Translation functor on parabolic Verma module

I want to prove that following proposition by using Theorems/propositions in Representations of Semisimple Lie Algebras in the BGG Category $ \mathcal{O}$ .

Define $ \Lambda := \{\nu \in \mathfrak{h}^* : \langle\nu,\alpha^\lor\rangle \in \mathbb{Z} \ \text{for all }\alpha \in \Phi^+\}. $

Define $ \Lambda^+ := \{\nu \in \mathfrak{h}^* : \langle\nu,\alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+\}. $

Define $ \Lambda^+_I := \{\nu \in \mathfrak{h}^* : \langle\nu,\alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+_I\}. $

The $ \mathbb{Z}$ -span $ \Lambda_r$ of $ \Phi$ is called the root lattice.

Recall that a weight $ \eta\in \mathfrak{h}^*$ is antidominant if $ \langle \eta+\rho,\alpha^{\lor}\rangle\not\in\mathbb{Z}^{>0}$ for all $ \alpha\in \Phi^+$ .

For $ w \in W$ and $ \eta\in \mathfrak{h}^*$ , define a shifted action of $ W$ (called the dot action) by $ w \cdot \eta := w(\eta + \rho) – \rho$ . If $ \eta, \nu \in \mathfrak{h}^*$ , then we say that $ \eta$ and $ \nu$ are linked if for some $ w \in W$ , we have $ \nu = w \cdot \eta$ .

The weight $ \eta \in \mathfrak{h}^*$ is regular if $ |W \cdot \eta| = |W|$ or, equivalently, if $ \langle \eta + \rho,\alpha^\lor\rangle \neq 0$ for all $ \alpha\in\Phi$

By exercise 1.13, $ M\in\mathcal{O}_{\chi_\lambda}$ has a direct sum decomposition $ M=\bigoplus M_i$ such that all weights of each $ M_i$ are contained in a single coset of the root lattice $ \Lambda_r$ in $ \mathfrak{h}^*$ . Therefore, the category $ \mathcal{O}_{\chi_\lambda}$ decomposes as a direct sum of full subcategories, which can be indexed by the nonempty intersection of the orbit $ W\cdot\lambda$ with the cosets $ \mathfrak{h}^*/\Lambda_r$ . We use the antidominant weight $ \mu$ in the intersection to parameterize the corresponding subcategory of $ \mathcal{O}_{\chi_\lambda}$ . We denote this subcategory by $ \mathcal{O}_{\mu}$ .

We say $ \eta$ and $ \nu$ are compatible if $ \nu – \eta \in\Lambda$ .

Fix $ \eta, \nu \in \mathfrak{h}^*$ such that $ \eta,\nu$ are compatible and write $ pr_\eta$ and $ pr_\nu$ for the natural projections of the category $ \mathcal{O}$ onto $ \mathcal{O}_{\chi_\eta}$ and $ \mathcal{O}_{\chi_\nu}$ . Let $ \overline{\nu-\eta}$ be the unique $ W$ -conjugate in $ \Lambda^+$ of $ \nu-\eta$ . If $ M \in \mathcal{O}$ , then $ M \mapsto pr_ \nu \left(L(\overline{\nu-\eta}) \otimes (pr_\eta M)\right)$ followed by inclusion into $ \mathcal{O}$ defines an exact functor $ \mathcal{O} \to \mathcal{O}$ . Its restriction to $ \mathcal{O}_{\chi_\eta}$ (without the inclusion) relates the two subcategories $ \mathcal{O}_{\chi_\eta}$ and $ \mathcal{O}_{\chi_\nu}$ . Write $ T^\nu_\eta$ for the resulting functor on $ \mathcal{O}$ (or on $ \mathcal{O}_{\chi_\eta}$ ). We call $ T^\nu_\eta$ a translation functor.

Proposition: Let $ \eta,\nu\in \Lambda_I^+$ , Suppose $ \eta,\nu\in \Lambda_I^+$ are integral, regular and antidominant. Then there is an equivalence of categories between $ \mathcal{O}_\nu$ and $ \mathcal{O}_\eta$ such that (i) $ T^{\eta}_{\nu}(L(x\cdot \nu))\cong L(x\cdot \eta)$ and $ T^{\eta}_{\nu}(M(x\cdot \nu))\cong M(x\cdot \eta)$ for $ x\in W$ . (ii) If $ x\cdot\nu$ is in $ \Lambda_I^+$ then $ T^{\eta}_{\nu}(M_I(x\cdot \nu))\cong M_I(x\cdot \eta)$ .

Proof: Since $ \eta,\nu\in \Lambda_I^+$ are integral, regular and antidominant, it holds that $ \eta,\nu$ are compatible. Since $ \eta,\nu$ are integral, we have $ \eta^\natural=\eta,\nu^\natural=\nu$ and hence $ \eta^\natural,\nu^\natural\in F$ where $ \Phi_F^-=\Phi^+$ , $ \Phi_F^0=\Phi_F^+=\emptyset$ . Applying Theorem 7.6, Proposition 7.7 and Theorem 7.8, we deduce that there is an equivalence of categories $ T^{\eta}_{\nu}$ between $ \mathcal{O}_{\nu}$ and $ \mathcal{O}_{\eta}$ satisfying (i).

For a proof of (ii) it suffices to show that $ T^{\eta}_{\nu}(M_I(x\cdot \nu))\cong M_I(x\cdot \eta)$ by the properties of the full subcategory. In fact, there is an exact sequence (Theorem 9.4) \begin{equation}\label{longexact1} \bigoplus_{\alpha\in I} M(s_\alpha x \cdot \nu) \xrightarrow{f} M(x \cdot \nu) \xrightarrow{g} M_I (x \cdot \nu) \to 0. \quad\quad(1) \end{equation} Applying $ T^{\eta}_{\nu}$ to (1), we get another exact sequence \begin{equation}\label{longexact2} \bigoplus_{\alpha\in I} M(s_\alpha x \cdot \eta) \xrightarrow{T^{\eta}_{\nu}(f)} M(x \cdot \eta) \xrightarrow{T^{\eta}_{\nu}(g)} T^{\eta}_{\nu}(M_I(x \cdot \nu)) \to 0. \quad\quad(2) \end{equation} On the other hand, by replacing $ \nu$ in (1) with $ \eta$ , we can get an exact sequence in $ \mathcal{O}_{\eta}$ : \begin{equation}\label{longexact3} \bigoplus_{\alpha\in I} M(s_\alpha x \cdot \eta) \xrightarrow{f’} M(x \cdot \eta) \xrightarrow{g’} M_I (x \cdot \eta) \to 0. \quad\quad(3) \end{equation}

Since (2) is exact, it holds that $ T^{\eta}_{\nu}(g)$ is surjective and $ \ker(T^{\eta}_{\nu}(g))=\mathrm{Im}(T^{\eta}_{\nu}(f))$ , we have $ T^{\eta}_{\nu}(M_I(x \cdot \nu))= \mathrm{Im}(T^{\eta}_{\nu}(g)) \cong M(x\cdot\eta)/\ker(T^{\eta}_{\nu}(g)) =M(x\cdot\eta)/\mathrm{Im}(T^{\eta}_{\nu}(f)).$

Since (3) is exact, it holds that $ g’$ is surjective and $ \ker(g’)=\mathrm{Im}(f’)$ , we have $ M_I(x \cdot \eta)=\mathrm{Im}(g’) \cong M(x\cdot\eta)/\ker(g’) =M(x\cdot\eta)/\mathrm{Im}(f’). $

My question: How to show $ T^{\eta}_{\nu}(f)=f’$ ?

Translation of diagnosis problem to SAT

I have the following diagnosis problem:
h(A): z1 = not(in1)
h(D): z2 = not(in2)
h(B): z3 = z1 or z2
h(C): out1 = not(z3)
h(E): out2 = not(z3)

This is an image of the system:
enter image description here

I have an observation where I know that:
in1 = false
in2 = false
out1 = true
out2 = true

I want to check if {B} is a diagnosis of the system, meaning that I assume that A,D,C,E are valid, while I don’t assume the same about B.
This I want to turn into a sat problem. So first I map each variable in my system to a literal:
in1: 1
in2: 2
z1 (the output of A): 3
z2 (the output of D): 4
z3 (the output of B): 5
out1: 6
out2: 7

So to check if {B} is a diagnosis of the system (meaning that if I remove the assumption about {B} the system will be OK) I create a CNF like so:

not(1) AND not(2) AND 6 AND 7 – this represents my observations, the variables whose value I know for sure.

Now I want to add a representation for my constraints. For example, I know that 6 = not(5), but I don’t know the value of 5 (because I can’t assume the validity of component B).
My question is how do I add this knowledge to my CNF representation? If I add “not(5)” to my CNF then I assume the output of B is always false, and this isn’t correct.

Thank you very much.

User registration email translation

I am working on an English/Arabic website, the user can switch between the two languages using the language switch block, I wanted to translate the email sent to the user after registration. So after some digging, I figured out I could do this using the account setting (translate tab), I added the Arabic translation of the email; then, after testing, I found out that the registration email is still sent in English.

Shouldn’t a user registered in the Arabic language get an Arabic registration email?

I also found out that the user’s preferred language is English, even after registering in the Arabic language.

These are my language detection settings.

enter image description here enter image description here

Do I need provide translation for an UI library

I have a library for a lockscreen in Android, that includes diferent dilogs for fingerprints, texts and all that stuff. My question is: What the best practice to provide localization. Should I have instuments that provides developers eho use this library to set up localization outside of library, which means if developer wants to support different languages it will be necessary to set up all the strings that being used in the library.

Or should I add localiation for those strings inside tha app?

To have better idea:

Not able to enable ‘Configuration Translation’ module gives error

I am trying to enable MULTILINGUAL -> Configuration Translation.

It gives below error.

Error: Call to a member function getPath() on null in Drupal\config_translation\ConfigNamesMapper->getOverviewRoute() (line 247 of /var/www/project/docroot/core/modules/config_translation/src/ConfigNamesMapper.php) #0 /var/www/project/docroot/core/modules/config_translation/src/Routing/RouteSubscriber.php(39): Drupal\config_translation\ConfigNamesMapper->getOverviewRoute() #1 /var/www/project/docroot/core/lib/Drupal/Core/Routing/RouteSubscriberBase.php(37): Drupal\config_translation\Routing\RouteSubscriber->alterRoutes(Object(Symfony\Component\Routing\RouteCollection)) #2 [internal function]: Drupal\Core\Routing\RouteSubscriberBase->onAlterRoutes(Object(Drupal\Core\Routing\RouteBuildEvent), ‘routing.route_a…’, Object(Drupal\Component\EventDispatcher\ContainerAwareEventDispatcher)) #3 /var/www/project/docroot/core/lib/Drupal/Component/EventDispatcher/ContainerAwareEventDispatcher.php(111): call_user_func(Array, Object(Drupal\Core\Routing\RouteBuildEvent), ‘routing.route_a…’, Object(Drupal\Component\EventDispatcher\ContainerAwareEventDispatcher)) #4 /var/www/project/docroot/core/lib/Drupal/Core/Routing/RouteBuilder.php(184): Drupal\Component\EventDispatcher\ContainerAwareEventDispatcher->dispatch(‘routing.route_a…’, Object(Drupal\Core\Routing\RouteBuildEvent)) #5 /var/www/project/docroot/core/lib/Drupal/Core/ProxyClass/Routing/RouteBuilder.php(83): Drupal\Core\Routing\RouteBuilder->rebuild() #6 /var/www/project/docroot/core/lib/Drupal/Core/Extension/ModuleInstaller.php(322): Drupal\Core\ProxyClass\Routing\RouteBuilder->rebuild() #7 /var/www/project/docroot/core/lib/Drupal/Core/ProxyClass/Extension/ModuleInstaller.php(83): Drupal\Core\Extension\ModuleInstaller->install(Array, true) #8 /var/www/project/docroot/core/modules/system/src/Form/ModulesListForm.php(458): Drupal\Core\ProxyClass\Extension\ModuleInstaller->install(Array) #9 [internal function]: Drupal\system\Form\ModulesListForm->submitForm(Array, Object(Drupal\Core\Form\FormState)) #10 /var/www/project/docroot/core/lib/Drupal/Core/Form/FormSubmitter.php(111): call_user_func_array(Array, Array) #11 /var/www/project/docroot/core/lib/Drupal/Core/Form/FormSubmitter.php(51): Drupal\Core\Form\FormSubmitter->executeSubmitHandlers(Array, Object(Drupal\Core\Form\FormState)) #12 /var/www/project/docroot/core/lib/Drupal/Core/Form/FormBuilder.php(589): Drupal\Core\Form\FormSubmitter->doSubmitForm(Array, Object(Drupal\Core\Form\FormState)) #13 /var/www/project/docroot/core/lib/Drupal/Core/Form/FormBuilder.php(318): Drupal\Core\Form\FormBuilder->processForm(‘system_modules’, Array, Object(Drupal\Core\Form\FormState)) #14 /var/www/project/docroot/core/lib/Drupal/Core/Controller/FormController.php(93): Drupal\Core\Form\FormBuilder->buildForm(‘system_modules’, Object(Drupal\Core\Form\FormState)) #15 [internal function]: Drupal\Core\Controller\FormController->getContentResult(Object(Symfony\Component\HttpFoundation\Request), Object(Drupal\Core\Routing\RouteMatch)) #16 /var/www/project/docroot/core/lib/Drupal/Core/EventSubscriber/EarlyRenderingControllerWrapperSubscriber.php(123): call_user_func_array(Array, Array) #17 /var/www/project/docroot/core/lib/Drupal/Core/Render/Renderer.php(582): Drupal\Core\EventSubscriber\EarlyRenderingControllerWrapperSubscriber->Drupal\Core\EventSubscriber{closure}() #18 /var/www/project/docroot/core/lib/Drupal/Core/EventSubscriber/EarlyRenderingControllerWrapperSubscriber.php(124): Drupal\Core\Render\Renderer->executeInRenderContext(Object(Drupal\Core\Render\RenderContext), Object(Closure)) #19 /var/www/project/docroot/core/lib/Drupal/Core/EventSubscriber/EarlyRenderingControllerWrapperSubscriber.php(97): Drupal\Core\EventSubscriber\EarlyRenderingControllerWrapperSubscriber->wrapControllerExecutionInRenderContext(Array, Array) #20 /var/www/project/vendor/symfony/http-kernel/HttpKernel.php(151): Drupal\Core\EventSubscriber\EarlyRenderingControllerWrapperSubscriber->Drupal\Core\EventSubscriber{closure}() #21 /var/www/project/vendor/symfony/http-kernel/HttpKernel.php(68): Symfony\Component\HttpKernel\HttpKernel->handleRaw(Object(Symfony\Component\HttpFoundation\Request), 1) #22 /var/www/project/docroot/core/lib/Drupal/Core/StackMiddleware/Session.php(57): Symfony\Component\HttpKernel\HttpKernel->handle(Object(Symfony\Component\HttpFoundation\Request), 1, true) #23 /var/www/project/docroot/core/lib/Drupal/Core/StackMiddleware/KernelPreHandle.php(47): Drupal\Core\StackMiddleware\Session->handle(Object(Symfony\Component\HttpFoundation\Request), 1, true) #24 /var/www/project/docroot/core/modules/page_cache/src/StackMiddleware/PageCache.php(99): Drupal\Core\StackMiddleware\KernelPreHandle->handle(Object(Symfony\Component\HttpFoundation\Request), 1, true) #25 /var/www/project/docroot/core/modules/page_cache/src/StackMiddleware/PageCache.php(78): Drupal\page_cache\StackMiddleware\PageCache->pass(Object(Symfony\Component\HttpFoundation\Request), 1, true) #26 /var/www/project/docroot/core/lib/Drupal/Core/StackMiddleware/ReverseProxyMiddleware.php(47): Drupal\page_cache\StackMiddleware\PageCache->handle(Object(Symfony\Component\HttpFoundation\Request), 1, true) #27 /var/www/project/docroot/core/lib/Drupal/Core/StackMiddleware/NegotiationMiddleware.php(52): Drupal\Core\StackMiddleware\ReverseProxyMiddleware->handle(Object(Symfony\Component\HttpFoundation\Request), 1, true) #28 /var/www/project/vendor/stack/builder/src/Stack/StackedHttpKernel.php(23): Drupal\Core\StackMiddleware\NegotiationMiddleware->handle(Object(Symfony\Component\HttpFoundation\Request), 1, true) #29 /var/www/project/docroot/core/lib/Drupal/Core/DrupalKernel.php(669): Stack\StackedHttpKernel->handle(Object(Symfony\Component\HttpFoundation\Request), 1, true) #30 /var/www/project/docroot/index.php(19): Drupal\Core\DrupalKernel->handle(Object(Symfony\Component\HttpFoundation\Request)) #31 {main}.

I am not sure what exactly cause of this. Anyone faced?

How to find the shortest route between (0,0) and (4,4) in a 5×5 matrix, given one horizontal or vertical translation per step [on hold]

Consider the following matrix:

enter image description here

Input: The entire list of coordinates ranging from (0,0) to (4,4).

A list indicating certain coordinates which cannot be used to generate a path reached.


  • each position on the grid is a node
  • nodes share an edge if exactly one coordinate differs by exactly 1
  • X nodes (and their associated edges) are deleted

The problem then becomes, find the shortest path in the graph between two nodes. If the nodes are not in the same connected component, there is no such path.

Output: The shortest path between S and E, considering the positions of the X nodes, given by a sequence of coordinates.

So in the above picture, a possible route may be the following: (0,0)->(0,1)->(0,2)->(0,3)->(1,3)->(2,3)->(3,3)->(4,3)->(4,4)

A total of eight steps.

If the Xs occur in a way that prevents any movement (i.e, Xs at (0,1),(1,1), and (1,0), it should return -1 or indicate in some other way no path is possible.)

I have received advice that the A* search algorithm (pseudocode in this link) is relevant to this problem, but I am having difficulty seeing how to apply it.

Issue with JPA statement “translation”

I have a JPA statement which is like this

select p from Proposal p where p.creationTime > :startDate AND p.creationTime < :stopDate AND ((p.owner = :owner) OR (:member MEMBER OF p.sharedWithTeam.members)) ORDER BY p.creationTime DESC 

But it never generates any result. I use EclipseLink.

After digging a bit into the generated SQL, I found out that it translates the query into something that can never be true!

WHERE ((((t0.CREATIONTIME > ?) AND (t0.CREATIONTIME < ?)) AND ((t0.OWNER_username = ?) OR (? = t1.username))) AND ((t2.ID = t0.SHAREDWITHTEAM_ID) AND ((t3.teams_ID = t2.ID) AND (t1.username = t3.members_username)))) ORDER BY t0.CREATIONTIME DESC LIMIT ? OFFSET ? 

(I only included the where clause to make it shorter…)

I can’t see what is wrong with my JPQL query. Could it be my use of “OR”? or my “bold” use of the member of condition?


Exporting content translation to similar server

I have 2 instances of the same site built using Drupal. Both need to be translated, and most of the work of the translation has been done on site A and we need to take the translations from site A to site B. Site B will have slightly newer content, but not a lot and a new version of Drupal (both sites are/will be 8.6+).

I can export translation but they’re just for the interface, the content is not getting exported. Is there a way to export or send the translations in site A to site B without having to translate everything again one by one in site B?


Offline translation files not recognized

When I try to use Google Translate offline, I get an error:

Translation failed. Offline translation file not available. Please check SD card is inserted.

In Settings-> Storage I could see that the Translate app was stored in the phone’s internal memory. I tried moving it to the SD card, and even after restarting the device, the same error was still there.

Using a file browser, I am able to look at the SD card and see that /storage/extSdCard/Android/data/ contains folders like dict.en_pt_25, lang.en_pt, and lang.pt_en where “pt” is also replaced by some other two-letter language codes. .../r12 contains the files for French.

How do I get Translate to work?