Bethe subalgebras in antidominantly shifted Yangians

The loop group $G((z^{-1}))$ of a simple complex Lie group $G$ has a natural Poisson structure. We introduce a natural family of Poisson commutative subalgebras $\overline{{\mathbf{B}}}(C) \subset \mathcal{O}(G((z^{-1}))$ depending on the parameter $C\in G$ called classical universal Bethe subalgebras. To every antidominant cocharacter $\mu$ of the maximal torus $T \subset G$ one can associate the closed Poisson subspace $\mathcal{W}_\mu$ of $G((z^{-1}))$ (the Poisson algebra $\mathcal{O}(\mathcal{W}_\mu)$ is the classical limit of so-called shifted Yangian $Y_\mu(\mathfrak{g})$). We consider the images of $\overline{{\mathbf{B}}}(C)$ in $\mathcal{O}(\mathcal{W}_\mu)$, that we denote by $\overline{B}_\mu(C)$, that should be considered as classical versions of (not yet defined in general) Bethe subalgebras in shifted Yangians. For regular $C$ centralizing $\mu$, we compute the Poincar\'e series of these subalgebras. For $\mathfrak{g}=\mathfrak{gl}_n$, we define the natural quantization ${\mathbf{Y}}^{\mathrm{rtt}}(\mathfrak{gl}_n)$ of $\mathcal{O}(\operatorname{Mat}_n((z^{-1}))))$ and universal Bethe subalgebras ${\mathbf{B}}(C) \subset {\mathbf{Y}}^{\mathrm{rtt}}(\mathfrak{gl}_n)$. Using the RTT realization of $Y_\mu(\mathfrak{gl}_n)$ (invented by Frassek, Pestun, and Tsymbaliuk), we obtain the natural surjections ${\mathbf{Y}}^{\mathrm{rtt}}(\mathfrak{gl}_n) \twoheadrightarrow Y_\mu(\mathfrak{gl}_n)$ which quantize the embedding $\mathcal{W}_\mu \subset \operatorname{Mat}_n((z^{-1}))$). Taking the images of ${\mathbf{B}}(C)$ in $Y_\mu(\mathfrak{gl}_n)$ we recover Bethe subalgebras $B_\mu(C) \subset Y_\mu(\mathfrak{gl}_n)$ proposed by Frassek, Pestun and Tsymbaliuk.