@MISC{Arthur_towardsa, author = {James Arthur}, title = {TOWARDS A LOCAL TRACE FORMULA}, year = {} }

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Abstract

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) on L2(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 automorphic forms. 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) of AF-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 L2(G(F)) to the function (g, g2) ' (gllg2) (g, g2) E G(F)\G(AF). In this way, the representation R becomes equivalent to the regular representation of G(AF) on L2(G(F)\G(AF)). As is well known, R may be interpreted as a representation of the convolution algebra E