Coxeter system of lines and planes are sets of injectivity for the twisted spherical means

It is well known that a line in R2R2 is not a set of injectivity for the spherical means for odd functions about that line. We prove that any line passing through the origin is a set of injectivity for the twisted spherical means (TSM) for functions f∈L2(C)f∈L2(C), whose each of spectral projection e14|z|2f×φk is a polynomial. Then, we prove that any Coxeter system of even number of lines is a set of injectivity for the TSM for Lp(C)Lp(C), 1⩽p⩽21⩽p⩽2. Further, we deduce that certain Coxeter system of even number of planes is a set of injectivity for the TSM for Lp(Cn)Lp(Cn), 1⩽p⩽21⩽p⩽2. We observe that a set SR2n−1×C is a set of injectivity for the TSM for a certain class of functions on Cn+1Cn+1.