Introduction

We will use the same notations as in chapters 4, 5 and 7.

The purpose of this chapter is to prove that the Euler–Poincaré function (5.1.2) is a pseudo-coefficient of the Steinberg representation. This result is due to Casselman and Kottwitz and we will follow [Bo–Wa] and [Ko 1].

The Steinberg representation

For any I ⊂ Δ, the induced representation

is nothing else than

where is the ℚ-vector space of locally constant functions on G(F) with values in ℚ which are left PI (F)-invariant and where ρI is the left action of G(F) on which is induced by the right translation on PI(F)\G(F).

For any I ⊂ J ⊂ Δ, we have a natural commutative diagram of monomorphismsin

in Reps(G(F)) (a left PJ (F)-invariant (resp. PI (F)-invariant) function is automatically left PI (F)-invariant (resp. B(F)-invariant) as.

For any, we have

in as is the subgroup of G(F) which is generated by and (for any I ⊂ Δ, PI(F) is the subgroup of G(F) which is generated by B(F) and.

For each I ⊂ Δ, let us denote by

the cokernel of the morphism

(sum of the natural monomorphisms) in Reps(G(F)). It is clear that

is isomorphic to the trivial representation. The smooth representation is the so-called Steinberg representation of G(F) and is also denoted by (StG(F)stG(F)) or simply (St, st).

THEOREM (8.1.2) (Casselman). — (i) For each is irreducible in Reps(G(F)) and, for any and are isomorphic if and only if I′ = I″.

1. (ii) For each I ⊂ Δ, the Jordan–Hölder subquotients of are exactly the for I ⊂ J ⊂ Δ, each of them occurring with multiplicity one.

2. […]