Two contributions to the representation theory of algebraic groups

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.