Holomorphic slices, symplectic reduction and multiplicities of representations

I prove the existence of slices for an action of a reductive complex Lie group on a K\"ahler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for the action of a compact real form of the group. I give applications of this result to symplectic reduction and geometric quantization at singular levels of the momentum map. In particular, I obtain a formula for the multiplicities of the irreducible representations occurring in the quantization in terms of symplectic invariants of reduced spaces, generalizing a result of Guillemin and Sternberg.

[1]  D. Snow.  Reductive group actions on Stein spaces , 1982 .

[2]  H. Grauert,et al.  Plurisubharmonische Funktionen in komplexen Räumen , 1956 .

[3]  S. Sternberg,et al.  A Normal Form for the Moment Map , 1984 .

[4]  George Kempf.  The Hochster-Roberts theorem of invariant theory. , 1979 .

[5]  Hubert Flenner.  Rationale quasihomogene Singularitäten , 1981 .

[6]  Gerald W. Schwarz,et al.  Inequalities defining orbit spaces , 1985 .

[7]  Reductive group actions on Kähler manifolds , 1992, alg-geom/9210006.

[8]  Linearization of reductive group actions , 1982 .

[9]  R. Schlafly,et al.  On equivariant isometric embeddings , 1980 .

[10]  J. Boutot.  Singularités rationnelles et quotients par les groupes réductifs , 1987 .

[11]  G. Mostow.  COVARIANT FIBERINGS OF KLEIN SPACES, II.* , 1962 .

[12]  D. Luna.  Fonctions différentiables invariantes sous l'opération d'un groupe réductif , 1976 .

[13]  E. Lerman,et al.  Stratified symplectic spaces and reduction , 1991 .

[14]  R. Wells,et al.  Holomorphic approximation and hyperfunction theory on aC1 totally real submanifold of a complex manifold , 1972 .

[15]  F. Kirwan.  Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31 , 1984 .

[16]  Yozô Matsushima.  Espaces homogènes de Stein des groupes de Lie complexes. II , 1960 .

[17]  Alan Weinstein,et al.  Lectures on Symplectic Manifolds , 1977 .

[18]  G. Mostow.  On Covariant Fiberings of Klein Spaces , 1955 .

[19]  F. Bruhat,et al.  Quelques propriétés fondamentales des ensembles analytiques-réels , 1959 .

[20]  T. Kawasaki.  The Riemann-Roch theorem for complex V -manifolds , 1979 .

[21]  I. Satake.  The Gauss-Bonnet Theorem for V-manifolds , 1957 .

[22]  J. Sampson,et al.  A Künneth formula for coherent algebraic sheaves , 1959 .

[23]  G. Kempf,et al.  The length of vectors in representation spaces , 1979 .

[24]  A. Sommese.  Extension theorems for reductive group actions on compact Kaehler manifolds , 1975 .

[25]  W. Fulton,et al.  Riemann-roch for singular varieties , 1975 .

[26]  L. Hörmander,et al.  An introduction to complex analysis in several variables , 1973 .

[27]  Linda Ness,et al.  A Stratification of the Null Cone Via the Moment Map , 1984 .

[28]  P. Heinzner.  Geometric invariant theory on Stein spaces , 1991 .

[29]  H. Grauert.  On Levi's Problem and the Imbedding of Real-Analytic Manifolds , 1958 .

[30]  A. Neeman.  The topology of quotient varieties , 1985 .

[31]  P. Lelong.  Plurisubharmonic Functions and Positive Differential Forms , 1969 .

[32]  Orbits that always have affine stable neighbourhoods , 1992 .

[33]  Klaus Jänich.  Differenzierbare G-Mannigfaltigkeiten , 1968 .

[34]  Gerald W. Schwarz.  The Topology of Algebraic Quotients , 1989 .

[35]  R. Richardson.  Deformations of lie subgroups and the variation of isotropy subgroups , 1972 .

[36]  Shlomo Sternberg,et al.  Geometric quantization and multiplicities of group representations , 1982 .

[37]  F. Kirwan.  Partial desingularisations of quotients of nonsingular varieties and their Betti numbers , 1985 .