On the product formula on noncompact Grassmannians (Q2866753)

The functional equation for the spherical functions on a Riemannian symmetric space \(G/K\) can be written NEWLINE\[NEWLINE\phi _{\lambda }(e^X)\phi _{\lambda }(e^Y)=\int _{\mathfrak a}\phi _{\lambda } (e^H)d\mu _{X,Y}(H),NEWLINE\]NEWLINE for \(X,Y\in {\mathfrak a}\), a Cartan subspace of the Lie algebra of the Lie group \(G\), where \(\mu _{X,Y}\) is a probability measure on \(\mathfrak a\). The authors consider the problem to characterize the pairs \((X,Y)\) for which the measure \(\mu _{X,Y}\) is absolutely continuous with respect to the Lebesgue measure. One considers in this paper the case of noncompact Grassmanians \(SO_0(p,q)/SO(p)\times SO(q)\). To state their results the authors define the configuration of an element \(X\) in \(\mathfrak a\) as the set of multiplicities of the eigenvalues of a certain projection of \(X\), and say that the pair \((X,Y)\) is eligible if a certain condition on the configurations of \(X\) and \(Y\) is satisfied. It says that the singularities of \(X\) and \(Y\) are limited. Then the main result says: the measure \(\mu _{X,Y}\) is absolutely continuous with respect to the Lebesgue measure if and only if the pair \(X,Y\) is eligible. This result holds for the spaces \(SU(p,q)/SU(p)\times SU(q)\) and \(Sp(p,q)/Sp(p)\times Sp(q)\) as well.