REFLECTION SUBGROUPS OF THE MONODROMY GROUPS OF APPELL’S F4

Abstract

Assume the system of differential equations E4(a, b, c, c′; X, Y) satisfied by Appell’s hypergeometric function F4(a, b, c, c′; X, Y) has a finite irreducible monodromy group M4(a, b, c, c′). The monodromy matrix Γ3∗ derived from a loop Γ3 surrounding once the irreducible component C ={(X, Y)|(X - Y)2 -2(X + Y) +1 =0} of the singular locus of E4 is a complex reflection. The minimal normal subgroup NC of M4 containing Γ3∗ is, by definition, a finite complex reflection group of rank four. Let P(G) be the projective monodromy group of the Gauss hypergeometric differential equation 2E1(a, b, c). It is known that NC is reducible if e :=c +c′-a -b -1 ∉Z or if e ∈Z and P(G) is a dihedral group. We prove that, if e ∈Z,then NC is the (irreducible) Coxeter group W(D4), W(F4)and W(H4) according as P(G) is the tetrahedral, octahedral and icosahedral group, respectively.