an analogue of the Langlands–Shelstad transfer conjecture for orbital integrals over any local field of characteristic zero, as well as the fundamental lemma for units of the Hecke algebra in the unramified case. This generalizes prior work of Adams and Renard in the real case and serves as a first step in studying the Arthur–Selberg trace formula for metaplectic groups.

Let $G=GL_n(F)$, where $F$ is a $p$-adic field of characteristic zero and residual characteristic $p$. Assuming that $p>2n$, we compare germs of characters of irreducible admissible representations of $G$ with germs of characters of unipotent representations of direct products of general linear groups over finite extensions of $F$. We show that the character of an irreducible admissible representation has an $s$-asymptotic germ expansion, for some semisimple $s$ in the Lie algebra of $G$. Furthermore, this expansion matches with the $0$-asymptotic expansion (that is, the local character expansion) of the character of a unipotent representation of the centralizer of $s$ in $G$.

AMS 2000 Mathematics subject classification: Primary 22E50; 22E35