Generalized Thue-Morse words and palindromic richness

We prove that the generalized Thue-Morse word ${\mathbf{t}}_{b,m}$ defined for $b\ge 2$ and $m\ge 1$ as $$\mathbf{t}${b,m} = (s_b(n) \mod m){n=0}^{+\infty}$,where$s_b(n)$denotesthesumofdigitsinthebase-$b$representationoftheinteger$n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering simultaneously antimorphisms $\Theta \in D_m$,weshowthat$$\mathbf{t}$_{b,m}$issaturatedby$\Theta$-palindromes up to the highest possible level. Using the terminology generalizing the notion of palindromic richness for more antimorphisms recently introduced by the author and E. Pelantov'a, we show that ${\mathbf{t}}_{b,m}$ is $D_m$-rich. We also calculate the factor complexity of ${\mathbf{t}}_{b,m}$.