I think I misunderstood your first comment, and also maybe now I'm not sure what you mean by "symmetric closure." By "D_1-equivalent" in the second paragraph I mean, "is connected by a chain of D_1-equivalences, $x \to x' \leftarrow \cdots \to y$," which I hope is the same as "$x$ and $y$ are isomorphic in a quotient category $C/D$." If that's enough to prove $D_1 \rtimes D_2$ is subcategory generated by $D_1$ and $D_2$, can you spoon feed me the explanation?

Hi Dragos. I think a $D$-equivalence $x \to y$ does not guarantee a $D$-equivalence $y \to x$. For example when $C$ is the derived category of abelian groups and $D$ is the subcategory of torsion groups, then $\mathbf{Z} \to \mathbf{Q}$ is a $D$-equivalence. I don't know if this disproves your conclusion, though.

Thanks Dmitri. I think I haven't understood your criterion for isomorphism of spectra yet, in the second paragraph. But before I dig in let me check something: in the third paragraph you take a supremum over a set of numbers $r$ that obey $r > q \geq t$, which implies $r > t$. Then in parentheses you argue that this set of numbers is nonempty because it contains $r = t$. Is that a problem?

I think I didn't put it together until now, that odd unimodular lattices make a genus, and that's the genus containing $\mathbf{Z}^n$.

Thanks for the O'Meara reference. The last footnote on the last page, referring to that Kneser work, is: "See M. Kneser (1957). For an example of the classical approach using 'reduction theory' we refer to BW Jones (1950)." I'd be interested to find out what O'M means, but I didn't find the Jones book yet.

Thanks Jeremy. How does magma do it? And is it the kind of thing that was known, or could have been, to Zolotareff?

Hi Pavel. I think I am exactly asking why the equivariance implies Gabber's theorem. It doesn't seem like you get it from just any kind of circle action. Is it a special case of a theorem about coherent sheaves on a more general class of derived stacks?

When $R = \mathbf{C}$, a line bundle on $M_G$ gives for each pair of commuting elements $(x,y)$ in $G$ a homomorphism $\rho:Z_G(x,y) \to \mathbf{C}^*$. If I represent $k$ as a $\mathbf{C}^*$-valued $3$-cocycle $k(g_1,g_2,g_3)$, is there an explicit formula for $\rho(g)$ in terms $k$? Perhaps $\rho(g) = k(x,y,g)$?

Is it the observation attributed to Kostant in the first two paragraphs here? mathoverflow.net/questions/14770/…