adverbs - What part of speech does “here” have in “I am here”?

I'm in agreement with John Lawler that here is not an adverb here: as he says, 'It does not modify the verb am.' One has to really stretch the meaning of modify for that not to be true (which some linguists do).

Intuitively, though, here is more closely associated with the noun group (a pronoun in this case) than with the verb - the speaker's location rather than 'how' he is existing. We could compare 'I am cold', where 'am' is obviously a delexical verbal link between subject and some attribute. But 'cold' is adjectival, describing an inherent characteristic, while 'here' expresses the reference of a noun or noun-phrase in the context, rather than attributes (which are expressed by adjectives). We're almost at the definition of determiners here, but there is more semantic content to 'here'. I think that words like 'here', 'there', 'home' (in 'Is he home?' 'He went home') need their own category (and a working name for this is locative / directional particles).