Two Contributions to the Representation Theory of Algebraic Groups

Abstract

Let V be a �nite dimensional complex vector space. A subset X in V has the separation property if the following holds: For any pair l, m of linearly independent linear functions on V there is a point x in X such that l(x) = 0 and m(x) 6= 0. We study the the case where V = C[x; y]n is an irreducible representation of SL2. The subsets we are interested in are the closures of SL2{orbits Of of forms in C[x; y]n. We give an explicit description of those orbits that have the separation property: The closure of Of has the separation property if and only if the form f contains a linear factor of multiplicity one. In the second part of this thesis we study tensor products V� V� of irreducible G{representations (where G is a reductive complex algebraic group). In general, such a tensor product is not irreducible anymore. It is a fundamental question how the irreducible components are embedded in the tensor product. A special component of the tensor product is the so-called Cartan component V�+� which is the component with the maximal highest weight. It appears exactly once in the decomposition. Another interesting subset of V� V� is the set of decomposable tensors. The following question arises in this context: Is the set of decomposable tensors in the Cartan component of such a tensor product given as the closure of the G{orbit of a highest weight vector? If this is the case we say that the Cartan component is small. We show that in general, Cartan components are small. We present what happens for G = SL2 and G = SL3 and discuss the representations of the special linear group in detail.