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XjG) over Q, whenever G aets on X with finite isotropy groups [22]. Let X be a G-seheme and let H C G be a closed subgroup. Then, for U as above, the quotient U -+ UjH exists and is a principal H-bundle. Thus, there is a smooth map X XH U -+ X XG U with fiber GjH, whieh induees a map A~(X) -+ A;r (X) of degree dim( G j H). If moreover H is a Levi subgroup of G, then G j H is the unipotent radieal of G and henee is isomorphie to affine spaee. Thus, A;;(X) is isomorphie to A;r (X). Our latter remark reduees many questions on equivariant Chow groups to the ease of reduetive groups.

Soc. ) 11 (1984), 363-366. [43] E. Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), 247-258. [44] P. Littelmann and C. Procesi, Equivariant cohomology of wonderful compactifications, in: Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Birkhäuser, Basel 1990, 219-262. [45] V. Mehta and V. Srinivas, Normality of Schubert varieties, Amer. J. Math. 109 (1987), 987-989. [46] W. Rossmann, Equivariant multiplicities on complex varieties, in: Orbites unipotentes et faisceatIX pervers, AsUrisque 173-174 (1989), 313-330.

They are smooth varieties, and the dosure of one is a disjoint union of decomposition dasses. In these lectures we study decomposition dasses. In the first lecture we restrict ourselves to n x n-matrices. The notion of decomposition dass was introduced by Borho-Kraft [8) in 1979, stimulated by some conjectures made by Dixmier in representation theory. But many of the basic properties in the special case of matrices can be derived by simple linear algebra. Our aim is to apply dassical linear algebra as developed by Frobenius, Jordan, Kronecker and Weierstrass, among others.

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