WIENER-TAUBERIAN-THEOREMS FOR SL2(R)

In this article we prove a Wiener Tauberian theorem for L-p(SL2(R), 1 less than or equal to p < 2. Let G be the group SL2(R) and K its maxim al compact subgroup SO(2, R). Let M be {+/-I}. We show that if the Fou rier transforms of a set of functions in LP(G) do not vanish simultane ously on any irreducible Lp-epsilon-tempered representation for some e psilon > 0, where they are assumed to be defined, and if for each M-ty pe at least one of the matrix coefficients of any of those Fourier tra nsforms does not 'decay too rapidly at infinity' in a certain sense, t hen this set of functions generate L-p(G) as a L-1(G)-bimodule. As a k ey step towards this main theorem we prove a W-T Theorem for LP-sectio ns of certain line bundles over G/K. W-T theorems on SL2(R) have been proved so far, for biinvariant L-1 functions and for L-1 functions on the symmetric space SL2(R)/SO(2,R), where the generator is left K-fini te. Our results are on the space of all Lp functions (resp. sections), p is an element of [1,2) of SL2(R) (resp. of line bundles over SL2(R) /SO(2, R)), without any restriction of K-finiteness on the generators.