Geodesic
Invariant Polynomials for Multiplicity Free Actions
Journal of Lie theory, Tome 19 (2009) no. 4, pp. 771-795.

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\def\C{{\Bbb C}} \def\R{{\Bbb R}} \def\HH{{\Bbb H}} This work concerns linear multiplicity free actions of the complex groups $G_\C=GL(n,\C)$, $GL(n,\C)\times GL(n,\C)$ and $GL(2n,\C)$ on the vector spaces $V=Sym(n,\C)$, $M_n(\C)$ and $Skew(2n,\C)$. We relate the canonical invariants in $\C[V \oplus V^*]$ to spherical functions for Riemannian symmetric pairs $(G,K)$ where $G=GL(n,\R)$, $GL(n,\C)$ or $GL(n,\HH)$ respectively. These in turn can be expressed using three families of classical zonal polynomials. We use this fact to derive a combinatorial algorithm for the generalized binomial coefficients in each case. Many of these results were obtained previously by Knop and Sahi using different methods.
Classification : 20G05, 13A50, 05E15
Mots-clés : Multiplicity free actions, invariant theory, symmetric functions 
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     author = {C. Benson and R. M. Howe and G. Ratcliff },
     title = {Invariant {Polynomials} for {Multiplicity} {Free} {Actions}},
     journal = {Journal of Lie theory},
     pages = {771--795},
     publisher = {mathdoc},
     volume = {19},
     number = {4},
     year = {2009},
     url = {http://geodesic.mathdoc.fr/item/JLT_2009_19_4_JLT_2009_19_4_a8/}
}
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C. Benson; R. M. Howe; G. Ratcliff . Invariant Polynomials for Multiplicity Free Actions. Journal of Lie theory, Tome 19 (2009) no. 4, pp. 771-795. http://geodesic.mathdoc.fr/item/JLT_2009_19_4_JLT_2009_19_4_a8/