on $X$ and on $\M$ via $\R$-automorphisms, then the induced $\G$-action on $\M^\vee$ will make the pairing $\langle\,,\,\rangle$ equivariant and so the Fourier transform $\calF$ is a $\G$-equivariant isomorphism. Since the dimension of the $\C$-vector space of $\G$-invariant functions agrees with the number of $\G$-orbits, we deduce the following lemma.

\[brauer\] Let $M$ be a finitely generated module over a finite Frobenius algebra $\R$. Let a finite group $\G$ act on $M$ by via $\R$-automorphisms, and also operate on a finite set $X$. Then we have $$\card((X\times\M)/\G)=\card((X\times \M^\vee)/\G),$$ i.e. the number of $\G$-orbits on $X\times \M$ agree with the number of $\G$-orbits on $X\times \M^\vee$.

When $\R$ is a finite field and $X$ is a point, this observation is attributed to Brauer [@kraft-riedtmann Remark 5.5]. Also the orbit structure could be very different on the two sides.

Fourier transform for group representations
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Moment maps associated to group scheme actions
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Let $\k$ be a field. We denote by $\Alg_\k$ and $\Gr$ the categories of $\k$-algebras and groups. Let $\G$ be an algebraic affine $\k$-group scheme. By notation abuse we will use the same letter $\G$ to denote the associated $\k$-group functor $\Alg_\k\rightarrow \Gr$, $A\mapsto \G(\R)={\rm Hom}_{\k-{\bf alg}}(\k[\G],\R)$.

Recall that, 