T.N. Venkataramana

Report for the Arbeitstagung

“Restriction maps in cohomology of arithmetic groups”

We give a criterion to determine when a Hecke translate of a cohomology class on a compact Shimura variety S(Γ) restricts non-trivially to a smaller Shimura variety.

In details, let G, H be connected, semi-simple simply connected algebraic groups over ℚ, such that all their ℚ-simple factors are noncompact at infinity (this is to ensure that G(ℚ) is dense in G(𝔸_{f}), 𝔸_{f} = the ring of finite adeles over ℚ). Let f : H → G be a morphism of ℚ-algebraic groups, with finite kernel. Fix a maximal compact subgroup K_{∞} ⊂ G(ℝ) so that K_{∞}^{H} := f^{−1}(K_{∞}) is a maximal compact subgroup of H(ℝ). One has the associated map f : Y := H(ℝ)/K_{∞}^{H} → G(ℝ)/K_{∞} := X; of symmetric spaces. For each congruence arithmetic group Γ ⊂ G(ℚ), write S(Γ) = Γ\X, S_{H}(Γ) = Γ ∩ H\Y, and j = j(Γ) : S_{H}(Γ) → S(Γ). If g ∈ G(ℚ), one has an isomorphism S(g^{+}Γg) ⟶^{g} S(Γ) induced by x ↦ g(z) on X, and j_{g} : S_{H}(g^{−1}Γg) ⟶^{j} S(Γ) ⟶^{g} S(Γ) is the composite. Consider the restriction map

Res = ∏ j_{g}^{∗} : H^{∘}(S(Γ)) → ∏_{g∈G(ℚ)}(S_{H}(g^{−1}Γg))

\underline{Theorem 1}: Suppose G, H are as in the foregoing, so that both Y, X are Hermitian symmetric and f : Y → X is holomorphic. Suppose S_{H}(Γ) is a divisor on S(Γ). Then

Res = H^{m}(S(Γ), ℂ) → ∏ H^{m}(S_{H}(Γ), ℂ)

is injective for m ≤ dim S_{H}(Γ) (= dim S(Γ) − 1).

\underline{Corollary (of the proof)}: Suppose G, H are as in Theorem 1. Suppose that up to compact factors, G(ℝ) and H(ℝ) are of the form

(1) G(ℝ) = SU(n, 1), H(ℝ) = SU(m, 1) included in the natural imbedding in SU(n, 1)

(2) G(ℝ) = Spin(n, 2), H(ℝ) = Spin(m, 2)

(3) G(ℝ) = Sp_{2}, H(ℝ) = Sp_{1} × Sp_{1}. Then Res : H^{i}(S(Γ)) → ∏ H^{i}(S_{H}(Γ)) (i ≤ dim Y) is injective.

Let g_{0}, k_{0} be the real Lie algebras of G(ℝ) and K_{∞}, and g_{0} = k_{0} ⊕ p_{0} the associated Cartan decomposition. Let g, k, p be the complexifications of g_{0}, k_{0} and p_{0}.

We view g_{0} and g_{u} = k_{0} ⊕ i p_{0} as real subalgebras of g. Let G_{ℂ} be the complexifications of G and G_{u} ⊂ G_{ℂ} the analytic subgroup with Lie algebra g_{u}. Let X̂ = G_{u}/K_{∞} be the “compact

dual” of X. One has the Matsushima formula H^{∘}(S(Γ)) = Hom_{K_{∞}}(Λ^{∘}_{p}, C^{∞}(Γ\G)(0)). Thus

H^{∘}(X̂) = Hom_{K_{∞}}(Λ_{p}, ℂ) ↪ H^{∘}(S(Γ)).

Here Λ_{p} is the exterior algebra of p and C^{∞}(Γ\G)(0) the space of smooth functions on Γ\G killed by the Casimir of g. Let Ŷ ↪ X̂ be the compact dual of Y, and [Ŷ] the Poincaré dual of the cycle Ŷ in X̂. Thus [Ŷ] ∈ H^{∘}(X̂) ⊂ H^{∘}(S(Γ)). Let [S_{H}(Γ)] be the Poincaré dual of S_{H}(Γ) in H^{∘}(S(Γ)).

Consider the direct limit \underrightarrow{lim}_{Γ} H^{∘}(S(Γ)) as Γ varies through congruence arithmetic subgroups

of G(ℚ). The density of G(ℚ) in G(𝔸_{f}) implies that

\underrightarrow{lim} H^{∘}(S(Γ)) = Hom_{K_{∞}}(Λp, C^{∞}(G(ℚ)\G(𝔸))(0))

=^{def} H^{∘}(Sh(G)).

Now G(𝔸_{f}) acts on H^{∘}(Sh(G)) by right translations, and it is known that H^{∘}(Sh(G)) may be written as a direct sum H^{∘}(Sh(G)) = ⨁ m(π_{f})π_{f} of irreducible representations π_{f}, each occurring with a finite multiplicity, m(π_{f}).

Now H^{∘}(S(Γ)) = H^{∘}(Sh(G))^{K}, where K is the closure of Γ(⊂ G(ℚ) ⊂ G(𝔸_{f})) in G(𝔸_{f}). Hence ξ_{Γ} = [S_{H}(Γ)] is a class in H^{∘}(Sh(G)). Let V_{Γ} be the ℂ-span of G(𝔸_{f})-translates of ξ_{Γ}.

\underline{Theorem 2}:

1) The space V_{Γ}^{G(𝔸_{f})} of G(𝔸_{f})-invariants in V_{Γ}, is spanned by [Ŷ].

2) If w ∈ H^{∘}(Sh(G)) such that Res(w) = 0, then w ∧ [Ŷ] = 0.

We now indicate a proof that Theorem 2 ⇒ Theorem 1. For simplicity assume that G(ℝ) = SU(n, 1), H(ℝ) = SU(n − 1, 1) up to compact factors. Then X = unit ball in ℂ^{n} ↩ Y = unit ball in ℂ^{n−1}, X̂ = ℙ^{n}(ℂ), and [Ŷ] = L ∈ H^{2}(ℙ^{n}(ℂ)) is a generator. If w ∈ H^{m}(Sh(G)), with m ≤ n − 1, then Res(w) = 0 ⇒ w ∧ L = 0 (by Th. 2). This implies w = 0 by Lefschetz’s theorem or hyperplane sections.

From now on we assume f : Y → X is a holomorphic imbedding of Hermitian symmetric domains. Let T ⊂ K_{∞} be a maximal torus in K_{∞} (also in G). Let t_{0}, t its real and complex Lie algebras, Φ^{+}(k, t) = roots of t in k which are ‘positive’. Fix X ∈ i Lie(T), s.t. α(X) ≥ 0

for all α ∈ Φ^{+}(k, t), set q = g^{x} ⨁_{α(X)>0}g_{α}, U = ⨁_{α(X)>0}g_{α}. Then u ∩ p = u ∩ p^{+} ⊕ u ∩ p^{−},

put m = dim(u ∩ p). Let u^{−} = ⨁_{α(x)<0}g_{α}, and consider w(q) = u ∩ p^{+} ⊕ u^{−} ∩ p^{+} ⊂ p^{+}. w(q)

is also m-dimensional. Let V^{+}(q) be the K_{∞}-module generated by Λ^{m} W (q) in Λ^{m} p^{+}. Write

E(G, H, m) = K_{∞}-module generated by Λ^{m} p_{H}^{+} in Λ^{m} p^{+}. Put V (q) = K_{∞}-span of Λ^{m} u ∩ p in Λ^{m}. Content-Length: 1476 X-Lines: 67 Status: RO \underline{Theorem} (Parthasarathy, Vogan-Zuckerman): Irreducible unitary representations π_{∞} of G(ℝ) with H^{∘}(g, K_{α}, π_{∞}) ≠ 0 are parametrised by the q’s as above. Denote the associated representation by A_{q}. Then

H^{i}(g, K_{∞}, π_{∞}) = Hom_{K_{∞}}(V (q)_{i}, π_{∞})

where V (q)_{i} is the isotypic of V (q) in Λ^{i} φ.

\underline{Definition}: We say that w ∈ H^{i}(Sh(G)) is of type A_{q} if the image of w (w thought of as a

homomorphism of Λ^{i} p into C^{∞}(G(ℚ)\G(𝔸))) generates a submodule isomorphic to A_{q}. If the degree i of w is the dimension of u ∩ p, then we say that w is \underline{strongly primitive}.

\underline{Theorem 3} Let w ∈ H^{m}(Sh(G)) be a strongly primitive class of type A_{q}. If Res(w) = 0, then

V^{+}(q) ∩ E(G, H, m) = 0.

Thus Theorem 3 gives a sufficient condition that a class w has nonzero restriction. By applying Theorem 3 to cup-products, we get:

\underline{Theorem 4}: Let w, w′ be strongly primitive classes of type A_{q}, A_{q′} and of degrees m, m′ on Sh(G), with m + m′ ≤ dim(Sh(G)). Then g^{∗}w ∧ w′ = 0 for all g ∈ G(𝔸_{f}) only if

V^{+}(q) ∧ V^{+}(q′) = 0.