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\title{Representation theory \\ Assignment 2 \\ Due Friday Friday February 18}
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\begin{enumerate}
\item
Show that there is a isomorphism of Lie groups $ \mathbb{R}^\times \rightarrow \mathbb{R} \times \mathbb{Z}/2 $ (where $ \mathbb{R}^\times $ is a group under multiplication and $ \mathbb{R} $ is a group under addition). Use this fact to show that $ \mathbb{R}^\times $ has two different structure of a real algebraic group. Show that these two different structures are non-isomorphic by showing that their complexifications are different.
\item
\begin{enumerate}
\item Show that every algebraic representation of $ GL_n(\mathbb{R}) $ on (finite-dimensional) complex vector space is complete reducible (i.e. is the direct sum of irreducible subrepresentations).
\item Give an example of a non-algebraic representation of $ GL_n(\mathbb{R}) $ on a (finite-dimensional) complex vector space which is not complete reducible.
\end{enumerate}
\item Consider $ G = SO_{2n}(\mathbb{C})$. Find the roots and coroots of $ G $ as well the $ \psi_\alpha : SL_2(\mathbb{C}) \rightarrow SO_{2n}(\mathbb{C}) $.
Here is a suggestion to help you get started. Recall that $ SO_{2n}(\mathbb{C}) $ is the automorphisms of $ \mathbb{C}^{2n} $ which preserve a non-degenerate symmetric bilinear form $\langle, \rangle $. Choose a basis $$ v_1, \dots, v_n, v_{-n}, \dots, v_{-1} $$ for $ \mathbb{C}^{2n} $ such that
$$ \langle v_i, v_j \rangle = \begin{cases}
1, \text{ if } i = -j \\
0, \text{ otherwise}
\end{cases}
$$
Then the maximal torus is given by those elements of $ SO_{2n}(\mathbb{C}) $ which are diagonal with respect to this basis.
\item Consider the group $GO_{2n}(\mathbb{C}) $ which is called the orthogonal similitude group. It consists of those automorphisms of $ \mathbb{C}^{2n} $ which preserve the bilinear form up to a scalar. In other words for each $ g \in G $, there exists a scalar $ a \in \mathbb{C}^\times $ such that $ \langle g v, g w \rangle = a \langle v, w \rangle $ for all $ v, w \in \mathbb{C}^{2n} $. Find the root datum of $ GO_{2n}(\mathbb{C}) $ and compare with $ SO_{2n}(\mathbb{C}) $.
\item Show that $ \Lambda^2 \mathbb{C}^4 $ carries a natural (though not completely canonical) non-degenerate symmetric bilinear form. Use this fact to define a 2-to-1 cover $ SL_4(\mathbb{C}) \rightarrow SO_6(\mathbb{C}) $. What does this map look like on the level of root data?
\end{enumerate}
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