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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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