Phyllida Lloyd's production at London's Donmar Warehouse sees Shakespeare's political conspiracy drama performed by the female inmates.
Janet McTeer and Harriet Walter are both shortlisted for best actress for the Donmar Warehouse production of Mary Stuart.
BBC: NEWS | Entertainment | Elliot show tops stage shortlist
Tom Hiddleston is to play the title role in Coriolanus as part of the autumn season at the Donmar Warehouse.
The 42-year-old has starred in several plays in London's West End, including Life is a Dream at the Donmar Warehouse.
Bierman, the former executive director of the Donmar Warehouse, left the theatre company for the Almeida Theatre earlier this year.
The Guardian's Michael Billington says the actor is "excellent" in the Donmar Warehouse's "outstanding" revival of Eugene O'Neill's Anna Christie.
Other winners included Sir Derek Jacobi, who won best best performance in a play for the Donmar Warehouse production of King Lear.
Gillan, who plays the Doctor's feisty assistant Amy Pond, will appear with Douglas Hodge in the show, which opens at the Donmar Warehouse in October.
In London he also has made commitments to the National Theatre of Great Britain, the Old Vic theater and the Donmar Warehouse, London's leading repertory theater.
Donmar Warehouse artistic director Josie Rourke will direct The Machine, a new play about chess player Garry Kasparov's battle with the supercomputer Deep Blue, written by Matt Charman.
Also in the arts world, Donmar Warehouse artist director Michael Grandage, National Portrait Gallery director Sandy Nairne, artist Gillian Ayres and textile designer Celia Birtwell were appointed CBEs.
Together they are joint chief executives of the Ambassador Theatre Group (ATG), which owns major independent theatres in the UK, including the Donmar Warehouse and the Edinburgh Playhouse.
The Royal Court Theatre's Now or Later, written by Christopher Shinn, competes with two Donmar Warehouse revivals - Othello, starring Ewan McGregor, and The Chalk Garden - for the theatre prize.
Bunny Christie's set, she observed, almost returns the Donmar space to its warehouse origins.
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