Our main object here is to analyse the process of induction of (two-sided) ideals in enveloping algebras, in particular the behaviour of their associated varieties, by means of the new concept of relative enveloping algebras. For any algebraic group $H$, and any $H$-principal fibration $X\rightarrow Y$, we define the relative enveloping algebra as the sheaf of algebras $\mathcal {U}$ on $Y$ made up by all $H$-invariant differential operators on $X$. If the principal fibration is trivial, then $\mathcal {U}$ identifies with the enveloping algebra $U(\mathfrak {h})$ of the Lie algebra $\mathfrak {h}$ of $H$, tensored by the sheaf $D_Y$ of differential operators on the base $Y$. One of our main constructions is the following. For an ideal $J$ of $U(\mathfrak {h})$ we construct an sheaf of ideals $\widetilde J$ in $\mathcal {U}$, which is $D_Y\otimes J$ in a local trivialization. In a special case ($H$ a torus, $J$ maximal), the sheaf $\mathcal {U}/{\widetilde J}$ is known as twisted differential operators in the literature [BeBe]. For our present main purpose, we apply this new concept to the principal fibration $G\rightarrow G/H$ of an algebraic group $G$ by a closed sugbroup $H$. Let $\mathfrak {g}={\rm Lie}\, G$. The left $G$ action gives rise to an algebra homomorphism $U(\mathfrak {g})\rightarrow \Gamma (G/H,\mathcal {U})$, and so an ideal $J$ of $U(\mathfrak {h})$ gives rise to a homomorphism $U(\mathfrak {g})\rightarrow \Gamma (G/H,\mathcal {U}/{\widetilde J})$. Now the kernel of the homomorphism turns out to be the ideal $I$ induced from $J$ in $U(\mathfrak {g})$, up to a certain twist (Corollary of Proposition 2). Our main result (Theorem 2) gives then the precise relation between the associated varieties of $J$ resp. $I$ in $\mathfrak {h}$ resp. $\mathfrak {g}$, in the case $H$ parabolic. The behaviour of associated varieties of ideals is analogous to that of wave front sets of representations, as studied by Barbasch and Vogan [BV]. We define the characteristic variety of a coherent $\mathcal {U}$-module $\mathcal {M}$ as a subvariety $\mathrm {Ch}(\mathcal {M})$ in $(T^*X)/H$, and we prove that its image under the momentum map contains the associated variety of the $U(\mathfrak {g})$-module $\Gamma (G/H,\mathcal {M})$, assuming $H$ parabolic. Application to $\mathcal {M}=\mathcal {U}/{\widetilde J}$ then yields our main result. This generalizes some of the results in [BB I] and [BB III]. We also use this opportunity to report on recent work of V. Ginsburg and B. Kostant concerning shifted cotangent bundles and polarizations, and to relate it to our present context. Finally, some applications to Dixmier sheets and to a conjecture of Gelfand and Kirillov are made.