ABSTRACT
In Chapter 4, we constructed three canonical nilpotent orbits in any simple Lie algebra g https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2056.tif"/> , essentially from nothing. In this chapter we show how to construct new nilpotent orbits from old ones (in smaller algebras), following Lusztig and Spaltenstein. More precisely, given a nilpotent orbit O l https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2057.tif"/> in a Levi subalgebra l https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2058.tif"/> of g https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2059.tif"/> , we will produce a nilpotent orbit O g https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2060.tif"/> in g https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2061.tif"/> called the orbit induced from O l https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2062.tif"/> . The definition of O g https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2063.tif"/> seems to depend on a choice of parabolic subalgebra p https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2064.tif"/> with Levi subalgebra l https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2065.tif"/> , but we will prove that O g https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2066.tif"/> is actually independent of this choice. In §7.2 we show that every nilpotent orbit s l n https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2067.tif"/> in is induced from the 0 orbit in some Levi subalgebra and give a formula for the partition of any induced orbit. This formula is generalized in the next section to any classical algebra g https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2068.tif"/> . We also give a simple partition criterion due to Kempken and Spaltenstein for a classical orbit to be rigid (that is, not induced from any other orbit).
