REFLECTION SUBGROUPS OF THE MONODROMY GROUPS OF APPELL'S F-4

摘要

Assume the system of differential equations E-4(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 M-4(a, 13, c, The monodromy matrix Gamma(3*) derived from a loop Gamma(3) surrounding once the irreducible component C = {(X, Y) vertical bar (X - Y)(2) - 2(X + Y) + 1 = 0} of the singular locus of E-4 is a complex reflection. The minimal normal subgroup N-C of M-4 containing Gamma 3(*) 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 E-2(1)(a, b, c). It is known that N-C is reducible if epsilon := c + c' a b-1 is not an element of Z or if epsilon is an element of Z and P(G) is a dihedral group. We prove that, if epsilon is an element of Z, then N-C is the (irreducible) Coxeter group W (D-4), W(F-4) and W (H-4) according as P(G) is the tetrahedral, octahedral and icosahedral group, respectively.