Exercise 6, § 6

Put $b_i(G) =$ dim$_{\mathbf{R}}H^i(G)$ for $i\geq 0$ and, if X denotes an indeterminate, $P_G(X) =\sum_{i\geq 0}b_i(G)X^i$.

a) Prove that $b_i(G) =\int_G$ Tr $\wedge \wedge^i$(Ad $g$)$dg$ and $P_G(X) =\int_G$ det(1 $+ X$.Ad $g$)$dg$ (cf. Appendix II, Lemma 1).

b) Deduce the equalities $\sum_ib_i(G) = 2^r$, and $\sum_i(-1)^ib_i(G) = 0$ if dim $G>0$ (use H. Weyl’s formula, or Exerc. 8 of §2).

c) Take $G =\mathbf{U}(n,\mathbf{C})$. Show that $P_G(X)$ is the coefficient of $(X_1. . .X_n)^{2n-2}$ in the polynomial $\frac{1}{n!}(1 + X)^n\prod_{_{1\leq i,j\leq n}}$(XX$_i+ X_j$)$(X_i-X_j)$ (with coefficients in $\mathbf{Z}[X]$)$.i_{\not=j}$