Odds and Ends
نویسنده
چکیده
We represent G2(n) ∼= SO(n)/O(2) × SO(n − 2). Then we may regard the dual object G2(n)̂ ∼= {(m1,m2);m1,m2 ≥ 0 and even}, as the highest weights (mi) of irreducible unitary representations of SO(n) which occur in G2(n). They satisfy mi = 0 for i ≥ 2 and m1,m2 are even. Given K on G2(n), we may then form the Fourier series of K. General theorems [8, 44.47] assure us of pointwise convergence except possibly when n = 3, and in that case it is easily checked directly. Rather than describing the most general results here, I give some when n = 3 and discuss the problems in this case. Now G2(3) ∼= P2 and the dual object consists of the nonnegative even integers. For each m in this dual, the representation space has dimension 2m + 1 and a basis may be indexed by h, −m ≤ h ≤ m. All the Fourier transforms K̂(m) are matrices with only one row possibly nonzero, so I give elements K̂(m;h). For all m, K̂(m;h) = 0 unless −2 ≤ h ≤ 2. Denote by Qm an associated Legendre function of the second kind and by (n)k the Pochhammer symbol, (n)k = n(n+ 1) · · · (n+ k − 1).
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