×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bass, H, Algebraic K-theory, (1968), Benjamin New York · Zbl 0174.30302
[2] Blackadar, B, K-theory for operator algebras, () · Zbl 0913.46054
[3] Blanc, P, (co)homologie différentiable et changement de groupes, Astérisque, 124-125, 13-30, (1985) · Zbl 0563.57023
[4] Blanc, P; Wigner, D, Homology of Lie groups and Poincaré duality, Lett. math. phys., 7, 259-270, (1983) · Zbl 0548.57026
[5] Brylinski, J.-L, Some examples of Hochschild and cyclic homology, (), 33-72
[6] Burghelea, D, The cyclic homology of group rings, Comment. math. helv., 60, 354-365, (1985) · Zbl 0595.16022
[7] Cartan, H; Eilenberg, S, Homological algebra, (1956), Princeton Univ. Press Princeton, NJ · Zbl 0075.24305
[8] Casselman, W, Characters and Jacquet modules, Math. ann., 230, 101-105, (1977) · Zbl 0337.22019
[9] Casselman, W, Nonunitary argument for p-adic representation, J. fac. sci. univ. Tokyo, 28, 907-908, (1982)
[10] Clozel, L, Orbital integrals on p-adic groups: A proof of the Howe conjecture, Ann. of math., 129, 237-251, (1989) · Zbl 0675.22007
[11] Connes, A, Non-commutative differential geometry, Inst. hautes études sci. publ. math., 62, 257-360, (1986)
[12] Connes, A, Cohomologie cyclique et foncteurs ext^n, C. R. acad. sci. Paris, 296, 953-960, (1983) · Zbl 0534.18009
[13] Deligne, P, Le support du caractère d’une représentation supercuspidale, C. R. acad. sci. Paris, 283, 155-157, (1976) · Zbl 0336.22009
[14] Eckmann, B, Cohomology of groups and transfer, Ann. of math., 58, 481-493, (1953) · Zbl 0052.02002
[15] Elashvili, A, Letter to J.-L. brylinski, (March 23, 1991)
[16] Goodwillie, T, Cyclic homology, derivations, and the free loop space, Topology, 24, 187-215, (1985) · Zbl 0569.16021
[17] Grothendieck, A, Produits tensoriels topologiques et espaces nucléaires, Mem. amer. math. soc., 16, (1955) · Zbl 0064.35501
[18] Harish-Chandra; van Dijk, Harmonic analysis on reductive p-adic groups, () · Zbl 0289.22018
[19] Hochschild, G; Kostant, B; Rosenberg, A, Differential forms on regular affine algebras, Trans. amer. math. soc., 102, 383-408, (1962) · Zbl 0102.27701
[20] Julg, P; Valette, A, Twisted coboundary operator on a tree and the Selberg principle, J. operator theory, 16, 285-304, (1986) · Zbl 0613.46058
[21] Julg, P; Valette, A, L’opérateur de cobord tordu sur un arbre et le principe de Selberg, J. operator theory, 17, 347-355, (1987) · Zbl 0639.46063
[22] Karoubi, M, Homologie cyclique et K-théorie, Astérique, 149, (1987) · Zbl 0648.18008
[23] Karoubi, M; Villamayor, O, Homologie cyclique d’algèbres de groupes, (1989), preprint
[24] Loday, J.-L; Quillen, D, Cyclic homology and the Lie algebra homology of matrices, Comment. math. helv., 59, 565-591, (1984) · Zbl 0565.17006
[25] Pichaud, J, G-homologie et (g, K)-homologie des G-modules différentiables, Ann. sci. école norm. sup., 16, 219-236, (1983) · Zbl 0539.57020
[26] Rogawski, J, Representations of GL(n) and division algebras over p-adic fields, Duke math. J., 50, 161-169, (1983) · Zbl 0523.22015
[27] serre, J.-P, Corps locaux, (1968), Herman Paris
[28] Shalika, J, The multiplicity one theorem for GL(n), Ann. of math., 100, 171-193, (1973) · Zbl 0316.12010
[29] Silberger, A, Introduction to harmonic analysis on reductive p-adic groups, (1979), Princeton Univ. Press Princeton, NJ
[30] Steinberg, R, Regular elements of semisimple algebraic groups, Inst. hautes études sci. publ. math., 25, 281-312, (1965) · Zbl 0136.30002
[31] Tits, J, Reductive groups over local fields, (), 29-69, Part I
[32] Tsygan, B, Homology of matrix Lie algebras over rings and the Hochschild homology, Russian math. surveys, 38, 198-199, (1983) · Zbl 0526.17006
[33] Vignéras, M.-F, Caractérisation des intégrales orbitales sur un groupe réductif p-adique, J. fac. sci. univ. Tokyo sect. IA math., 29, 945-962, (1981) · Zbl 0499.22011
[34] Wodzicki, M, Excision in cyclic homology, Ann. of math., 129, 591-640, (1989) · Zbl 0689.16013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.