Timeline for Twisted D-module structure on pushfoward of structure sheaf of Bruhat cell
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 21, 2023 at 18:32 | comment | added | Qixian Zhao | I didn't check carefully your sl2 calculation, but you may want to look at section 4 of math.utah.edu/~milicic/Eprints/hmsw2.pdf | |
| Mar 21, 2023 at 18:25 | comment | added | Qixian Zhao | When we are defining $\iota_{w\star} \mathcal O_{X_w}$ as a $\mathcal D_X^\lambda$-module, we are secretly viewing $\mathcal O_{X_w}$ as a $\mathcal D_X^{\lambda,\iota_w}$-module and use this structure to define push foward. The resulting direct image has the same underlying $\mathcal O_X$-module structure (at least locally). It would probably be helpful to look at the definition of direct image functor for twisted modules. | |
| Mar 21, 2023 at 18:17 | comment | added | Qixian Zhao | For every morphism of smooth varieties $f: Y \to X$ and every twisted sheaf of differential operator (tdo) $\mathcal D$ on $X$, one can define a tdo $\mathcal D^f$ on $Y$, the "pullback" of $\mathcal D$. Then one defines pushforward of $\mathcal D^f$-modules as $\mathcal D$-modules. In the current setting $Y = X_w$, $f = \iota_w$, $\mathcal D = \mathcal D_X^\lambda$, and you can check relatively easily that $\mathcal D_X^{\lambda, \iota_w}$ is naturally isomorphic to the ordinary differential operators $\mathcal D_{X_w}$. | |
| Mar 17, 2023 at 23:48 | comment | added | Exit path | Twisted D-modules make sense on any space equipped with a torus bundle (see the accepted answer from this question). To twist D-modules on Bruhat cells just pull $G/N \to G/B$ back along the inclusion. To compute for $SL_2$, in principle you just need a concrete model of the line bundle as well as trivializations over both cells | |
| Mar 17, 2023 at 21:58 | comment | added | Fan Zhou | Sorry, I’m not sure I understand, could you please elaborate? I know that $\lambda$-twisted D-modules on $G/B$ are the same as weakly $B$-equivariant D-modules on $G$ with the difference between the equivariant and the D-module structures being $-\lambda$, but I'm not sure how to do this for the Bruhat cells (indeed, what are twisted D-modules over Bruhat cells?). Also, can one actually compute this action from your point of view, say for this $\mathfrak{sl}_2$ example? I feel like I need to actually be able to compute it here to understand it. | |
| Mar 13, 2023 at 3:13 | comment | added | Exit path | One can think of twisted D-modules as D-modules on the $T$-bundle $G/N \to G/B$ with fixed monodromy on each fiber. Since the Bruhat cells are affine, every $T$-bundle over them trivializes, so one can think of pushforward as going from ordinary D-modules to twisted ones. Obviously, this depends on a trivialization for each cell, so I'm not sure that's the right way to think about it | |
| Mar 13, 2023 at 0:08 | history | edited | Fan Zhou | CC BY-SA 4.0 |
added example
|
| Mar 11, 2023 at 2:13 | history | edited | Fan Zhou | CC BY-SA 4.0 |
said Beilinson-Drinfeld instead of Beilinson-Bernstein for some reason
|
| Mar 11, 2023 at 2:11 | history | edited | LSpice | CC BY-SA 4.0 |
PDF -> abs, and name of reference
|
| Mar 11, 2023 at 2:03 | history | asked | Fan Zhou | CC BY-SA 4.0 |