Wiener Tauberian Theorems for Sl 2 ( R )

In this article we prove a Wiener Tauberian theorem for L(SL2(R)), 1 ≤ p < 2. Let G be the group SL2(R) and K its maximal compact subgroup SO(2,R). Let M be {±I}. We show that if the Fourier transforms of a set of functions in L(G) do not vanish simultaneously on any irreducible Lp− -tempered representation for some > 0, where they are assumed to be defined, and if for each M-type at least one of the matrix coefficients of any of those Fourier transforms does not ‘decay too rapidly at ∞’ in a certain sense, then this set of functions generate L(G) as a L(G)-bimodule. As a key step towards this main theorem we prove a W-T Theorem for L-sections of certain line bundles over G/K. W-T theorems on SL2(R) have been proved so far, for biinvariant L functions and for L functions on the symmetric space SL2(R)/SO(2,R), where the generator is left K-finite. Our results are on the space of all L functions (resp. sections), p ∈ [1, 2) of SL2(R) (resp. of line bundles over SL2(R)/SO(2,R)), without any restriction of K-finiteness on the generators.