Beijing Institute of Mathematical Sciences and Applications

This course is devoted to Jens Franke’s paper “Harmonic Analysis in Weighted $L^{2}$-Spaces”, published in 1998 in the ”Annales scientifiques de l’École Normale Supérieure“. Franke’s work lies at the intersection of harmonic analysis on arithmetic quotients, the spectral theory of Eisenstein series, and the cohomology of arithmetic groups.

Let $G$ be a connected reductive group over a number field $F$, and consider an arithmetic quotient of the form $$S_{K_f}=G(F)\backslash G(\mathbb A)/(K_\infty K_f),$$ with the usual modification involving the split center. Such a space is generally noncompact but has finite volume. Its cohomology can be described by differential forms, or equivalently by relative Lie algebra cohomology with coefficients in the large representation $$
C^\infty\bigl(G(F)\backslash G(\mathbb A)\bigr).$$ Although this model is natural, it is too large to be directly useful from the viewpoint of automorphic representation theory. On the other hand, the Hilbert space $$L^{2}\bigl(G(F)\backslash G(\mathbb A)\bigr)$$ has a powerful spectral decomposition, but it excludes many non-square-integrable Eisenstein series that contribute essentially to the full cohomology.

Franke’s central idea is that weighted $L^{2}$-spaces and weighted Sobolev spaces provide a bridge between these two settings. The weights control growth in the different cuspidal directions. Schematically, if $H(g)$ is a reduction-theoretic height function, one considers norms of the form $$\|f\|_{\lambda}^{2}=\int_{G(F)\backslash G(\mathbb A)} |f(g)|^{2}e^{-2\langle\lambda,H(g)\rangle}\,dg.$$ Varying the parameter $\lambda$ changes which asymptotic terms are square-integrable. The critical walls are determined by the exponents appearing in constant terms of automorphic forms.

Let $\mathcal A(G)$ denote the space of automorphic forms: smooth functions of moderate growth that are $K_\infty$-finite, finite under the center of the universal enveloping algebra, and invariant under a suitable compact open subgroup at the finite places. Franke proves that the natural inclusion >$$\mathcal A(G) \hookrightarrow C^\infty\bigl(G(F)\backslash G(\mathbb A)\bigr)$$ induces an isomorphism in relative Lie algebra cohomology. At a fixed level $K_f$, this gives, up to the standard conventions concerning the split center and coefficient duals, $$H^\bullet\!\left(\mathfrak g_\infty,K_\infty; \mathcal A(G)^{K_f}\otimes E \right) \cong H^\bullet\!\left(\mathfrak g_\infty,K_\infty;C^\infty\bigl(G(F)\backslash G(\mathbb A)\bigr)^{K_f}\otimes E \right).$$ The right-hand side identifies with the cohomology of the corresponding locally symmetric space. Thus the full cohomology may be computed using automorphic forms rather than arbitrary smooth functions. This is the conjecture of Borel proved in Franke’s paper.

Franke also proves that every automorphic form is generated by Eisenstein series. More precisely, if $P=MN$ is a parabolic subgroup, $\phi$ is cuspidal automorphic data on the Levi subgroup $M$, and $E_P(g,\phi,\lambda)$ is the associated meromorphic Eisenstein series, then $$\mathcal A(G)=\operatorname{span} \left\{\text{Laurent coefficients of }E_P(g,\phi,\lambda)\right\},$$ as the parabolic subgroup, the cuspidal datum, the evaluation point, and the Laurent coefficient vary. Consequently, the cohomology of an arithmetic quotient can be studied recursively in terms of cuspidal automorphic representations on Levi subgroups and the analytic behavior of their Eisenstein series.