Student seminar exercises

Some exercises from Yuhan
Determine the dimension of $H_k$ , the space of spherical harmonics of degree k on $S^{n-1}$ .
Write down some examples of spherical harmonics.
Explain the decomposition $P_k=\hat{H}_k\oplus$ $r^2\hat{H}_{k-2} \oplus$ $r^4\hat{H}_{k-4}\oplus$ $\cdots$ , or equivalently, $\mathbb{C}[x_1,\ldots,x_n]=\oplus_{k\geq 0}E_k\otimes H_k$ .
Explain why $E_k$ is an irreducible representation of $\mathfrak{sl}_2(\mathbb{R})$ .

2022 S2

Describe the Bruhat decomposition for $GL_n\mathbb{C}$ explicitly for small $n$ .
Explain why almost all $g\in GL_n\mathbb{C}$ have a unique factorisation $g=n^-b$ , where $n^-\in N^-$ and $b\in B$ . Here $n^-\in N^-$ is the subgroup of lower-triangular matrices with 1’s on the diagonal, and $B$ is the subgroup of upper-triangular matrices.
List a few examples of locally convex, complete, topological vector spaces.
Explain the completed direct sum in Theorem 6.4.
Explain why the Heisenberg group $N/Z$ is not a matrix group.
Explain why the generic fiber of the map $T\times G/T\to G$ consists of $|W|$ many points in the case of $G=U(n)$ and in general.
Compute the Jacobian of the map $T\times G/T\to G$ in the case of $G=U(n)$ and in general.

2022 S1

What are the groups $E_2,E_3$ ?
Compute the dimension of $SO(n,\mathbb{R})$ .
What are the Lie algebras of the matrix groups $GL(n,\mathbb{R}),SL(n,\mathbb{R}), SO(n,\mathbb{R})$ ?
What is $SO(3,\mathbb{R})$ as a manifold?
Verify that the map $T_g$ , $g=\cos\theta+u\sin\theta$ indeed is rotation about the axis $u$ through the angle $2\theta$ .
Show that the Lie algebras of $SU_2$ and $SO_3$ are isomorphic (as Lie algebras).
Construct the double coverings $SU_2\times SU_2\to SO_4$ , $SL_2\mathbb{C}\to SO_{1,3}^+,\ SL_2\mathbb{R}\to SO_{1,2}^+$ and $SO_4\to SO_3\times SO_3$ .
Consider the action of $SO(4)$ on $\wedge^2\mathbb{R}^4$ . Find a decomposition of $\wedge^2\mathbb{R}^4$ into the direct sum of two stable subspaces.
Explain the picture of $SL_2\mathbb{R}$ . What is the simply connected covering group of $SL_2\mathbb{R}$ ?
Show that the space of complex structures on $\mathbb{R}^{2n}$ compatible with the inner product is isomorphic to the isotropic Grassmannian of $\mathbb{C}^{2n}$ .
For the sphere $S^{n-1}$ , find an isometry $f_x$ which reverses geodesics through $x$ . Let $G=O(n)$ be the isometry group of $S^{n-1}$ . Consider the automorphism $\alpha:G\to G,\,g\mapsto f_{x_0}\circ g\circ f_{x_0}$ , where $x_0$ is a fixed point on the sphere. What is the fixed point subgroup $G^\alpha$ ? What is the stabiliser group $H$ of $x_0$ in $G$ ? What is the relation between $G^\alpha$ and $H$ ?
Show that the positive-definite symmetric matrices form an open subset of the vector space of symmetric matrices.
Prove that an open convex subset in $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$ .
Describe the polar decomposition of $GL(n,\mathbb{C})$ . That is, every invertible complex $n\times n$ matrix has a unique factorisation into a positive-definite Hermitian matrix and a unitary matrix (or the other way around).
Describe the analogous decomposition in Theorem 4.2 for $GL(n,\mathbb{R})$ .
Describe the cell decomposition of $GL_n\mathbb{C}/B$ explicitly for small $n$ .
Find a maximal torus for the Lie groups $SO_{n}\mathbb{R}, U_n,SU_n,Sp_{2n}\mathbb{R}$ .
Define a smooth structure on $O_3$ , the projective space $\mathbb{P}_{\mathbb{R}}^n$ , the Grassmannian $Gr_k(\mathbb{R}^n)$ .
Find the tangent space at the identity (Lie algebras) of the Lie groups $SO_{n}\mathbb{R}, U_n,SU_n,Sp_{2n}\mathbb{R}$ .