The soul lives on eternally, and it maintains its connection with family members, in particular those who are especially dear. The soul suffers in their pain and rejoices in their joy. * Source materials compiled by Rabbi Shloma Majeski. Translations are in bold. Underlining is the emphasis of the compiler.
Translated by Boruch Merkur
The first chapter of Rabbi Shloma Majeski’s compilation Likkutei Mekoros establishes that the Rebbe MH”M’s focus after the histalkus of the Rebbe Rayatz on Yud Shvat 5710 was on publically articulating that the Rebbe is, in fact, still alive, making this point repeatedly and empathetically during that period. Chapter Two sets out to explain how that can possibly be – first, discussing the eternal nature of the soul, something that is common to every Jew; then pointing out the manner by which this eternality is of a different caliber among tzaddikim; third, after the histalkus of a tzaddik, he lives on even in the physical world; and finally, among tzaddikim themselves there are those who stand out and are particularly noted as possessing eternal life.
Chapter Two: 
The Eternal Soul
Appearing in the hosafos of Likkutei Sichos Vol. 10, pg. 212-213, is a letter of condolence the Rebbe sent to a woman who had apparently lost her husband in a terrorist attack, rachmana litzlan:
With pain I learned of the tragic death of your husband, may Hashem avenge his blood. May it be G-d’s will that from now on your entire family only experiences goodness and kindness – goodness that is overt and revealed.
Although you have perhaps heard the following sentiments from several people already, I too shall express my thoughts on the matter, at least in brief. But first I would like to point out that the intent of my letter is not to resolve difficult questions, but to simply ease the pain, even if just slightly.
The fact is that the main bond between loved ones – especially the bond between husband and wife or between children and their father – is not on account of the physical body, which is comprised of flesh, sinews, and bones, but on account of the character of the soul and spirit, which is the main aspect of a person, constituting his very being and essence. It is just that relationships develop through the medium of the body – through its eyes, ears, hands, the speech organs, and so on. Within and by means of the various parts of the body, the person’s ideas and feelings, as well as his character, find expression. Indeed, the spiritual makeup of a person, his character, is his true identity, as stated above.
Although the bullet of a gun, or shrapnel from a rocket shell, or disease and the like, may damage and harm the body, they do not have the capacity to harm or even detract from the soul or the neshama. Death caused by a bullet or by disease entails an interruption between body and soul. However, the soul lives on eternally, and it maintains its connection with family members, in particular those who are especially dear. The soul suffers in their pain and rejoices in all the kinds of joy the family experiences. It is just that the living family members, those who remain alive in the physical world, do not perceive this [emotional] response of the soul with their human eyes; it is not something that can be felt with their hands. The reason for this is because the physical connection has been severed.
From this it is also understood that for the soul that resides in the World of Truth there is an especial pleasure in seeing that the members of the family recover from tragedy and strive in all their areas of influence to organize their lives in a positive manner, even being a source of encouragement to those around them. In general, the soul that resides in the World of Truth sees and knows about all the good deeds done by members of the family, and it derives pleasure therefrom.
The nachas ruach the soul experiences is particularly pronounced when it sees the children being raised with a proper chinuch, not having despaired [in the face of tragedy] and not becoming downtrodden, G-d forbid. But on the contrary, as in the traditional blessing, “To raise them to Torah, chuppa (marriage), and good deeds.” For it is certain that the necessary capabilities to accomplish this have been granted [from On High] to all those who were issued this task; ample capacity has been given to them to wholly fulfill it.
May it be G-d’s will that you do all this with alacrity and vigor, and that you should be a living example to all those around you. May the merit of your husband, z”l, stand for you and for all members of your household, sh’yichyu.
With blessings that things go smoothly and good tidings in all matters discussed.