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Cyclic homology and the Selberg principle. (English) Zbl 0783.55004
We cite from the introduction: “This paper gives an algebraic proof of the following abstract Selberg principle for a connected reductive group \(G\) over a nonarchimedean local field \(F\). Let \({\mathcal C}_ c^ \infty(G)\) be the convolution algebra of compactly supported smooth (=locally constant) functions on \(G\). We prove that for any regular semisimple element \(\gamma\) of \(G\), which is noncompact, the orbital integral \(\int_{G/G_ \gamma}Tr(e)(g\gamma g^{-1})d\overline g\) is zero. If \(F\) is of characteristic zero, this vanishing holds for any noncompact element of \(G\).”
As the orbital integral defines a trace \(I_ \gamma\) on \({\mathcal C}_ c^ \infty(G)\), the Selberg principle states the vanishing of \(\langle ch_ 0(e),I_ \gamma\rangle\), the zeroth order component of \(ch(e)\in PHC_ 0\bigl({\mathcal C}_ c^ \infty(G)\bigr)\), evaluated on \(I_ \gamma\in HC^ 0\bigl({\mathcal C}_ c^ \infty(G)\bigr)\). The proof is based on a careful study of the Hochschild complex (cyclic bicomplex) of \({\mathcal C}_ c^ \infty(G)\) and the Connes-Gysin sequence relating Hochschild- and cyclic homology.
For discrete groups \(\Gamma\), group homology is well known to be isomorphic to the Hochschild homology of the group algebra. Moreover, the cyclic homology of the group algebra decomposes into a direct sum of contributions from the different conjugacy classes of \(\Gamma\).
In the case of \(p\)-adic Lie groups one finds an analogous isomorphism \(C_*\bigl({\mathcal C}_ c^ \infty(G),V\bigr)\simeq C_*(G,V^{ad})\) between the smooth Hochschild complex of the differentiable \(G\)-bimodule \(V\) and the standard complex calculating the differentiable homology of \(G\) with coefficients in \(V^{ad}\). Replacing \(V={\mathcal C}_ c^ \infty(G)\) by \({\mathcal C}_ c^ \infty(U)\), the \(G\)-module of smooth compactly supported functions on an Ad-invariant open subset \(U\) of \(G\), the above isomorphism allows to define \(U\)-localized Hochschild complexes and cyclic bicomplexes whose homology injects into the global one, (and which becomes thus the direct limit of the local homology groups). \[ HH_*\bigl({\mathcal C}_ c^ \infty(G)\bigr)_ U\hookrightarrow HH_*\bigl({\mathcal C}_ c^ \infty(G)\bigr),\quad HC_*\bigl({\mathcal C}_ c^ \infty(G)\bigr)_ U\hookrightarrow HC_*\bigl({\mathcal C}_ c^ \infty(G)\bigr). \] The main new tool introduced in this paper allows to analyze these local homologies in the case \(U=Ad(G)(H^{\text{reg}})\), \(H^{\text{reg}}\) the set of regular elements of a Cartan subgroup \(H\). This is the construction of the “higher orbital integrals” \[ \Phi:{\mathcal C}_ c^ \infty(G^{n+1})_ U\to{\mathcal C}_ c^ \infty(H^{n+1}) \] \[ \Phi(f)(h_ 0,\ldots,h_ n)=\int_{(G/H)^{n+1}}f\bigl(\sigma(y_ 0)h_ 0\sigma (y_ 1)^{- 1},\ldots,\sigma(y_ n)h_ n\sigma(y_ 0)^{-1}\bigr)dy_ 0\ldots dy_ n \] \((\sigma\) any continuous section of \(\pi:G\to G/H)\), which define chain maps \(C_*\bigl({\mathcal C}_ c^ \infty(G)\bigr)_ U\to C_*\bigl({\mathcal C}_ c^ \infty(H)\bigr)\), \(CC_*\bigl({\mathcal C}_ c^ \infty(G)\bigr)_ U\to CC_*\bigl({\mathcal C}_ c^ \infty(H)\bigr)\) of Hochschild- and cyclic complexes. (For \(n=0\) one recovers the ordinary orbital integral.) The homology of the latter complexes is calculated easily: the Fourier transform allows to identify them with the Hochschild (cyclic) complexes of certain algebras of smooth functions (under ordinary multiplication) on the Pontryagin dual of \(H\) (a commutative real Lie group), i.e. with ordinary de Rham complexes.
Summing up, the authors have established a commutative diagram \[ \begin{matrix} HH_ 0 \bigl({\mathcal C}_ 0^ \infty(G)\bigr)&\hookleftarrow HH_ 0 \bigl({\mathcal C}_ c^ \infty(G)\bigr)_ U&@>\Phi>> HH_ 0\bigl( {\mathcal C}_ c^ \infty(H) \bigr)&\simeq \Omega ^ 0_{dR} (\hat H) \\ B'\downarrow & B'\downarrow & \downarrow & \downarrow\delta \\ HH_ 1 \bigl({\mathcal C}_ c^ \infty(G)\bigr)&\hookleftarrow HH_ 1 \bigl({\mathcal C}_ c^ \infty(G)\bigr)_ U&@>\Phi>> HH_ 1\bigl({\mathcal C}_ c^ \infty(H)\bigr)&\simeq \Omega ^ 1_{dR} (\hat H)\end{matrix} \] representing a part of the Connes-Gysin sequence. This sequence reads \[ \begin{matrix}\quad &Tr(e) = ch_ 0(e) & \leftarrow &ch_ 2 (e) \\ \quad& \in &\quad & \in \\ HH_ 1\bigl({\mathcal C}_ c^ \infty(G)\bigr)@<B'<<&HH_ 0\bigl({\mathcal C}_ c^ \infty(G)\bigr)= HC_ 0\bigl({\mathcal C}_ c^ \infty(G)\bigr) & @<S<< &HC_ 2\bigl({\mathcal C}_ c^ \infty(G)\bigr) \end{matrix} \] so that especially \(B'\bigl(Tr(e)\bigr)=0\). The operators in the Connes-Gysin sequence commuting with localization, one also has \(B'\bigl(Tr(e)|_ U\bigr)=0\). Since an element \(\gamma\in H\), viewed as function (character) on \(\hat H\), is annulated by \(\delta\) if and only if \(\gamma\) is contained in a compact subgroup of \(H\), the commutative diagram shows \[ \Phi\bigl(Tr(e)|_ U\bigr)(\gamma)=\int_{G/G_ \gamma}Tr(e)(\overline g\gamma\overline g^{-1})d\overline g=0 \] for all non-compact regular elements \(\gamma\in H\).
The authors conjecture furthermore that the periodic localized cyclic homology of \({\mathcal C}_ c^ \infty(G)\) vanishes. The paper ends with an explicit treatment of the case \(G=SL_ 2(Q_ p)\) and an extension of the above to modular representations in positive characteristic.

55N15 Topological \(K\)-theory
22E35 Analysis on \(p\)-adic Lie groups
18G60 Other (co)homology theories (MSC2010)
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