A construction of residues of Eisenstein series and related
square-integrable classes in the cohomology of arithmetic groups of low k-rank

Neven Grbac; Joachim Schwermer

doi:10.1515/forum-2019-0029

Article

A construction of residues of Eisenstein series and related square-integrable classes in the cohomology of arithmetic groups of low k-rank

Neven Grbac and Joachim Schwermer

Published/Copyright: June 13, 2019

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From the journal Forum Mathematicum Volume 31 Issue 5

Abstract

The cohomology of an arithmetic congruence subgroup of a connected reductive algebraic group defined over a number field is captured in the automorphic cohomology of that group. The residual Eisenstein cohomology is by definition the part of the automorphic cohomology represented by square-integrable residues of Eisenstein series. The existence of residual Eisenstein cohomology classes depends on a subtle combination of geometric conditions (coming from cohomological reasons) and arithmetic conditions in terms of analytic properties of automorphic L-functions (coming from the study of poles of Eisenstein series). Hence, there are almost no unconditional results in the literature regarding the very existence of non-trivial residual Eisenstein cohomology classes. In this paper, we show the existence of certain non-trivial residual cohomology classes in the case of the split symplectic, and odd and even special orthogonal groups of rank two, as well as the exceptional group of type G2, defined over a totally real number field. The construction of cuspidal automorphic representations of GL2 with prescribed local and global properties is decisive in this context.

Keywords: Eisenstein cohomology; square-integrable cohomology classes; construction of non-trivial classes; automorphic forms; Eisenstein series; split groups of rank two

MSC 2010: 11F75; 11F67; 11F70; 22E40; 22E55

Communicated by Freydoon Shahidi

Funding source: Hrvatska Zaklada za Znanost

Award Identifier / Grant number: 3628

Award Identifier / Grant number: 9364

Funding statement: The first named author was supported in part by the Croatian Science Foundation (projects 3628 and 9364) and by the University of Rijeka (research grant 13.14.1.2.02). Both authors acknowledge the support obtained within the frame work of the Croatian-Austrian Scientific agreement (HR 17/2014).

A Unitary representations with non-zero cohomology

It is a fundamental problem to determine (up to infinitesimal equivalence) all irreducible unitary representations (π,Hπ) of a real Lie group G with non-vanishing Lie algebra cohomology. A complete solution to this classification problem was given in a constructive approach by Vogan and Zuckerman [66]. An outgrowth of this is the computation of the relative Lie algebra cohomology groups H∗⁢(𝔤,K,Hπ,K⊗F), where 𝔤 denotes the complexified Lie algebra of the given connected real reductive Lie group, K⊂G a maximal compact subgroup.

Following [45, 68], we briefly review in this appendix the classification in the case where G is the exceptional split real Lie group of type G2. It is a connected group of rkℝ⁡G=2. The Weyl group WG of G is isomorphic to the dihedral group D6 of order 12. Let K be a maximal compact subgroup of G; its Lie algebra 𝔨0 is isomorphic to 𝔰⁢𝔭⁢(1)⊕𝔰⁢𝔭⁢(1).

Let θK be the corresponding Cartan involution and let 𝔤0=𝔨0⊕𝔭0 be the corresponding Cartan decomposition of the Lie algebra 𝔤0 of G2. Given an irreducible unitary representation (π,Hπ) of G with non-vanishing cohomology with respect to a finite-dimensional representation space F, there is a θK-stable parabolic subalgebra 𝔮 of 𝔤. By definition, 𝔮 is a parabolic subalgebra of 𝔤 such that θK⁢𝔮=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 𝔤0 of 𝔤. Write 𝔲 for the nilradical of 𝔮. Then 𝔩 is the complexification of a real subalgebra 𝔩0 of 𝔤0. 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 𝔩0. Then F/𝔲⁢F is a one-dimensional unitary representation of L. Write -λ:𝔩→ℂ for its differential. Via cohomological induction, the data (𝔮,λ) determine a unique irreducible unitary representation A𝔮⁢(λ) of G so that the Harish-Chandra module of (π,Hπ) is equivalent to the one of A𝔮⁢(λ).

It is worth noting that the Levi subgroup L has the same rank as G, is preserved by the Cartan involution θK, and the restriction of θK to L is a Cartan involution. Moreover, the group L contains a maximal torus T⊂K. This result serves as a guideline to construct all possible θK-stable parabolic subalgebras 𝔮 in 𝔤 up to conjugation by K. There are only finitely many K-conjugacy classes of θK-stable parabolic subalgebras 𝔮 in 𝔤.

In the given case the construction runs as follows: Fix non-zero elements x,y in 𝔨0, the first one belonging to the first summand, the second to the second, and let i⁢𝔱 be the real vector space spanned by i⁢x,i⁢y. Then 𝔱 is a Cartan subalgebra of 𝔨0, and 𝔱≅ℂ2. We denote the evaluation in the first and second coordinate by e1 and e2, respectively, and we write α1=e2-e1 and α2=3⁢e1-e2. Taking αi,i=1,2, as simple roots, the set Δ+⁢(𝔤,𝔱) of positive roots of 𝔤 with respect to 𝔱 is given as the set

Δ+⁢(𝔤,𝔱)=Δ+⁢(𝔨,𝔱)∪Δ+⁢(𝔭,𝔱),

where

Δ+⁢(𝔨,𝔱)={α1+α2,3⁢α1+α2},Δ+⁢(𝔭,𝔱)={α1,α2,2⁢α1+α2,3⁢α1+2⁢α2}.

Note that α1 is the short simple root, and α2 is the long simple root. The fundamental dominant weights are Λ1:=2⁢α1+α2 and Λ2:=3⁢α1+α2.

Starting off from an element z∈𝔱, there is an associated θ-stable parabolic subalgebra 𝔮 of 𝔤ℂ with Levi decomposition 𝔮=𝔩ℂ⊕𝔲ℂ defined by 𝔮ℂ = sum of non-negative eigenspaces of ad⁡(z), 𝔩ℂ = centralizer of z, and 𝔲ℂ = sum of positive eigenspaces of ad⁡(z). Let λ be the differential of a unitary character of L, the connected subgroup of G with Lie algebra 𝔩ℂ∩𝔤, such that 〈α,λ|𝔱ℂ〉≥0 for each root α of 𝔲 with respect to 𝔱ℂ. One refers to such a one-dimensional representation λ:𝔩ℂ→ℂ as an admissible character. A pair (𝔮,λ) of a θ-stable parabolic subalgebra 𝔮 of 𝔤ℂ and an admissible character λ determines a unique irreducible unitary representation A𝔮⁢(λ) of G with non-vanishing cohomology with respect to a suitable finite-dimensional representation (ν,F) of G.

Up to infinitesimal equivalence, if 𝔩ℂ⊂𝔨ℂ, one obtains discrete series representations, and there are exactly three of them up to infinitesimal equivalence having the same infinitesimal character for a given admissible character λ. Recall that this number is generally given as the ratio |WG/WK|, where WK denotes the Weyl group of K. The only degree in which these three discrete series representations πi,i=1,2,3, have Hj⁢(𝔤,K,Hπi⊗F)≠0 with a suitable coefficient system is j=4.

The trivial representation of G only matters if the coefficient system F is trivial as well. Note that one has Hj⁢(𝔤,K,ℂ)=ℂ if j=0,4,8 and Hj⁢(𝔤,K,ℂ)=0 otherwise.

The most interesting irreducible unitary representations of G=G2 are (up to infinitesimal equivalence) the ones originating in the following way: Consider two elements zj∈𝔱, j=1,2, with αj⁢(zj)>0 and αk⁢(zj)=0 for k≠j. We denote the corresponding θ-stable parabolic subalgebra as constructed by 𝔮j,j=1,2. The connected subgroup Lj, j=1,2, is isomorphic to SL2⁢(ℝ)×U⁢(1). These two algebras 𝔮j, j=1,2, are the only θ-stable parabolic subalgebras of 𝔤 with R⁢(𝔮j)=3. Let λ:𝔩j→ℂ be an admissible character. Then the corresponding irreducible unitary representation A𝔮j⁢(λ) of G is non-tempered. We summarize this classification result in the case of an arbitrary coefficient system, see [45, 67, 68].

Proposition A.1.

Let G be the split simple real Lie group of type G2, g its complexified Lie algebra, and K⊂G a maximal compact subgroup. Let (ν,F) be an irreducible finite-dimensional representation of G with highest weight Λ=c1⁢Λ1+c2⁢Λ2, c1,c2 non-negative integers. Then we have:

Fix the index j∈{1,2}. If the integral coefficient ci=0,i≠j, then there exists an admissible character χj:𝔩j→ℂ with regard to 𝔮j such that the corresponding irreducible non-tempered representation A𝔮j⁢(χj), as constructed above, occurs with
Hq⁢(𝔤,K,A𝔮j⁢(χj)⊗F)={ℂif ⁢q=3,5,0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
If both integral coefficients c1≠0,c2≠0, then there is no irreducible unitary representation (π,H) of G with Hq⁢(𝔤,K,π⊗F)≠0 for q=3,5.

Remark A.2.

Observe the shift in indices: This occurs as well if we describe the two non-tempered representation as Langlands quotients of principal series representations (see [45, 7.7.(3)]). We have

J⁢(P2,σ,12⁢ρP2~)=A𝔮1⁢(χ1),J⁢(P1,σ,12⁢ρP1~)=A𝔮2⁢(χ2).

Here we use the notation used in Section 7 for the principal series representations.

Acknowledgements

We thank Don Blasius for a helpful discussion concerning monomial representations. A large part of the work on this paper has been done during several visits of the first named author to the Erwin Schrödinger Institute in Vienna. He would like to thank the Institute for the hospitality and wonderful working environment. Both authors benefited from a stay at the MPI Mathematik in Bonn in the fall 2014. Some research related to this paper was carried through by the second named author during a stay at the Department of Mathematics, Ohio State University. He thanks James Cogdell for his generous hospitality, and the Institute of Mathematics.

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Received: 2019-01-31

Revised: 2019-03-15

Published Online: 2019-06-13

Published in Print: 2019-09-01

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Keywords for this article

Eisenstein cohomology; square-integrable cohomology classes; construction of non-trivial classes; automorphic forms; Eisenstein series; split groups of rank two