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Roger Plymen

Publications and source records attributed to Roger Plymen.

At least 19 recordsLinked to original sources
arXiv2026

Geometry and topology of the tempered Iwahori-spherical representations of a split semisimple $p$-adic group

Let $\mathfrak{G}$ be a connected split $p$-adic group of type $B_n$, $C_n$ or $D_n$. Amongst the tempered representations of $\mathfrak{G}$ a key role is played by the Iwahori-spherical block. We provide a compact Hausdorff model for this space which allows us to compute the $K$-theory ranks for the corresponding $C^*$-algebra. The model, an extended quotient, is stratified by sectors. Underlying the approach of this paper is the interplay between the geometric extended quotient and the spectral extended quotient in the context of the ABPS conjecture. We give geometric descriptions of every sector thus equipping the spectrum with a cellular structure. We classify the geometric structures arising in our model, and specifically discover real projective spaces along with cones and suspensions of these. These examples require a minor modification to our homotopy sector conjecture: we show that Langlands dual sectors are rationally (indeed dyadically) homotopy equivalent in all cases ($A_n, B_n, C_n, D_n, E_6, E_7$ and $E_8$).

math.RT↗
arXiv2026

A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations

Let $F$ be a non-archimedean local field such that $4|q-1$, with $q$ the order of the residue field of $F$, and let $(M^0,\sigma^0)$ be the depth-zero cuspidal pair for the twisted Levi subgroup $G^0$ of $\mathrm{SL}_8$ arising from quadratic and quartic field extensions, as defined in the recent article by Adler-Fintzen-Ohara [AFO]. Then the corresponding Bernstein block is described by a twisted Hecke algebra $\mathcal{H}^0$. We describe $\mathcal{H}^0$ explicitly as a noncommutative $\mathbb{C}$-algebra with generators and relations. We describe explicitly the simple modules of $\mathcal{H}^0$. All the simple modules are $2$-dimensional. The primitive spectrum of $\mathcal{H}^0$ is then an explicit complex algebraic variety $\mathfrak{X}$. The maximal compact real form of $\mathfrak{X}$ is homeomorphic to a Klein bottle. This Klein bottle is a model of the unitary principal series of $G^0$ attached to the cuspidal pair $(M^0, \sigma^0)$. We make a full comparison with the classical situation in which $G = \mathrm{SL}_8$ and $(M,\sigma)$ is a cuspidal pair for $G$. The supercuspidal representation $\sigma$ is constructed from the same quadratic and quartic extensions of $F$. Let $\mathfrak{s}$ be the point in the Bernstein spectrum $\mathfrak{B} G$ determined by $(M,\sigma)$ and let $\mathfrak{s}^0$ be the point in the Bernstein spectrum $\mathfrak{B} G^0$ determined by $(M^0, \sigma^0)$. We compare the two points $\mathfrak{s}$ and $\mathfrak{s}^0$ and show explicitly that the corresponding Bernstein varieties are isomorphic. In that case, the Klein bottle re-appears, this time floating in the tempered dual of $\mathrm{SL}_8$.

math.RT↗
arXiv2024

Comparison of the Sally-Shalika character formulas with the endoscopic character identities for $\mathrm{SL}_2$

We consider the depth-zero supercuspidal $L$-packets of $\mathrm{SL}_2(F)$ where $F$ is a non-archimedean local field of characteristic zero. We compare the explicit endoscopic character identities for $\mathrm{SL}_2(F)$ with the classical character formulas of Sally-Shalika. Our main result concerns the supercuspidal $L$-packet of size $4$. For this $L$-packet, we show how the norm $1$ groups $H_1, H_2, H_3$ in the three quadratic extensions of $F$ play a crucial role in the endoscopic character identities for $\mathrm{SL}_2(F)$.

math.RT↗
arXiv2022

Centralisers, complex reflection groups and actions in the Weyl group $E_6$

The compact, connected Lie group $E_6$ 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 $A_n$ case showed that this duality persists at the level of homotopy, not just homology. In this paper we compute the extended quotients of maximal tori for the two forms of $E_6$, showing that the homotopy equivalences of sectors established in the $A_n$ case also exist here, leading to a conjecture that the homotopy equivalences always exist for Langlands dual pairs. In computing these sectors we show that centralisers in the $E_6$ Weyl group decompose as direct products of reflection groups, generalising Springer's results for regular elements, and we develop a pairing between the component groups of fixed sets generalising Reeder's results. As a further application we compute the $K$-theory of the reduced Iwahori-spherical $C^*$-algebra of the p-adic group $E_6$, which may be of adjoint type or simply connected.

math.GR↗
arXiv2022

$2$-Spinors via Linear Algebra

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 Dirac equation determines a pair of conjugate $2$-spinor fields over the mass shell $X_m$.

math-ph↗
arXiv2020

Comparison of the depths on both sides of the local Langlands correspondence for Weil-restricted groups (with appendix by Jessica Fintzen)

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 $M(F)$. We obtain a depth-comparison formula for Weil-restricted groups.

math.RT↗
arXiv2018

Note on the non-preservation of depth

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.

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↗
arXiv2017

Poincaré duality and Langlands duality for extended affine Weyl groups

In this paper we construct an equivariant Poincaré 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.

math.KT↗
arXiv2017

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↗
arXiv2016

Stratified Langlands duality in the $A_n$ tower

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 of the extended quotient $\mathbf{S}_k // W$ as an algebraic variety and of $T_k // W$ as a topological space, in both cases describing them as bundles over unions of tori. Corresponding to the invariance of $K$-theory under Langlands duality, this calculation provides a homotopy equivalence between $T_k // W$ and its dual $T_{\frac{n}{k}} // W$. Hence there is an isomorphism in cohomology for the extended quotients which is stratified as a direct sum over conjugacy classes of the Weyl group. We use our formula to compute a number of examples.

math.KT↗
arXiv2016

On $L$-packets and depth for $SL_2(K)$ and its inner form

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 $L$-packets for $SL_2 (K)$ and for its non-split inner form, and we provide explicit formulae for the depths of their $L$-parameters.

math.RT↗
arXiv2016

Epsilon factors as algebraic characters on the smooth dual of $\mathrm{GL}_n$

Let $K$ be a non-archimedean local field and let $G = \mathrm{GL}_n(K)$. We have shown in previous work that the smooth dual $\mathbf{Irr}(G)$ admits a complex structure: in this article we show how the epsilon factors interface with this complex structure. The epsilon factors, up to a constant term, factor as invariant characters through the corresponding complex tori. For the arithmetically unramified smooth dual of $\mathrm{GL}_n$, we provide explicit formulas for the invariant characters.

math.RT↗
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↗
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

Functoriality and K-theory for $GL_n(\mathbb{R})$

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 $GL_n(\mathbb{R})$ and $GL_n(\mathbb{C})$. This article is the archimedean companion of our previous article in the Journal of Noncommutative Geometry.

math.KT↗