On the convergence of the trace of a power of the Laplace-Beltrami operator with a potential on the sphereS n

Abstract

Example 3. Let the one-dimensional torus act on the surface x 2 = yz 3 according to the formula (x, y, z) --+ (tx, t2y, z). Then the fibers of the morphism of passage to the quotient are affine lines that are normal, and the quotient is normal as well, but the surface itself is not normal.Finally, let us list the linear groups from two classes indicated below that satisfy the conditions of the theorem.The irreducible representations of semisimple groups: SO,~, S2SL=, A2SL2k, Spin7, Spin9, G2, E6, SL, | SLn , SL2 | Sp, , SL4 | Sp4.Reducible representations of simple groups: SLn @ SL', SL, ~9 A2SL,~ , SL'2k ~9 A2SL2k, where ' stands for the dual representation.One can also show that a reducible representation of a simple group cannot be spherical. The list of irreducible actions can be obtained from Theorem 3 of [4], and one can find the desired reducible representations of simple groups by the Elashvili tables [5].The author thanks E. B. Vinberg and D. I. Panyushev for their attentio~ to the research and their numerous helpful remarks.