In this chapter we consider the compactly supported étale ℓ-adic cohomology of period domains. We will write down in the basic case a recursive formula for the Euler–Poincaré characteristic in a Grothendieck group of JF(ℚp)-representations (possibly infinite-dimensional ones). Once again, we use the Langlands Lemma to resolve the corresponding recursion formula. The general argument is essentially the same as for finite fields, but the representation theory of p-adic reductive groups is somewhat different from that of finite reductive groups. We start with a review of the necessary background on smooth representations of p-adic groups.

Generalized Steinberg representations

In this section, we let G denote a reductive group over ℚp. In contrast to the first two parts of this book, where representations of finite groups were considered, we are interested here in smooth representations of G(ℚp) on vector spaces over the field. Here smooth means that the stabilizer of any vector has to be open in G(ℚp). Such representations are generally infinite-dimensional, but we do not need to consider any topology on the underlying vector space. They form in an obvious way an abelian category Rep∞(G(ℚp)).

We note that the subcategory of finitely generated objects is much bigger than that of finite length objects (unless G is anisotropic). Moreover, both subcategories are known to be abelian, but the canonical map between their Grothendieck groups is far from being injective, see.