Lie Group and Representation Theory Seminar, Kyoto 2006
Lie Group and Representation Theory Seminar, Kyoto 2006| Date: | September 1 (Fri), 2006, 11:00--12:00 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | 菊地 克彦氏 (Katsuhiko Kikuchi)(Kyoto University) |
| Title: | Invariant polynomials and invariant differential operators for multiplicity-free actions of rank 3 |
| Abstract: |
Let $V$ be a finite-dimensional vector space
over $\mathbb{C}$, and $K$ a compact Lie group
acting on $V$ linearly.
We call $(K, V)$ a {\it multiplicity-free action}
if each irreducible component appears at most one
in the (holomorphic) polynomial ring ${\cal P}(V)$ on $V$.
If $(K, V)$ is multiplicity-free, then there exists
a number $r$ such that the ring
${\cal P}(V_{\mathbb{R}})^K
={\cal P}(V)\otimes \overline{{\cal P}(V)}$ of
$K$-invariant polynomials on the underlying
real vector space $V_{\mathbb{R}}$ of $V$
is isomorphic to the polynomial ring of $r$ variables.
The number $r$ is called the {\it rank} of $(K, V)$.
For each highest weight $\lambda$ which appears in the irreducible decomposition of ${\cal P}(V)$ there exist, up to a scalar, a unique $K$-invariant polynomial $p_{\lambda}(z, \overline{z})$ and a unique $K$-invariant differential operator $p_{\lambda}(z, \partial )$. In this talk, we describe all $K$-invariant polynomials $\{p_{\lambda}(z, \overline{z})\}$ and $K$-invariant differential operators $\{p_{\lambda}(z, \partial )\}$ for a rank 3 multiplicity-free action $(K, V)$ which is not derived from a Hermitian symmetric space. Moreover, we give two `symmetric' slices for visibility of the action $(K, V)$. We show that the action of the stabilizer of one indicates the symmetry of the $K$-invariant polynomials, and that of the other indicates the symmetry of the eigenvalues of the $K$-invariant differential operators. |
| Date: | September 29 (Fri), 2006, 16:30--18:00 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Pierre PANSU (Paris-Sud) |
| Title: | L^p -cohomology and negative curvature |
| Abstract: |
L^p -cohomology of a Riemannian manifold is the cohomology
of the (de Rham) complex of differential forms which are L^p
-integrable. We explain the role played by L^p -cohomology in three
problems related to negatively curved manifolds and groups.
- Hopf's conjecture on the sign of Euler characteristic of compact negatively curved manifolds (specificly, the Kahler case). - Cannon's conjecture on hyperbolic groups whose ideal boundary is a 2-sphere. - Optimal sectional curvature pinching for rank one symmetric spaces. |