Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb C[V]^G$. We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$ over the mapping of invariants $\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in $\mathcal C$, e.g., the real analytic class, then $f$ admits a lift of the same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation. If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.

In this paper we construct a flat smooth section of an induced space $I(s,\eta)$ of $SL_2(\mathbb{R})$ so that the attached Whittaker function is not of finite order. An asymptotic method of classical analysis is used.

Let $\pi$ be an irreducible generalized principal series representation of $G = \Sp(2,\mathbb{R})$ induced from its Jacobi parabolic subgroup. We show that the space of algebraic intertwining operators from $\pi$ to the representation induced from an irreducible admissible representation of $\SL(2,\mathbb{C})$ in $G$ is at most one dimensional. Spherical functions in the title are the images of $K$-finite vectors by this intertwining operator. We obtain an integral expression of Mellin-Barnes type for the radial part of our spherical function.

Let $G$ be a connected reductive $p$-adic group and let $\frakg$ be its Lie algebra. Let $\calO$ be any $G$-orbit in $\frakg$. Then the orbital integral $\mu_{\calO}$ corresponding to $\calO$ is an invariant distribution on $\frakg $, and Harish-Chandra proved that its Fourier transform $\hat \mu_{\calO}$ is a locally constant function on the set $\frakg'$ of regular semisimple elements of $\frakg$. If $\frakh$ is a Cartan subalgebra of $\frakg$, and $\omega$ is a compact subset of $\frakh\cap\frakg'$, we give a formula for $\hat \mu_{\calO}(tH)$ for $H\in\omega$ and $t\in F^{\times}$ sufficiently large. In the case that $\calO$ is a regular semisimple orbit, the formula is already known by work of Waldspurger. In the case that $\calO$ is a nilpotent orbit, the behavior of $\hat\mu_{\calO}$ at infinity is already known because of its homogeneity properties. The general case combines aspects of these two extreme cases. The formula for $\hat\mu _{\calO}$ at infinity can be used to formulate a ``theory of the constant term'' for the space of distributions spanned by the Fourier transforms of orbital integrals. It can also be used to show that the Fourier transforms of orbital integrals are ``linearly independent at infinity.''

The trace formula contains terms on the spectral side that are constructed from unramified automorphic $L$-functions. We shall establish an identify that relates these terms with corresponding terms attached to endoscopic groups of $G$. In the process, we shall show that the $L$-functions of $G$ that come from automorphic representations of endoscopic groups have meromorphic continuation.

We give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of $\SU(2, 2)$ with its conjugate.