Given a D-module $\mathcal{M}$ on $\mathbb{C}^n$, its Fourier transform $\widehat{\mathcal{M}}$ is equal to $\mathcal{M}$ as a set, but its module structure over $\mathbb{C}[x_1,...,x_n,\partial_1,...,\partial_n]$ is given by $$x_i \cdot m \ = \ -\partial_i m \ \ \ \text{ and } \ \ \ \partial_i \cdot m \ = \ x_im$$ This definition comes from how the classical Fourier transform acts on ODEs, see *.
But now consider a compact Lie group $G$. For each $\rho \in \widehat{G}$ there is a ``Fourier transform'' $$\text{functions on }G \ \longrightarrow \ \mathbb{C}$$ $$f \ \longrightarrow \ \int f(g)\rho(g) dg$$
Is there a ``Fourier transform'' of D-modules on $G$ (or maybe some flag variety of $G$) associated to each $\rho \in \widehat{G}$?
The reason I suspect this might be true is that, though one might initially protest that this is probably an affine phenonemon, Lie groups (or rather their flags) are D-affine.
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*On $\mathbb{C}$, the Fourier transform take an ODEs with polynomial coefficients $$a_n(x)\partial^nf+\cdots+a_0(x)f \ = \ 0$$ to another such ODE $$a_n(-\partial)k^n\widehat{f}+\cdots+a_0(-\partial)\widehat{f}\ = \ 0$$ because the Fourier transform of $x^m$ is $\delta^{(m)}(k)$. The same is true for $\mathbb{C}^n$.
