Differential equations and the algebra of confluent spherical functions on semisimple Lie groups (Q2825761)

The author studies confluent spherical functions on a connected semisimple Lie group \(G\), with finite center and of real rank \(1\) and the relationship of its algebra with the well-known Schwartz algebra of spherical functions on \(G\). It is well known that the differential system satisfied by the Harish-Chandra (zonal) spherical functions can be replaced with \(\delta^\prime (\omega). \varphi_\lambda = \gamma (\omega)(\lambda).\varphi_\lambda\), where \(\omega\) is the Casimir operator of \(G\) and \(\delta^\prime (\omega)\) denotes the radial component of the differential operator associated with \(\omega\). In the case of a real rank 1 semisimple Lie group \(G\), every spherical function is expressible in terms of a hypergeometric function NEWLINE\[NEWLINEF_1(a,c: z)NEWLINE\]NEWLINE called a confluent spherical function and denoted by \(\varphi_\lambda^\sigma(\exp t H_0)\), where \(z=-(\sinh t)^2\) and \(H_0\) is a certain element in a maximal abelian subspace of the Lie algebra of \(G\). The author studies the properties of the function \(\varphi_\lambda(\exp t H_0)\) for small values of \(t\). He finds the Stanton-Thomas expansion of \(\varphi_\lambda(\exp t H_0)\) which is appropriate to define the general notion of a confluent spherical function.