Serre functors for quotient stacks

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\BGL{BGL}\newcommand{\ss}{\mathrm{ss}}$In the sequel, all schemes will be assumed to be over $\mathbb{C}$ for simplicity. For a smooth projective variety $X$, one has that the derived category $D^b(X)$ admits a Serre functor $S(-) := (-) \otimes \omega_X [\dim X]$ which generalises usual Serre duality. However, we now consider a quotient stack $\mathfrak{X} := [X/G]$ and study the derived category of $G$-equivariant coherent sheaves over $X$, $D^b_G(X) = D^b(\mathfrak{X})$. There are many examples of $X,G$ affine schemes where $\mathfrak{X}$ is already interesting. For instance, take $X = \operatorname{Hom}(\mathbb{C}^N,\mathbb{C}^d)$ and $G = \GL(d)$ acting by left multiplication; then by the choice of a linearisation ($g \mapsto \det(g)$), the associated semistable locus is precisely $\operatorname{Grass}(\mathbb{C}^N,d)$ of $d$-dimensional quotients of an $N$-dimensional space. Moreover, these semistable loci can be (as in the previous example) are often projective smooth varieties. Finally, we know by a theorem of Halpern-Leistner that we can see $D^b(\mathfrak{X}^{\ss}(\mathcal{L})) \subset D^b(\mathfrak{X})$ as a full subcategory via windows, where $\mathcal{L}$ is a choice of a linearisation of the $G$-action on $X$.

Question: When do $D^b(\mathfrak{X})$ admits a Serre functor for $X,G$ affine? In the cases in which $D^b(\mathfrak{X})$ exist, how is the Serre functor computed?

My guess is that somehow this Serre functor also has to be related to the Serre functor of the projective GIT quotient $\mathfrak{X}^{\ss}(\mathcal{L})$ whenever it is a smooth projective variety (as in the example above). I would be satisfied with an answer to the simpler question:

Question: Let $G$ be an algebraic group. Does (or when) $D^b(BG)$ admit a Serre functor? What is this Serre functor?

As a toy example which I am aiming to understand, consider the morphisms of stacks $p' : \BGL(1,d) \to B\mathbb{G}_m \times \BGL(d)$ and $q' : \BGL(1,d) \to \BGL(d+1)$. Here $\GL(1,d)$ denotes the parabolic subgroup of $\GL(d+1)$ consisting of block diagonal matrices of block diagonal length $2$ and dimensions of the blocks of the diagonal $1x1$ and $dxd$. Then one can consider the well-defined functor $ (q')_* \circ (p')^* : D^b(B\mathbb{G}_m \times \BGL(d)) \to D^b(\BGL(d+1)) $ essentially given by the Borel-Weil-Bott theorem. Using basic representation theory, one can conclude that $(p')_* \circ (q')^* : D^b(\BGL(d+1)) \to D^b(B\mathbb{G}_m \times \BGL(d))$ is well-defined and given by computing Lie algebra cohomology of the Lie algebra $\mathfrak{u}$ of the unipotent radical $U \subset \GL(1,d)$. In a similar way, the left adjoint to $(p')_* (q')^*$ exists and is given by computing Lie algebra homology of $\mathfrak{u}$. In both cases, of course, one has to track down the action of the Levi subgroup on $\mathfrak{u}$. Then these two functors are essentially given by tensoring with an Eilenberg-Chevalley complex (or its dual), and hence must be related by a shift (and potentially some modification on the action coming from the Levi group), as some kind of Poincaré duality (since $\mathfrak{u} \cong \mathbb{C}^d$ is an abelian simply-connected complex Lie group). My guess is that you can relate both functors using Serre functors of the corresponding classifying spaces involved.