arXiv subjects

Paul Baum

Publications and source records attributed to Paul Baum.

At least 19 recordsLinked to original sources

arXiv2019

Morita equivalence for $k$-algebras

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 extended affine Weyl groups.

math.RT↗

arXiv2019

Smooth Duals of Inner Forms of $GL_n$ and $SL_n$

Let $F$ be a non-archimedean local field. We prove that every Bernstein component in the smooth dual of each inner form of the general linear group $GL_n(F)$ is canonically in bijection with the extended quotient for the action, given by Bernstein, of a finite group on a complex torus. For inner forms of $SL_n(F)$ we prove that each Bernstein component is canonically in bijection with the associated twisted extended quotient. In both cases, the bijections satisfy naturality properties with respect to the tempered dual, parabolic induction, central character, and the local Langlands correspondence.

math.RT↗

arXiv2018

Conjectures about p-adic groups and their noncommutative geometry

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.

math.RT↗

arXiv2016

K-homology and Fredholm Operators II: Elliptic Operators

This is an expository paper which gives a proof of the Atiyah-Singer index theorem for elliptic operators. Specifcally, we compute the geometric K-cycle that corresponds to the analytic K-cycle determined by the operator. This paper and its companion ("K-homology and index theory II: Dirac Operators") was written to clear up basic points about index theory that are generally accepted as valid, but for which no proof has been published. Some of these points are needed for the solution of the Heisenberg-elliptic index problem in our paper "K-homology and index theory on contact manifolds" Acta. Math. 2014.

math.DG↗

arXiv2016

Geometric structure for the principal series of a split reductive $p$-adic group with connected centre

Let $\mathcal{G}$ be a split reductive $p$-adic group with connected centre. We show that each Bernstein block in the principal series of $\mathcal{G}$ admits a definite geometric structure, namely that of an extended quotient. For the Iwahori-spherical block, this extended quotient has the form $T//W$ where $T$ is a maximal torus in the Langlands dual group of $\mathcal{G}$ and $W$ is the Weyl group of $\mathcal{G}$.

math.RT↗

arXiv2016

K-homology and Fredholm operators I: Dirac Operators

This is an expository paper which gives a proof of the Atiyah-Singer index theorem for Dirac operators, presenting the theorem as a computation of the K-homology of a point. This paper and its follow up ("K-homology and index theory II: Elliptic Operators") was written to clear up basic points about index theory that are generally accepted as valid, but for which no proof has been published. Some of these points are needed for the solution of the Heisenberg-elliptic index problem in our paper "K-homology and index theory on contact manifolds" Acta. Math. 2014.

math.DG↗

arXiv2015

The principal series of $p$-adic groups with disconnected centre

Let G be a split connected reductive group over a local non-archimedean field. We classify all irreducible complex G-representations in the principal series, irrespective of the (dis)connectedness of the centre of G. This leads to a local Langlands correspondence for principal series representations, which satisfies all expected properties. We also prove that the ABPS conjecture about the geometric structure of Bernstein components is valid throughout the principal series of G.

math.RT↗

arXiv2015

Expanders, exact crossed products, and the Baum-Connes conjecture

We reformulate the Baum-Connes conjecture with coefficients by introducing a new crossed product functor for C*-algebras. All confirming examples for the original Baum-Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to the original Baum-Connes conjecture become confirming examples for our reformulated conjecture.

math.KT↗

arXiv2015

Hecke algebras for inner forms of p-adic special linear groups

Let F be a non-archimedean local field and let $G^\sharp$ be the group of F-rational points of an inner form of $SL_n$. We study Hecke algebras for all Bernstein components of $G^\sharp$, via restriction from an inner form G of $GL_n (F)$. For any packet of 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.

math.RT↗

arXiv2014

Depth and the local Langlands correspondence

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

math.RT↗

arXiv2014

The local Langlands correspondence for inner forms of $SL_n$

Let F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group $SL_n (F)$. It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for $SL_n (F)$ enhanced with an irreducible representation of an S-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of $SL_n (F)$. An analogous result is shown in the archimedean case. 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 $GL_n (F)$, when the fields are close enough compared to the depth of the representations.

math.RT↗

arXiv2014

On the local Langlands correspondence for non-tempered representations

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 R-groups associated to non-tempered essentially square-integrable representations of Levi subgroups of G. We establish the basic properties of these new R-groups, which generalize Knapp--Stein R-groups.

math.RT↗

arXiv2013

Geometric structure and the local Langlands conjecture

We prove that a strengthened form of the local Langlands conjecture is valid throughout the principal series of any connected split reductive $p$-adic group. The method of proof is to establish the presence of a very simple geometric structure, in both the smooth dual and the Langlands parameters. We prove that this geometric structure is present, in the same way, for the general linear group, including all of its inner forms. With these results as evidence, we give a detailed formulation of a general geometric structure conjecture.

math.RT↗

arXiv2011

Extended quotients in the principal series of reductive p-adic groups

The geometric conjecture developed by the authors in [1,2,3,4] applies to the smooth dual Irr(G) of any reductive p-adic group G. It predicts a definite geometric structure - the structure of an extended quotient - for each component in the Bernstein decomposition of Irr(G). In this article, we prove the geometric conjecture for the principal series in any split connected reductive p-adic group G. The proof proceeds via Springer parameters and Langlands parameters. As a consequence of this approach, we establish strong links with the local Langlands correspondence. One important feature of our approach is the emphasis on two-sided cells in extended affine Weyl groups.

math.RT↗

arXiv2011

Kazhdan-Lusztig parameters and extended quotients

The Kazhdan-Lusztig parameters are important parameters in the representation theory of $p$-adic groups and affine Hecke algebras. We show that the Kazhdan-Lusztig parameters have a definite geometric structure, namely that of the extended quotient $T//W$ of a complex torus $T$ by a finite Weyl group $W$. More generally, we show that the corresponding parameters, in the principal series of a reductive $p$-adic group with connected centre, admit such a geometric structure. This confirms, in a special case, our recently formulated geometric conjecture. In the course of this study, we provide a unified framework for Kazhdan-Lusztig parameters on the one hand, and Springer parameters on the other hand. Our framework contains a complex parameter $s$, and allows us to interpolate between $s = 1$ and $s = \sqrt q$. When $s = 1$, we recover the parameters which occur in the Springer correspondence; when $s = \sqrt q$, we recover the Kazhdan-Lusztig parameters.

math.RT↗

arXiv2010

Geometric structure in the principal series of the p-adic group G_2

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 [3], 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 $G_2$. A feature of this article is the role played by cocharacters $h_c$ attached to two-sided cells $c$ 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 $A(G)$ of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space $A(G)$ is a model of the smooth dual $Irr(G)$. In this respect, our programme is a conjectural refinement of the Bernstein programme. The algebraic deformation is controlled by cocharacters $h_c$, one for each two-sided cell $c$ in certain extended affine Weyl groups. The cocharacters themselves appear to be closely related to Langlands parameters.

math.RT↗

arXiv2010

Equivariant geometric K-homology for compact Lie group actions

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "official" equivariant K-homology groups) and show that these are isomorphism.

math.KT↗

arXiv2009

A Geometric Description of Equivariant K-Homology for Proper Actions

Let G be a discrete group and let X be a G-finite, proper G-CW-complex. We prove that Kasparov's equivariant K-homology groups KK^G(C_0(X),\C) are isomorphic to the geometric equivariant K-homology groups of X that are obtained by making the geometric K-homology theory of Baum and Douglas equivariant in the natural way. This reconciles the original and current formulations of the Baum-Connes conjecture for discrete groups.

math.KT↗