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\title{Representation theory \\ Assignment 3 \\ Due Friday March 18}



\begin{document}
\maketitle
\begin{enumerate}
\item
Consider the flag variety $G/B $ of the group $ G = SO_{2n}(\mathbb{C})$.  
\begin{enumerate}
\item Define an inclusion from $ G/B $ into the set of orthogonal flags in $ \mathbb{C}^{2n} $ and identify the image. 
\item Recall that the compact form is $ K = SO_{2n}(\mathbb{R}) $.  Give a linear algebra description of $ K/T $ and find a bijection between this set and the set of orthogonal flags described above.  (Note that the maximal torus of $ K $ consists of $ 2\times 2 $ blocks of rotation matrices placed down the diagonal.)
\end{enumerate}
\item
Take $ K = U(n) $ and let $ \lambda = (\lambda_1, \dots, \lambda_n) $ be a non-generic point in $ \mathfrak{t}^* $ (so some of the $\lambda_i $ are allowed to be equal). 
\begin{enumerate}
\item Describe $ \mathcal{H}_\lambda $ in terms of orthogonal decompositions of $ \mathbb{C}^n $ and also in terms of partial flags.
\item Find a subgroup $P \subset GL_n(\mathbb{C}) $ (depending on $ \lambda$) such that $ G/P \cong \mathcal{H}_\lambda $.
\item Specialize to the case where $ \lambda = (1,\dots, 1, 0, \dots, 0) $ (there are $ k$ 1s and $ n-k$ 0s).  Describe the $ B $ orbits on $ G/P $.  In particular find their dimensions.
\end{enumerate}

\item Let $ V $ be a vector space.  Consider the line bundle $ \mathcal{O}(k) := \mathcal{O}(1)^{\otimes k} $ on $ \mathbb{P}(V) $.  Show that there is an injective map 
$$
Sym^k (V^*) \rightarrow \Gamma(\mathbb{P}(V), \mathcal{O}(k))
$$
and show that this map is an isomorphism when $ \dim V = 2 $ (i.e. for $\mathbb{P}^1$) by using the usual open cover of $ \mathbb{P}^1 $ (actually it's always an isomorphism).

\item Take $ G = GL_n $ and take $ \lambda = (k,0,\dots, 0) $.  Describe the line bundle $L(\lambda) $ on $G/B $ and the resulting map $G/B \rightarrow \mathbb{P}^N$.  Describe $\Gamma(G/B, L(\lambda))^* $.  Do the same thing when $ \lambda = (1, \dots, 1, 0, \dots, 0) $.

\item
Let $ \mathbb{G}_a $ denote the complex numbers, viewed as a group under addition (the additive group).  A unipotent group is a an algebraic group $ G $ such that either $ G \cong \mathbb{G}_a $ or $ G $ contains a central subgroup $Z$, with $Z \cong \mathbb{G}_a $, and such that $ G/Z $ is also unipotent (it is a recursive definition).
\begin{enumerate}
\item Show that the group of uni-uppertriangular matrices $N $ is a unipotent group.  
\item Show that the unipotent subgroup $ N $ of any reductive group is a unipotent group.  (Hint: define subgroups of $ N $ using subalgebras of $ \mathfrak{n} $.)
\item Show that every irreducible representation of a unipotent group $ G $ is trivial.  (Hint: first show that $Z$ acts by a scalar by Schur's lemma and hence trivially and then continue.)
\item Show that if $ G $ is unipotent, then its Lie algebra is nilpotent.  Is the converse true?
\item (optional) Usually, unipotent groups are defined as groups $ G $ which admit a filtration $ G_0 = \{1\} \subset G_1 \subset \cdots \subset G_n = G $, where each $ G_i $ is normal in $ G $ and $ G_{i+1}/G_i $ is isomorphic to $ \mathbb{G}_a $ for all $ i$.  Prove or disprove that this matches my definition above.
\end{enumerate}
\end{enumerate}
\end{document}
