Decoding the character table The following is character table of water molecule

I would like to know
(1)What does the last 2 column in the table indicate?
(2) How to deduce the functions in the last column?
 A: Looking at page 81 and 82 of Applications of Group Theory To the Physics of Solids. by A. Dresselhaus it seems that the second last column are the possible basis functions of that irreducible representation. The last column contains non-basis functions that transform under that irrep.
I think the $A_1$ is the trivial irrep, therefore $z$ is left invariant (basis function) and also the functions $x^2$, $y^2$, $z^2$ are invariant.
Then the $A_2$ is the one which takes care of the $z$-reflection parity. When we reflect through the $xy$-plane, both $x$ and $y$ will incur a sign change and the function $xy$ as well as the angular momentum $R_z$ (spherical harmonic) is invariant.
The $B_1$ and $B_2$ are also one dimensional irreps, they seem to encode the $y$ and $x$ parities. Therefore they leave functions $xy$ and $yz$ invariant, as well as their respective angular momentum. It is just not clear why $x$ is a basis function when $xz$ is conserved, whereas for the $A_2$, $z$ has not been listed as a basis vector.
I suppose you have to read a bit more in those group theory notes in order to find a more precise definition of those two columns.
