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We study general symmetric powers $S^k L^{[n]}$ of a tautological bundle$L^{[n]}$ on the Hilbert scheme $X^{[n]}$ of $n$ points over a smoothquasi-projective surface $X$, associated to a line bundle $L$ on $X$. Let $V_L$be the $\mathfrak{S}_n$-vector bundle on $X^n$ defined as the exterior directsum $L \boxplus \cdots \boxplus L$. We prove that the Bridgeland-King-Reidtransform $\mathbf{\Phi}(S^k L^{[n]})$ of symmetric powers $S^k L^{[n]}$ isquasi isomorphic to the last term of a finite decreasing filtration on thenatural vector bundle $S^k V_L$, defined by kernels of operators $D^l_L$, whichoperate locally as higher order restrictions to pairwise diagonals. We use thisdescription and the natural filtration on $(S^k V_L)^{\mathfrak{S}_n}$ inducedby the decomposition in direct sum, to obtain, for $n =2$ or $k \leq 4$, afinite decreasing filtration $\mathcal{W}^\bullet$ on the direct image$\mu_*(S^k L^{[n]})$ for the Hilbert-Chow morphism whose graded sheaves wecontrol completely. As a consequence of this structural result, we obtain achain of cohomological consequences, like a spectral sequence abutting to thecohomology of symmetric powers $S^k L^{[n]}$, an effective vanishing theoremfor the cohomology of symmetric powers $S^k L^{[n]} \otimes \mathcal{D}_A$twisted by the determinant, in presence of adequate positivity hypothesis on$L$ and $A$, as well as universal formulas for their Euler-Poincar\'echaracteristic.

Higher symmetric powers of tautological bundles on Hilbert schemes of points on a surface