Roger Plymen

• The University of Manchester

We consider the depth-zero supercuspidal $L$-packets of $\mathrm{SL}_2$. With the aid of the classical character formulas of Sally-Shalika, we prove the endoscopic character identities. For the depth-zero $L$-packet of cardinality $4$, we find that, in order to comply with the Sally-Shalika formulas, the three endoscopic groups must be paired caref...

The compact, connected Lie group $$E_6$$ E 6 admits two forms: simply connected and adjoint type. As we previously established, the Baum–Connes isomorphism relates the two Langlands dual forms, giving a duality between the equivariant K-theory of the Weyl group acting on the corresponding maximal tori. Our study of the $$A_n$$ A n case showed that...

In this paper we elucidate the structure of centralisers in the Weyl group of type $E_6$ and exhibit these as complex reflection groups. We determine the action of the centraliser on the corresponding fixed set in the maximal torus of both the simply connected $E_6$ Lie group and its Langlands dual. The component groups of the fixed sets are finite...

We give a streamlined account of $2$-spinors, up to and including the Dirac equation, using little more than the resources of linear algebra. We prove that the Dirac bundle is isomorphic to the associated bundles $\mathrm{SL}_2(\mathbb{C}) \times_{\mathrm{SU}_2} S$ and $\mathrm{SL}_2(\mathbb{C}) \times_{\mathrm{SU}_2} \bar{S}$. A solution of the Di...

In this paper we elucidate the structure of centralisers in the Weyl group of type $E_6$ and exhibit these as complex reflection groups. We also determine the action of the centraliser on the corresponding fixed set in the maximal torus of both the simply connected $E_6$ Lie group and its Langlands dual. The component groups of the fixed sets are f...

Let E/F be a finite and Galois extension of non-archimedean local fields. Let G be a connected reductive group defined over E and let M:=RE/FG be the reductive group over F obtained by Weil restriction of scalars. We investigate depth, and the enhanced local Langlands correspondence, in the transition from G(E) to M(F). We obtain a depth-comparison...

Let $E/F$ be a finite and Galois extension of non-archimedean local fields. Let $G$ be a connected reductive group defined over $E$ and let $M: = \mathfrak{R}_{E/F}\, G$ be the reductive group over $F$ obtained by Weil restriction of scalars. We investigate depth, and the enhanced local Langlands correspondence, in the transition from $G(E)$ to $M(...

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 \mathrm{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 \mathrm{SL}_n(F) we prove that each Ber...

Let $K$ be a non-archimedean local field. In the local Langlands correspondence for tori over $K$, we prove an asymptotic result for the depths.

Let $K$ be a local field of characteristic $p$. We consider the local Langlands correspondence for tori, and construct examples for which depth is not preserved.

Let S_k denote a maximal torus in the complex Lie group G = SL_n(C)/C_k and let T_k denote a maximal torus in its compact real form SU_n(C)/C_k, where k divides n. Let W denote the Weyl group of G, namely the symmetric group S_n. We elucidate the structure of the extended quotient S_k//W as an algebraic variety and of T_k//W as a topological space,...

In this paper we construct an equivariant Poincar\'e duality between dual tori equipped with finite group actions. We use this to demonstrate that Langlands duality induces a rational isomorphism between the group $C^*$-algebras of extended affine Weyl groups at the level of $K$-theory.

In this paper we construct an equivariant Poincar\'e duality between dual tori equipped with finite group actions. We use this to demonstrate that Langlands duality induces a rational isomorphism between the group $C^*$-algebras of extended affine Weyl groups at the level of $K$-theory.

We review Morita equivalence for finite type $k$-algebras $A$ and also a weakening of Morita equivalence which we call spectral 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 spectral...

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 assoc...

Let $\mathbf{S}_k$ denote a maximal torus in the complex Lie group $\mathbf{G} = \mathrm{SL}_n(\mathbb{C})/C_k$ and let $T_k$ denote a maximal torus in its compact real form $\mathrm{SU}_n(\mathbb{C})/C_k$, where $k$ divides $n$. Let $W$ denote the Weyl group of $\mathbf{G}$, namely the symmetric group $\mathfrak{S}_n$. We elucidate the structure o...

Let $\mathbf{S}_k$ denote a maximal torus in the complex Lie group $\mathbf{G} = \mathrm{SL}_n(\mathbb{C})/C_k$ and let $T_k$ denote a maximal torus in its compact real form $\mathrm{SU}_n(\mathbb{C})/C_k$, where $k$ divides $n$. Let $W$ denote the Weyl group of $\mathbf{G}$, namely the symmetric group $\mathfrak{S}_n$. We elucidate the structure o...

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.

We consider the group SL2(K), where K is a local non-archimedean field of characteristic two. We prove that the depth of any irreducible representation of SL2(K) is larger than the depth of the corresponding Langlands parameter, with equality if and only if the L-parameter is essentially tame. We also work out a classification of all L-packets for...

We consider the group $SL_2(K)$, where $K$ is a local non-archimedean field of characteristic two. We prove that the depth of any irreducible representation of $SL_2 (K)$ is larger than the depth of the corresponding Langlands parameter, with equality if and only if the L-parameter is essentially tame. We also work out a classification of all $L$-p...

We show how the epsilon factors for $GL_n$ factor, as finite morphisms of algebraic varieties, through the corresponding extended quotients. The finite morphisms are, up to a constant, rational characters of complex tori.

In this paper we consider the Baum-Connes correspondence for the affine and extended affine Weyl groups of a compact connected semisimple Lie group. We show that the Baum-Connes correspondence in this context arises from Langlands duality for the Lie group.

We investigate base change and automorphic induction $\mathbb{C}/\mathbb{R}$ at the level of K-theory for the general linear group $GL_n(\mathbb{R})$. In the course of this study, we compute in detail the C*-algebra K-theory of this disconnected group. We investigate the interaction of base change with the Baum-Connes correspondence for $GL_n(\math...

Let G be a split reductive p-adic group with connected centre. We show that each Bernstein block in the principal series of 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 G and W is the...

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 assoc...

Let G be any reductive p-adic group. We conjecture that every Bernstein component in the space of irreducible smooth G-representations can be described as a "twisted extended quotient" of the associated Bernstein torus by the asssociated finite group. We also pose some conjectures about L-packets and about the structure of the Schwartz algebra of G...

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 center of G. This leads to a local Langlands correspondence for principal series representations of G. It satisfies all expected properties, in pa...

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 canon...

We consider blocks in the representation theory of reductive p-adic groups. On each such block we conjecture a definite geometric structure, that of an extended quotient. We prove that this geometric structure is present for each block in the representation theory of any inner form of GL_n(F), and also for each block in the principal series of a co...

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 catego...

Using representation theory, we compute the spectrum of the Dirac operator on the universal covering group of $SL_2(\mathbb R)$, exhibiting it as the generator of $KK^1(\mathbb C, \mathfrak A)$, where $\mathfrak A$ is the reduced $C^*$-algebra of the group. This yields a new and direct computation of the $K$-theory of $\mathfrak A$. A fundamental r...

Let G = SL_2(K) with K a local function field of characteristic 2. We review Artin-Schreier theory for the field K, and show that this leads to a parametrization of certain L-packets in the smooth dual of G. We relate this to a recent geometric conjecture. The L-packets in the principal series are parametrized by quadratic extensions, and the super...

Let G be an inner form of a general linear group or a special linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths.

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...

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....

Let G = SL_2(K) with K a local function field of characteristic 2. We review Artin-Schreier theory for the field K, and show that this leads to a parametrization of L-packets in the smooth dual of G. We relate this to a recent geometric conjecture. The L-packets in the principal series are parametrized by quadratic extensions, and the supercuspidal...

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,...

Let G denote a split simply connected almost simple p-adic group. We study the unramified C*-algebra of G and prove that the rank of the K-theory group K_0 is the connection index f(G).

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 serie...

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 correspond...

This expository note will state the ABP (Aubert-Baum-Plymen) conjecture. The conjecture can be stated at four levels: 1. K-theory of C*-algebras 2. Periodic cyclic homology of finite type algebras 3. Geometric equivalence of finite type algebras 4. Representation theory. The emphasis in this note will be on representation theory.

This expository note will state the ABP (Aubert-Baum-Plymen) conjecture. The conjecture can be stated at four levels: 1. K-theory of C*-algebras 2. Periodic cyclic homology of finite type algebras 3. Geometric equivalence of finite type algebras 4. Representation theory. The emphasis in this note will be on representation theory. This is v2.

Let $F$ be a nonarchimedean local field of characteristic zero and let SL(N) = SL(N,F). This article is devoted to studying the influence of the elliptic representations of SL(N) on the $K$-theory. We provide full arithmetic details. This study reveals an intricate geometric structure. One point of interest is that the $R$-group is realized as an i...

We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson [2]. Entering 2,000,000 Riemann zeros, we prove that there exists x in the interval [exp (727.951858), exp (727.952178)] for which π(x) - li(x) > 3.2 × 10¹⁵¹. There are at least 10¹⁵⁴ successive integers x in this interval...

We survey Artin-Schreier theory, adapted to the local function field F_2((x)). This leads to a neat parametrization of the L-packets in the principal series of SL_2(F_2((x))).

In the representation theory of reductive p p -adic groups G 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 co...

Without Abstract

We conjecture the existence of a simple geometric structure underlying questions of reducibility of parabolically induced representations of reductive p-adic groups.

We investigate base change $C/R$ at the level of $K$-theory for the general linear group $GL(n,R)$. In the course of this study, we compute in detail the $C*$-algebra $K$-theory of this disconnected group. We investigate the interaction of base change with the Baum-Connes correspondence for $GL(n,R)$ and $GL(n,C)$. This article is the archimedean c...

Let F be a nonarchimedean local field and let G = GL(n) = GL(n,F). Let E/F be a finite Galois extension. We investigate base change E/F at two levels: at the level of algebraic varieties, and at the level of K-theory. We put special emphasis on the representations with Iwahori fixed vectors, and the tempered spectrum of GL(1) and GL(2). In this con...

Let 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 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...

We compute the entire cyclic homology and cohomology of the Schatten ideals fl^p.

We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson. Entering $2,000,000$ Riemann zeros, we prove that there exists $x$ in the interval $[exp(727.951858), exp(727.952178)]$ for which $\pi(x)-\li(x) > 3.2 \times 10^{151}$. There are at least $10^{154}$ successive integers $x$...

Let F be a nonarchimedean local field, let D be a division algebra over F, let GL(n)=GL(n,F). Let ν denote Plancherel measure for GL(n). Let Ω be a component in the Bernstein variety Ω(GL(n)). Then Ω yields its fundamental invariants: the cardinality q of the residue field of F, the sizes m1,…,mt, exponents e1,…,et, torsion numbers r1,…,rt, formal...

Let 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 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...

Let F be a nonarchimedean local field and let GL(N) = GL(N,F). We prove the existence of parahoric types for GL(N). We construct representative cycles in all the homology classes of the chamber homology of GL(3).

Certain cocycles constructed by Connes are characters of $p$-summable Fredholm modules. In this article, we establish some consequences of the universal properties which these characters enjoy. Our main technical result is that the entire cyclic cohomology of the p-th Schatten ideal L^p (respectively, homology) is independent of p and isomorphic to...

We provide an explicit Plancherel formula for the p-adic group GL(n). We determine explicitly the Bernstein decomposition of Plancherel measure, including all numerical constants. We also prove a transfer-of-measure formula for GL(n). To cite this article: A.-M. Aubert, R. Plymen, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Let F be a nonarchimedean local field and let G = GL(N) = GL(N, F). In this article, we reveal an unexpected connection be-tween the arithmetic of the field F and the K-theory of the tempered dual of G. The bridge between these two aspects is the enlarged building β 1 G. The K-theory of the reduced C*-algebra of GL(N) is isomorphic (after tensoring...

Let F be a nonarchimedean local field, let D be a division algebra over F, let GL(n) = GL(n,F). Let \nu denote Plancherel measure for GL(n). Each component \Omega in the Bernstein variety \Omega(GL(n)) has several numerical invariants attached to it. We provide explicit formulas for the Bernstein component \nu_{\Omega} of Plancherel measure in term...

We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group $GL(2,F)$ where $F$ is any global field.

We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group $GL(2,F)$ where $F$ is any global field.

Let G denote the p-adic group GL(n). With the aid of Langlands parameters, we equip each Bernstein component Z in the smooth dual of G with the structure of complex algebraic variety. We prove that the periodic cyclic homology of the corresponding ideal in the Hecke algebra H(G) is isomorphic to the de Rham cohomology of Z. We show how the structur...

Let G denote the p-adic group GL(n), and let S(G) denote the Schwartz algebra of G. We construct a Chern character from the K-theory of the reduced C*-algebra of G to the periodic cyclic homology of S(G) which becomes an isomorphism after tensoring over Z with C.

We describe a geometric counterpart of the Baum-Connes map for the p-adic group GL(n).

The reduced C*-algebra of the p-adic group GL(n) admits a Bernstein decomposition. We give a minimal refinement of this decomposition, and provide structure theorems for the reduced Iwahori-Hecke C*-algebra and the reduced spherical C*-algebra. This leads to a very explicit description of the tempered dual of GL(n) in terms of Bernstein parameters...

Let G be a compact connected Lie group. We prove that the Fourier algebra A(G) is weakly amenable if and only if G is abelian.

We give a proof of the Baum-Connes conjecture for p-adic GL(n). . . Version Francaise Abregee Soit F un corps local non archimedien de caracteristique 0. Soit GL(n, F ) le groupe des matrices inversibles n

Let F be a global field, A its ring of adeles, G a reductive group over F. We prove the Baum-Connes conjecture for the adelic group G(A). Comment: 9 pages

Let A(n) be the smooth dual of the p-adic group G = GL(n). We create on A(n) the structure of a complex algebraic variety. There is a morphism of A(n) onto the Bernstein variety ΩG which is injective on each component of A(n). The tempered dual of G is a deformation retract of A(n). The periodic cyclic homology of the Hecke algebra of G is isomorph...

We construct a Chern character map from the K-theory of the reduced C^* algebra of the p-adic GL(n) with values in the periodic cyclic homology of the Schwartz algebra of this group. We prove that this map is an isomorphism after tensoring with C by comparing an explicit formula, stated in the algebraic case by Cuntz and Quillen, with the classical...

Over the past several years, operator algebraists have become increasingly interested in the problem of calculating the K-theory of group C∗-algebras. The focal point of research in this area ist he BaumConnesConjecture[BCH],which proposes a description of K-theory for the C∗-algebra of a group in terms of homology and the representation theory of...

Relying on properties of the inductive tensor product, we construct cyclic type homology theories for certain nuclear algebras. In this context, we establish continuity theorems. We compute the periodic cyclic homology of the Schwartz algebra of p-adic GL(n) in terms of compactly supported de Rham cohomology of the tempered dual of GL(n).RésumeNous...

Relying of properties of the inductive tensor product, we construct cyclic type homology theories for certain nuclear algebras. In this context we establish continuity theorems. We compute the periodic cyclic homology of the Schwartz algebra of p-adic GL(n) in terms of compactly supported de Rham cohomology of the tempered dual of GL(n).

We give a proof of the Baum-Connes conjecture for p-adic GL(n).RésuméNous donnons une démonstration de la conjecture de Baum-Connes pour le groupe p-adique GL(n).

We give a proof of the Baum-Connes conjecture for p-adic GL(n). . .

In this article we cover an episode in the representation theory of GL(n) defined over a p-adic field with finite residue class field. We concentrate on the irreducible tempered representations admitting non-zero Iwahori-fixed vectors. We describe the space of these representations in terms of Deligne-Langlands parameters. In [6], Kazdhan and Luszt...

Infinite-dimensional Clifford algebras and their Fock representations originated in the quantum mechanical study of electrons. In this book, the authors give a definitive account of the various Clifford algebras over a real Hilbert space and of their Fock representations. A careful consideration of the latter's transformation properties under Bogol...

This is a view from noncommutative geometry of the representation theory of GL(n). Topics covered include: Morita equivalence, the Iwahori-Hecke C*-algebra, the spherical C*-algebra, C*-Plancherel Theorem, Brylinski quotient (now called the extended quotient).

Let F be a p-adic field and let G = SL(2) be the group of unimodular 2 × 2 matrices over F. The aim of this paper is to calculate certain equivariant homology groups attached to the action of G on its tree. They arise in connection with a theorem of M. Pimsner on the K-theory of the C *-algebra of G [12], and our purpose is to explore the represent...

A number of texts have recently become available which provide good general introductions to p-Adic numbers and p-Adic analysis. However, there is at present a gap between such books and the sophisticated applications in the research literature. The aim of this book is to bridge this gulf by providing a collection of intermediate level articles on...

A number of texts have recently become available which provide good general introductions to p-Adic numbers and p-Adic analysis. However, there is at present a gap between such books and the sophisticated applications in the research literature. The aim of this book is to bridge this gulf by providing a collection of intermediate level articles on...

p>The reduced C^*-algebra of the p-adic group GL(n) is Morita equivalent to an abelian C^*-algebra. The structure of this abelian C^*-algebra is described in terms of unramified unitary characters of Levi subgroups. The K-groups K₀ and K₁ are both free abelian of infinite rank. Generators are esse...

The Dirac operator plays a fundamental role in the geometric construction of the discrete series for semisimple Lie groups. We show that, at the level of K-theory, the Dirac operator also plays a central role in connection with the principal series for complex connected semisimple Lie groups. This proves the Connes-Kasparov conjecture for such grou...

Let F be a symplectic vector bundle over a space X. We construct a bundle of elementary C∗-algebras over X, and prove that the Dixmier-Douady invariant of this bundle is zero. The underlying Hilbert bundles, with their associated module structures, determine a characteristic class: we prove that this class is the second Stiefel-Whitney class of F.