Some remarks on multiplicity free spaces

A finite dimensional representation V of a connected reductive group G is a multiplicity free space if its symmetric algebra S^*(V) is a multiplicity free G-module. First, we establish a simple criterion for a module to be a multiplicity free space. Then we determine all invariant differential operators on a multiplicity free space in terms of a finite reflection group, the little Weyl group of V. Actually, this is a special case of a much more general result. The point is that for the important case of multiplicity free spaces the proof becomes significantly simpler.

Subsequently we study Capelli operators which form a canonical linear basis of the ring of invariant differential operators. We prove the characterization theorem which allows to compute the spectrum of a Capelli operator. At the end, we reproduce the classification of multiplicity free spaces due to Kac, Brion, Benson-Ratcliff, and Leahy.

Historical remark: Every hermitian symmetric space gives rise to a multiplicity free space(namely K acting on p⁺). For these spaces, the characterization theorem has been conjectured by Kostant (guided by his similar characterization of Schubert polynomials). Subsequently, this conjecture was proved by Sahi (in the Kostant volume PM123). Upon learning of the little Weyl group (first constructed by Brion), Sahi suggested to me that the characterization theorem should be valid for any multiplicity free space. The present paper contains a proof thereof.

Appeared in: Proc. NATO Adv. Study Inst. on Representation Theory and Algebraic Geometry (A. Broer, A. Daigneault, G. Sabidussi, eds.), Nato ASI Series C, Vol. 514, Dortrecht: Kluwer (1998), 301-317

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Last updated: August 18, 2005