[Paper Review] Boundary behavior of special cohomology classes arising from the Weil representation

Jens Funke, John J. Millson|arXiv (Cornell University)|Nov 6, 2008

Advanced Algebra and Geometry15 references7 citations

TL;DR

This paper investigates the boundary behavior of theta series associated with special cohomology classes arising from the Weil representation on non-compact locally symmetric spaces for orthogonal groups of signature $(p,q)$. It proves that these theta functions extend to the Borel-Serre compactification, with the restriction to each face being a theta series for a smaller orthogonal group and larger coefficient system. In the $\mathbb{Q}$-split case of signature $(p,p)$, the small Borel-Serre compactification fails to support the extension due to non-unique boundary limits; instead, the 'big' Borel-Serre compactification resolves this issue, enabling the construction of well-defined boundary restrictions and establishing the nonvanishing of special (co)homology classes via finite covers.

ABSTRACT

In our previous paper [math.NT/0408050], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p,q)$. This correspondence is realized using theta functions associated to explicitly constructed "special" Schwartz forms. Furthermore, the theta functions give rise to generating series of certain "special cycles" in $X$ with coefficients. In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel-Sere compactification $\bar{X}$ of $X$. However, for the $\Q$-split case for signature $(p,p)$, we have to construct and consider a slightly larger compactification, the "big" Borel-Serre compactification. The restriction to each face of $\bar{X}$ is again a theta series as in [math.NT/0408050], now for a smaller orthogonal group and a larger coefficient system. As application we establish the cohomological nonvanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish.

Motivation & Objective

• To understand the boundary behavior of theta functions associated with special cohomology classes arising from the Weil representation on non-compact locally symmetric spaces for orthogonal groups of signature $(p,q)$.
• To analyze the obstruction to extending these theta functions to the small Borel-Serre compactification in the $\mathbb{Q}$-split case of signature $(p,p)$, where boundary limits depend on the approach path.
• To construct and utilize the 'big' Borel-Serre compactification to resolve the extension problem and ensure well-defined boundary restrictions.
• To establish the cohomological nonvanishing of special cycles by passing to an appropriate finite cover of the locally symmetric space.
• To generalize the geometric theta correspondence to nontrivial coefficient systems $\mathbb{S}_{[\lambda]}(V_{\mathbb{C}})$ via explicit Schwartz forms and theta distributions.

Proposed method

• The authors use the Weil representation to construct explicit $(\mathfrak{g},K)$-cocycles with values in $\mathcal{S}(V^{n}_{\mathbb{R}}) \otimes \mathbb{S}_{[\lambda]}(V_{\mathbb{C}})$, which generate closed differential forms on the symmetric space $D$.
• They define a theta distribution $\Theta_{\mathcal{L}} = \sum_{\ell \in \mathcal{L}} \delta_\ell$ for a lattice $\mathcal{L} \subset V^n$, and pair it with the cocycle to obtain a closed $nq$-form $\theta_{\varphi^{V}_{nq,[\lambda]}}$ on the quotient $X = \Gamma \backslash D$ with coefficients in $\mathbb{S}_{[\lambda]}(V_{\mathbb{C}})$.
• The boundary behavior is analyzed by studying the asymptotic limits of the theta function as the symmetric space approaches its Borel-Serre boundary, particularly near corners corresponding to parabolic subgroups.
• For the $\mathbb{Q}$-split case $\operatorname{SO}(p,p)$, the small Borel-Serre compactification fails to support a well-defined boundary restriction due to path-dependent limits; this is resolved by introducing the 'big' Borel-Serre compactification with an extra $\mathbb{R}_+$-factor.
• The restriction of the theta function to each face of the compactification is shown to be a theta series for a smaller orthogonal group and a larger coefficient system, generalizing the results of [12].
• Poisson summation and partial Fourier transforms are applied to analyze the convergence and dependence on boundary parameters, particularly in the $\operatorname{SO}(2,2)$ case, to demonstrate non-extendability to the small compactification.

Experimental results

Research questions

• RQ1 Why does the theta function $\theta(\varphi_{2,0})$ fail to extend to the small Borel-Serre compactification of $\operatorname{SO}(2,2)$?
• RQ2 What is the role of the 'big' Borel-Serre compactification in resolving the non-extendability of theta functions in the $\mathbb{Q}$-split case of signature $(p,p)$?
• RQ3 How does the restriction of the theta function to each face of the compactified space relate to theta series for smaller orthogonal groups and modified coefficient systems?
• RQ4 What is the cohomological significance of the special cycles $Z_{T,[\lambda]}$ when restricted to finite covers of $X$?
• RQ5 How do the boundary limits of the theta function depend on the path of approach, and why does this break well-definedness in the small compactification?

Key findings

• The theta function $\theta_{\varphi^{V}_{nq,[\lambda]}}$ extends to the Borel-Serre compactification $\overline{X}$ of $X$ for non-compact symmetric spaces of type $(p,q)$, with well-defined restrictions to each face.
• In the $\mathbb{Q}$-split case of signature $(p,p)$, the small Borel-Serre compactification fails to support a well-defined extension of the theta function due to path-dependent boundary limits, particularly at the corner corresponding to the $2$-torus $e'(P')$.
• The 'big' Borel-Serre compactification resolves the extension problem by introducing an extra $\mathbb{R}_+$-factor, making the boundary limits independent of the approach path.
• The restriction of the theta function to each face of the big Borel-Serre compactification is again a theta series, now for a smaller orthogonal group and a larger coefficient system.
• The cohomological nonvanishing of the special cycles $[Z_{T,[\lambda]}]$ is established by passing to an appropriate finite cover of $X$, proving that the relevant (co)homology groups do not vanish.
• In the $\operatorname{SO}(2,2)$ case, the limit of $\theta(\varphi_{2,0})$ as $t_1, t_2 \to \infty$ depends on $t_2$, which is not a coordinate on the boundary torus, thus violating well-definedness and confirming the failure of extension to the small compactification.