Upon the conclusion of Pesukei d’Zimrah and Yishtabach, the chazan recites Half-Kaddish. As we know, the virtue of the Kaddish is very great, and the Chachamim praise those who respond Amen to it with kavanah (see further in this book 23:6). The Chachamim instituted its recital at the conclusion of every stage of the prayer service in order to end each step in supreme sanctity, and from that holiness to continue on to the next stage of prayer. This Half-Kaddish creates a break between Pesukei d’Zimrah (corresponding to the world of yetzirah) and Keriat Shema and its berachot (which correspond to the world of beriah).
One may not interrupt by speaking between Pesukei d’Zimrah and Birkot Keriat Shema. Someone who does interrupt commits a sin, since Pesukei d’Zimrah serves as a preparation for prayer (Shulchan Aruch 54:3).
If a group of people praying reaches the end of Pesukei d’Zimrah and they do not yet have a minyan, they should stop after Yishtabach and wait for a minyan. They may learn Torah in the meantime. L’chatchilah, they should study silently, so they do not interrupt their prayer by speaking. However, one who cannot learn without actually uttering the words may learn by saying them, since, for the sake of a mitzvah, it is permissible to interrupt between Yishtabach and Barchu. The chazan should wait before Yishtabach for ten men to assemble so that he can say Yishtabach, and afterwards, Half-Kaddish. If he already concluded Yishtabach, after ten men have already gathered he should recite three verses of Scripture so that the Kaddish will apply to them and subsequently recite Half-Kaddish (Mishnah Berurah 53:10-11).
When the gabbai must relay an urgent message to the congregation, either pertaining to the needs of the community or for the sake of a mitzvah, and it is impossible to delay the announcement until after the prayer service, he may announce it between Yishtabach and Kaddish. Afterwards, the chazan recites a few verses and Kaddish. However, if the Kaddish was already recited, it is forbidden to interrupt even for the sake of a mitzvah, and he must wait until after Kaddish titkabal to make the announcements (Shulchan Aruch, Rama 54:3; 57:2; Mishnah Berurah 54:6).