 
Summary: JOURNAL OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000000
S 08940347(XX)00000
UNITARY SHIMURA CORRESPONDENCES
FOR SPLIT REAL GROUPS
J. ADAMS, D. BARBASCH, A. PAUL, P. TRAPA, AND D. A. VOGAN, JR.
1. Introduction
Let G be the split real form of a connected reductive algebraic group G. Let
be a Cartan involution of G and K = G
the corresponding maximal compact
subgroup. Let B be a Borel subgroup of G with unipotent radical N. Then
A = B B
is a stable split Cartan subgroup of G. Notice the departure from standard
notation for real reductive groups, in which A often refers to the identity component
of a maximal split torus. We will write this identity component as A0
. In our
notation, the Iwasawa decomposition is G = KA0
N. Write
M = B K = A K Zn
