We show that the Laplacian acting on differential forms on a sphere can be lifted to an operator on its rotation group which is intrinsically equivalent to the Laplacian acting on functions on the Lie group. Further, using the result and the Urakawa summation formula for the heat kernel of the latter Laplacian and the Weyl integration formula, we get a summation formula for the kernel of the former.
References
K. Abe and I. Yokota, Volumes of compact symmetric spaces, Tokyo J. Math. 20 (1997), 87--105. MR1451862 0884.53040 10.3836/tjm/1270042402 euclid.tjm/1270042402
K. Abe and I. Yokota, Volumes of compact symmetric spaces, Tokyo J. Math. 20 (1997), 87--105. MR1451862 0884.53040 10.3836/tjm/1270042402 euclid.tjm/1270042402
I. Chavel, Eigenvalues in Riemannian geometry, Pure Appl. Math. 115, Academic Press, Inc., Orlando, FL, 1984. MR768584 0551.53001I. Chavel, Eigenvalues in Riemannian geometry, Pure Appl. Math. 115, Academic Press, Inc., Orlando, FL, 1984. MR768584 0551.53001
A. Ikeda and Y. Taniguchi, Spectra and eigenforms of the Laplacian on $S^n$ and $P^n(C)$, Osaka J. Math. 15 (1978), 515--546. MR510492 0392.53033 euclid.ojm/1200771570
A. Ikeda and Y. Taniguchi, Spectra and eigenforms of the Laplacian on $S^n$ and $P^n(C)$, Osaka J. Math. 15 (1978), 515--546. MR510492 0392.53033 euclid.ojm/1200771570
E. Kaneda, The spectra of 1-forms on simply connected compact irreducible Riemannian symmetric spaces, J. Math. Kyoto Univ. 23 (1983), 369--395. MR706170 euclid.kjm/1250521565
E. Kaneda, The spectra of 1-forms on simply connected compact irreducible Riemannian symmetric spaces, J. Math. Kyoto Univ. 23 (1983), 369--395. MR706170 euclid.kjm/1250521565
E. Kaneda, The spectra of 1-forms on simply connected compact irreducible Riemannian symmetric spaces II, J. Math. Kyoto Univ. 24 (1984), 141--162. MR737830 0595.58045 euclid.kjm/1250521390
E. Kaneda, The spectra of 1-forms on simply connected compact irreducible Riemannian symmetric spaces II, J. Math. Kyoto Univ. 24 (1984), 141--162. MR737830 0595.58045 euclid.kjm/1250521390
M. Nagase, Twistor spaces and the general adiabatic expansions, J. Funct. Anal. 251 (2007), 680--737. MR2356426 1130.58017 10.1016/j.jfa.2007.06.012M. Nagase, Twistor spaces and the general adiabatic expansions, J. Funct. Anal. 251 (2007), 680--737. MR2356426 1130.58017 10.1016/j.jfa.2007.06.012
H. Urakawa, The heat equation on a compact Lie group, Osaka J. Math. 12 (1975), 285--297. MR404526 0335.22009 euclid.ojm/1200757856
H. Urakawa, The heat equation on a compact Lie group, Osaka J. Math. 12 (1975), 285--297. MR404526 0335.22009 euclid.ojm/1200757856
H. Urakawa, Analytic torsion of space forms of certain compact symmetric spaces, Nagoya Math. J. 67 (1977), 65--88. MR478243 0368.58012 euclid.nmj/1118796472
H. Urakawa, Analytic torsion of space forms of certain compact symmetric spaces, Nagoya Math. J. 67 (1977), 65--88. MR478243 0368.58012 euclid.nmj/1118796472