Direct systems of spherical functions and representations (Q2856383)

For an increasing sequence of compact symmetric spaces \(M_n=G_n/K_n\), one considers, for each \(n\), a spherical representation \((\pi _n,V_n)\) of \(G_n\): \(\pi _n\) is an irreducible representation of \(G_n\) on the space \(V_n\) and the subspace \(V_n^{K_n}\) of \(K_n\)-invariant vectors in \(V_n\) is one dimensional. Let \(u_n\) be a highest weight vector in \(V_n\). One assumes that the representation \((\pi _n,V_n)\) is realized as the representation of \(G_n\) on the space generated by \(\pi _{n+1}(G_n)u_{n+1}\). The authors study the infinite dimensional homogeneous space NEWLINE\[NEWLINEM_{\infty } =G_{\infty }/K_{\infty }=\lim _{\rightarrow } G_n/K_n,NEWLINE\]NEWLINE and the representation NEWLINE\[NEWLINE(\pi _{\infty } ,V_{\infty })=\lim _{\rightarrow} (\pi _n,V_n).NEWLINE\]NEWLINE They prove that the representation \((\pi _{\infty },V_{\infty })\) is spherical if and only if the ranks of the symmetric spaces \(M_n=G_n/K_n\) are bounded, and this holds if and only if \(G_{\infty }/K_{\infty }\) is the Grassmann manifold of \(p\)-planes in \({\mathbb F}^{\infty }\) (\({\mathbb F}={\mathbb R,\mathbb C}\) or \(\mathbb H\)). The proof amounts to studying whether there is a Cauchy sequence \((e_n)\) such that \(e_n\) is a unit vector in \(V_n^{K_n}\). This condition can be expressed in terms of the \(c\)-functions of the dual symmetric spaces.