Projective representations and spin characters of complex reflection groups $G(m,p,n)$ and $G(m,p,\mathrm{\infty })$, III

This paper is a continuation of two previous papers in MSJ Memoirs, Vol.,29 (Math. Soc. Japan, 2013) with the same title and numbered as I and II. Based on the hereditary property given there, from mother groups $G(m,1,n)$, the generalized symmetric groups, to child groups $G(m,p,n)$, the complex reflection groups, we study in detail classification and construction of irreducible projective representations (= spin representations) and their characters of $G(m,1,n)$ for $n$ finite. Then, taking limits as $n$ tends to infinity, we obtain spin characters of the inductive limit groups $G(m,1,\mathrm{\infty })$. By the heredity studied further, this gives the main kernel of the results for $G(m,p,\mathrm{\infty })$ with $p\mathrm{\mid }m,p>1$.