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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.

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