On Bernstein polynomials for compact Lie groups
For a compact connected Lie group \(G\), let \(A(G)\) be the ring of functions on \(G\) generated by all matrix coefficients of all irreducible finite dimensional representations of \(G\). For \(f\in A(G)\), let \(A_f\) be the ring of rational functions whose denominators are nonnegative powers of \(f\). For an independent variable \(s\), we consider the ring of polynomials \(A_f[s]\). Let \(A_f[s]\cdot f^s\) be the \(A_f[s]\) free module generated by \(f^s\). We extend formally the action of the Lie algebra of \(G\) on \(f^s\) as \(X(f^s)={s\over f}X(f)\cdot f^s\). Then we extend this action to \(A_f[s]\cdot f^s\). Let \({\mathcal D}(G)\) be the ring of differential operators generated by \(A(G)\) and the Lie algebra of \(G\). The action of the Lie algebra of \(G\) can be extended to \({\mathcal D}(G)\). Allowing \(s\) to act multiplicatively, we have an action of \({\mathcal D}(G)[s]\). By the theory of \({\mathcal D}\)-modules, we have the functional equation \(D(g,s)(f(g))^{s+1}=b(s)(f(g))^s\), where \(D(g,s)\) belongs to the ring of differential operators \({\mathcal D}(G)[s]\). The polynomial \(b(s)\) is not unique. There exists a monic polynomial of minimal degree that satisfies the above equation. This polynomial is called the Bernstein polynomial attached to \(f\). The explicit calculation of Bernstein polynomials has been the goal of many papers. The author gives the Bernstein polynomials for basic matrix entries of irreducible unitary representations of the compact Lie group \(G=SU(2)\). Their application to the analytic continuation of certain distributions on \(SU(2)\) is given. The Bernstein polynomials for \(B_-\times B_-\)-semi-invariant functions on a semisimple complex Lie group are briefly described.
matrix entries of irreducible representations
scientific article; zbMATH DE number 1247577
scientific article