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On the Riemann-Hilbert problem for a compact Lie group. (English. Russian original) Zbl 0731.30033

Sov. Math., Dokl. 41, No. 1, 165-169 (1990); translation from Dokl. Akad. Nauk SSSR 310, No. 5, 1055-1058 (1990).
Let G be a compact connected Lie group of rank r and denote its complexification by \(G_{{\mathbb{C}}}\). A loop in G of a given smoothness class is a mapping f of that smoothness class from the unit circle (in \({\mathbb{C}})\) T to G; it is said to be marked in f(1) is the unit in G. The set of all loops in G forms a group LG under pointwise multiplication. The marked loops form a normal subgroup \(\Omega\) G of LG. The subgroup of LG whose elements are boundary values of holomorphic maps of the interior \(B_+\) of the unit disk (resp. interior \(B_-\) of the complement of \(B_+\) in P\({\mathbb{C}})\) into \(G_{{\mathbb{C}}}\) is denoted by \(L^+G_{{\mathbb{C}}}\) (resp. \(L^-G_{{\mathbb{C}}}).\)
Let \(\gamma\) be a representation of G in a complex vector space V. Then the homogeneous generalized Riemann-Hilbert problem \(R_ f\) with coefficient \(f\in LG\) is the problem of existence and number of pairs of maps \(X_+: B_+\to V\), \(X_-: B_-\to V\), with \(X_{-(\infty)}=0\), that have boundary value of the stated smoothness class and satisfy \(X_+(z)=\gamma (f(z))\cdot X_ -(z)\), \(z\in T\); if h is a loop in V, then the corresponding non-homogeneous g.R.H.P. \(R_{f,h}\) is given by \(X_+(z)=\gamma (f(x))\cdot X_ -(z)+h(z)\), \(z\in T\). Following the reviewer’s scheme [Math. Ann. 151, 365-423 (1963), correction ibid. 153, 350 (1964; Zbl 0115.087)] it can be shown that each Hölder loop f in a classical simple Lie group admits a factorization \(f=f_+\cdot H\cdot f_ -\), where \(f_{\pm}\in L^{\pm}G_{{\mathbb{C}}}\), and H is a homomorphism from T into some maximal torus \(T^ r\) of G; in fact, there is a r-tuple \(k(f)=(k_ 1(f),...,k_ r(f))\) of integers, called partial indices, such that \(H(\exp (2\pi it))=\exp_{T^ r}(k_ 1t,...,k_ rt).\) Calling the Birkhoff stratum \(M_ k\) of \(k\in {\mathbb{Z}}^ r\) the set of all Hölder loops f with \(k(f)=k\), the author shows for a classical simple Lie group that the Birkhoff strata are contractible Fredholm submanifolds of \(\Omega\) G. In case \(G=U(n)\) and \(j=\hat k^ p\), \(1\leq p\leq n\), is an exterior power of the standard representation, the index of \(R_ f\) is shown to be equal to \[ {n-1\choose p- 1}\sum^{n}_{i=1}k_ i(f). \] Moreover it is shown that the dimension of the base of versal deformation of the principal holomorphism \(G_{{\mathbb{C}}}\)-bundle over P\({\mathbb{C}}\) given by a loop f is equal to \(\sum \{k_ i(f)-k_ j(f)-1:\) \(k_ i(f)>k_ j(f)\}\) and that \(M_ k\) is a closed analytic submanifold of LG of codimension \(\sum \{k_ i(f)- k_ j(f)-1:\) \(k_ i(f)>k_ j(f)\}\).

MSC:

30E25 Boundary value problems in the complex plane
22E99 Lie groups

Citations:

Zbl 0115.087
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