Learn Russian: Это первый указ. Пусть министр подпишет документ. - This is the first decree. Let the minister sign the document.

The particle пусть (let) is followed by either the present or future tense. We use the future tense here because подписать (to sign) is a perfective verb, focusing on the successful completion of the action. Perfective verbs do not have a present tense form, so their conjugated form is technically the future tense (подпишет), even when making an immediate command.

Could I use пускай instead of пусть in this sentence?

Yes, пускай and пусть mean the exact same thing here and are used the exact same way. However, пусть is more standard and neutral, making it a better fit for a formal context involving a minister and an official decree. Пускай sounds slightly more conversational or colloquial.

Документ is the direct object being signed. Why doesn't its ending change?

It actually is in the accusative case! Because документ is an inanimate masculine noun, its accusative form is identical to its nominative (dictionary) form. If the object were a feminine noun, like бумага (paper), you would see the ending change to show the accusative case: Пусть министр подпишет бумагу.

Is the word указ (decree) related to other common Russian words?