By contrast, one might argue that emotions are (proto)-volitional acts. After all, emotions often motivate one to act and in this sense, they seem to be strongly linked to volitions.

Or are they something completely different than cognitions and volitions? And which part, function or organ of the self is responsible for emotions?

What would be the “standard” South Asian view about emotions? And how does the Viśiṣṭādvaita Vedanta view differ from it?

Well, the first thing to say is that there is no “standard” view, but at least two. The Sāṅkhya model is very authoritative and has emotions as cittavṛtti `affections/perturbances of the mind’, completely distinct from the self, which is a pure observer, unaffected by emotions. The Nyāya and the Advaita Vedānta model inherit the basic idea of the self as a pure observer and therefore imagine that in the state of liberation, no emotion is experienced. This stage might be nonetheless described by Advaita Vedānta authors as blissful since bliss would be the inner nature of the self.

By contrast, the Mīmāṃsā model sees the self as inherently an agent and a knower. It acknowledges the sequence, originally discussed by Nyāya authors, moving from cognitions to volitions and then to efforts and actions, however it considers that one and the same actor is responsible throughout the process. Volitions are described as having the form of desire to obtain or desire to avoid, thus including an emotional colouring. In this sense, one would imagine that emotions are implicitly considered to be (proto)-volitional acts. This point is particularly explicit whenever Mīmāṃsā authors make fun of the claim of “desireless actions” and claim that in order to undertake any action one needs desire (rāga) or aversion (dveṣa). The term desire (rāga) has a strong emotional connotation and includes one’s strong attachment to something or inclination towards it, and the same applies to aversion (dveṣa).
Viśiṣṭādvaita Vedānta authors inherit the Mīmāṃsā model and can therefore state that the liberated subject will continue to experience emotions.

The picture is however further complicated by the fact that Viśiṣṭādvaita authors need more emotions than the couple desire-aversion. Since they do not find in Mīmāṃsā the conceptual resources to deal with complex emotions such as desperation, which is essential for their soteriology, they turn to aesthetics. This discipline had evolved complex theories of emotions based on its original link to theater and to the psychology of actors and audience. Already in its foundational text, the Nāṭyaśāstra, there was a clear distinction of fundamental emotions, linked with their physical epiphenomena (such as goose bumps or blushing) and with the kinds of auxiliary emotions for each of them. Since the Nāṭyaśāstra is meant for theater professionists, it also discusses how to solicit such emotions —something of key importance for thinkers aiming at using emotions for soteriological purposes. The interaction of aesthetics and soteriology is paramount in another school of Vedānta, namely the Gauḍīya Vedānta founded by Caitanya and developed by Jīva and Rūpa Gosvāmin.