Summary: TOWARDS A LOCAL TRACE FORMULA
By JAMES ARTHUR*
1. Suppose that G is a connected, reductive algebraic group over a
local field F. We assume that F is of characteristic 0. We can form the
Hilbert space L2(G(F)) of functions on G(F) which are square integrable
with respect to the Haar measure. The regular representation
(R(yl, y2)4)(x) = ((yi1xy2), E L2(, G(F)),, G(F),
is then a unitary representation of G(F) X G(F) onL2(G(F)). Kazhdan has
suggested that there should be a local trace formula attached to R which is
analogous to the global trace formula for automorphicforms. The purpose
of this note is to discuss how one might go about proving such an identity,
and to describe the ultimate form the identity is likely to take.
To see the analogy with automorphic forms more clearly, consider the
diagonal embedding of F into the ring
AF = FOF.
The group G(AF) ofAF-valued points in G is just G(F) X G(F). The group
G(F) embeds into G(AF) as the diagonal subgroup. Observe that we can
map L2(G(F)) isomorphically onto L2(G(F)\G(AF)) by sending any 4 E
L2(G(F)) to the function
(g, g2) ' (gllg2) (g, g2) E G(F)\G(AF).