On Macdonald's formula for the volume of a compact Lie group

Abstract

Abstract. We give a simple proof of Macdonald's formula for the volume of a compact Lie group. Mathematics Subject Classification (1991). 22E15. Keywords. Compact Lie group, Todd class.It is well-known that a compact Lie group G of rank r is rational homotopy equivalent to a product of spheresMacdonald calculated the volume of G with respect to the Haar measure µ induced from a Lebesgue measure λ on the Lie algebra g of G. The result is as follows:where g Z is the Chevalley lattice and σ is the product of the superficial measure of the unit spheres S 2mi+1 in R 2mi+2 , that isIn this note we shall give a simpler proof of this formula. Let T be a maximal subgroup of G, t its Lie algebra and t Z a lattice in t such that the kernel of the exponential map from t to T is 2πt Z . We fix a positive definite inner product on g which is invariant under the adjoint action of G such that the induced volume form coincides with λ. This inner product induces Lebesgue measures on t and g/t (also denoted by λ), which determine a Haar measure on T and a G-invariant measure on the flag manifold G/T (also denoted by µ). Then clearly µ(T ) = (2π) r λ(t/t Z ), µ(G) = µ(T )µ(G/T ).

Mapping the geometry of the F4 group

“…In section 2 we have found three Spin(9) subgroups. As we said there, we choose H = Spin(9) 1 which we will call simply Spin (9). Then P is the 16 dimensional real vector space generated by the matrices c i , with i = 22, .…”

“…We will call it so(7), so that K = exp( so (7)). At this point let us note that in order to construct the Euler parametrization we proceed by induction: Together with B one needs to give the parametrization of the maximal subgroup H = Spin (9). Again, this can be done applying the generalized Euler parametrization with respect to the maximal subgroup Spin (8).…”

“…V is one dimensional and we can take c 30 as generator. r = 15 and K = Spin(6) generated by c α , α = 3, 5,6,8,9,10,12,13,14,15,17,18,19,20,21. Up to now we have proceeded in a systematical way.…”

Mapping the geometry of the E6 group

“…They have exactly the structure given in [8], with L i = e i , the canonical base of R 6 . The Macdonald formula [9], [11] gives for the volume of the compact form of E 6 V ol(E 6 ) = √ 3 · 2 17 · π 42 3 10 · 5 5 · 7 3 · 11 .…”

“…QQ [7,1] = e [7]; QQ [7,2] = e [6]; QQ [7,3] = −e [8]; QQ [7,4] = e [5]; QQ [7,5] = −e [4]; QQ [7,6] = −e [2]; QQ [7,7] = −e [1]; QQ [7,8] %%%% construction of the matrices MT = { {{mt [1], 0, 0, 0, 0, 0, 0, 0}, {mt [2], mt [3], mt [4], mt [5], mt [6], mt [7], mt [8], mt [9]}, {mt [10], mt [11] C The matrices.…”