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Monodromy for the hypergeometric function \(_ nF_{n-1}\). (English) Zbl 0663.30044
Consider the \(n\)-dimensional complex vector space of analytic solutions of the differential equation \(D(\alpha,\beta)f(z)=0\), where \[ D(\alpha,\beta):=(\theta +\beta_ 1-1)\dots(\theta +\beta_ n-1)-z(\theta +\alpha_ 1)\dots(\theta +\alpha_ n),\quad \theta:=z(d/dz), \] and \(\alpha_ 1,\dots,\alpha_ n,\beta_ 1,\dots,\beta_ n\in\mathbb C\). This is the hypergeometric equation with regular singularities at \(0, 1, \infty\).
More specifically, let \(V(\alpha,\beta)\) be the space of solutions holomorphic around \(z=1/2\); this space has a basis of solutions expressed as generalized hypergeometric series \({}_ nF_{n-1}\) provided the numbers \(\beta_ 1,\dots,\beta_ n\) are distinct mod \(\mathbb Z\). The fundamental group G of the punctured complex plane \(\mathbb C\setminus \{0,1\}\) (base point at \(z=1/2)\) acts on \(V(\alpha,\beta)\), inducing a homomorphism of \(G\) into \(\text{GL}(n,\mathbb C)\) whose image is called the monodromy group. This paper gives a complete analysis of which subgroups of \(\text{GL}(n,\mathbb C)\) are thus obtained, as a function of the parameters \(\alpha_ 1,\dots,\beta_ 1,\dots \). The group \(G\) is generated by \(g_ 0\), \(g_ 1\) (positively oriented simple closed paths around \(0, 1\), respectively). Let \(g_{\infty}=(g_ 1b_ 0)^{- 1}\). The monodromy groups are called “hypergeometric groups” and are subgroups \(H\) of \(\text{GL}(n,\mathbb C)\) generated by \(h_ 0\), \(h_ 1\), \(h_{\infty}\) satisfying: \(h_{\infty}h_ 1h_ 0=I\), the eigenvalues of \(h_{\infty}\), \(h_ 0\) are \(e^{2\pi i\alpha_ j}\), \(e^{-2\pi i\beta_ j}\) \((1\leq j\leq n)\), respectively, and \(h_ 1\) is a pseudo-reflection (rank of \(h_ 1-I\) is 1). Under the assumption \(\alpha_ j- \beta_ k\not\in\mathbb Z\) (all \(j,k\)), \(H\) is an irreducible subgroup of \(\text{GL}(n,\mathbb C)\).
If all the \(\alpha_ j\), \(\beta_ k\) are real, then there is an Hermitian form invariant under \(H\) (an explicit formula for the signature is derived). The situation of \(H\) being finite is described in detail. There are tables of the parameter values corresponding to the classification of finite unitary reflection groups by G. C. Shephard and J. A. Todd [Can. J. Math. 6, 274–304 (1954; Zbl 0055.14305)].

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H15 Other geometric groups, including crystallographic groups
33C05 Classical hypergeometric functions, \({}_2F_1\)
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