Exercise 8, § 2

Let G be a connected compact Lie group, $dg$ the normalized Haar measure on G.

a) Let $f: G\rightarrow G$ be a morphism of manifolds; for $g\in G$, the differential $T_g(f)$ can be identified by left translations with a linear map from L(G) to L(G). Prove the formulas

deg $f.\int_G\varphi (g)dg=\int_G\varphi (f(g$))det $T_g(f)dg$

for every integrable (complex-valued) function $\varphi$ on G, and

deg $f=\int_G$ det $T_g(f)dg$.

b) Let $\psi_k: G\rightarrow G$ be the map $g \rightarrow g^k$. Prove the formula

deg $\psi_k=\int_G$ det(1 + Ad $g+\cdots +$ (Ad $g$)$^{k-1}$)$dg$.

c) Let T be a maximal torus of G, and $T_r$ the set of regular elements of T. Show that exp$^{-1}_G(T_r)\subset L(T)$ and $\psi_k^{-1}(T_r)\subset T_r$ for all $k\geq 1$.

d) Show that deg $\psi_k=k^{dim(T)}$; deduce the equality $\int_G$ det(1 + Ad $g+\cdots +$ (Ad $g$)$^{k-1}$)$dg=k^{dim(T)}$.