On the cohomology of uniform arithmetically defined subgroups in SU (2n)

Transcription

1 Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1 On the cohomology of uniform arithmetically defined subgroups in SU (2n) By JOACHIM SCHWERMER and CHRISTOPH WALDNER Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria (Received ) Abstract We study the cohomology of compact locally symmetric spaces attached to arithmetically defined subgroups of the real Lie group G = SU (2n). Our focus is on constructing totally geodesic cycles which originate with reductive subgroups in G. We prove that these cycles, also called geometric cycles, are non-bounding. Thus this geometric construction yields nonvanishing (co)homology classes. In view of the interpretation of these cohomology groups in terms of automorphic forms, the existence of non-vanishing geometric cycles implies the existence of certain automorphic forms. In the case at hand, we substantiate this close relation between geometry and automorphic theory by discussing the classification of irreducible unitary representations of G with non-zero cohomology in some detail. This permits a comparison between geometric constructions and automorphic forms. Introduction Let G be a connected semi-simple real Lie group with finite center, and let K be a maximal compact subgroup. The homogenous space K \ G = X attached to the Riemannian symmetric pair (G, K) is a Riemannian symmetric space, diffeomorphic to some R n. Suppose that Γ G is a torsion free discrete subgroup (such that G/Γ has finite volume with respect to some non-zero G-invariant measure). Then Γ acts freely on X, and the quotient space X/Γ is a Riemannian locally symmetric space. Our object of concern is the cohomology space H (X/Γ, C), viewed as singular cohomology or, by use of the derham theorem, as the cohomology attached to the complex of Γ-invariant differential forms on X. We have to distinguish two cases: Γ is uniform (or cocompact) if G/Γ is compact, nonuniform otherwise. For example, the principal congruence subgroups Γ(m) SL n (Z) of level m are torsion free subgroups of finite index for m > 4, and Γ(m) are nonuniform discrete subgroups of SL n (R). It is a more difficult task to construct uniform discrete subgroups in a given G. However, by a number theoretical approach, a connected semi-simple Lie group always has discrete subgroups Γ so that G/Γ is compact [1]. Arithmetically defined subgroups in semi-simple algebraic groups defined over some algebraic number field give rise to such uniform examples. In this paper, in the case of uniform discrete subgroups in the real Lie group G = SU (2n), we focus on constructing totally geodesic cycles in X/Γ which originate with reductive subgroups of G. In many cases, it can be shown that these cycles, to be called geometric cycles,

2 2 Joachim Schwermer and Christoph Waldner yield non-vanishing (co)homology classes. In particular, such geometric cycles naturally occur as fixed point components of an automorphism of finite order which is induced by a rational automorphism of the algebraic group which underlies G. In the case at hand, we consider uniform discrete subgroups Γ G which arise as arithmetically defined subgroups in suitable algebraic groups G defined over some totally real algebraic number field F. These groups admit various families of rational F -automorphisms of finite order, one to be denoted by {ν k } k=1,...,n 1 where ν k is an automorphisms of G of order two, and another one consisting of two involutions µ s, s = 1, 2. Then the main result of this paper is the following Theorem. There exists a uniform discrete arithmetically defined subgroup Γ of the real Lie group SU (2n) so that the cohomology H j (X/Γ, C) contains a non-trival cohomology class for any integer j = 4k(n k), and j = dim X/Γ 4k(n k), [k = 1,..., n 1] (with dim X/Γ = (n 1)(2n + 1)) respectively j = (n + 1)(n 1) and j = n(n 1). By duality, these classes are detected by the fundamental classes of totally geodesic submanifolds, so called geometric cycles, of the form C(ν k ) resp. C (ν k ) in the first case, and C(µ s ) resp. C (µ s ) in the second case. These classes cannot be obtained as the restriction of a continuous class from the underlying Lie group SU (2n). One origin for this approach to understand the geometry of compact locally symmetric spaces of the form X/Γ and their cohomology lies in the work of Millson-Raghunathan [12]. There they construct non-bounding geometric cycles in the case of groups of units of certain quadratic or Hermitian forms which do not represent zero over their field of definition. These results are now superseded by the treatment given in [16] where, as one aspect in the study of geometric cycles and corresponding intersection numbers, the systematic use of non-abelian Galois cohomology serves as a suitable frame work to analyze the intersections of geometric cycles and the questions of orientability involved. The present case-study is an application of the general results in [16]. Since the cohomology of an arithmetically defined group Γ is strongly related to the theory of automorphic forms with respect to Γ this geometric construction of non-vanishing classes leads to results concerning the existence of specific automorphic forms. The derham cohomology groups H (X/Γ, C) can be interpreted as the relative Lie algebra cohomology groups H (g, K, C (G/Γ) K C) where g denotes the complexified Lie algebra of G. Since Γ is a uniform discrete group, by a result of Matsushima, the study of this cohomology amounts to the study of the finite algebraic sum π Ĝm(π, Γ) H (g, K, H π,k C) where the sum ranges over all irreducible unitary representations (π, H π ) of G which occur with non-zero multiplicity m(π, Γ) in the spectral decomposition of the space of square integrable functions L 2 (G/Γ) = ˆ π Ĝm(π, Γ) H π and have non-vanishing relative Lie algebra cohomology. Thus, we make explicit the general classification, due to Vogan-Zuckerman, of unitary representations with non-vanishing cohomology in the case of the real Lie group SU (2n).

3 On the cohomology of uniform arithmetic subgroups in SU (2n) 3 In view of this representation theoretical interpretation of the cohomology groups, the existence of non-vanishing geometric cycles implies the existence of certain automorphic forms. However, on one hand, a direct comparison of the various families of non-vanishing classes for X/Γ with the family {A q } q of irreducible unitary representations of SU (2n) with non-zero cohomology shows that the cardinality of the latter exceeds by far the number of geometrically constructed cycles. Therefore the geometric construction misses possible cohomological degrees. On the other hand, in some cases one can identify an automorphic form which corresponds to a non-bounding geometric cycle but, in all generality, this is an enticing open problem. It might be that the theory of period integrals is of some help in a structural characterization. We outline the content of the paper: In Section 1 we give an account of the necessary background material for the group SU (2n). This group is of type 2 A 2n 1, and may also be viewed as the special linear group over the division algebra H of real quaternions. Then, in Section 2, given a discrete subgroup Γ of a Lie group G we briefly review how the derham cohomology groups are related to the relative Lie algebra cohomology groups. It is because of this isomorphism that representation theoretic methods can be used to describe the cohomology of X/Γ. In Section 3, based on a brief review of the constructive approach to the classification [22] of irreducible unitary representations of a real semi-simple Lie group with non-zero cohomology, we enumerate these representations (up to infinitesimal equivalence) in the case at hand and give their cohomology. In Section 4, mainly to fix notations, we survey the general construction of geometric cycles in arithmetic quotients. Then, in Section 5, we construct suitable Q-anisotropic algebraic groups of type 2 A 2n 1, introduce various families of rational automorphisms on these and determine the corresponding fixed pints sets. In section 6, we construct non-bounding cycles and give a proof of the main result above. We conclude with some remarks regarding the geometric construction of non-vanishing cohomology classes and their interpretation in terms of automorphic forms. Notations (1) Let k/q be an arbitrary finite extension of the field Q, and denote by O k its ring of integers. The set of places will be denoted by V, and V (resp. V f ) refers to the set of archimedean (resp. non archimedean) places of k. The completion of k at a place v V is denoted by k v ; its ring of integers by O v (v V f ). (2) The algebraic groups considered are linear. If H is an algebraic group defined over a field k, and k is a commutative k algebra, we denote by H(k ) the group of k valued points of H. When k is a field we denote by H/k the k algebraic group H k k obtained from H by extending the ground field from k to k. 1. The real Lie group SU (2n) In this section, mainly to fix notations, we recollect some background material regarding the real Lie group SU (2n), its realization as SL(n, H), and the corresponding root systems The group The real quaternions, denoted by H, is an algebra over R with a basis denoted by e 0, e 1, e 2, e 3, and multiplication defined so that e 0 is the multiplicative identity element, and e 2 1 = e 2 2 = e 2 3 = e 0, and e 1 e 2 = e 2 e 1 = e 3. These equations determine how basis elements are multiplied, and thus how any elements in H are multiplied. This is the archetypical example of

4 4 Joachim Schwermer and Christoph Waldner a central simple algebra over R; it is a division algebra whose center is R = Re 0. Given a quaternion x = x 0 e 0 + x 1 e 1 + x 2 e 2 + x 3 e 3, the assignment x x = x 0 e 0 x 1 e 1 x 2 e 2 x 3 e 3 defines an involution on H, usually called the quaternionic conjugation or canonical involution. The algebra H contains the field C as a subfield via the embedding a + bi ae 0 + be 1. Thus, a finite dimensional vector space V over H can be viewed as a vector space over C. A quaternion can be uniquely written in the form (x 0 e 0 + x 1 e 1 )e 0 + (x 2 e 0 + x 3 e 1 )e 2. It follows that the vector space H n, viewed as a C-vector space, is canonically isomorphic to C 2n, via the assignment (a 1,, a n, a n+1,, a 2n ) (a 1 e 0 + a n+1 e 2,, a n e 0 + a 2n e 2 ). We identify H n with C 2n via this isomorphism. A given H-linear endomorphism φ of C 2n is also C-linear. Conversely, a C-linear endomorphism of C 2n is H-linear if and only if l q φ = φ l q for all q H where l q denotes the map given by left multiplication with q. One sees that this condition is satisfied if and only if l e2 φ = φ l e2. We denote by GL(n, H) the subgroup of H-linear automorphisms in GL(C 2n ). This group coincides with the group of fixed points under the involutive semiautomorphism of GL(C 2n ), defined by φ l e2 φ l e2. Thus, the group GL(n, H) is a real form of GL(2n, C), and the Lie algebra of GL(n, H) is real form of gl(2n, C). Notice that the groups GL(2n, C) and GL(n, H) are connected, and we have that the special linear group SL(2n, C) is the derived group of the group GL(C 2n ). We denote the derived group of GL(n, H) by SL(n, H). This group is also called SU (2n). The Lie algebra of SL(n, H) is the derived Lie algebra of gl(2n, C), thus a real form of a Lie algebra of type A 2n 1. The intersection of the group SL(n, H) with the maximal compact subgroup U(2n) in GL(2n, C) is the group Sp(n) Cartan decomposition and root system Let G be the group SL(n, H) with its maximal compact subgroup K = Sp(n). Let g 0 = sl(n, H) be the Lie algebra of SL(n, H) and θ : g 0 g 0, X t X the corresponding Cartan involution of g 0. In the Cartan decomposition g 0 = k 0 p 0 of g 0, k 0 = sp(n) is the Lie algebra of Sp(n), and p 0 = {X g 0 t X = X}. The space h0 = n {H = diag(x 1 + iy 1,..., x n + iy n ) s=1 x s = 0} is a θ-stable Cartan subalgebra of g 0 with Cartan decomposition h 0 = t 0 a 0, where t 0 = h 0 k 0 = {diag(iy 1,..., iy n )} and n a 0 = h 0 p 0 = {diag(x 1,..., x n ) s=1 x s = 0}. We define maps e s, f s : h 0 C, by e s (H) = iy s, f s (H) = x s, s = 1,..., n. Then Φ(g, h) = {±e j ± e k ± (f j f k ) 1 j < k n} {±2e s s = 1,..., n} is the root system of g relative h. The map θ acts by +1 on e s and by 1 on f s. We define α i = e i e i+1 + (f i f i+1 ), i = 1,..., n 1 and β n = 2e n. We take as θ-stable system of simple roots. Then Π(g, h) = {α i, θ(α i ), i = 1,..., n 1, β n } Φ + (g, h) = {e j ± e k ± (f j f k ) 1 j < k n} {2e s s = 1,..., n}. Occasionally, it is more tractable to use the initial formulation of G resp. g 0 as SU (2n) resp. su (2n). This frame work in place, writing 1 := e 0, i := e 1, j := e 2, k := e 3 in the following,

5 On the cohomology of uniform arithmetic subgroups in SU (2n) 5 ( SU z1 z (2n) = { 2 z 2 z 1 ) ( SL(2n, C)} and su Z1 (2n) = { Z 2 Z 2 Z1 ) sl(2n, C)}. Write v = a+bi+cj+dij M n (H), ( a, b, c, d M n (R) as v = Z 1 +Z 2 j, where Z 1 = a+bi, Z 2 = Z1 c di M n (C). The map v Z ) 2 induces the isomorphism g 0 = su (2n). Let Z 2 Z1 ( ) I J = n. In this formulation g 0 is a real form of g = sl(2n, C), obtained as the I n fixed points of the involution σ : g g, σ(x) = J XJ and the Cartan involution is again θ : g g, θ(x) = t X. Cleary, σ and θ commute with one another, and k = sp(2n, C), sp(2n, C) su (2n) = sp(n). In this formulation the vector spaces k, k 0, p,... have the form: ( ) A t C k = { sl(2n, C) A, D gl(n, C), A = t Ā, D = t D} C D ( Z1 k 0 = { Z ) 2 g 0 Z 1 = t Z1, Z 2 = t Z 2 } Z 2 Z1 ( ) A t C p = { sl(2n, C) A, D gl(n, C), A = t Ā, D = t D} C D ( Z1 p 0 = { Z ) 2 g 0 Z 1 = t Z1, Z 2 = t Z 2 } Z 2 Z1 h = {diag(z 1,..., z 2n ) 2n s=1 z s = 0} h 0 = {diag(x 1 + iy 1,..., x n + iy n, x 1 iy 1,..., x n iy n ) n x s = 0} s=1 t 0 = {diag(iy 1,..., iy n, iy 1,..., iy n ) h 0 y 1,..., y n R} n a 0 = {diag(x 1,..., x n, x 1,..., x n ) x s = 0, x 1,..., x n R} s= The symmetric space and its compact dual The symmetric space X = Sp(n) \ SU (2n) attached to the Riemannian symmetric pair (G, K) of non-compact type is of type AII. Its dimension is (n 1)(2n + 1). The compact dual of X is X u := Sp(n) \ SU(2n). By [6, p. 493], the Poincaré polynomial of X u is given by n n P (X u, t) = (1 + t 4i 3 )/(1 + t) = (1 + t 4i 3 ). i=1 (Here the term 1/(1 + t) comes from the fact that in [6], they actually compute the cohomology of Sp(n) \ U(2n) = Sp(n) \ SU(2n) U(1).) Unfortunatally, there is no general method to determine the coefficients of these Poincaré polynomials directly out of this form. i=2 2. Cohomology of discrete groups and Lie algebra cohomology The derham cohomology groups attached to a torsion free discrete subgroup of a reductive real Lie group are related in a natural way to certain relative Lie algebra cohomology groups.

6 6 Joachim Schwermer and Christoph Waldner It is this transition by which some questions on the cohomology of discrete groups are turned into questions about cohomological properties of unitary representations of the underlying Lie group An interpretation in Lie algebra cohomology Let G be a real Lie group with finitely many connected components, g its Lie algebra, let K be a compact subgroup of G, k its Lie algebra. We denote the natural projection map G K\G by π. Let (ν, E) be a finite dimensional irreducible representation of G on a real or complex vector space E. We want to study the complex Ω (K\G, E) of smooth E-valued differential forms in terms of representation theory. There is a natural identification of the tangent space at the point e G with g. This gives rise to an identification of complexes Ω (G, E) Hom(Λ g, C (G) E). The pullback map π : Ω (K\G, E) Ω (G, E) of the C -map π is compatible with differentials thus an inclusion of complexes. We endow C (G) E with the G-module structure given as the tensor product of the left regular representation l of G on C (G) and of (ν, E). Then a q-form ω in Hom(Λ g, C (G) E) is in the image of π if ω is annihilated by the interior products i Y, Y k, and ω lies in Hom K (Λ g, C (G) E) where K acts on Λ g by the adjoint action. Then the space C (G) K of all C -vectors f for which l(k)f spans a finite-dimensional subspace of C (G) is preserved by the action of g (obtained by differentiation of l) and compatible with the action of K. Moreover C (G) K is locally finite as a K-module. Thus, C (G) K is a (g, K)-module. Then there is an isomorphism of graded complexes of Ω (K\G, E) onto C (g, K, C (G) K E) Given any discrete torsion free subgroup Γ of G the space of functions invariant by Γ acting on the right is a (g, K)-submodule of C (G) K. One obtains an isomorphism of Ω (K\G, E) Γ onto C (g, K, C (G/Γ) K E). Thus, there is a canonical isomorphism H (K\G/Γ, Ẽ) = H (Ω(K\G, E) Γ ) H (g, K, C (G/Γ) K E). We refer, for example, to [4, Chapter VII] for a more thorough treatment A result of Matsushima Suppose G is a real reductive Lie group with finitely many connected components, K G is a maximal compact subgroup and Γ G is a torsion free discrete subgroup so that the quotient G/Γ is compact. In that case the left regular representation of G on the space L 2 (G/Γ) of square integrable functions (modulo the center) on G/Γ decomposes as a direct Hilbert sum of irreducible unitary representations (π, H π ) of G with finite multiplicities L 2 (G/Γ) = ˆ π Ĝm(π, Γ) H π. Here Ĝ denotes the unitary dual of G, and the multiplicity m(π, Γ) with which (π, H π) occurs in L 2 (G/Γ) is a non-negative integer for each π. Given such an irreducible unitary Hilbert space representation (π, H π ) the space H π,k of all C -vectors v H π such that π(k)v spans a finite-dimensional subspace of H π carries a natural (g, K)-module structure. Since (π, H π ) is irreducible, H π,k is irreducible as a (g, K)-module. The space H π,k of K-finite vectors is dense in the space Hπ of C -vectors for H π, and there is an inclusion ˆ π Ĝm(π, Γ) H π,k C (G/Γ) K.

7 On the cohomology of uniform arithmetic subgroups in SU (2n) 7 Let (ν, E) be a finite-dimensional irreducible real of complex representation of G. Then this inclusion induces an isomorphism H (K\G/Γ, Ẽ) = H (g, K, C (G/Γ) K E) π Ĝm(π, Γ) H (g, K, H π,k E) where the right hand side is a finite direct algebraic sum. This result is due to Matsushima [10]. The representations which can possibly contribute to the sum on the right hand side are usually called representations with non-vanishing (Lie algebra) cohomology. By [4, I, 4.2], the cohomology H (g, K, H π,k E) can only be non-zero if the center of the enveloping algebra of g acts on H π E as in the trivial representation. As a consequence, given (ν, E) there are (up to infinitesimal equivalence) only finitely many irreducible representations (π, H π ) of G with H (g, K, H π,k E) 0. Only those (π, H π ) might occur whose infinitesimal character χ π coincides with the one of the contragredient representation of (ν, E). If one drops the assumption that the quotient G/Γ is compact these results are not true any more. However, by replacing the coefficient module C (G/Γ) by appropriate spaces of functions which satisfy certain growth conditions, they hold in a modified form. 3. Unitary representations with non-vanishing cohomology In this section, we briefly discuss the constructive approach to the classification [22] of irreducible unitary representations of a connected real semi-simple Lie group with non-vanishing relative Lie algebra cohomology. This general result allows us to enumerate (up to infinitesimal equivalence) the irreducible unitary (g, K) - modules with non-vanishing Lie algebra cohomology in the case G = SU (2n) in an explicit way. They are parametrized by θ-stable parabolic subalgebras q of g. Given an irreducible unitary representation (π, H π ) of G, we denote the Harish Chandra module of H π (i.e., the set of K finite vectors in the space of C vectors of H π ) by the same letter or by H π,k The classification Let G be connected real semi-simple Lie group G with finite center, K G a maximal compact subgroup, We denote by g 0 (resp. k 0 ) the Lie algebra of G (resp. K) and g (resp. k) its complexification. Let θ the corresponding Cartan involution, with Cartan decomposition g 0 = k 0 p 0. Let t 0 k 0 be a Cartan subalgebra, Φ + (k, t) a positive root system, h 0 = Z g0 (t 0 ) a θ-stable Cartan subalgebra of g 0 and extend Φ + (k, t) to a positive root system Φ + (g, h). Let ρ be the half sum of all roots in Φ + (g, h). By definition a θ-stable parabolic subalgebra q of g 0 is a parabolic subalgebra of g such that θq = q, and q q = l is a Levi subalgebra of q where q refers to the image of q under complex conjugation with respect to the real form g 0 of g. Write u for the nilradical of q. Then l is the complexification of a real subalgebra l 0 of g 0. The normalizer of q in G is connected since G is, and it coincides with the connected Lie subgroup L of G with Lie algebra l 0. The Levi subgroup L has the same rank as G, is preserved by the Cartan involution θ, and the restriction of θ to L is a Cartan involution. Moreover, the group L contains a maximal torus T K. We will indicate below a construction of all possible θ-stable parabolic subalgebras q in g 0 up to conjugation by K. There are only finitely many K-conjugacy classes of θ-stable parabolic subalgebras q in g 0. Via cohomological induction, a θ-stable parabolic subalgebra q gives rise to an irreducible

8 8 Joachim Schwermer and Christoph Waldner unitary representation R S q (C) = A q of G with S = dim(u k) [22, Thm. 2.5]. It is uniquely determined up to infinitesimal equivalence by the K-conjugacy class of q. One has H j (g, K, A q,k ) = Hom L K ( j R (l p), C) where R = R(q) := dim(u p). In particular, note that the Lie algebra cohomology with respect to the representation A q vanishes in degrees below dim(u p) and above dim(u p) + dim(l p). One may reformulate this result in a sightly different way. Let L u the compact form of the real Lie group L and X L,u the compact dual to the symmetric space X L = (K L)\L. Then the right hand side is isomorphic to the cohomology H j R (L 0 u/(l u K) 0 ; C). By virtue of Poincare duality, we conclude 1 2 (dim X dim X L) = R(q). We denote by P (q, t) the Poincaré polynomial of H (g, K; A q,k ). Then this result implies that P (q, t) = t R(q) P (X L,u, t). If l k, the representation A q belongs to the discrete series of G. Then L = L K, and the cohomology H j (g, K, A q,k ) vanishes in all degrees j R(q) = dim(u p), that is, j 1 2 dim (G/K). One has HR(q) (g, K, A q,k ) = C (see e.g. [21, Sect. 3]). If the θ-stable parabolic subalgebra q is g then L coincides with G; we take A q = C. Suppose (π, H π ) is an irreducible unitary representation (π, H π ) of G with H (g, K, H π,k ) 0. Then there is a θ-stable parabolic subalgebra q of g so that π = A q. This construction [22, Thm.4.1] is essentially a consequence of results of Parthasarathy [13] One finds a proof of the unitarity of the representation A q in [20] Open polyhedral cones Fix an element x it 0 that is dominant for K. Then the θ-stable parabolic subalgebra q with Levi decomposition q = l + u associated to x is defined by l = h α(x)=0 g α and u = α(x)>0 g α (α(x) R for all α Φ). Let L be the connected subgroup of G with Lie algebra l 0 = l g 0. The connected group K acts on the set of θ-stable parabolic subalgebras via the Weyl group N K (t)/z K (t) = W K and on it 0. Hence one gets an induced surjective map it 0 /W K { θ-stable parabolic q } /K, x q x. We denote the W K -conjugacy class of x it 0, by the same letter x. If q x is K-conjugate to q x and x, x it 0 /W K, then q x = q x. We call the set C x := {x it 0 /W K q x = q x } an open polyhedral cone of it 0 /W K. One has the following one-to-one correspondence between the set of K-conjugacy classes of θ-stable parabolic subalgebras and the open poyhedral cones of it 0 /W K { } 1:1 θ-stable parabolic q /K { } open poyhedral cones of it 0 /W K On the set { θ-stable parabolic q } /K one has the equivalence q = u l q = u l : Φ(u p) = Φ(u p) u p = u p. Now it is known (see e.g. [17, Prop. 4.5]): { Aq q θ-stable parabolic }/ 1:1 { θ-stable parabolic q } / /K where on the left hand side denotes the relation given by means of infinitesimal equivalence of the corresponding Harish-Chandra modules.

9 On the cohomology of uniform arithmetic subgroups in SU (2n) The case of the real Lie group SU (2n) In making explicit this classification in the case of the real Lie group SU (2n) we use the notation introduced in Section 1.. We denote an ordered partition (i.e. one which depends on the order of the parts) n = k i=1 m i (resp. n 1 = k i=1 m i) of n (resp. n 1) into k parts, simply by {m 1 ; m 2 ;... ; m k } or m. The number of all ordered partitions of n (resp. n 1 ) into k parts is simply ( n 1 k 1 (resp. ( n 2 k 1) ). To a given partition m of n or n 1 into k parts, choose y 1,..., y k R such that y i > y i+1 > 0, i = 1,..., k 1 and define Y i = (y i, y i,..., y i ) R mi, for i = 1,..., k 1. { (yk,..., y Define Y k = k ) R m k, if k i=1 m i = n (0,..., 0) R mk+1, if k i=1 m i = n 1 and diag(y 1,..., Y k, Y 1,..., Y k ) =: x it 0 The associated θ-stable parabolic subalgebra q x is denoted by q m. Of course x depends on the choosen y i s but as we will see later, q x just depends on m. If u m is the unipotent radical of q m we write R m = dim(u m p). Theorem If (π, H π ) is an irreducible unitary representation of G = SU (2n) with nonvanishing cohomology H (g, K; H π,k ) then there is an ordered partition m = {m 1 ;... ; m k } of n or n 1 such that the Harish-Chandra module of (π, H π ) is the representation A qm. Furthermore, two representations A qm, A qm are equivalent if and only if m = m. Thus, one has a one-to-one correspondence / {A q q θ-stable parabolic } 1:1 { ordered partition of n} { ordered partition of n 1} The number of irreducible unitary representations (up to infinitesimal equivalence) with non zero cohomology is 3 2 n 2. The real Levi subalgebra l 0 of q m is given by { l 0 gl(m1, C) gl(m 2, C)... gl(m k 1, C) sl(m k, C) ir = gl(m 1, C) gl(m k 1, C) sl(m k + 1, H), k i=1 m i = n k i=1 m i = n 1 and the shift is R m = (n 1)(2n+1) dim X L 2, where { m 2 dim X L = 1 + m m 2 k 1, k i=1 m i = n m m 2 k 1 + m k k(2m k + 3), i=1 m i = n 1. In particular, all representations A q are non-tempered, except the unique fundamental series representation which corresponds to the partition {1;... ; 1} of n. The Poincaré polynomial of H (g, K; A qm ) is given by ) P (q m, t) = t Rm { 1 k ms 1+t s=1 i=1 (1 + t2i 1 ), mk +1 i=2 (1 + t 4i 3 ) k 1 s=1 ms i=1 (1 + t2i 1 ), k i=1 m i = n k i=1 m i = n 1. Proof. The Poincaré polynomial P (q m, t) can be easily computed by using the Poincaré polynomial of Sp(n) \ SU(2n) (see 1) and the Poincaré polynomial of the compact dual of X L = U(n) \ GL(n, C), i.e., of the space X L,u = U(n) \ U(n) U(n). Inspecting again [6, p. 493], we get P (X L,u, t) = n i=1 (1 + t2i 1 ). The Weyl group W K = N K (t)/z K (t) of K = Sp(n) is isomorphic to the semidirect product

10 10 Joachim Schwermer and Christoph Waldner S n {±1} n. The action of w W K on it 0 = {diag(y, y) y = (y 1,..., y n ) R n } is induced by w.diag(y, y) = { diag(yπ, y π ), if w = π S n, where y π = y π(1),..., y π(n) diag(ɛ i y, ɛ i y), if w = (1,..., 1, 1, 1,... ), where ɛ i = diag(1,..., 1, 1, 1,..., 1) is the diagonal matrix with 1 in the i-th entry. Any given element diag(y, y ) it 0 is conjugate via W K to an element x = diag(y, y), where y = diag(y 1,..., Y k, Y 1,..., Y k ), Y i = (y i, y i,..., y i ) R mi, i = 1,..., k for some positive integers m 1,..., m k with k i=1 m i = n. Further, we can assume that y 1 > y 2 > > y k 1 > y k 0. We have to distinguish the two cases y k > 0 and y k = 0. In each case we can assign to x the ordered partition m = {m 1 ; m 2 ;... ; m k }. Two θ-stable parabolic subalgebras q x, q x are equal, if and only if the corresponding ordered partition m, m are equal and (y k = y k = 0 y k, y k > 0). Hence { } 1:1 open poyhedral cones in it0 /W K { ordered partition of n} { ordered partition of n} { (m, ), if yk > 0 q x (, m), if y k = 0 First assume that y k > 0. Then {( ) A } A, B are block diagonal matrices, l = sl(2n, C) B with blocks M i gl(m i, C), i = 1,..., k {( ) } A A is a block diagonal matrix, l 0 = sl(2n, C) Ā with blocks M i gl(m i, C), i = 1,..., k = gl(m 1, C) gl(m 2, C)... gl(m k 1, C) sl(m k, C) ir ) A is a block upper triangular matrix, A B u p = ( t sl(2n, C) with all diagonal blocks M A i gl(m i, C), i = 1,..., k zero, and B = t B 1 a s 1, s 1 < b n, or s 1 < a s 2, s 2 < b n, or Φ(u p) = {0} e a e b s 2 < a s 3, s 3 < b n, or {e a + e b 1 a < b n}.... s k 2 < a s k 1, s k 1 < b n

11 On the cohomology of uniform arithmetic subgroups in SU (2n) 11 Second assume that y k = 0. Then ( ) A E l = sl(2n, C) F B ( ) A F l 0 = sl(2n, C) F Ā A, B, E, F are block diagonal matrices, with blocks M i gl(m i, C), i = 1,..., k, the M i s for E, F are all zero except M k A, F is a block diagonal matrix, with blocks M i gl(m i, C), i = 1,..., k, the M i s for F are all zero except M k = gl(m 1, C) gl(m k 1, C) sl(m k, H) A is a block upper triangular matrix, ) A B with all diagonal blocks M u p = ( t sl(2n, C) i gl(m i, C), A i = 1,..., k zero, B = t B, and the k th block matrix in the diagonal of B is 0 1 a s 1, s 1 < b n, or s 1 < a s 2, s 2 < b n, or Φ(u p) = {0} e a e b s 2 < a s 3, s 3 < b n, or... s k 2 < a s k 1, s k 1 < b n { e a + e b 1 a s k 1, a < b n }. Let m and m be ordered partitions of n. Assume that q resp. q is the θ-stable parabolic subalgebra, associated to a (m, ) or (, m) (resp. (m, ) or (, m )). By the sets Φ(u p) we can read off that these two θ-stable parabolic subalgebras q, q (if they are assumed to be not conjugate via K), are equivalent if and only if q is associated to a (m, ), q is associated to a (, m ) and m k = m k = 1. So we have to consider the ordered partition {m 1 ; m 2 ;... ; m k 1 ; 1} in the list of infinitesimal equivalence classes only once. But the set of ordered partitions of n with m k 1 is exactly the set of ordered partition of n 1 (via m k m k 1). This shows the bijection in the assertion of the proposition. The number of all irreducible unitary representation with non zero cohomology is therefore n ( n 1 ) n 1 ( k=1 k 1 + n 2 k=1 k 1) = 2 n n 2 = 3 2 n 2. Remark. Note that the trivial partition {n 1} of n 1 yields the trivial representation The fundamental series representation The partition {1;... ; 1} of n yields the following data for the corresponding irreducible unitary representation A q : l 0 = gl(1, C) n 1 ir, L = GL(1, C) n 1 U(1) R(q) = dim(u p) = n(n 1) n 1 ( ) n 1 P (q, t) = t n(n 1) (1 + t) n 1 = t k+n(n 1). k Hence we obtain for the relative Lie algebra cohomology the following result: dim H i (g, K; A q ) = k=0 { ( n 1 i n(n 1)), n(n 1) i (n + 1)(n 1) 0, otherwise

12 12 Joachim Schwermer and Christoph Waldner The resulting representation A q is, because [l, l] = 0 k, a fundamental series representation of G. 4. Geometric construction of cohomology classes In this section, we give a brief overview of the general construction of geometric cycles in arithmetic quotients X/Γ as outlined in [18, Sections 6 and 9]. In the next section, in the specific case of interest for us, we use one of the techniques developed in [16] to show that certain geometric cycles exist and represent non-zero homology classes for the underlying manifold X/Γ. This relies on the approach via excess intersections Generalities Let G be a connected reductive algebraic group defined over an algebraic number field k. We choose an embedding ρ : G GL N and write G Ok = G(k) GL N (O k ) for the group of integral points with respect to ρ. For every archimedean place v V corresponding to the embedding σ v : k k there are given a local field k v = R or C and a real Lie group G v = G σv (k v ). The group G = G v, v V viewed as the topological product of the groups G v, v V, is isomorphic to the group of real points G (R) of the algebraic Q-group G = Res k/q G obtained from G by restriction of scalars. In G, we identify G(k) resp. G Ok with the set of elements (g σv ) v V with g G(k) resp. g G Ok. If Γ is an arithmetic subgroup of G then Γ is a discrete subgroup in G. Each of the groups G v has finitely many connected components. The factor G v has maximal compact subgroups, and any two of these are conjugate by an inner automorphism. Thus, if K v is one of them, the homogeneous space K v \G v = X v may be viewed as the space of maximal compact subgroups of G v. Since X v is diffeomorphic to R d(gv), where d(g v ) = dim G v dim K v, the space X v is contractible. Notice that, if G is semi-simple, the space X v is the symmetric space associated to G v. We let X = X v v V (or we write X G emphasizing the underlying k-group G) resp. d(g) = v V d(g v ). A torsion-free arithmetic subgroup Γ of G acts properly discontinously and freely on X and the quotient X/Γ is a smooth manifold of dimension d(g). The space X/Γ has finite volume if and only if G has no non-trivial rational character, and it is compact if and only if, in addition, every rational unipotent element belongs to the radical of G [2, Th. 12.3]. If E is a Γ-module, we denote the corresponding local system on X/Γ by Ẽ. Then there are canonical isomorphisms H q (Γ, E) = H q (X/Γ, Ẽ) resp. Hq (Γ, E) = H q (X/Γ, Ẽ), for any degree q between the (co-)homology of X/Γ and the Eilenberg-MacLane cohomology of Γ. Thus, the cohomological dimension cd(γ) of Γ is at most d(g). If X/Γ is compact, we have cd(γ) = d(g), otherwise cd(γ) < d(g). In fact, by [3], cd(γ) = d(g) rk k G in the latter case. On the other hand, let (ν, E) be a finite dimensional irreducible representation of the real Lie group G on a real or complex vector space E.

13 On the cohomology of uniform arithmetic subgroups in SU (2n) 13 The group G operates on X and on the complex (Ω (X, E), d) of smooth E-valued forms on X. Given a torsion free arithmetic subgroup Γ of G, the cohomology H (X/Γ, Ẽ) of the manifold X/Γ with coefficients in the local system defined by (ν, E) is canonically isomorphic to the derham cohomology H (Ω(X, E) Γ ) The construction of geometric cycles Let G denote a connected semi-simple algebraic group defined over an algebraic number field k, Γ G(k) an arithmetic subgroup. Let H be a reductive k - subgroup of G, let K H be a maximal compact subgroup of the real Lie group H, and let X H = K H \H be the space associated to H. If x 0 X is fixed under the natural action of K H G on X, then the assignment h x 0 h defines a closed embedding X H = K H \H X, that is, the orbit map identifies X H with a totally geodesic submanifold of X. Thus, we also have a natural map j H Γ : X H /Γ H X/Γ, where Γ H = Γ H(k). It is known [18, Sect. 6] that the map j H Γ is proper. Now we are interested in situations in which for a given subgroup H and a torsion free arithmetic subgroup Γ of G, the corresponding map j H Γ is an injective immersion. Thus, by being proper, j H Γ is an embedding, and the image j H (X H /Γ H ) of X H /Γ H is a submanifold in X/Γ. This submanifold is totally geodesic, to be called a geometric cycle in X/Γ. The following Theorem, stated in [18, Sect. 6, Thm. D] with an outline of its proof, is a combination of a result by Raghunathan [5, Sect. 2] and a result in [16]. Theorem Let G be a connected semi-simple algebraic k-group, let H G be a connected reductive k-subgroup, and let Γ be an arithmetic subgroup of G(k). Then there exists a subgroup Γ of finite index in Γ such that if Γ is replaced by Γ the map j H Γ : X H /Γ H X/Γ is a proper, injective, closed embedding, and so that each connected component of the image is an orientable, totally geodesic submanifold of X/Γ. For example, such geometric cycles naturally arise as fixed point components of an automorphism µ of finite order on X/Γ which is induced by a rational automorphism of G. It is known (see e.g. [18, 6.4]) that the connected components of the fixed point set F ix( µ, X/Γ) are totally geodesic closed submanifolds in X/Γ of the form F (γ) = X(γ)/Γ(γ) where γ ranges over a set of representatives for the classes in the non-abelian cohomology set H 1 ( µ, Γ). Such a connected compenent is of the form X(γ)/Γ(γ) where X(γ) is the set of fixed points of the action of µ on X twisted by the cocycle γ. The component originates with the group G(γ) of elements fixed by the γ-twisted µ-action on G. Occasionally one also writes X G(γ) for X(γ). As first noted in [14] resp. [15] in specific cases, the map j G(γ) Γ is injective in such a case. In general, we are interested in cases where a geometric cycle Y is orientable and its fundamental class is not homologous to zero in X/Γ, in singular homology or homology with closed supports, as necessary. As stated, there exists a subgroup of finite index in Γ such that the corresponding cycles are orientable. One way to go about the second question is to construct an orientable submanifold Y of complementary dimension such that the intersection product (if defined) of its fundamental class with that of Y is non-zero. However,

14 14 Joachim Schwermer and Christoph Waldner geometric cycles of complementary dimension usually intersect in a quite complicated set, possibly of dimension greater than zero. The theory of excess intersections as developed in [16, Sects. 3 and 4], is helpful in such a situation. In particular, it provides a formula for the intersection number of a pair of two such geometric cycles Y and Y which intersect perfectly. We will use this technique in the specific case we are interested in. 5. The case of arithmetic subgroups in algebraic groups of type 2 A 2n 1 Starting off with a totally real number field F, a quaternion algebra Q /F, a specific quadratic extension field L of F, the associated quaternion algebra Q = Q L with admits an involution τ of the second kind we attach to a suitable Hermitian form on Q the corresponding simple connected algebraic group G = SU(H, Q, τ) defined over F. For an appropriate choice of our data the group G (R) is the real Lie group SU (2n). Arithmetic subgroups of G give rise to discrete subgroups in SU (2n). Then we define various (families of) rational F -automorphisms of finite order on G and determine their corresponding groups of fixed points The algebraic group Let F a totally real number field of degree [F : Q] =: r 1. Denote by V = {s : F R} the set of real places of F. Instead of s(f ) we write F s. Let d F, d > 0 such that s(d) < 0, for all s V {id}. It is a consequence of the weak approximation theorem for number fields that such a number exists. Let L = F ( d) and let σ denote the unique non-trivial Galoisautomorphism of the extension L/F. Furthermore, let Q be a quaternion algebra over F, endowed with the canonical conjugation τ c. We suppose that Q does not split over R. The quaternion algebra Q := Q L admits the involution τ := τ c σ of the second kind, that is, the restriction of τ to the center Z(Q) = L of Q acts as the Galoisautomorphism σ. All involutions of the second kind on a quaternion algebra are obtained in this way (cf. [8, 2.22]). We denote the involution τ c 1 (resp. 1 σ) on Q simply by τ c (resp. σ). Note that σ induces an F -algebra automorphism, also denoted by σ, of the matrix algebra M n (Q) of order two. By our choice of L we see that Q does not split over R and that all other conjugates s(q), s V {id}, split over R. Now, we choose h 1,..., h n F, such that s(h i ) > 0, for all i = 1,..., n, s V and define H = diag(h 1,..., h n ), H k = diag(h 1,..., h k ) and H k = diag(h k+1,..., h n ), k = 1,..., n 1. Then we have the unitary group and the special unitary group U(H, Q, τ) = {g GL(n, Q) τ( t g)hg = H} SU(H, Q, τ) = U(Q, H) SL(n, Q). The algebraic group G = SU(H, Q, τ) is a simple, simply connected, connected F -group of type 2 A 2n 1. Let G := Res F/Q G be the algebraic Q-group obtained from G by restriction of scalars. By our choice of Q and L, we have G := G (R) = SU (2n) and G := G(R) = SU (2n) SU(2n) SU(2n). Hence G is F -anisotropic resp. G is Q-anisotropic. Note, that in the case F = Q, G = SU (2n) and G = G is Q-anisotropic too, since H is positive definite.

15 On the cohomology of uniform arithmetic subgroups in SU (2n) Rational automorphisms of order two The F -rational involution θ : G G, g H 1 τ c ( t g) 1 H induces the ordinary Cartan involution θ : g t ḡ 1 on SL(n, H), because the real group G (θ)(r) given by the fixed points of θ in G is Sp(n) (see also 5.3). Given an index k = 1,..., n 1, one has the F -rational involution ν k : G G, g I k,n k gi k,n k, where I k,n k denotes the diagonalmatrix with 1 at the first k entries and 1 at the last n k entries. The involutions ν k and θ commute with one another. Let Skew(Q, τ c ) = {x Q τ c (x) = x} be the set of skew-symmetric elements in Q. For any invertible element x Skew(Q, τ c ), we get an involution conj u τ c on Q (cf. [8, 2.21]). The algebra Q has an L-basis given by elements 1, i, j, k such that i 2 =: a 0, j 2 =: b 0 F and ij = k = ji, for short, Q = Q(a 0, b 0 L). Obviously, i, j Skew(Q, τ c ). Define J 1 := ji n, J 2 := ii n,. Then µ s : G G, g J s gjs 1, s = 1, 2 is an F -rational automorphism. Moreover, µ s is an involution and commutes with θ. Note that τ rs : Q Q, τ r1 (x) = jτ c (x)j 1, τ r2 (x) = iτ c (x)i 1 is an antiinvolution of Q (called the reversion) Reductive subgroups We are going to determine the groups of fixed points under the various rational automorphisms of finite order introduced before. One immediately sees that and that SU(H, Q, τ)(ν k ) = S(U(H k, Q, τ) U(H k, Q, τ)), SU(H, Q, τ)(ν k θ) = {g SU(H, Q, τ) H 1 I k,n k τ c ( t g) 1 HI k,n k = g} = {g SU(H, Q, τ) H 1 I k,n k τ c ( t g) 1 HI k,n k = H 1 τ( t g) 1 H} = {g SU(H, Q, τ) I k,n k gi k,n k = σ(g)} { ( ) A B = SU(H, Q, τ) C D A M k (Q ), B M k,n k ( dq ) C M n k,k ( dq ), D M n k (Q ) } = SU(HI k,n k, Q, τ c ), SU(H, Q, τ)(ν k θ)(r) = Sp(k, n k). Note that for x = x x 1 i + x 2 j + x 3 k Q, one has jxj 1 = x 0 1 x 1 i + x 2 j x 3 k and ixi 1 = x x 1 i x 2 j x 3 k. Thus we obtain SU(H, Q, τ)(µ s ) = {g SU(H, Q, τ) J s gjs 1 = g} { SU(H, L[j], τ), s = 1 = SU(H, L[i], τ), s = 2. and SU(H, Q, τ)(µ s )(R) = SL(n, C), s = 1, 2. Next define Q 1 = Q(a 0 d, b 0 F ) and Q 2 = Q(a 0, b 0 d F ). Since d > 0, these quaternion F - algebras ramify over R. Note that an x Q is in Q 1 (resp. Q 2 ) if and only if σ(x) = jxj 1

16 16 Joachim Schwermer and Christoph Waldner (resp. σ(x) = ixi 1 ). It easy to see that τ Qs = τ rs, s = 1, 2. Consequently we obtain SU(H, Q, τ)(µ s θ) = {g SU(H, Q, τ) J s H 1 τ c ( t g) 1 HJ 1 s = g} = {g SU(H, Q, τ) J s H 1 τ c ( t g) 1 HJs 1 = H 1 τ( t g) 1 H} = {g SU(H, Q, τ) J s τ c ( t g) 1 Js 1 = τ( t g) 1 } = {h SU(H, Q, τ) J s hjs 1 = σ(h)} = SU(H, Q s, τ Qs ) = SU(H, Q s, τ rs ), s = 1, 2. and SU(H, Q, τ)(µ s θ)(r) = SO(n, H) for the group of real points. As a summary the following table lists all the subgroups of G constructed above. It also indicates the notation for the corresponding geometric cycles discussed in the next section. Recall that the range of the index k is k = 1,..., n 1. B B = B(R) = C G SU(H, Q, τ) SL(n, H) X/Γ G (ν k ) S(U(H k, Q, τ) U(H k, Q, τ)) S(GL(k, H) GL(n k, H)) C(ν k ) G (ν k θ) SU(HI k,n k, Q, τ c ) Sp(k, n k) C(ν k θ) G (µ 1 ) SU(H, L[j], τ) SL(n, C) C(µ 1 ) G (µ 2 ) SU(H, L[i], τ) SL(n, C) C(µ 2 ) G (µ 1 θ) SU(H, Q 1, τ r1 ) SO(n, H) C(µ 1 θ) G (µ 2 θ) SU(H, Q 2, τ r2 ) SO(n, H) C(µ 2 θ) 6. Non-bounding cycles We retain the notation of the previous section Special cycles We consider an algebraic F -group G = SU(H, Q, τ) and the corresponding algebraic Q- group G := Res F/Q G obtained from G by restriction of scalars. By our choice of Q and L, we have G := G (R) = SU (2n) and G := G(R) = SU (2n) SU(2n) SU(2n). A torsion free arithmetic subgroup of G gives rise to a discrete subgroup of the real Lie group SU (2n). By the very construction of G, using the compactness criterion of Borel and Harish-Chandra [2, Th. 12.3], these discrete subgroups are cocompact. Consequently, the arithmetic quotient X/Γ is compact. On one hand, we have the family {ν k } k=1,...,n 1 of F - rational automorphisms of G of order two. On the other, we have the two F - rational automorphisms µ s : G G. In both cases the automorphisms commute with the Cartan involution θ. Therefore, following the general construction of geometric cycles, a torsion free arithmetic subgroup of G gives rise to the family {C(ν k )} k=1,...,n 1 of cycles. They come with the family {C(ν k θ)} k=1,...,n 1 where the cycles C(ν k ) and C(ν k θ), k = 1,..., n 1, have complementary dimension in X/Γ. Similarly, there are cycles C(µ s ), s = 1, 2, with the cycles C(µ 1 θ), s = 1, 2 of complementary dimension. From the final list in the previous section one reads off the following dimension formulas for the geometric cycles as constructed:

17 On the cohomology of uniform arithmetic subgroups in SU (2n) A non-vanishing result dim X/Γ = (n 1)(2n + 1) dim C(ν k ) = (k 1)(2k + 1) + (n k 1)(2n 2k + 1) + 1 dim C(ν k θ) = 4k(n k), [k = 1,..., n 1] dim C(µ s ) = (n + 1)(n 1), [s = 1, 2] dim C(µ s θ) = n(n 1), [s = 1, 2]. Theorem There exists a uniform discrete arithmetically defined subgroup Γ of the real Lie group SU (2n) so that the cohomology H j (X/Γ, R) contains a non-trival cohomology class for any integer j = 4k(n k), and j = dim X/Γ 4k(n k), [k = 1,..., n 1] respectively j = (n + 1)(n 1) and j = n(n 1). By duality, these classes are detected by the fundamental classes of a totally geodesic submanifold, a so called geometric cycle, of the form C(ν k ) resp. C(ν k θ) in the first case, and C(µ s ) resp. C(µ s θ) in the second case. These classes cannot be obtained as the restriction of a continuous class from the underlying Lie group SU (2n). Proof. By the discussion in the previous subsection it is enough to show that there is an arithmetic subgroup of the simple algebraic Q-group G, subject to the choices made there, so that the cycles {C(ν k )} k=1,...,n 1, the complementary cycles {C(ν k θ)} k=1,...,n 1 and the cycles C(µ s ), s = 1, 2, together with the cycles C(µ s θ), s = 1, 2, are non-bounding in X/Γ. Let be a torsion free subgroup of G. For a fixed integer k, we consider the cycle Y k = C(ν k ) together with the cycle Y k = C(ν kθ) of complementary dimension in X/. Both cycles arise as fixed point components of two involutions on X/ which commute with one another. Thus, by [16, Lemma 1.4], they intersect perfectly, that is, the connected components of the intersection are immersed submanifolds in X/ and for each of the components F of Y k Y k the tangent bundle T F of F coincides with the intersection of the restriction of the tangent bundles of Y k and Y k to F, T F = T Y F T Y F. Since G(ν k )(R) and G(ν k θ)(r) are connected groups we may assume (by passing to a subgroup of finite index if necessary) that the connected components F are orientable (see [16]). Thus, since X(ν k ) and X(ν k θ) are of complementary dimension and intersect in exactly one point, the assumptions of [16, Thm. 4.11] are satisfied. This general result implies that there is an arithmetic subgroup Γ of G, Γ of finite index, so that the intersection number of the two cycles relative Γ in question is non-zero. Consequently the geometric cycles [C(ν k, Γ)] and [C(ν k θ, Γ)] give rise via duality to non-vanishing classes in the cohomology of the compact arithmetic quotient X/Γ. By passing to a subgroup of finite index if necessary, it follows from a general result of Millson-Raghunathan [12, Thm. 2.1], that we can achieve that the classes so obtained are not in the image of the map βγ : I H (Ω(X, R) Γ ) H (X/Γ, R). (6 1) where I denotes the space of R - valued forms on X which are invariant under the natural

18 18 Joachim Schwermer and Christoph Waldner action of the Lie group G associated with X. We conclude, that the classes cannot be represented by a G - invariant R - valued form on X. This proves our claim. Of course, the same line of argument applies to the other cycles originating with the two F - rational automorphisms µ s : G G. Remark. Note that if A q is the fundamental series representation as introduced in 3.3, then R(q) = n(n 1) = dim C(µ s θ) and dim X R(q) = (n + 1)(n 1) = dim C(µ). 7. Examples and comments Let A q be one of the irreducible unitary representation of G = SU (2n) with non-vanishing relative Lie algebra cohomology H i (g, K, A q ). Suppose that A q is not the trivial representation. By the actual computation of the cohomology as given in Section 3, H i (g, K, A q ) vanishes for j < R(q) := dim(u p). We denote by r G the minimum of the values R(q) taken over all proper θ-stable subalgebras q of g. In the case at hand, one has r G = 2(n 1) if n > 3 whereas the rank of the real Lie algebra g 0 is n 1. If n = 3, then r G = 3. In the following tables, in the cases n = 2, 3, we give an overview of the representations A q and the dimensions of the vector spaces H i (g, K, A q ). The θ-stable parabolic subalgebras q are denoted by their corresponding parameter m, an ordered partition of n (resp. (n 1)). Furthermore, these lists also contain the degrees in which a non-bounding geometric cycle as constructed in the previous section contributes to the cohomology. The degree i in which a geometric cycle C contributes non trivially is indicated by. By the main result, the lowest possible degree in which one of the geometric cycles can contribute is the degree 4(n 1). This value is always greater than the constant r G. It is also evident that the geometric cycles have dimensions which are close by and centered around the middle dimension 1 2 dim X/Γ of the locally symmetric space. The question arises if there is a way to have a geometric counterpart to classes which might be constructed within the frame work of the automorphic theory in some of the missing degrees. In turn, in view of the representation theoretical interpretation of the cohomology groups, the existence of nonvanishing geometric cycles implies the existence of certain automorphic forms. However, it is not at all clear how to describe (or understand) these automorphic representations in a better way. This is even an open problem in the case of the exceptional group G 2 [23] where the geometric construction of cohomology classes detects all cohomological degrees in which non-vanishing possibly occurs. It might be that the theory of period integrals is of some help in a structural characterization. The case n = 2: The case n = 3: q \ i {2} 1 1 {1; 1} 1 1 {1} 1 1 C(ν 1 ) C(ν 1 θ) C(µ s ) C(µ s θ)