Peter-Weyl Theorem

Let $G$ be a compact Lie group. Then:

1. Every continuous irreducible representation is finite-dimensional
2. Every continuous finite-dimensional representation is completely reducible
3. The matrix coefficients of irreducible representations form an orthonormal basis of $L^2(G)$ with respect to Haar measure
4. As a $G \times G$-representation, $L^2(G) \cong \bigoplus_{\lambda} V_\lambda \otimes V_\lambda^*$ where the sum is over all irreducible representations

Weyl's Theorem on Complete Reducibility

Every continuous finite-dimensional representation $\rho: G \to GL(V)$ of a compact Lie group decomposes as: $V \cong \bigoplus_{\lambda} V_\lambda^{\oplus m_\lambda}$ where $V_\lambda$ are irreducible representations and $m_\lambda \geq 0$ are multiplicities.

Classification of Compact Connected Lie Groups

Every compact connected Lie group $G$ has a finite central cover: $\tilde{G} = T^m \times G_1 \times \cdots \times G_k$ where $T^m$ is a torus and $G_i$ are compact simple Lie groups. The compact simple groups are classified by:

• $SU(n+1)$ (type $A_n$)
• $SO(2n+1)$ (type $B_n$) and $Sp(n)$ (type $C_n$)
• $SO(2n)$ (type $D_n$)
• Exceptional groups $G_2, F_4, E_6, E_7, E_8$

Maximal Torus and Weyl Group

For compact connected $G$:

1. All maximal tori are conjugate with dimension equal to the rank
2. The Weyl group $W = N_G(T)/T$ is a finite group acting on the torus
3. Conjugacy classes in $G$ correspond bijectively to $W$-orbits in $T$
4. Characters (traces of representations) are determined by their values on $T$

Weyl Character Formula

The character $\chi_\lambda$ of irreducible representation with highest weight $\lambda$ is: $\chi_\lambda(t) = \frac{\sum_{w \in W} \epsilon(w) e^{w(\lambda + \rho)(t)}}{\sum_{w \in W} \epsilon(w) e^{w\rho(t)}}$ for $t \in T$, where $\rho$ is half the sum of positive roots and $\epsilon(w) = \pm 1$ is the sign of $w$.