Dual pairs and Kostant-Sekiguchi correspondence. II. Classification of nilpotent elements

We classify the homogeneous nilpotent orbits in certain Lie color algebras and specialize the results to the setting of a real reductive dual pair.For any member of a dual pair, we prove the bijectivity of the two Kostant-Sekiguchi maps by straightforward argument. For a dual pair we determine the correspondence of the real orbits, the correspondence of the complex orbits and explain how these two relations behave under the Kostant-Sekiguchi maps. In particular we prove that for a dual pair in the stable range there is a Kostant-Sekiguchi map such that the conjecture formulated in [6] holds. We also show that the conjecture cannot be true in general.

[1]  Kyo Nishiyama,et al.  Theta lifting of unitary lowest weight modules and their associated cycles , 2004 .

[2]  Hiroyuki Ochiai,et al.  Theta lifting of nilpotent orbits for symmetric pairs , 2003, math/0312453.

[3]  T. Przebinda,et al.  Dual Pairs and Kostant–Sekiguchi Correspondence, I , 2002 .

[4]  W. Schmid,et al.  Characteristic cycles and wave front cycles of representations of reductive Lie groups , 2000, math/0005305.

[5]  T. Przebinda,et al.  Nilpotent Orbits and Complex Dual Pairs , 1997 .

[6]  Arun Ram,et al.  Tensor product representations for orthosymplectic Lie superalgebras , 1996, math/9607232.

[7]  T. Przebinda,et al.  The Oscillator Character Formula, for isometry groups of split forms in deep stable range , 1996 .

[8]  T. Przebinda.  Characters, dual pairs, and unitary representations , 1993 .

[9]  Takuya Ohta.  THE CLOSURES OF NILPOTENT ORBITS IN THE CLASSICAL SYMMETRIC PAIRS AND THEIR SINGULARITIES , 1991 .

[10]  D. Djoković.  Closures of conjugacy classes in classical real linear Lie groups , 1982 .

[11]  R. Cushman,et al.  Conjugacy classes in linear groups , 1977 .

[12]  Harish-Chandra.  Invariant Distributions on Lie Algebras , 1964 .

[13]  D. Barbasch,et al.  Closure ordering and the Kostant-Sekiguchi correspondence , 1998 .

[14]  W. Schmid,et al.  Representations of reductive lie groups , 1988 .

[15]  J. Sekiguchi.  Remarks on real nilpotent orbits of a symmetric pair , 1987 .

[16]  Takuya Ohta.  THE SINGULARITIES OF THE CLOSURES OF NILPOTENT ORBITS IN CERTAIN SYMMETRIC PAIRS , 1986 .

[17]  関口次郎.  The Nilpotent Subvariety of the Vector Space Associated to a Symmetric Pair(対称対に附随するベクトル空間のベキ零部分多様体) , 1984 .

[18]  R. Howe.  Series and invariant theory , 1977 .