$M$-spherical $K$-modules of a rank one semisimple Lie group

L. CAGLIERO; J. TIRAO

MANUSCRIPTA MATHEMATICA

Springer - Verlag

Lugar: Berlin; Año: 2004 vol. 113 p. 107 - 124

0025-2611

Let $G_circ=K_circ A_circ N_circ$ be the Iwasawa decomposition of a connected non-compact real semisimple Lie group with finite centre and let $M_circ$ be the centralizer of $A_circ$ in $M_circ$. Let ${rak g}={rak k}oplus {rak p}$ be the complexification of the corresponding Cartan decomposition of $ ext{ Lie,}G_circ$. It was proved by B.~Kostant that, for any $M$-spherical $K$-module $V$, there exists a unique $d$ (the Kostant degree of $V$) such that $V$ can be realized as a submodule of the space of all $rak k$-harmonic polynomials of degree $d$ on $rak p$. par In the paper under review, the authors give an algorithm for obtaining a highest weight vector from any $M$-invariant vector in an irreducible $M$-spherical $K$-module. This algorithm allows to compute a sharp bound for the Kostant degree, $d(v)$, of any $M$-invariant vector $v$ in a locally finite $M$-spherical $K$-module $V$. This method computes $d(v)$ effectively if the real rank of $G_circ$ is equal to one.