8 - Euler–Poincaré functions as pseudocoefficients of the Steinberg representation

Summary

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 P_I (F)-invariant and where ρ_I is the left action of G(F) on which is induced by the right translation on P_I(F)\G(F).

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

in Rep_s(G(F)) (a left P_J (F)-invariant (resp. P_I (F)-invariant) function is automatically left P_I (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, P_I(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 Rep_s(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 (St_G(F)st_G(F)) or simply (St, st).

THEOREM (8.1.2) (Casselman). — (i) For each is irreducible in Rep_s(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. […]