Why limit it to jazz? I'd love to see it bring out a younger crowd, and be a platform for all music. Emerson has a little times square-type thing going on across the street, with live performers in the window daily, it would be awesome if they could stretch it across the street.
From my eyes the common works tremendously well in terms of circulation and use, the diagram is solid through the middle, but the edges need work, and a bit around the big monument. The built infrastructure of the park needs a thorough overhaul. Hire a competent local firm with respect for historical renovation and tasteful detail work (this eliminates Halvorson thankfully), perhaps Michael Van Valkenbergh Associates (Harvard Yard, Wellesley College) or Reed Hilderbrand (Arnold Arboretum, Mount Auburn Cemetery)
I understand the paltriness of the vegetation along tremont as being necessary due to the green line tunnel, but really it can be handled better than that. Also the scale of the trees at the Boylston/Tremont corner really fails. I'm assuming also that is because of the curve of the tunnel. It would be great to bring in West 8 or some amazing landscape/urban design firm to do some sort of edge/plaza intervention there. Think crown fountain in chicago. West 8 has done some projects like this where a vital, modern, urban place is reconciled in a historical setting, without being deferential (they are contemporary designers in Europe, its kind of necessary to do that to survive). Really play off of the movie theater and the glitz and glamour of that stretch of tremont through the theater district (both existing and planned) and as the real 24 hour edge of the park.
And probably the most controversial suggestion: cut down that god awful tree that is in front of the statehouse, behind the Black Civil War unit's memorial. Am I missing something here? Why is that there? It is stunted, contorted, and violates the axis of that sloping lawn down to the fountain--er excuse, homeless urinal. Also drop a few dimes and get some real trees along that thing, and pay attention to where you plant them!