Expand all Collapse all | Results 1 - 6 of 6 |
1. CJM 2011 (vol 64 pp. 409)
Lifting Quasianalytic Mappings over Invariants 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.
Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variation Categories:14L24, 14L30, 20G20, 22E45 |
2. CJM 2009 (vol 61 pp. 373)
An Infinite Order Whittaker Function 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.
Categories:11F70, 22E45, 41A60, 11M99, 30D15, 33C15 |
3. CJM 2002 (vol 54 pp. 828)
Spherical Functions for the Semisimple Symmetric Pair $\bigl( \Sp(2,\mathbb{R}), \SL(2,\mathbb{C}) \bigr)$ |
Spherical Functions for the Semisimple Symmetric Pair $\bigl( \Sp(2,\mathbb{R}), \SL(2,\mathbb{C}) \bigr)$ 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.
Categories:22E45, 11F70 |
4. CJM 2000 (vol 52 pp. 1192)
Orbital Integrals on $p$-Adic Lie Algebras 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.''
Categories:22E30, 22E45 |
5. CJM 1999 (vol 51 pp. 1135)
Endoscopic $L$-Functions and a Combinatorial Identity 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.
Categories:22E45, 22E46 |
6. CJM 1997 (vol 49 pp. 1224)
Tensor products of analytic continuations of holomorphic discrete series 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.
Keywords:Holomorphic discrete series, tensor product, spherical function, Clebsch-Gordan coefficient, Plancherel theorem Categories:22E45, 43A85, 32M15, 33A65 |