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                <title xml:lang="en">Parabolic restrictions and double deformations of weight multiplicities</title>
                <title xml:lang="fr">Restriction paraboliques et double-déformations des multiplicités des poids</title>
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              <p>We introduce some (p,q)-deformations of the weight multiplicities for the representations of any simple Lie algebra g over the complex numbers. This is done by associating the indeterminate q to the positive roots of a parabolic subsystem of g and the indeterminate p to the remaining positive roots. When p=q, we just recover the usual Lusztig analogues of weight multiplicities. We then study the positivity of the coefficients in these double deformations. In particular, the positivity holdswhen p=1 in which case the polynomials have a natural algebraic interpretation in terms of a parabolic Brylinski filtration. For the parabolic restriction from type C to type A, this positivity result was conjectured by Lee. We also establish this positivity, in any finite type and for any p, for a stabilized version of our double deformation.  In addition, we study the double deformation obtained by replacing the pair (p,q) by (p+1,q+1), show it has nonnegative coefficients and admits a combinatorial description in terms of crystals.</p>
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