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1. CJM 2012 (vol 64 pp. 721)
Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations We revisit with another view point the construction by R. Brylinski
and B. Kostant of minimal representations of simple Lie groups. We
start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$
a homogeneous polynomial of degree 4 on $V$.
The manifold $\Xi $ is an orbit of a covering of ${\rm Conf}(V,Q)$,
the conformal group of the pair $(V,Q)$, in a finite dimensional
representation space.
By a generalized Kantor-Koecher-Tits construction we obtain a complex
simple Lie algebra $\mathfrak g$, and furthermore a real
form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie
group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak
g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic
functions defined on the manifold $\Xi $.
Keywords:minimal representation, Kantor-Koecher-Tits construction, Jordan algebra, Bernstein identity, Meijer $G$-function Categories:17C36, 22E46, 32M15, 33C80 |
2. CJM 2009 (vol 61 pp. 1407)
Traces, Cross-Ratios and 2-Generator Subgroups of $\SU(2,1)$ In this work, we investigate how to decompose a pair $(A,B)$ of
loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under
the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are
involutions. The main result is a decomposability criterion, which
is expressed in terms of traces of elements of the group $\langle
A,B\rangle$.
Categories:14L24, 22E40, 32M15, 51M10 |
3. CJM 2000 (vol 52 pp. 982)
Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds Let $Y$ be an infinite covering space of a projective manifold
$M$ in $\P^N$ of dimension $n\geq 2$. Let $C$ be the intersection with
$M$ of at most $n-1$ generic hypersurfaces of degree $d$ in $\mathbb{P}^N$.
The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$
be the smoothed distance from a fixed point in $Y$ in a metric pulled up
from $M$. Let $\O_\phi(X)$ be the Hilbert space of holomorphic
functions $f$ on $X$ such that $f^2 e^{-\phi}$ is integrable on $X$, and
define $\O_\phi(Y)$ similarly. Our main result is that (under more
general hypotheses than described here) the restriction $\O_\phi(Y)
\to \O_\phi(X)$ is an isomorphism for $d$ large enough.
This yields new examples of Riemann surfaces and domains of holomorphy
in $\C^n$ with corona. We consider the important special case when $Y$
is the unit ball $\B$ in $\C^n$, and show that for $d$ large enough,
every bounded holomorphic function on $X$ extends to a unique function
in the intersection of all the nontrivial weighted Bergman spaces on
$\B$. Finally, assuming that the covering group is arithmetic, we
establish three dichotomies concerning the extension of bounded
holomorphic and harmonic functions from $X$ to $\B$.
Categories:32A10, 14E20, 30F99, 32M15 |
4. 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 |