Scott Adams, the Dilbert cartoon creator and diehard Trump promoter, has taken to the semi-jocular practice of adopting the mishearing of Trump's much-loved adjunct big-league, and using bigly as if it were a real adverb ("I just watched the debate on replay. Trump won bigly. This one wasn't close"). Adams is kidding, I think, but the mishearing is very common: by May 5, bigly was getting over 70,000 hits in the Google News index. I'm worried it may catch on, and we'll wake up some morning not only with the orange-quiffed sexist boor in the White House but with bigly added to the stock of adverbs in standard English.
Believe it or not, there is potential theoretical relevance. If the formation of adverbs like carefully from adjective stems like careful is a matter of lexical word formation (in other words, if it is derivational morphology, hence appropriately discussed in Chapter 18 of The Cambridge Grammar of the English Language), then we should expect to find arbitrary lexical exceptions, and the absence of lexemes such as *bigly, *littlely, *bluely, *oldly, *youngly, etc., is to be expected. Lexical word-formation is riddled with such gaps: take a look at the table I gave in this post, comparing the strikingly irregular pattern seen in horror / horrify / horrific / horrid with terror / terrify / terrific / *terrid and stupor / stupefy / *stupific / stupid and fervor / *fervify / *fervific / fervid and so on.
But if adjective and adverb are the same category and -ly is just an inflectional affix that the words in that category have to take in some syntactic contexts (for example, when they modify verbs), then we should expect much broader application of the suffix. Inflectional affixes show a strong tendency to apply very widely across items in a given category, not picking and choosing at random the way that -ify and -ic and -id seem to do.
In short, every new case of an adjective taking the -ly suffix to form an adverb is a data point that might undercut the idea that -ly is a derivational affix that forms adverbs (not an inflectional affix sometimes required on adjectives).
Of course, the adjectives of English already happen to have a core subset that take inflections that most other adjectives reject. Compare big / bigger / biggest, small / smaller / smallest, cool / cooler / coolest, etc., with intelligent / *intelligenter / *intelligentest, fascinating / *fascinatinger / *fascinatingest, important / *importanter / *importantest, cantankerous / *cantankerouser / *cantankerousest, etc. So a defender of the view that -ly is an inflection would simply say that adjectives are generally pickier than verbs or nouns when it comes to acceptance of inflectional suffixes, but we should expect a word like big to move over time into the long list of adjectives that take the -ly inflection in addition to the -er and -est inflections.
The differentiators and the collapsers can both cover the facts. We are still looking for a clincher. It may surprise you (it surprises me) that 400 years of work on describing the grammar of standard English, including the past 50 years of quite intensive investigation, have not managed to settle an issue as basic as whether a distinct lexical category of adverbs should be posited in English.
Tradition favors the differentiators. It is traditional to assume that the two categories are distinct, as a glance at almost any grammar at all will confirm. But that doesn't mean all those grammars are right; on identifying the members of the class of prepositions, for example, I claim they are all wrong.
However, I currently think that positing a separate category of adverbs is not wrong. I'm a differentiator. A long paper in 2010 entitled "The distribution and category status of adjectives and adverbs" (Word Structure 3(1), 31–81) argues that the traditional view on this point is indeed correct, and the -ly adverb-forming suffix is derivational. I'm sure you will trust me when I tell you that the paper is truly magisterial, a triumph of human intellect, the definitive work on this topic. Though when later you discover that I am one of its authors (it is by John Payne, Rodney Huddleston, and me), that may undermine your trust a little ("He would say that, wouldn't he?").
The opposing view — that in some Indo-European languages the adverbs should be collapsed with the adjectives into a single category — has been advocated by a number of linguists. Jerzy Kurylowicz proposed it for French (in 1936); Gerard Moignet developed the case further (in 1963); Joseph Emonds championed it for English (in 1976); Andrew Radford pushed for it too (in 1988); so did several other linguists. (The relevant references can be found in the bibliography of the Payne/Huddleston/Pullum paper.)
Recently the collapsers have attracted support through a number of new arguments presented in a 2012 paper by Heinz Giegerich: "The morphology of -ly and the categorial status of 'adverbs' in English" (English Language & Linguistics 16(3), 341–359). Yes, Giegerich has the effrontery not just to disagree with Payne, Huddleston, and me, but in addition to put the word "adverb" in scare quotes in his title as if it were disreputable! Scoundrel.
Naturally I would like to see Giegerich's paper reduced to a laughing-stock, retracted by the publishers as worthless, fraudulent, and a disservice to science. But it's not as easy as that. Personally, I face a real problem. Heinz Giegerich works just down the hall in my building. He's an important member of my department, and one of my favorite colleagues. His work is respected. Dammit, I like the guy. And even more relevantly, his arguments are perfectly intelligible; I don't disagree with any of his facts. He just sees different implications emerging from the facts. So now what?
The Payne/Huddleston/Pullum paper and the Giegerich paper added together occupy 68 pages of learned journal space. Other respectable and crucially relevant articles on the topic that I know about take up far more than that (and there will be yet more contributions that I still haven't found out about). The arguments are complex and both sides make sound and persuasive points.
The truth is that linguists cannot at present settle the question of whether the (traditionally so-called) adverbs of English should be segregated from the adjectives in a separate category, or whether there should be one syntactic category that includes them all, -ly being merely a suffix that some of them have to bear in some contexts. It's just one indication that grammatical analysis is really quite a bit harder than you might think.