What kinds of 'splits' characterise diasporic literature and why? How does diasporic literature negotiate the space 'in between'--be it geographical, cultural and/or personal?
Diasporic literature is a genre characterized by the gulf between a homeland and an adopted country. In these works, characters often face the isolation of belonging neither to the cultures they have left behind, nor to the ones they have migrated into. Salman Rushdie is an author of diasporic literature, and you can look to his novel Grimus as a strong example of that. Other Indian authors who have written on the same subject are Monica Ali (with Brick Lane) and Jhumpa Lahiri (Interpreter of Maldies). The splits that occur in these works are all geographical, cultural, and personal; there is a schism that is caused not only by physical migration, but by the adoption of new cultural customs, and the ensuing personal rifts between old family members and friends.
The way that these authors explore the "space in between" as you put it is by placing their characters in a very isolating space. Often they cannot be completely understood by people of their old world or their new one. Think of the immigrant experiences you've witnessed in real life; perhaps you've seen the children of first-generation immigrants struggle through some of the same emotions. There is a conflict in diasporic literature over how much to assimilate and how much to hold on to one's origins and cultural customs.
Another example might be found in the short story "Sandpiper" by Ahdaf Soueif, which tells of an English woman's experiences in Egypt. The narrator knows that she will never be a true member of the Egyptian culture, but she feels that she has made a life there, having had her child there. She thus likens herself to a sandpiper that must roam the narrow shores, straddling the vast oceans and deserts that surround her on either side.