Shintani relation for base change: unitary and elliptic representations

Abstract: Let

E / F

be a cyclic extension of

p

-adic fields and

n

a positive integer. Arthur and Clozel constructed a base change process

p i m a p s t o p i_{E}

which associates to a smooth irreducible representation of

G L_{n} (F)

a smooth irreducible representation of

G L_{n} (E)

, invariant under

G a l (E / F)

. When

p i

is tempered,

p i_{E}

is tempered and is characterized by an identity (the Shintani character relation) relating the character of

p i

to the character of

p i_{E}

twisted by the action of

G a l (E / F)

. In this paper we show that the Shintani relation also holds when

p i

is unitary or elliptic. We prove similar results for the extension

C / R

. As a corollary we show that for a cyclic extension

E / F

of number fields the base change for automorphic residual representations of the ad`ele group

G L_{n} (A_{F})

respects the Shintani relation at each place of

F

.