Celebratio Mathematica — Baum — Complete Bibliography

[1] P. F. Baum: “Cohomology of homogeneous spaces,” Bull. Am. Math. Soc. 69 : 4 (1963), pp. 531–533. Based on the author’s PhD thesis (1963). MR 148804 Zbl 0152.40503 article

[2] P. F. Baum: Cohomology of homogeneous spaces. Ph.D. thesis, Princeton University, 1963. Advised by J. C. Moore and N. Steenrod. An article based on this was published in Bull. Am. Math. Soc. 69:4 (1963). MR 2613915 phdthesis

N. E. Steenrod	Related
John Coleman Moore	Related

[3] P. F. Baum and W. Browder: “The cohomology of quotients of classical groups,” Topology 3 : 4 (June 1965), pp. 305–336. MR 189063 Zbl 0152.22101 article

[4] P. F. Baum: “Local isomorphism of compact connected Lie groups,” Pac. J. Math. 22 : 2 (February 1967), pp. 197–204. MR 213470 Zbl 0178.02802 article

[5] P. F. Baum: “Quadratic maps and stable homotopy groups of spheres,” Illinois J. Math. 11 : 4 (1967), pp. 586–595. MR 220285 Zbl 0166.19102 article

The original proof [Bott 1959] of the Bott periodicity theorem used Morse theory. Recent work on this theorem has, however, been algebraic in nature [Atiyah and Bott 1964], [Atiyah et al. 1964], [Bass 1965], [Wood 1966]. The new proofs of the Bott periodicity theorem center around showing that a stable homotopy class can be represented by a specially simple sort of polynomial map. In [1964] Atiyah and Bott ask whether it might be possible to use this approach on other homotopy problems. Is there, for example, some specially simple class of polynomial maps which carries the stable homotopy of spheres? As a possible first step towards selecting such a class we shall indicate that probably one wants to examine the properties of quadratic maps. In detail, we shall show that:

The stable \( J \)-homomorphism can be interpreted as an algebraic operation which converts a linear map into a quadratic map.
Any element of a \( k \)-stem can be represented by a quadratic map \[ q: R^n\to R^l \quad\text{such that}\quad q(S^{n-1})\subset R^l -\{0\} .\]

[6] P. Baum and L. Smith: “The real cohomology of differentiable fibre bundles,” Comment. Math. Helv. 42 : 1 (December 1967), pp. 171–179. MR 221522 Zbl 0166.19302 article

[7] P. F. Baum: “On the cohomology of homogeneous spaces,” Topology 7 : 1 (January 1968), pp. 15–38. MR 219085 Zbl 0158.42002 article

[8] P. Baum and J. Cheeger: “Infinitesimal isometries and Pontryagin numbers,” Topology 8 : 2 (April 1969), pp. 173–193. MR 238351 Zbl 0179.28802 article

[9] P. F. Baum and R. Bott: “On the zeroes of meromorphic vector-fields,” pp. 29–47 in Essays on topology and related topics: Mémoires dédiés à Georges de Rham [Essays on topology and related topics: Memoirs dedicated to Georges de Rham] (Geneva, 26–28 March 1969). Edited by A. Haefliger and R. Narasimhan. Springer (Berlin), 1970. MR 261635 Zbl 0193.52201 incollection

Let \( M \) be a compact complex analytic manifold and let \( x \) be a holomorphic vector-field on \( M \). In an earlier paper by one of us (see [Bott 1967]) it was shown that the behavior of \( x \) near its zeroes determined all the Chern numbers of \( M \) and the nature of this determination was explicitly given where \( x \) had only nondegenerate zeroes. The primary purpose of this note is to extend this result to meromorphic fields, or equivalently to sections \( s \) of \( T\otimes L \) where \( T \) is the holomorphic tangent bundle to \( M \) and \( L \) is a holomorphic line bundle. We will also drop the non-degeneracy assumption of the zeroes of \( s \), but we treat only the case where \( s \) vanishes at isolated points \( \{p\} \).

Georges de Rham	Related
Raghavan Narasimhan	Related
Raoul H. Bott	Related	Volume
André Haefliger	Related

[10] P. F. Baum: “Vector fields and Gauss–Bonnet,” Bull. Am. Math. Soc. 76 : 6 (1970), pp. 1202–1211. Based on an invited address given at AMS Summer Meeting in Eugene, OR. MR 266255 Zbl 0203.54102 article

[11] F. Hirzebruch: “Lectures on \( K \)-theory,” pp. 223–238 in Algebraic topology: A student’s guide (Seattle, 1963). Edited by J. F. Adams. London Mathematical Society Lecture Note Series 4. Cambridge University Press, 1972. Notes prepared by Paul Baum. Lecture notes of the AMS Summer Topology Institute. incollection

F. Hirzebruch	Related	Volume
John Frank Adams	Related

[12] P. Baum and R. Bott: “Singularities of holomorphic foliations,” J. Diff. Geom. 7 : 3–4 (1972), pp. 279–342. To S. S. Chern and D. C. Spencer on their 60th birthdays. MR 377923 Zbl 0268.57011 article

Raoul H. Bott	Related	Volume
Donald Clayton Spencer	Related
S. S. Chern	Related	Volume

[13] P. Baum: Chern classes and singularities of complex foliations. Preprint, Brown University, 1973. techreport

[14] P. Baum: “Structure of foliation singularities,” Advances in Math. 15 : 3 (March 1975), pp. 361–374. Dedicated to Mark Baum on his seventieth birthday. MR 377125 Zbl 0296.57007 article

In the study of vector-fields, vector-field singularities play a central role. Similarly, when studying foliations, a key part should be played by foliation singularities (e.g., see [Baum and Bott 1972]). To examine foliation singularities, a first question is: “What is the ‘generic’ singularity of a foliation?” Put otherwise: “What sort of a singularity is it reasonable to expect?” The theorem stated in Section 2 and proved in Section 5 gives a partial answer to this question. The theorem asserts that a foliation singularity which satisfies certain natural dimension conditions must have, almost everywhere, a very simple structure.

The work of this note will be done in the complex-analytic framework. But after suitable minor modifications everything done here is also valid in the \( C^{\infty} \) category. Given a complex-analytic manifold \( M \), sheaves will be used to give a precise definition of “holomorphic foliation with singularities”. This notion is, in a natural fashion, contravariant. That is, if \( f:M_1 \to M_2 \) is a holomorphic map, then \( f \) “pulls-back” a holomorphic foliation with singularities on \( M_2 \) to such an object on \( M_1 \). (Compare [Haefliger 1970, 1971].) The theorem of Section 2 then asserts that a foliation singularity satisfying certain dimension conditions is the. pull-back via a holomorphic submersion of an isolated zero of a holomorphic vectorfield. This describes the singularity.

[15] P. Baum: “Riemann–Roch theorem for singular varieties,” pp. 3–16 in Differential geometry (Stanford, CA, 30 July–17 August 1973), part 2. Edited by S. S. Chern and R. Osserman. Proceedings of Symposia in Pure Mathematics 27. American Mathematical Society (Providence, RI), 1975. MR 389907 Zbl 0344.32021 incollection

S. S. Chern	Related	Volume
Robert Osserman	Related

[16] P. Baum, W. Fulton, and R. MacPherson: “Riemann–Roch for singular varieties,” Inst. Hautes Études Sci. Publ. Math. 45 (1975), pp. 101–145. MR 412190 Zbl 0332.14003 article

William Fulton	Related
Robert Duncan MacPherson	Related

[17] P. Baum, W. Fulton, and R. MacPherson: Riemann–Roch and topological \( K \)-theory for singular varieties. Preprint 42, Aarhus University Mathematics Institute, 1977. A version of this was later published in Acta Math. 143:3–4 (1979). Zbl 0355.14008 techreport

William Fulton	Related
Robert Duncan MacPherson	Related

[18] P. Baum, W. Fulton, and G. Quart: Lefschetz–Riemann–Roch for singular varieties. Preprint 41, Aarhus University Mathematics Institute, 1977. A version of this was later published in Acta Math. 143:3–4 (1979). Zbl 0357.14004 techreport

George J. Quart	Related
William Fulton	Related

[19] P. Baum, W. Fulton, and R. MacPherson: “Riemann–Roch and topological \( K \) theory for singular varieties,” Acta Math. 143 : 3–4 (1979), pp. 155–192. A preprint was published in 1977. MR 549773 Zbl 0474.14004 article

The basic Riemann–Roch problem is to give, for any sheaf \( \mathcal{S} \) of \( O_X \) modules on an algebraic variety \( X \), a formula for \( \chi(X,\mathcal{S}) \), the alternating sum of the ranks of the sheaf cohomology groups \( H_i(X,\mathcal{S}) \). Perhaps the most striking fact about \( \chi(X,\mathcal{S}) \) is that it is constant in a flat family: while the individual ranks of the \( H^i(X,\mathcal{S}) \) may vary, their alternating sum does not. This invariance under deformation leads one to suspect that \( \chi(X,\mathcal{S}) \) may be a topological invariant. In this paper we will present the Riemann–Roch Theorem as a transition from algebra to topology; one consequence will be a topological formula for \( \chi(X,\mathcal{S}) \).

William Fulton	Related
Robert Duncan MacPherson	Related

[20] P. Baum, W. Fulton, and G. Quart: “Lefschetz–Riemann–Roch for singular varieties,” Acta Math. 143 : 3–4 (1979), pp. 193–211. A preprint version was published in 1977. MR 549774 Zbl 0454.14009 article

George J. Quart	Related
William Fulton	Related

[21] P. Baum and R. G. Douglas: “Index theory, bordism, and \( K \)-homology,” pp. 1–31 in Operator algebras and \( K \)-theory (San Francisco, 7–8 January 1981). Edited by R. G. Douglas and C. Schochet. Contemporary Mathematics 10. American Mathematical Society (Providence, RI), 1982. MR 658506 Zbl 0507.55004 incollection

Our purpose in this note is threefold. After giving brief and selective descriptions of the topological and analytical realizations of \( K \)-homology, we present our view of index theory. Sedond, we shall say more about our proof of the isomorphism of topological and analytical realizations of \( K \)-homology since that lies at the heart of matters. In particular, we want to discuss the bordism step of the proof since this makes contact with a number of interesting topics including relative \( K \)-homology and elliptic boundary value problems. Lastly, we want to work out a key step in the bordism proof in two special cases where the result may have independent interest.

Claude L. Schochet	Related
Ronald George Douglas	Related

[22] P. Baum and R. G. Douglas: “Toeplitz operators and Poincaré duality,” pp. 137–166 in Toeplitz centennial: Toeplitz memorial conference in operator theory, dedicated to the 100th anniversary of the birth of Otto Toeplitz (Tel Aviv, 11–15 May 1981). Edited by I. Gohberg. Operator Theory: Advances and Applications 4. Birkhäuser (Basel), 1982. MR 669904 Zbl 0517.55001 incollection

Otto Toeplitz	Related
Israel Gohberg	Related
Ronald George Douglas	Related

[23] P. Baum and R. G. Douglas: “\( K \) homology and index theory,” pp. 117–173 in Operator algebras and applications (Kingston, ON, 14 July–2 August 1980), part 1. Edited by R. V. Kadison. Proceedings of Symposia in Pure Mathematics 38. American Mathematical Society (Providence, RI), 1982. MR 679698 Zbl 0532.55004 incollection

Ronald George Douglas	Related
Richard V. Kadison	Related

[24] P. Baum: “Fixed point formula for singular varieties,” pp. 3–22 in Current trends in algebraic topology (London, ON, 29 June–10 July 1981), part 2. Edited by R. M. Kane. CMS Conference Proceedings 2. American Mathematical Society (Providence, RI), 1982. MR 686136 Zbl 0574.14018 incollection

[25] P. Baum, J.-L. Brylinski, and R. MacPherson: “Cohomologie équivariante délocalisée” [Delocalized equivariant cohomology], C. R. Acad. Sci. Paris Sér. I Math. 300 : 17 (1985), pp. 605–608. MR 791098 Zbl 0589.55003 article

Jean-Luc Brylinski	Related
Robert Duncan MacPherson	Related

[26] P. Baum and A. Connes: “Leafwise homotopy equivalence and rational Pontrjagin classes,” pp. 1–14 in Foliations (Tokyo, 14–18 July 1983). Edited by I. Tamura. Advanced Studies in Pure Mathematics 5. North-Holland (Amsterdam), 1985. MR 877325 Zbl 0641.57008 incollection

Itiro Tamura	Related
Alain Connes	Related

[27] P. Baum and A. Connes: “\( K \) theory for actions of discrete groups,” pp. 1–12 in Conferencias del taller de topología algebraica [Conferences of the algebraic topology workshop]. Edited by L. Astey and E. Micha. VI Coloquio del Departamento de Matemáticas. CINVESTAV-IPN (Mexico City), 1986. incollection

Elías Micha	Related
Luis Astey	Related
Alain Connes	Related

[28] P. Baum and A. Connes: “Chern character for discrete groups,” pp. 163–232 in A fête of topology: Papers dedicated to Itiro Tamura. Edited by Y. Matsumoto, T. Mizutani, and S. Morita. Academic Press (Boston), 1988. MR 928402 Zbl 0656.55005 incollection

Alain Connes	Related
Itiro Tamura	Related
Yukio Matsumoto	Related
Shigeyuki Morita	Related
Tomohiko Mizutani	Related

[29] P. Baum and A. Connes: “\( K \)-theory for discrete groups,” pp. 1–20 in Operator algebras and applications (Warwick, UK, 20–25 July 1987), vol. 1: Structure theory; \( K \)-theory, geometry and topology. Edited by D. Evans and M. Takesaki. London Mathematical Society Lecture Note Series 135. Cambridge University Press, 1988. MR 996437 Zbl 0685.46041 incollection

Masamichi Takesaki	Related
Alain Connes	Related

[30] P. Baum, R. G. Douglas, and M. E. Taylor: “Cycles and relative cycles in analytic \( K \)-homology,” J. Diff. Geom. 30 : 3 (1989), pp. 761–804. MR 1021372 Zbl 0697.58050 article

Michael Eugene Taylor	Related
Ronald George Douglas	Related

[31] P. Baum and J. Block: “Equivariant bicycles on singular spaces,” C. R. Acad. Sci. Paris Sér. I Math. 311 : 2 (1990), pp. 115–120. MR 1065441 Zbl 0719.19003 article

[32] P. Baum and R. G. Douglas: “Relative \( K \) homology and \( C^* \) algebras,” \( K \)-Theory 5 : 1 (1991), pp. 1–46. MR 1141333 Zbl 0755.46035 article

[33] P. Baum: “The Dirac operator,” pp. 163–167 in F. Hirzebruch, T. Berger, and R. Jung: Manifolds and modular forms, 2nd edition. Aspects of Mathematics 20. Vieweg (Wiesbaden), 1992. Appendix II. incollection

F. Hirzebruch	Related	Volume
Rainer Jung	Related
Thomas Berger	Related

[34] P. Baum and J. Block: “Excess intersection in equivariant bivariant \( K \)-theory,” C. R. Acad. Sci. Paris Sér. I Math. 314 : 5 (1992), pp. 387–392. With abridged French version. MR 1153721 Zbl 0762.19008 article

[35] P. Baum, N. Higson, and R. Plymen: “Equivariant homology for \( \mathrm{SL}(2) \) of a \( p \)-adic field,” pp. 1–18 in Index theory and operator algebras (Boulder, CO, 6–10 August 1991). Edited by J. Fox and P. Haskell. Contemporary Mathematics 148. American Mathematical Society (Providence, RI), 1993. MR 1228497 Zbl 0844.46043 incollection

Peter Evarts Haskell	Related
Roger J. Plymen	Related
Nigel David Higson	Related

[36] P. Baum, A. Connes, and N. Higson: “Classifying space for proper actions and \( K \)-theory of group \( C^* \)-algebras,” pp. 241–291 in \( C^* \)-algebras: 1943–1993 (San Antonio, TX, 13–14 January 1993). Edited by R. S. Doran. Contemporary Mathematics 167. American Mathematical Society (Providence, RI), 1994. MR 1292018 Zbl 0830.46061 incollection

Nigel David Higson	Related
Robert Stuart Doran	Related
Alain Connes	Related

[37] P. Baum, N. Higson, and R. Plymen: Cosheaf homology and \( K \) theory for \( p \)-adic groups. Preprint, Pennsylvania State University, 1995. techreport

Roger J. Plymen	Related
Nigel David Higson	Related

[38] P. Baum: “Working with Bott (rocking and rolling with Raoul),” pp. xxii–xxiii in Raoul Bott: Collected papers, vol. 3: Foliations. Edited by R. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston), 1995. MR 1321887 incollection

Raoul H. Bott	Related	Volume
Robert Duncan MacPherson	Related

[39] P. Baum, N. Higson, and R. Plymen: “A proof of the Baum–Connes conjecture for \( p \)-adic \( \mathrm{GL}(n) \),” C. R. Acad. Sci. Paris Sér. I Math. 325 : 2 (July 1997), pp. 171–176. MR 1467072 Zbl 0918.46061 article

Roger J. Plymen	Related
Nigel David Higson	Related

[40] P. F. Baum, N. Higson, and R. J. Plymen: “Representation theory of \( p \)-adic groups: A view from operator algebras,” pp. 111–149 in The mathematical legacy of Harish-Chandra: A celebration of representation theory and harmonic analysis (Baltimore, MD, 9–10 January 1998). Edited by R. Doran and V. Varadarajan. Proceedings of Symposia in Pure Mathematics 68. American Mathematical Society (Providence, RI), 2000. MR 1767895 Zbl 0982.19006 incollection

Robert Stuart Doran	Related
Veeravalli S. Varadarajan	Related
Roger J. Plymen	Related
Harish-Chandra	Related
Nigel David Higson	Related

[41] P. Baum and A. Connes: “Geometric \( K \)-theory for Lie groups and foliations,” Enseign. Math. (2) 46 : 1–2 (2000), pp. 3–42. MR 1769535 Zbl 0985.46042 article

[42] P. Baum, S. Millington, and R. Plymen: “A proof of the Baum–Connes conjecture for reductive adelic groups,” C. R. Acad. Sci. Paris Sér. I Math. 332 : 3 (February 2001), pp. 195–200. MR 1817360 Zbl 1105.19300 article

Roger J. Plymen	Related
Stephen David Millington	Related

[43] P. Baum and V. Nistor: “Periodic cyclic homology of Iwahori–Hecke algebras,” C. R. Acad. Sci. Paris Sér. I Math. 332 : 9 (May 2001), pp. 783–788. MR 1836086 Zbl 1013.16003 article

[44] P. Baum and P. Schneider: “Equivariant-bivariant Chern character for profinite groups,” \( K \)-Theory 25 : 4 (2002), pp. 313–353. MR 1914452 Zbl 0997.55010 article

For the action of a locally compact and totally disconnected group \( G \) on a pair of locally compact spaces \( X \) and \( Y \) we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space \( Y \) an universal proper \( G \)-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant \( K \)-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov \( KK \)-theory. Under certain basic finiteness conditions on the first space \( Y \) we conjecture the existence of a Chern character from the equivariant Kasparov \( KK \)-theory of \( Y \) and \( X \) into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the \( KK \) theory with the complex numbers. This conjecture is proved for profinite groups \( G \). An essential role in our construction is played by a bivariant version of Segal localization which we establish for \( KK \)-theory.

[45] P. Baum and V. Nistor: “Periodic cyclic homology of Iwahori–Hecke algebras,” \( K \)-Theory 27 : 4 (December 2002), pp. 329–357. MR 1962907 Zbl 1056.16005 article

We determine the periodic cyclic homology of the Iwahori–Hecke algebras \( H_q \), for \( q\in\mathbb{C}^* \) not a ‘proper root of unity’. (In this paper, by a proper root of unity we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a ‘weakly spectrum preserving’ morphism of finite type algebras induces an isomorphism in periodic cyclic homology. The concept of a weakly spectrum preserving morphism is defined in this paper, and most of our work is devoted to understanding this class of morphisms. Results of Kazhdan and Lusztig and Lusztig show that, for the indicated values of \( q \), there exists a weakly spectrum preserving morphism \[ \phi_q:H_q\to J ,\] to a fixed finite type algebra \( J \). This proves that \( \phi_q \) induces an isomorphism in periodic cyclic homology and, in particular, that all algebras \( H_q \) have the same periodic cyclic homology, for the indicated values of \( q \). The periodic cyclic homology groups of the algebra \( H_1 \) can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group.

[46] P. Baum, S. Millington, and R. Plymen: “Local-global principle for the Baum–Connes conjecture with coefficients,” \( K \)-Theory 28 : 1 (2003), pp. 1–18. MR 1988816 Zbl 1034.46073 article

Roger J. Plymen	Related
Stephen David Millington	Related

[47] P. Baum and J. Brodzki: Equivariant \( KK \)-theory and noncommutative index theory, 2004. Part VI of e-book “Lecture notes on noncommutative geometry and quantum groups” (European Mathematical Society, ed. Piotr M. Hajac). misc

Piotr Mieczysław Hajac	Related
Jacek Brodzki	Related

[48] P. Baum and R. Meyer: The Baum–Connes conjecture, localization of categories and quantum groups, 2004. Part VIII of e-book “Lecture notes on noncommutative geometry and quantum groups” (European Mathematical Society, ed. Piotr M. Hajac). misc

Ralf Meyer	Related
Piotr Mieczysław Hajac	Related

[49] P. Baum and H. Moscovici: Foliations, \( C^* \)-algebras and index theory, 2004. Part II of e-book “Lecture notes on noncommutative geometry and quantum groups” (European Mathematical Society, ed. Piotr M. Hajac). misc

Piotr Mieczysław Hajac	Related
Henri Moscovici	Related

[50] P. Baum and M. Karoubi: “On the Baum–Connes conjecture in the real case,” Q. J. Math. 55 : 3 (September 2004), pp. 231–235. MR 2082090 Zbl 1064.19003 ArXiv math/0509495 article

[51] P. Baum: “On the index of equivariant elliptic operators,” pp. 41–49 in Operator algebras, quantization, and noncommutative geometry: A centennial celebration honoring John von Neumann and Marshall H. Stone. Edited by R. Doran and R. V. Kadison. Contemporary Mathematics 365. American Mathematical Society (Providence, RI), 2004. MR 2106816 Zbl 1081.46047 incollection

Robert Stuart Doran	Related
Marshall Harvey Stone	Related	Volume
Richard V. Kadison	Related
John von Neumann	Related

[52] P. F. Baum, P. M. Hajac, R. Matthes, and W. Szymański: “The \( K \)-theory of Heegaard-type quantum 3-spheres,” \( K \)-Theory 35 : 1–2 (2005), pp. 159–186. Dedicated to the memory of Olaf Richter. An erratum to this was published in K-Theory 37:1–2 (2006). MR 2240219 Zbl 1111.46051 article

Olaf Richter	Related
Wojciech Szymański	Related
Piotr Mieczysław Hajac	Related
Rainer Matthes	Related

[53] P. F. Baum, P. M. Hajac, R. Matthes, and W. Szymański: “Erratum: ‘The \( K \)-theory of Heegaard-type quantum 3-spheres’,” \( K \)-Theory 37 : 1–2 (2006), pp. 211. Erratum to an article published in K-Theory 35:1–2 (2005). MR 2274673 Zbl 1210.46054 article

Rainer Matthes	Related
Piotr Mieczysław Hajac	Related
Wojciech Szymański	Related

[54] A.-M. Aubert, P. Baum, and R. Plymen: “The Hecke algebra of a reductive \( p \)-adic group: A geometric conjecture,” pp. 1–34 in Noncommutative geometry and number theory: Where arithmetic meets geometry and physics (Bonn, Germany, August 2003 and June 2004). Edited by C. Consani and M. Marcolli. Aspects of Mathematics 37. Vieweg (Wiesbaden, Germany), 2006. MR 2327297 Zbl 1120.14001 incollection

Let \( \mathcal{H}(G) \) be the Hecke algebra of a reductive \( p \)-adic group \( G \). We formulate a conjecture for the ideals in the Bernstein decomposition of \( \mathcal{H}(G) \). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig’s conjecture on the asymptotic Hecke algebra. We prove our conjecture for \( \mathrm{SL}(2) \) and \( \mathrm{GL}(n) \). We also prove part (1) of the conjecture for the Iwahori ideals of the groups \( \mathrm{PGL}(n) \) and \( \mathrm{SO}(5) \). The conjecture, if true, leads to a parametrization of the smooth dual of \( G \) by the points in a complex affine locally algebraic variety.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Caterina Consani	Related
Matilde Marcolli	Related

[55] P. Baum, P. M. Hajac, R. Matthes, and W. Szymanski: Non-commutative geometry approach to principal and associated bundles. Preprint, 2007. ArXiv math/0701033v2 techreport

We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) \( C^* \)-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of local triviality to fibre products of \( C^* \)-algebras. In the context of principal actions, we study in detail an example of a non-proper free action with continuous translation map, and examples of compact principal bundles which are piecewise trivial but not locally trivial, and neither piecewise trivial nor locally trivial, respectively. We show that the module of continuous sections of a vector bundle associated to a compact principal bundle is a cotensor product of the algebra of functions defined on the total space (that are continuous along the base and polynomial along the fibres) with the vector space of the representation. On the algebraic side, we review the formalism of connections for the universal differential algebras. In the differential geometry framework, we consider smooth connections on principal bundles as equivariant splittings of the cotangent bundle, as 1-form-valued derivations of the algebra of smooth functions on the structure group, and as axiomatically given covariant differentiations of functions defined on the total space. Finally, we use the Dirac monopole connection to compute the pairing of the line bundles associated to the Hopf fibration with the cyclic cocycle of integration over \( S^2 \).

Rainer Matthes	Related
Piotr Mieczysław Hajac	Related
Wojciech Szymański	Related

[56] P. Baum, N. Higson, and T. Schick: “On the equivalence of geometric and analytic \( K \)-homology,” pp. 1–24 in Special issue: In honor of Robert D. MacPherson, Part 3, published as Pure Appl. Math. Q. 3 : 1. International Press (Sommerville, MA), 2007. MR 2330153 Zbl 1146.19004 incollection

Thomas Schick	Related
Nigel David Higson	Related
Robert Duncan MacPherson	Related

[57] A.-M. Aubert, P. Baum, and R. Plymen: “Geometric structure in the representation theory of \( p \)-adic groups,” C. R. Math. Acad. Sci. Paris 345 : 10 (2007), pp. 573–578. Part II was published in Harmonic analysis on reductive, \( p \)-adic groups (2011), but with “reductive” in the title. MR 2374467 Zbl 1128.22009 article

Anne-Marie Aubert	Related
Roger J. Plymen	Related

[58] P. Baum: “The extended quotient,” pp. 23–26 in Guido’s Book of Conjectures: A gift to Guido Mislin on the occasion of his retirement from ETHZ, June 2006. Edited by I. Chatterji. Monographies de L’Enseignement Mathématique 40. Enseignement Mathématique (Geneva), 2008. incollection

Guido Mislin	Related
Indira Lara Chatterji	Related

[59] P. Baum: “Dirac operator and \( K \)-theory for discrete groups,” pp. 97–107 in A celebration of the mathematical legacy of Raoul Bott (Montreal, 9–13 June 2008). Edited by P. R. Kotiuga. CRM Proceedings & Lecture Notes 50. American Mathematical Society (Providence, RI), 2010. MR 2648889 Zbl 1201.19001 incollection

Let \( G \) be a locally compact, Hausdorff, and second countable (i.e. the topology has a countable base) topological group. Examples are Lie groups, \( p \)-adic groups, adelic groups, and discrete groups. In 1980, P. Baum and A. Connes proposed an answer to the problem of calculating the \( K \)-theory of the reduced \( C^* \) algebra of \( G \). When true, the conjecture has several corollaries. Among these are exhaustion of the discrete series by Dirac induction, Novikov conjecture on homotopy invariance of higher signatures, amd Gromov–Lawson–Rosenberg conjecture on Riemannian metrics of positive scalar curvature. Since 1980, the conjecture has been proved for many examples. Although it seems quite possible that eventually there will be a discrete group which is a counter-example, at the present moment there is no really good candidate for a counter-example. This talk will explain the conjecture, concentrating on the case of discrete groups from the point of view of the index of elliptic operators. The talk is intended for non-specialists. All the basic definitions (\( C^* \) algebra, \( K \)-theory etc) will be carefully stated.

Raoul H. Bott	Related	Volume
Peter Robert Kotiuga	Related

[60] P. Baum: “\( K \)-homology and D-branes,” pp. 81–94 in Superstrings, geometry, topology, and \( C^* \)-algebras (Fort Worth, TX, 18–22 May 2009). Edited by R. S. Doran, G. Friedman, and J. Rosenberg. Proceedings of Symposia in Pure Mathematics 81. American Mathematical Society (Providence, RI), 2010. MR 2681759 Zbl 1210.81079 incollection

\( K \)-homology is the dual theory to \( K \)-theory. In algebraic geometry [14], [7], the \( K \)-homology of a (possibly singular) projective variety \( X \) is the Grothendieck group of coherent algebraic sheaves on \( X \). In topology there are three ways to define \( K \)-homology. First \( K \)-homology is the homology theory determined by the Bott spectrum. Second, \( K \)-homology is the group of geometric \( K \)-cycles introduced by Baum–Douglas [6]. Third, using funtional analysis, \( K \)-homology is the group of abstract elliptic operators as in the work of M. .F. Atiyah [1], Brown–Douglas–Filmore [15], and G. Kasparov [21].

The D-branes of string theory [31] are twisted geometric \( K \)-cycles which are endowed with some additional structure. The charge of a D-brane is the element in the twisted \( K \)-homology of spacetime determined by the underlying twisted \( K \)-cycle of the D-brane. Essentially, the Baum–Douglas theory [6] was rediscovered in terms of constraints on open strings. The aim of this expository note is to briefly describe this development.

Greg Brian Friedman	Related
Robert Stuart Doran	Related
Jonathan Micah Rosenberg	Related

[61] P. Baum, N. Higson, and T. Schick: “A geometric description of equivariant \( K \)-homology for proper actions,” pp. 1–22 in Quanta of maths: Proceedings of meeting in honor of Alain Connes’ 60th birthday. Edited by E. Blanchard, D. Ellwood, M. Khalkhali, M. Marcolli, H. Moscovici, and S. Popa. Clay Mathematics Proceedings 11. American Mathematical Society (Providence, RI), 2010. MR 2732043 Zbl 1216.19006 ArXiv 0907.2066 incollection

Sorin Teodor Popa	Related
Étienne Blanchard	Related
Alain Connes	Related
Thomas Schick	Related
Masoud Khalkhali	Related
David Alexandre Ellwood	Related
Nigel David Higson	Related
Matilde Marcolli	Related
Henri Moscovici	Related

[62] P. Baum, H. Oyono-Oyono, T. Schick, and M. Walter: “Equivariant geometric \( K \)-homology for compact Lie group actions,” Abh. Math. Semin. Univ. Hambg. 80 : 2 (2010), pp. 149–173. MR 2734682 Zbl 1242.19006 article

Thomas Schick	Related
Hervé Oyono-Oyono	Related
Michael Walter	Related

[63] P. F. Baum, G. Cortiñas, R. Meyer, R. Sánchez-García, M. Schlichting, and B. Toën: Topics in algebraic and topological \( K \)-theory (Sedano, Spain, 22–27 January 2007). Edited by G. Cortiñas. Lecture Notes in Mathematics 2008. Springer (Berlin), 2011. MR 2761828 Zbl 1202.19001 book

Rubén José Sánchez-García	Related
Guillermo Cortiñas	Related
Bertrand Toën	Related
Marco Schlichting	Related
Ralf Meyer	Related

[64] P. F. Baum and R. J. Sánchez-García: “\( K \)-theory for group \( C^* \)-algebras,” pp. 1–43 in Topics in algebraic and topological \( K \)-theory (Sedano, Spain, 22–27 January 2007). Edited by G. Cortiñas. Lecture Notes in Mathematics 2008. Springer (Berlin), 2011. MR 2762553 Zbl 1216.19001 incollection

Rubén José Sánchez-García	Related
Guillermo Cortiñas	Related

[65] A.-M. Aubert, P. Baum, and R. Plymen: “Geometric structure in the principal series of the \( p \)-adic group \( \textrm{G}_2 \),” Represent. Theory 15 (2011), pp. 126–169. MR 2772586 Zbl 1268.22015 article

In the representation theory of reductive \( p \)-adic groups \( G \), the issue of reducibility of induced representations is an issue of great intricacy. It is our contention, expressed as a conjecture in [2007], that there exists a simple geometric structure underlying this intricate theory.

We will illustrate here the conjecture with some detailed computations in the principal series of \( \mathrm{G}_2 \).

A feature of this article is the role played by cocharacters \( h_{\mathbf{c}} \) attached to two-sided cells \( \mathbf{c} \) in certain extended affine Weyl groups.

The quotient varieties which occur in the Bernstein programme are replaced by extended quotients. We form the disjoint union \( \mathfrak{A}(G) \) of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space \( \mathfrak{A}(G) \) is a model of the smooth dual \( \textrm{Irr}(G) \). In this respect, our programme is a conjectural refinement of the Bernstein programme.

The algebraic deformation is controlled by the cocharacters \( h_{\mathbf{c}} \). The cocharacters themselves appear to be closely related to Langlands parameters.

Anne-Marie Aubert	Related
Roger J. Plymen	Related

[66] A.-M. Aubert, P. Baum, and R. Plymen: “Geometric structure in the representation theory of reductive \( p \)-adic groups, II,” pp. 71–90 in Harmonic analysis on reductive, \( p \)-adic groups (San Francisco, 16 January 2010). Edited by R. S. Doran, P. J. Sally, Jr., and L. Spice. Contemporary Mathematics 543. American Mathematical Society (Providence, RI), 2011. Part I was published in C. R. Math. Acad. Sci. Paris 345:10 (2007), doesn’t include “reductive” in title. MR 2798423 Zbl 1246.22019 incollection

Paul J. Sally	Related
Robert Stuart Doran	Related
Anne-Marie Aubert	Related
Roger J. Plymen	Related

@incollection {key2798423m,
	AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
		 Plymen, Roger},
	TITLE = {Geometric structure in the representation
		 theory of reductive \$p\$-adic groups,
		 {II}},
	BOOKTITLE = {Harmonic analysis on reductive, \$p\$-adic
		 groups},
	EDITOR = {Doran, Robert S. and Sally, Jr., Paul
		 J. and Spice, Loren},
	SERIES = {Contemporary Mathematics},
	NUMBER = {543},
	PUBLISHER = {American Mathematical Society},
	ADDRESS = {Providence, RI},
	YEAR = {2011},
	PAGES = {71--90},
	DOI = {10.1090/conm/543/10730},
	NOTE = {(San Francisco, 16 January 2010). Part
		 I was published in \textit{C. R. Math.
		 Acad. Sci. Paris} \textbf{345}:10 (2007),
		 doesn't include ``reductive'' in title.
		 MR:2798423. Zbl:1246.22019.},
	ISSN = {0271-4132},
	ISBN = {9780821849859},
}

[67] P. Baum, A.-M. Aubert, R. Plymen, and M. Solleveld: Geometric structure and the local Langlands conjecture. Preprint, 2012. ArXiv 1211.0180 techreport

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

[68] P. Baum, A. Carey, and B.-L. Wang: “\( K \)-cycles for twisted \( K \)-homology,” pp. 69–98 in Nanjing special issue on K-theory, number theory and geometry, published as J. K-Theory 12 : 1. Issue edited by X. Guo, H. Qin, and G. Tang. Cambridge University Press, August 2013. MR 3126635 Zbl 1300.19003 incollection

Hourong Qin	Related
Bai-Ling Wang	Related
Alan Lawrence Carey	Related
Guoping Tang	Related
Xuejun Guo	Related

[69] P. F. Baum and P. M. Hajac: “Local proof of algebraic characterization of free actions” in Special issue on noncommutative geometry and quantum groups in honor of Marc A. Rieffel, published as SIGMA 10. Issue edited by G. Elliott, P. M. Hajac, H. Li, and J. Rosenberg. 2014. paper no. 060. MR 3226990 Zbl 1295.22010 ArXiv 1402.3024 incollection

Let \( G \) be a compact Hausdorff topological group acting on a compact Hausdorff topological space \( X \). Within the \( C^* \)-algebra \( C(X) \) of all continuous complex-valued functions on \( X \), there is the Peter–Weyl algebra \( \mathcal{P}_G(X) \) which is the (purely algebraic) direct sum of the isotypical components for the action of \( G \) on \( C(X) \). We prove that the action of \( G \) on \( X \) is free if and only if the canonical map \[ \mathcal{P}_G(X)\otimes^{}_{C(X/G)}\mathcal{P}_G(X)\to\mathcal{P}_G(X)\otimes\mathcal{O}(G) \] is bijective. Here both tensor products are purely algebraic, and \( \mathcal{O}(G) \) denotes the Hopf algebra of “polynomial” functions on \( G \).

George Arthur Elliott	Related
Jonathan Micah Rosenberg	Related
Marc A. Rieffel	Related
Piotr Mieczysław Hajac	Related
Hanfeng Li	Related

[70] A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Geometric structure in smooth dual and local Langlands conjecture,” Jpn. J. Math. 9 : 2 (September 2014), pp. 99–136. Expository article based on the Takagi lectures. MR 3258616 Zbl 1371.11097 article

This expository paper first reviews some basic facts about \( p \)-adic fields, reductive \( p \)-adic groups, and the local Langlands conjecture. If \( G \) is a reductive \( p \)-adic group, then the smooth dual of \( G \) is the set of equivalence classes of smooth irreducible representations of \( G \). The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of subsets known as the Bernstein components. According to a conjecture due to ABPS (Aubert–Baum–Plymen–Solleveld), each Bernstein component has a geometric structure given by an appropriate extended quotient. The paper states this ABPS conjecture and then indicates evidence for the conjecture, and its connection to the local Langlands conjecture.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

[71] P. F. Baum and E. van Erp: “\( K \)-homology and index theory on contact manifolds,” Acta Math. 213 : 1 (2014), pp. 1–48. Dedicated to Sir Michael Atiyah on the occasion of his 85th birthday with admiration and affection. MR 3261009 Zbl 1323.58017 article

Erik van Erp	Related
M. F. Atiyah	Related	Volume

[72] A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “On the local Langlands correspondence for non-tempered representations,” Münster J. Math. 7 : 1 (2014), pp. 27–50. Dedicated to Peter Schneider on the occasion of his 60th birthday. MR 3271238 Zbl 06382808 ArXiv 1303.0828 article

Let \( G \) be a reductive \( p \)-adic group. We study how a local Langlands correspondence for irreducible tempered \( G \)-representations can be extended to a local Langlands correspondence for all irreducible smooth representations of \( G \). We prove that, under a natural condition involving compatibility with unramified twists, this is possible in a canonical way.

To this end we introduce analytic \( \mathrm{R} \)-groups associated to non-tempered essentially square-integrable representations of Levi subgroups of \( G \). We establish the basic properties of these new \( \mathrm{R} \)-groups, which generalize Knapp–Stein \( \mathrm{R} \)-groups.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Peter Schneider	Related
Maarten Solleveld	Related

[73] P. F. Baum, L. Dąbrowski, and P. M. Hajac: “Noncommutative Borsuk–Ulam-type conjectures,” pp. 9–18 in From Poisson brackets to universal quantum symmetries (Warsaw, 18–22 August 2014). Edited by N. Ciccoli and A. Sitarz. Banach Center Publications 106. Instytut Matematyczny PAN (Warsaw), 2015. MR 3469159 Zbl 1343.46064 incollection

Within the framework of free actions of compact quantum groups on unital \( C^* \)-algebras, we propose two conjectures. The first one states that, if \[ \delta:A\to A\otimes_{\min}H \] is a free coaction of the \( C^* \)-algebra \( H \) of a non-trivial compact quantum group on a unital \( C^* \)-algebra \( A \), then there is no \( H \)-equivariant \( * \)-homomorphism from \( A \) to the equivariant join \( C^* \)-algebra \( A\circledast_\delta H \). For \( A \) being the \( C^* \)-algebra of continuous functions on a sphere with the antipodal coaction of the \( C^* \)-algebra of functions on \( \mathbb{Z}/2\mathbb{Z} \), we recover the celebrated Borsuk–Ulam Theorem. The second conjecture states that there is no \( H \)-equivariant \( * \)-homomorphism from \( H \) to the equivariant join \( C^* \)-algebra \( A\circledast_\delta H \). We show how to prove the conjecture in the special case \[ A=C(SU_q(2))=H ,\] which is tantamount to showing the non-trivializability of Pflaum’s quantum instanton fibration built from \( SU_q(2) \).

Ludwik Dąbrowski	Related
Nicola Ciccoli	Related
Andrzej Sitarz	Related
Piotr Mieczysław Hajac	Related

[74] P. Baum, Carey, A., and B. Wang: \( K \)-homology and Fredholm operators I: Dirac operators. Preprint, 2016. ArXiv 1604.03502 techreport

Alan Lawrence Carey	Related
Bai-Ling Wang	Related

[75] P. Baum, Carey, A., and B. Wang: \( K \)-homology and Fredholm operators II: Elliptic operators. Preprint, 2016. ArXiv 1604.03535 techreport

Alan Lawrence Carey	Related
Bai-Ling Wang	Related

[76] P. Baum, E. Guentner, and R. Willett: “Expanders, exact crossed products, and the Baum–Connes conjecture,” Ann. K-Theory 1 : 2 (2016), pp. 155–208. MR 3514939 Zbl 1331.46064 article

Erik Paul Guentner	Related
Rufus Willett	Related

[77] A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Geometric structure for the principal series of a split reductive \( p \)-adic group with connected centre,” J. Noncommut. Geom. 10 : 2 (2016), pp. 663–680. MR 3519048 Zbl 1347.22013 article

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

[78] P. Baum, E. Guentner, and R. Willett: “Exactness and the Kadison–Kaplansky conjecture,” pp. 1–33 in Operator algebras and their applications: A tribute to Richard V. Kadison (San Antonio, TX, 10–11 January 2015). Edited by R. S. Doran and E. Park. Contemporary Mathematics 671. American Mathematical Society (Providence, RI), 2016. Dedicated to Richard Kadison on the occasion of his ninetieth birthday with affection and admiration. MR 3546676 Zbl 1366.46045 incollection

We survey results connecting exactness in the sense of \( C^* \)-algebra theory, coarse geometry, geometric group theory, and expander graphs. We summarize the construction of the (in)famous non-exact monster groups whose Cayley graphs contain expanders, following Gromov, Arzhantseva, Delzant, Sapir, and Osajda. We explain how failures of exactness for expanders and these monsters lead to counterexamples to Baum–Connes type conjectures: the recent work of Osajda allows us to give a more streamlined approach than currently exists elsewhere in the literature.

We then summarize our work on reformulating the Baum–Connes conjecture using exotic crossed products, and show that many counterexamples to the old conjecture give confirming examples to the reformulated one; our results in this direction are a little stronger than those in our earlier work. Finally, we give an application of the reformulated Baum–Connes conjecture to a version of the Kadison–Kaplansky conjecture on idempotents in group algebras.

Robert Stuart Doran	Related
Richard V. Kadison	Related
Rufus Willett	Related
Erik Paul Guentner	Related
Efton Park	Related

[79] A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “The local Langlands correspondence for inner forms of \( \mathrm{SL}_n \),” Res. Math. Sci. 3 (2016). paper no. 32. MR 3579297 Zbl 06663301 article

Let \( F \) be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group \( \mathrm{SL}_n(F) \). It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for \( \mathrm{SL}_n(F) \) enhanced with an irreducible representation of an \( \mathrm{S} \)-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of \( \mathrm{SL}_n(F) \) up to equivalence. An analogous result is shown in the archimedean case. For \( p \)-adic fields, this is based on the work of Hiraga and Saito. To settle the case where \( F \) has positive characteristic, we employ the method of close fields. We prove that this method is compatible with the local Langlands correspondence for inner forms of \( \mathrm{GL}_n(F) \), when the fields are close enough compared to the depth of the representations.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

[80] A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Depth and the local Langlands correspondence,” pp. 17–41 in Arbeitstagung Bonn 2013: In memory of Friedrich Hirzebruch (Bonn, Germany, 22–28 May 2013). Edited by W. Ballmann, C. Blohmann, G. Faltings, P. Teichner, and D. Zagier. Progress in Mathematics 319. Birkhäuser/Springer International (Cham, Switzerland), 2016. MR 3618046 Zbl 06748683 incollection

F. Hirzebruch	Related	Volume
Gerd Faltings	Related
Christian Blohmann	Related
Hans Werner Ballmann	Related
Peter Teichner	Related
Don Bernard Zagier	Related
Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

[81] P. F. Baum and E. van Erp: “\( K \)-homology and Fredholm operators, II: Elliptic operators,” Pure Appl. Math. Q. 12 : 2 (2016), pp. 225–241. MR 3767216 article

[82] P. Baum, Carey, A., and B. Wang: On the spectra of finite type algebras. Preprint, 2017. ArXiv 1705.01404 techreport

We review Morita equivalence for finite type \( k \)-algebras \( A \) and also a weakening of Morita equivalence which we call stratified equivalence. The spectrum of \( A \) is the set of equivalence classes of irreducible \( A \)-modules. For any finite type \( k \)-algebra \( A \), the spectrum of \( A \) is in bijection with the set of primitive ideals of \( A \). The stratified equivalence relation preserves the spectrum of \( A \) and also preserves the periodic cyclic homology of \( A \). However, the stratified equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence. A key example illustrating the distinction between Morita equivalence and stratified equivalence is provided by affine Hecke algebras associated to affine Weyl groups. Stratified equivalences lie at the heart of the ABPS conjecture, and lead to an explicit description of geometric structure in the smooth dual of a connected split reductive \( p \)-adic group.

Alan Lawrence Carey	Related
Bai-Ling Wang	Related

[83] A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Hecke algebras for inner forms of \( p \)-adic special linear groups,” J. Inst. Math. Jussieu 16 : 2 (2017), pp. 351–419. MR 3615412 Zbl 06704330 article

Let \( F \) be a non-Archimedean local field, and let \( G^{\sharp} \) be the group of \( F \)-rational points of an inner form of \( \mathrm{SL}_n \). We study Hecke algebras for all Bernstein components of \( G^{\sharp} \), via restriction from an inner form \( G \) of \( \mathrm{GL}_n(F) \).

For any packet of \( \mathrm{L} \)-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth \( G^{\sharp} \)-representations. This algebra comes from an idempotent in the full Hecke algebra of \( G^{\sharp} \), and the idempotent is derived from a type for \( G \). We show that the Hecke algebras for Bernstein components of \( G^{\sharp} \) are similar to affine Hecke algebras of type \( A \), yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

[84] A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “The principal series of \( p \)-adic groups with disconnected center,” Proc. Lond. Math. Soc. (3) 114 : 5 (2017), pp. 798–854. MR 3653247 Zbl 06778792 article

Let \( \mathscr{G} \) be a split connected reductive group over a local non-Archimedean field. We classify all irreducible complex \( \mathscr{G} \)-representations in the principal series, irrespective of the (dis)connectedness of the center of \( \mathscr{G} \). This leads to a local Langlands correspondence for principal series representations of \( \mathscr{G} \). It satisfies all expected properties, in particular it is functorial with respect to homomorphisms of reductive groups. At the same time, we show that every Bernstein component \( \mathfrak{s} \) in the principal series has the structure of an extended quotient of Bernstein’s torus by Bernstein’s finite group (both attached to \( \mathfrak{s} \)).

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

[85] P. F. Baum, K. De Commer, and P. M. Hajac: “Free actions of compact quantum groups on unital \( C^* \)-algebras,” Doc. Math. 22 (2017), pp. 825–849. MR 3665403 Zbl 06810396 ArXiv 1304.2812v1 article

Let \( F \) be a field, \( \Gamma \) a finite group, and \( \operatorname{Map}(\Gamma,F) \) the Hopf algebra of all set-theoretic maps \( \Gamma\to F \). If \( E \) is a finite field extension of \( F \) and \( \Gamma \) is its Galois group, the extension is Galois if and only if the canonical map \[ E\otimes_F E\to E\otimes_F \operatorname{Map}(\Gamma,F) \] resulting from viewing \( E \) as a \( \operatorname{Map}(\Gamma,F) \)-comodule is an isomorphism. Similarly, a finite covering space is regular if and only if the analogous canonical map is an isomorphism. In this paper, we extend this point of view to actions of compact quantum groups on unital \( C^* \)-algebras. We prove that such an action is free if and only if the canonical map (obtained using the underlying Hopf algebra of the compact quantum group) is an isomorphism. In particular, we are able to express the freeness of a compact Hausdorff topological group action on a compact Hausdorff topological space in algebraic terms. As an application, we show that a field of free actions on unital \( C^* \)-algebras yields a global free action.

Piotr Mieczysław Hajac	Related
Kenny De Commer	Related

[86] A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Conjectures about \( p \)-adic groups and their noncommutative geometry,” pp. 15–51 in Around Langlands correspondences (Orsay, France, 17–20 June 2015). Edited by F. Brumley, M. P. Gómez Aparicio, and A. Minguez. Contemporary Mathematics 691. American Mathematical Society (Providence, RI), 2017. MR 3666049 ArXiv 1508.02837 incollection

Let \( G \) be any reductive \( p \)-adic group. We discuss several conjectures, some of them new, that involve the representation theory and the geometry of \( G \).

At the heart of these conjectures are statements about the geometric structure of Bernstein components for \( G \), both at the level of the space of irreducible representations and at the level of the associated Hecke algebras. We relate this to two well-known conjectures: the local Langlands correspondence and the Baum–Connes conjecture for \( G \). In particular, we present a strategy to reduce the local Langlands correspondence for irreducible \( G \)-representations to the local Langlands correspondence for supercuspidal representations of Levi subgroups.

Farrell Brumley	Related
Alberto Mínguez	Related
Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

[87] P. F. Baum and E. van Erp: “\( K \)-homology and Fredholm operators, I: Dirac operators,” J. Geom. Phys. 134 (2018), pp. 101–118. MR 3886929 article

[88] A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Smooth duals of inner forms of \( \mathrm{ GL}_n \) and \( \mathrm{ SL}_n \),” Doc. Math. 24 (2019), pp. 373–420. MR 3960124 article

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related