West Bengalese state had been governed since a few decades ago by the Left Front, which is lead by the Communist Party of India (Marxist). CPI Marxist is one of the many political parties in India that use the name of a communist party. Among them are CPI Marxist-Leninist, CPI Maoist and (of course!) CPI.
CPI Marxist's rule had been mostly conflicting with both illegal and legal activities of CPI Maoist, which had recently been declared a terrorist organisation by the Indian state (preventing CPI Maoist from participating in further legal activities) as well as other communist parties. In an occasion, CPI Marxist wanted to evict some poor peasants out of their lands by pursuing a pro-Chinese (CPI Marxist is perhaps the only pro-China communist party of India) policy, trying to industrialise West Bengal fast on the cost of leaving poor farmers homeless and landless, and joining a capitalist pro-globalisation economic program. (Today, West Bengal has numerous special economic zones, which they acquired at the cost of oppressing peasants.)
In a recent event (which was mentioned in David Harvey's work Right To The City) there was a conflict between the state forces (in West Bengal, the state which was controled by the CPI Marxist) and the peasants who wanted to resist eviction and mistreatment. There was bloodshed due to state officers firing at the peasants. To justify its actions, CPI Marxist claimed the peasants to be organised by the CPI Maoist - this would make sense to the majority of the India, because CPI Maoist was a genuine peasant revolt that later evolved into a class-conscious communist movement. However, it was false from what appears to be the official report on the event and CPI Maoist was not involved in the resistance. (It would appear it was a different group of parties including CPI Marxist-Leninist.)
Overall, CPI Marxist rule had been a largely capitalistic one, even more so today than it was before. Because of such anti-popular leanings, CPI Marxist is slowly but surely falling out of favor in West Bengal.
P.S.: Communism is an economic system that can happen only on a global, international scale. When you said "which aspects of communism," I believe you wanted a comparable answer to pre-Khrushchev Soviet Union, Enver Hoxha's Albania, Mao's China, etc., i.e. socialism, i.e. dictatorship of the proletariat. And perceivable from my overall answer, my response to your question is a very simple one: None.
P.P.S.: Socialism (or, as described by Karl Marx, dictatorship of the proletariat) can not co-exist with parliamantary democracy. (Described by Karl Marx as bourgeois democracy.) This makes CPI Marxist, along with Hugo Chavez, Evo Morales, etc., a revisionist clique, i.e. they are not Marxists. As you can see, the name of CPI Marxist is fraudulent and it is vain to expect anything Marxist (that is, anything comparable to the dictatorship of the proletariat that existed in China, Albania, Soviet Union, etc., etc.) from them. (On the other hand, one could argue the socialistic management of Cuban, Venezuelan, Bolivian economies - socialistic but non-Marxist nature of Latin American countries, but I am yet to see anyone do so for the CPI Marxist.)