We compute the $K$-theory of the ${C}^{*}$-category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group $G$. Our result uses the Connes-Kasparov isomorphism for $G$, and in fact it is equivalent to the Connes-Kasparov isomorphism. We relate our computation to David Vogan’s well-known parametrization of the tempered irreducible representations of $G$ with real infinitesimal character. When the reductive group $G$ has real rank one, we formulate and prove a Fourier isomorphism theorem for equivariant order zero pseudodifferential operators on the symmetric space, and use it to prove a $K$-theoretic version of Vogan’s theorem.

Let $G(\mathbb {R})$ be a real reductive group. Suppose $\pi $ is an irreducible representation of $G(\mathbb {R})$ having a Whittaker model, and consider three invariants of $\pi $ related to nilpotent elements of the Lie algebra of G (or its dual): the associated variety, the wave-front set, and the set of Whittaker data for which $\pi $ has a Whittaker model. If $\pi $ is a discrete series representation, these invariants are known to determine each other. We provide a self-contained account of this and related matters. Many of the results were known: we give simplified proofs for several of them, for instance a simple proof (for generic discrete series) that the associated variety and the wave-front set are related by the Kostant–Sekiguchi correspondence.

We establish a derived geometric Satake equivalence for the quaternionic general linear group ${\textrm{GL}}_{n}({\mathbb H})$. By applying the real–symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the symmetric variety ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$. We explain how these equivalences fit into the general framework of a geometric Langlands correspondence for real groups and the relative Langlands duality conjecture. As an application, we compute the stalks of the IC-complexes for spherical orbit closures in the quaternionic affine Grassmannian and the loop space of ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$. We show that the stalks are given by the Kostka–Foulkes polynomials for ${\textrm{GL}}_n$ but with all degrees doubled.

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$-orbit with a bundle $Q\times _{L}S$. Here $Q$ is a parabolic subgroup with Levi decomposition $L\ltimes U$, and $S$ is a homogeneous space for a quotient $D=L/L_{n}$ of $L$, where $L_{n}\subseteq L$ is normal, such that $D$ is compact modulo center.

The admissible representations of a real reductive group G are known by work of Langlands, Knapp, Zuckerman and Vogan. This paper describes an effective algorithm for computing the irreducible representations of G with regular integral infinitesimal character. The algorithm also describes structure theory of G, including the orbits of K(ℂ) (a complexified maximal compact subgroup) on the flag variety. This algorithm has been implemented on a computer by the second author, as part of the ‘Atlas of Lie Groups and Representations’ project.