Exercise 10, § 2

In this exercise, we use the following result from algebraic topology (Lefschetz’s formula$^{11}$)$:$ let X be a finite dimensional compact manifold, and $f$ a morphism from X to itself. Assume that the set F of points $x$ of X such that $f(x) =x$ is finite, and that for all $x\in F$ the number $\delta (x) =$ det(1 $-T_x(f)$) is non-zero. If $H^i(f)$ denotes the endomorphism of the vector space $H^i(X,\mathbf{R})$ induced by $f$ (for $i\geq 0$), then

$\sum_{x\in F}\delta (x)/|\delta (x)|=\sum_{i\geq 0}(-1)^i$Tr $H^i(f)$.

Let G be a connected compact Lie group, T a maximal torus of G. For $g\in G$, denote by $\tau (g)$ the automorphism of the manifold $G/T$ induced by left multiplication by $g$.

a) Let $t$ be an element of T such that the subgroup generated by $t$ is dense in T. Show that the fixed points of $\tau (t)$ are the classes $nT$ for $n\in N_G(T)$. Deduce that $(N_G(T) : T) =\sum_i(-1)^i$ dim$_{\mathbf{R}}H^i(G/T,\mathbf{R})$.

b) Let $g$ be an arbitrary element of G; show that the set of fixed points of $\tau (g)$ is non-empty, and deduce another proof of Th. 2.\*