The Graduale Romanum indicates two Alleluias for the 4th Sunday of Easter, but no Gradual. Would it be liturgically correct to sing both Alleluias and omit the Responsorial Psalm? To sing the Greater Alleluia plus the Responsorial Psalm?
It's my understanding that when there's a greater and lesser alleluia, the lesser takes the place of the responsorial psalm or gradual, and the greater is in the normal place of the alleluia.
In Easter season, the Gradual is omitted and instead there are two Alleluias. It would indeed be correct to sing the first after the first reading and the second after the second reading (according to the instructions on page 10 of the GR 1974).
The instructions don't even make a distinction about greater or lesser Alleluias. On a weekday with one reading before the Gospel, one is instructed to sing either one of the chants.
As a matter of opinion, I think you'd be free to sing a responsorial psalm after the first reading and then take your pick of the two alleluias after the second reading.
Certainly the lesser (first) Alleluia psalm in the Gradual is the one which occupies the same "slot" as the Responsorial Psalm or "responsorial gradual"; and the greater (second) Alleluia is in the "Gospel Acclamation" slot. However, the Graduale Romanum and also the Graduale Simplex (cf. Missa I Tempus Paschale, p 166) make clear (as chonak says above) that the two Alleluias are pretty much interchangeable as choices for "an Alleluia psalm". The GIRM as usual reflects a certain confusion about the contents of the Graduales, but it's pretty clear that you can mix and match there too (GIRM 62-63).
Throughout the Easter season, of course, the proper Responsorial Psalm in the Lectionary has "Alleluia" as an alternative response, so you can use that instead if you want to be more traditional and have two Alleluia psalms.
(And on weekdays if you are not going to sing the Gospel Acclamation, you can omit it altogether, right, Adam? ;-) )
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