Reduced $C^*$-algebras and $K$-theory for reductive $p$-adic groups

Abstract: We calculate the $K$-theory of the reduced $C^*$-algebra $C^*_r(G)$ of a reductive $p$-adic group $G$. To do so, we show that each direct summand in Plymen's Plancherel decomposition of $C^*_r(G)$ is Morita equivalent to a twisted crossed product for an action of a finite group on the blow-up of a compact torus along the zero-locus of a certain Plancherel density. It follows that the $K$-theory of $C^*_r(G)$ is the direct sum of the twisted equivariant $K$-theory groups of these blow-ups, which can be computed using an Atiyah-Hirzebruch spectral sequence. As an illustration, the case of $\operatorname{Sp}_4$ is treated in some detail. Our main result is obtained from a more general study of $C^*$-algebras of compact operators on twisted equivariant Hilbert modules, from which we also recover results due to Wassermann for real groups, and to Afgoustidis and Aubert in the $p$-adic case.

• The paper develops a computable description of the K-theory for reduced C*-algebras by linking Plancherel decomposition to twisted crossed products via Morita equivalence.
• It employs the Atiyah-Hirzebruch spectral sequence for twisted equivariant K-theory, yielding explicit calculations, as demonstrated in the Sp4 case.
• The study refines p-adic representation theory by using geometric blow-ups along singular loci, bridging harmonic analysis and noncommutative topology.

Reduced $C^*$-Algebras and $K$-Theory for Reductive $p$-adic Groups

Overview and Objectives

The paper "Reduced $C^*$-algebras and $K$-theory for reductive $p$-adic groups" (2603.29965) develops a computable description of the $K$-theory for the reduced $C^*$-algebra $C^*_r(G)$ associated to a reductive $p$-adic group $K$0. This is accomplished by examining the direct summands in Plymen's Plancherel decomposition and relating each, up to Morita equivalence, to a twisted crossed product determined by a finite group action on a blow-up of a compact torus. These blow-ups are constructed along the zero sets of Plancherel density functions, which encode singularities related to representation theoretic multiplicities and intertwining operators.

The $K$1-theory computation leverages techniques from equivariant topology—specifically, the Atiyah-Hirzebruch spectral sequence for twisted equivariant $K$2-theory—and is illustrated by explicit calculations for $K$3. The study additionally recovers prior results for real reductive groups and $K$4-adic groups, placing them in an abstract algebraic framework based on compact operators on twisted equivariant Hilbert modules.

Plancherel Decomposition and Morita Equivalence

At the core of the paper is the decomposition of $K$5 for a reductive $K$6-adic group $K$7 via the Plancherel theorem, which separates the algebra into direct summands labeled by discrete-series representations $K$8 of Levi subgroups $K$9 and their torus of unramified unitary characters $p$0. The summands are of the form $p$1, with $p$2 a Weyl group acting both geometrically (on $p$3) and operatorially (via normalized intertwining operators), $p$4 the compact operators, and $p$5 the Hilbert space of the induced representation.

The central technical result is the identification of each such direct summand with a twisted crossed product $p$6, where $p$7 is a blow-up of $p$8 along hypersurfaces determined by the vanishing loci of Plancherel density functions, and $p$9 is a two-cocycle capturing the twist from intertwining operators. The main implication is that Morita equivalence reduces $C^*$0-theory computations for $C^*$1 to twisted equivariant $C^*$2-theory for these modified spaces.

Twisted Equivariant Hilbert Modules: Abstract Framework

To generalize these identifications, the authors introduce an abstract theory of twisted equivariant Hilbert modules. For a finite group $C^*$3 acting via a twisted (cocycle-modified) automorphism $C^*$4 and cocycle $C^*$5 on a commutative $C^*$6-algebra $C^*$7, and a twisted equivariant Hilbert module $C^*$8, the algebra of compact operators $C^*$9 is Morita equivalent to an explicitly described ideal in $K$0. The completeness condition necessary for this equivalence is articulated representation-theoretically, linking it to projective representations of isotropy subgroups.

This construction subsumes several prior results: it recovers Wassermann’s identification for real reductive groups, generalizes to $K$1-adic group cases considered by Afgoustidis, Aubert, Plymen, and Leung, and illuminates the role of scalars arising from intertwining operators (the "R-groups" and their projective characters).

Geometry of the Blow-Up and Representation Theory

A significant contribution is the geometric refinement of the parameter space. Instead of working directly with the torus $K$2, whose structure is singular at zeros of Plancherel density, the authors define a blow-up $K$3, which splits $K$4 along these singular loci. The action of $K$5 on $K$6 reflects stabilizer variability and the intricacies of intertwining operators; this enables the direct summand's $K$7-theory to be interpreted as twisted equivariant $K$8-theory on $K$9.

The construction is canonical and compatible with the representation-theoretic classification of tempered duals, particularly involving Knapp-Stein R-groups. The fibre structure above points in $p$0 corresponds precisely to choices of Weyl chambers, and the stabilizer analysis is systematically addressed in terms of scalar-intertwiner conditions.

$p$1-Theory Computation via Spectral Sequence

The $p$2-theory of $p$3, and hence of each summand, is computable via the Atiyah-Hirzebruch spectral sequence for twisted equivariant $p$4-theory. The $p$5-page is described in terms of equivariant cohomology of a coefficient system determined by projective representations of stabilizer groups encoded by the twist $p$6 and scalar conditions from intertwining operators. The spectral sequence collapses rationally, yielding explicit $p$7-group computations for low-dimensional cases.

The authors illustrate these computations with the complete analysis of $p$8, demonstrating the approach’s effectiveness: for instance, they calculate $p$9 and $K$0 for certain Plancherel components.

Numerical Results and Contradictory Claims

The main strong result is that the $K$1-theory of $K$2 is precisely the direct sum of twisted equivariant $K$3-groups of blow-ups $K$4 for each Plancherel block. The authors explicitly claim that the ordinary crossed product picture is insufficient unless the completeness condition is met, highlighting that scalar intertwiners (as determined by R-groups and cocycle data) invalidate standard results in many cases. Instead, one must work with the appropriate ideals and geometric modifications. This invalidates the naive identification with crossed products in general $K$5-adic settings.

Practical and Theoretical Implications

Practically, the approach provides an explicit pathway for $K$6-theory calculations in the representation theory of $K$7-adic groups, which is relevant for harmonic analysis, non-commutative geometry, and applications to the Baum-Connes and Connes-Kasparov conjectures. Theoretically, the identification of Plancherel summands with twisted crossed products via geometric blow-ups refines the understanding of the tempered dual and its connection to the equivariant topological invariants of buildings associated to $K$8.

The results also clarify the representation-theoretic content of $K$9-theory isomorphisms—bridging the gap between index-theoretic and harmonic analytic perspectives—and provide a new framework for connecting the right-hand side computations to geometric models (e.g., Bruhat-Tits buildings).

Future Directions

Anticipated future developments include:

• Extensions to arbitrary locally compact groups and more general twists, potentially encompassing non-reductive cases.
• Connections of these $C^*$0-theory computations to explicit models for $C^*$1-homology of buildings (left-hand side of Baum-Connes).
• Algorithmic and computational refinements for higher-rank groups and explicit calculations of cohomological invariants.
• Study of torsion phenomena in $C^*$2-theory, especially as they relate to singularities in Plancherel densities and non-triviality of cocycles.

Conclusion

The paper provides a rigorous and computable algebraic-topological framework for the $C^*$3-theory of reduced $C^*$4-algebras of reductive $C^*$5-adic groups, refining and generalizing prior results for real groups and specific $C^*$6-adic cases. By combining advanced representation theory, operator algebra, and equivariant topology, the authors deliver explicit Morita equivalences and spectral sequence constructions, illuminating essential subtleties in the structure and computation of $C^*$7-groups for these algebras. The extension to arbitrary Plancherel components and the blow-up construction constitutes a critical advance in the program connecting noncommutative geometry, harmonic analysis, and topological classification.