# Satake-Furstenberg compactifications, the moment map and \lambda_1

@article{Biliotti2010SatakeFurstenbergCT, title={Satake-Furstenberg compactifications, the moment map and \lambda\_1}, author={Leonardo Biliotti and Alessandro Ghigi}, journal={arXiv: Differential Geometry}, year={2010} }

Let G be a complex semisimple Lie group, K a maximal compact subgroup and V an irreducible representation of K. Denote by M the unique closed orbit of G in P(V) and by O its image via the moment map. For any measure on M we construct a map from the Satake compactification of G/K (associated to V) to the Lie algebra of K. For the K-invariant measure, this map is a homeomorphism of the Satake compactification onto the convex envelope of O. For a large class of measures the image of the map is the…

## References

SHOWING 1-10 OF 39 REFERENCES

### Convexity and Commuting Hamiltonians

- Mathematics
- 1982

The converse was proved by A. Horn [5], so that all points in this convex hull occur as diagonals of some matrix A with the given eigenvalues. Kostant [7] generalized these results to any compact Lie…

### ON REPRESENTATIONS AND COMPACTIFICATIONS OF SYMMETRIC RIEMANNIAN SPACES

- Mathematics
- 1960

In the recent study of automorphic functions [2], [4], [7], it has become necessary to take boundaries of symmetric bounded domains (or more generally, of symmetric Riemannian spaces) into account…

### A POISSON FORMULA FOR SEMI-SIMPLE LIE GROUPS*

- Mathematics
- 1963

for some bounded function h on the boundary of the disc. The function h(z) determines a function h(g) on G by setting h(g) = h(g(O)). If h(z) is harmonic, it may be shown that h(g) is annihilated by…

### Upper bound for the first eigenvalue of algebraic submanifolds

- Mathematics
- 1994

1. Statement of results Let M be a compact manifold endowed with a Riemannian metric. The spectrum of the Laplacian, A, acting on functions form a discrete set of the form {0 < ),~ < 22 < �9 �9 �9 <…

### The spectrum of a Riemannian manifold with a unit Killing vector field

- Mathematics
- 1983

Let (P, g) be a compact, connected, C°° Riemannian (n + l)-manifold (n > 1) with a unit Killing vector field with dual 1-form tj. For t > 0, let g, = i~'g + (f" — f"')t)|8>r),a family of metrics of…

### RIEMANNIAN METRICS WITH LARGE Xx

- Mathematics
- 2010

We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian. Let (Mn, g) be a compact,…

### Riemannian metrics with large

- Mathematics
- 1994

We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian. Let (Mn, g) be a compact,…

### Remarks on the Satake Compactifications

- Mathematics
- 2005

This article has three independent parts. The first one is a simplification, using some old results of the author, of a construction of the compactifications recently given by A. Borel and L. Ji. The…

### Torus Actions on Symplectic Manifolds

- Mathematics
- 1991

Introductory preface.- How I have (re-)written this book.- Acknowledgements.- What I have written in this book.- I. Smooth Lie group actions on manifolds.- I.1. Generalities.- I.2. Equivariant…

### An introduction to symplectic topology

- Mathematics
- 1991

Proposition 1.4. (1) Any symplectic vector space has even dimension (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the…