Some Homological Properties in Geometric Invariant Theory

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A representation (V,G) of a reductive algebraic group G over the complex number field C is said to be coregular, if V//G is non-singular. For a semisimple G, irreducible coregular representations are determined by P.Littelemann, and for a simple G, all coregular representations are classified by V.L. Popov, G.W. Schwarz, O.M. Adomovich and E.O. Golovina. In this research, we have determined coregular representations of non-semisimple reductive groups G with simple semisimple parts having enouch closed orbits. This is based on the decomposition of actions of algebraic tori on normal varieties into no-blowing-up actions of codimension one and blowing-up actions of codimension 2. The Chow groups preserve under quotient morphisms in the latter actions. Moreover, in the relaion with this, we have studied relative equidimensionalities and relative stabilities of actions of non-semisimple reductive groups and obtain some results which are useful in classifying coregular or equidimensional rep … More resentaions.We generalize a part of the classical ramification theory of finite Galois groups to one of quotient morphisms under affine group actions and give a criterion the result similar to in finite covering cases to hold in affine groups case, which is related to an extension of some results on semi-invariants of finite groups to in the case of centric diconnected tori.In order to study on invariant theory of classical groups over local rings, we give a nice criterion for a set of symplectic trasvections to be a genrating system of the sympectic group Sp(V) defined over local rings (due to Ishibashi).On representaion theory of finite groups: We determine essential ideals and primary decompositions of mod 2-cohomology ring of finite abelian 2-groups (due to Ogawa) and obtain partial results on extensions of some 2-groups which preserve the irresucibilities of induced characters (due to Sekiguchi). These results seem to be useful in studying functor properties in invariant theory of finite groups. Less