Automorphic
Distributions

•Suppose G = real points of a split reductive group defined over Q.

• G ˝ G = arithmetically defined
subgroup

–e.g. G =
SL(n,Z) ˝ SL(n,R)

–or G = GL(n,Z) ˝ GL(n,R) (if center taken into account
appropriately)

•An automorphic
representation is an embedding of a unitary
irreducible representation j : (p,V) ! L2(GnG)

•Under this G-invariant embedding j, the smooth vectors V1 are sent to C1(GnG).

•Consider the “evaluation at the identity” map

– **t****: v ****a**** j(v)(e)
**

–which is a continuous linear functional on V1 (with its natural
Frechet topology).

–Upshot: t 2 ((V’)-1)G - a G-invariant
distribution vector for the dual representation.

•Because (p,V) and (p’,V’) play symmetric roles, we may switch them and henceforth assume t 2 (V-1)G.