When I read the paper Article The volume of a compact Lie group
, I have come crossed some questions, I cannot find any details about it. I would like to ask for help from you.Ref: Page 56-61 from https://arxiv.org/abs/1509.00537
The following questions will be asked for pertinence to The volume of a compact Lie group. All notations are taken from this paper.
Q1: In the paper The volume of a compact Lie group, Mcdonald wrote that it is well-known that the manifold G (i.e., compact Lie group) has the same cohomology, apart from torsion, as a product of odd-dimensional spheres... How do I understand it? Why? Is there any references, books? Thanks.
Q2: Let = \sum_i x_iy_i, where x,y in Rn , the n-dimensional real vector space This extends to a scalar product on the algebra S(Rn) of polynomial functions on Rn , such that
=\alpha!\delta_{\alpha,\beta}
for any two multi-indices \alpha,\beta in Nn . How does the standard inner product on Rn extends to the scalar product over the algebra of polynomial functions on Rn ?
Q3: For P=\prod_{\alpha>0}\alpha, where \alpha is the positive root of G with respect to its maximal torus T. Mcdonald shown that
= |W|P(\rho),
where W is the Weyl group of G, and \rho = 1/2 \sum_{\alpha>0}\alpha, that is, the half the sum of all positive roots of G. Furthermore, he claimed that
= |W|P(\rho) = 2^{-N}\prod^n_{j=1}m_j!\prod_{\alpha>0}.
where m_j are the exponents of G, N is the number of all positive roots.
Remarks: I try to find the proof about it from many books about Lie groups, unfortunately I cannot success.
My questions is: how to prove the above last formula? Although I find it in the paper http://eudml.org/doc/81886, there is still ambiguity. Harder wrote in the Appendix of his paper, from Weyl's denominator formula:
\sum_{w\in W} sgn(w) e^{w\rho} = \prod_{\alpha>0}(e^{\alpha/2} - e^{-\alpha/2}), it follows that
\sum_{w\in W}sgn(w)(w\rho)^N = N! P. From this, =|W|P(\rho). In the following, I do not know how to get
\prod_{\alpha>0} \frac{2}{} = \prod^n_{j=1}(m_j-1)!. And why |W| = \prod^n_{j=1}m_j?
Please help me understand these 3 questions. Thanks a lot.
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