Hydrocarbon potential in Ganges Brahmaputra delta
There have been a few discoveries in the Bay of Bengal.

Santos made a gas discovery in 2012, and depending on what you'd call part of the delta complex (all of the Bengal Fan?), there are others. One field, Sangu, saw several years of production, but seems to be shut-in today. Another, Kutubdia, is awaiting development.

It's true though that there have not been many at all, if you compare with some other big deltas, like Niger or the Gulf of Mexico. Blakely (2010) suggested there are border disputes, which definitely slows exploration. I'm just guessing, but there may be other factors, such as regulatory issues, slow government licensing, or political pressures. Reading around, it does seem fair to say that the region has not been fully explored, and clearly there is a working hydrocarbon system.

References

Blakeley, I (2010). Bay of Bengal. GEO Expro 7 (6). Available online