Period relations for cusp forms of GSp₄

A.1 Two relative Lie algebra cohomology theories related to Shimura varieties

In this appendix, we let (G,X) be the datum defining a Shimura variety in the sense made precise in [12, Section 1.1]. For the sake of completeness, we recall that this means that G is a connected reductive linear algebraic group over ℚ and X is a G⁢(ℝ)-conjugacy class of homomorphisms h:Rℂ/ℝ⁢(GL1)→G×ℚℝ such that the following conditions hold:

1. The Hodge structure on the Lie algebra 𝔤ℚ of G given by Ad∘h is of type (0,0)+(1,-1)+(-1,1).

2. The automorphism Ad⁢(h⁢(i)) induces a Cartan involution on G𝗌𝗌⁢(ℝ), with G𝗌𝗌 being the derived group of G. The ℝ-group G𝗌𝗌×ℚℝ has no anisotropic factors over ℚ.

3. The weight map h∘w:GL1×ℚℝ→G×ℚℝ, where w:GL1×ℚℝ→Rℂ/ℝ⁢(GL1) is the canonical co-norm map, is defined over ℚ.

4. For a maximal ℚ-split torus Z′⊂ZG, the quotient ZG⁢(ℝ)/Z′⁢(ℝ) is compact.

With these assumptions, X is the finite disjoint union of Hermitian symmetric spaces of the form G𝗌𝗌⁢(ℝ)∘/K𝗌𝗌, where K𝗌𝗌 is a maximal connected compact subgroup of G𝗌𝗌⁢(ℝ). We let K be the centralizer of a fixed point h∈X in G⁢(ℝ). It contains the product of K𝗌𝗌 and ZG⁢(ℝ). Let 𝔨 be the Lie algebra of K. It operates by the adjoint action on 𝔤ℂ, and we obtain a 𝔨-invariant decomposition

Here, 𝔭- (resp. 𝔭+) is the holomorphic (resp. anti-holomorphic) tangent space of X at h. We let

This is a parabolic subalgebra of 𝔤ℂ with Levi subalgebra 𝔨ℂ and nilpotent, even abelian, radical 𝔭+. Observe that 𝔭h lies somewhat “skew” to the real structure of 𝔤ℂ=𝔤⊕i⁢𝔤 as 𝔭h∩𝔤=𝔨.

For us, a 𝔤-module V, which is also a representation of K, is called a (𝔤,K)-module if it is a (𝔤𝗌𝗌,K𝗌𝗌)-module in the sense of Borel–Wallach [4, Section 0.2] by restriction. Mainly to set notation and for the sake of precision, we will now rapidly recall the definition of two relative Lie algebra cohomology theories.

The relative Lie algebra cohomology Hq⁢(𝔤,𝔨,V) of V was defined in [4, Section I.1.2]. In [4, Section I.5.1], also the (𝔤,K)-cohomology of V was defined. It is the cohomology Hq⁢(𝔤,K,V) of the complex

If K is connected, then Hq⁢(𝔤,𝔨,V)=Hq⁢(𝔤,K,V). We recall that an irreducible representation Vλ of the real Lie group G⁢(ℝ) on a finite-dimensional complex vector space is called algebraic if the (extended) action of the complex Lie group G⁢(ℂ) on Vλ is a representation of the linear algebraic group G×ℚℂ over ℂ. Finally, we say that a (𝔤,K)-module V is (𝔤,K)-cohomological if there is an irreducible finite-dimensional algebraic G⁢(ℝ)-module Vλ such that Hq⁢(𝔤,K,V⊗Vλ)≠0 for some degree q.

The (𝔭h,K)-cohomology of a (𝔤,K)-module V is the cohomology of the complex

with df defined as above. Following [13], we say that a (𝔤,K)-module V is (𝔭h,K)-cohomological if there is an irreducible finite-dimensional K-module Vτ such that the space Hq⁢(𝔭h,K,V⊗Vτ)≠0 for some degree q.

A.2 A general result on the relation of (𝔤,K)-cohomology and (𝔭h,K)-cohomology

We denote by 𝒞G (resp. 𝒞K) the collection of all equivalence classes of irreducible algebraic representations of G⁢(ℝ) (resp. irreducible finite-dimensional representations of K). We assume to have fixed a Cartan subalgebra 𝔱ℂ of 𝔨ℂ and a set of positive roots Δ+⁢(𝔨ℂ,𝔱ℂ). Because of (A.1), 𝔱ℂ is a Cartan subalgebra of 𝔤ℂ, too, and we assume that Δ+⁢(𝔤ℂ,𝔱ℂ) is a choice of positive roots for 𝔤ℂ with respect to 𝔱ℂ, extending the given choice Δ+⁢(𝔨ℂ,𝔱ℂ) for 𝔨ℂ. We assume that 𝔭h is a standard parabolic subalgebra of 𝔤ℂ with respect to Δ+⁢(𝔤ℂ,𝔱ℂ), i.e., all roots in 𝔭+ are positive (which also explains the notation).

Clearly, 𝔭¯h:=𝔨ℂ⊕𝔭- is the parabolic subalgebra of 𝔤ℂ, which is opposite to 𝔭h. If Vλ∈𝒞G, then Vλ is determined by its highest weight λ with respect to Δ+⁢(𝔤ℂ,𝔱ℂ). Similarly, if Vτ∈𝒞K, then Vτ is determined by its highest weight τ with respect to Δ+⁢(𝔨ℂ,𝔱ℂ). Let

be the half-sum of roots in Δ+⁢(𝔤ℂ,𝔱ℂ) (resp. Δ+⁢(𝔨ℂ,𝔱ℂ)), and let W (resp. W𝔨) be the Weyl group of 𝔤ℂ (resp. 𝔨ℂ) with respect to 𝔱ℂ. Then the infinitesimal characters χVλ (resp. χVτ) of Vλ (resp. Vτ) are determined by λ+ρ (resp. τ+ρc) up to the action of W (resp. W𝔨). We will use the notation χVλ=χλ+ρ and χVτ=χτ+ρc. Recall that there is the obvious surjection

cf. [13, p. 31], mapping χΛ onto ξ⁢(χΛ)=χΛ+ρn. Here, Z⁢(𝔨ℂ) (resp. Z⁢(𝔤ℂ)) denotes the center of the universal enveloping algebra of 𝔨ℂ (resp. 𝔤ℂ), and ρn=ρ-ρc is the half-sum of non-compact roots in Δ+⁢(𝔤ℂ,𝔱ℂ) (i.e., the roots appearing in 𝔭+). The following is the main result of our appendix.

We conclude this appendix by the following corollary.