## Asymptotic behaviour of the dimensions of covariants. (Comportement asymptotique des dimensions des covariants.)(French)Zbl 0769.14016

Let $$X$$ be an affine algebraic variety defined over an algebraically closed field $$k$$ of characteristic 0 and let $$A=\bigoplus A_ n$$ be the graded algebra of regular functions on $$X$$. Let $$G$$ be a reductive algebraic group over $$k$$ acting algebraically and effectively on $$X$$. The authors study the multiplicities, $$m_{\lambda,n}$$, of irreducible representations $$\lambda$$ of $$G$$ in $$A_ n$$. They prove several results including the following: Assume the generic orbit of $$G$$ in $$X$$ is closed and the stabilizer of a generic point in $$X$$ is trivial. Assume also that $$m_{0,n}\neq 0$$ for large $$n$$, where 0 denotes the trivial 1-dimensional representation of $$G$$. Then, for any irreducible representation $$\lambda$$ of $$G$$, $$m_{\lambda,n}/m_{0,n}\to\dim\lambda$$ as $$n\to\infty$$.

### MSC:

 14L30 Group actions on varieties or schemes (quotients) 14L24 Geometric invariant theory 20G05 Representation theory for linear algebraic groups
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### References:

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