A combinatorial identity for the Jacobian of $t$-shifted invariants

Let $\mathfrak g$ be a simple Lie algebra. There are classical formulas for the Jacobians of the generating invariants of the Weyl group of $\mathfrak g$ and of the images under the Harich-Chandra projection of the generators of $ZU(\mathfrak g)$. We present a modification of these formulas related to Takiff Lie algebras.

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