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On page 311 of Dale Husemöller's book Fibre Bundles in Theorem 11.6 he has the following commutative diagram $$\begin{array} & & K(BG)\\ &\nearrow &\downarrow\\ R(G)&\longrightarrow & H^*(BG,\mathbb Q) \end{array} $$ Here $G$ is a compact connected Lie group, $BG$ is its classifying space, and $R(G)$ the representation ring. The map from $R(G) \rightarrow K(BG)$ is the map which takes a representation $\rho: G \rightarrow GL_n(\mathbb C)$ to the associated vector bundle $EG\times_{\rho}\mathbb C^n$. The map from $K(BG)\rightarrow H^*(BG,\mathbb Q)$ is the usual Chern character and the map $R(G)\rightarrow H^*(BG,\mathbb Q)$ is a map he calls $ch^*$ which he says has the following property: given a $T$-module ($T\subset G$ the maximal torus) $M$ then $$ch^*M=e^{-\tau (M)}$$ where $\tau: H^1(T,\mathbb Z)\rightarrow H^2(BG, \mathbb Z)$ is the transgression map.

My question is: why the minus sign in the definition of $ch^*$?


Edit: (TB) here's a screen shot of the relevant page from Google Books

Husemöller, page 311

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Thank you for adding the screen shot. –  clh Sep 5 '11 at 17:30

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