arXiv subjects

Daniel Skodlerack

Publications and source records attributed to Daniel Skodlerack.

11 recordsLinked to original sources

arXiv2026

Semisimple types for quaternionic forms of p-adic classical groups and compatible beta-extensions

Let $G$ be a quaternionic form of a $p$-adic classical group ($p$ odd). We construct a Bushnell-Kutzko-Stevens type for every Bernstein block of the category of smooth complex representations of $G$. Further we construct a system of compatible $\beta$-extensions, i.e. a family of $\beta$-extensions parametrised by the points of a chamber of the Bruhat-Tits building of the centralizer $G_\beta$ which are related via transfer.

math.RT↗

arXiv2026

Cuspidal endo-support and strong beta extensions

Let $G$ be an inner form of a general linear group or classical group over a non-archimedean local field of residual characteristic $p$, assumed odd in the classical case. We prove that every smooth representation of $G$ over an algebraically closed field $R$ of characteristic $\ell\neq p$ contains a maximal semisimple character, i.e., one for which the point in the building of the corresponding centralizer is a vertex. Further, for every endo-parameter adapted to $G$, we define its support, which leads also to the notion of cuspidal endo-support of an irreducible representation, and we relate this to its cuspidal support. We also introduce beta extensions for strong facets in the building of a centralizer, and show these are sufficient for the construction of types. These results are used in a subsequent paper to decompose the category of smooth $R$-representations of $G$.

math.RT↗

arXiv2024

Block decompositions for $p$-adic classical groups and their inner forms

For an inner form $\mathrm{G}$ of a general linear group or classical group over a non-archimedean local field of odd residue characteristic, we decompose the category of smooth representations on $\mathbb{Z}[\mu_{p^{\infty}},1/p]$-modules by endo-parameter. We prove that parabolic induction preserves these decompositions, and hence that it preserves endo-parameters. Moreover, we show that the decomposition by endo-parameter is the $\overline{\mathbb{Z}}[1/p]$-block decomposition; and, for $\mathrm{R}$ an integral domain, introduce a graph whose connected components parameterize the $\mathrm{R}$-blocks, in particular including the cases $\mathrm{R}=\overline{\mathbb{Z}}_{\ell}$ and $\mathrm{R}=\overline{\mathbb{F}}_\ell$ for $\ell\neq p$. From our description, we deduce that the $\overline{\mathbb{Z}_\ell}$-blocks and $\overline{\mathbb{F}_\ell}$-blocks of $\mathrm{G}$ are in natural bijection, as had long been expected. Our methods also apply to the trivial endo-parameter (i.e., the depth zero subcategory) of any connected reductive $p$-adic group, providing an alternative approach to results of Dat and Lanard in depth zero. Finally, under a technical assumption (known for inner forms of general linear groups) we reduce the $\mathrm{R}$-block decomposition of $\mathrm{G}$ to depth zero.

math.RT↗

arXiv2022

Cuspidal irreducible complex or l-modular representations of quaternionic forms of p-adic classical groups for odd p

Given a quaternionic form G of a p-adic classical group (p odd) we classify all cuspidal irreducible representations of G with coefficients in an algebraically closed field of characteristic different from p. We prove two theorems: At first: Every irreducible cuspidal representation of G is induced from a cuspidal type, i.e. from a certain irreducible representation of a compact open subgroup of G, constructed from a beta-extension and a cuspidal representation of a finite group. Secondly we show that two intertwining cuspidal types of G are up to equivalence conjugate under some element of G.

math.RT↗

arXiv2021

Semisimple characters for inner froms I: GL_n(D)

The article is about the representation theory of an inner form~$G$ of a general linear group over a non-archimedean local field. We introduce semisimple characters for~$G$ whose intertwining classes describe conjecturally via Local Langlands correspondence the behavior on wild inertia. These characters also play a potential role to understand the classification of irreducible smooth representations of inner forms of classical groups. We prove the intertwining formula for semisimple characters and an intertwining implies conjugacy like theorem. Further we show that endo-parameters for~$G$, i.e. invariants consisting of simple endo-classes and a numerical part, classify the intertwining classes of semisimple characters for~$G$. They should be the counter part for restrictions of Langlands-parameters to wild inertia under Local Langlands correspondence.

math.RT↗

arXiv2020

Endo-parameters for p-adic classical groups

For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal C-representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart's notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.

math.RT↗

arXiv2017

Semisimple characters for inner forms II: Quaternionic inner forms of classical groups

In this article we consider a quaternionic inner form $G$ of a $p$-adic classical group defined over a non-archimedian local field of odd residue characteristic. We construct all full self-dual semisimple characters for $G$ and we classify their intertwining classes using endo-parameters. Further we prove an intertwining and conjugacy theorem for self-dual semisimple characters. We give the formulas for the set of intertwiners between self-dual semisimple characters. We count all $G$-intertwining classes of self-dual semisimple characters which lift to the same $\tilde{G}$-intertwining class of a semisimple character for the ambient general linear group $\tilde{G}$ for $G$.

math.RT↗

arXiv2016

Intertwining semisimple characters for p-adic classical groups

Let~$G$ be a unitary group of an~$ε$-hermitian form~$h$ given over a nonarchimedean local field~$F_0$ of odd residue characteristic. We introduce a geometric combinatoric condition under which we prove "Intertwining implies Conjugacy" for semisimple characters of~$G$ and the general linear group of the ambient vector space of~$G$. Further we prove a Skolem-Noether result for the action of~$G$ on its Lie algebra, more precisely two Lie algebra elements of~$G$ which have the same characteristic polynomial over~$F$ must be conjugate under an element of~$G$ if there are corresponding semisimple characters which intertwine over an element of~$G$ } Let~$G$ be a unitary group over a nonarchimedean local field of odd residual characteristic. This paper concerns the study of the "wild part" of the irreducible smooth representations of~$G$, encoded in a so-called "semisimple character". We prove two fundamental results concerning them, which are crucial steps towards a classification of the cuspidal representations of~$G$. First we introduce a geometric combinatoric condition under which we prove an "intertwining implies conjugacy" theorem for semisimple characters, both in~$G$ and in the ambient general linear group. Second, we prove a Skolem--Noether theorem for the action of~$G$ on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of~$G$ which have the same characteristic polynomial must be conjugate under an element of~$G$ if there are corresponding semisimple strata which are intertwined by an element of~$G$.

math.NT↗

arXiv2015

On intertwining implies conjugacy for classical groups

Let G be a unitary group of a signed-Hermitian form h given over a non-Archimedian local field k of residue characteristic not two. Let V be the vector space on which h is defined. We consider minimal skew-strata, more precisely pairs (b,a) consisting of a Lie algebra element b and a hereditary order $a$ stable under the adjoint involution of h, such that b generates a field whose multiplicative group is a subset of the normalizer of $a$, and some more conditions. We prove that if two minimal skew-strata (b_i,a), i=1,2 interwine by an element of G, then they are conjugate under G, and we give a natural generalization for minimal semisimple skew-strata.

math.RT↗

arXiv2013

Field Embeddings which are conjugate under a unit of a p-adic classical Group

Let (V,h) be a Hermitian space over a division algebra D which is of index at most two over a non-Archimedean local field k of residue characteristic not 2. Let G be the unitary group defined by h and let σbe the adjoint involution. Suppose we are given two σ-invariant but not σ-fixed field extensions E1 and E2 of k in End_D(V) which are isomorphic under conjugation by an element g of G and suppose that there is a point x in the Bruhat-Tits building of G which is fixed by the action of E1\{0} and E2\{0} on the reduced building of Aut_D(V). Then E1 is conjugate to E2 under an element of the stabilizer of x in G if E1 and E2 are conjugate under an element of the stabilizer of x in Aut_D(V) and a weak extra condition. In addition in many cases the conjugation by g from E1 to E2 can be realized as conjugation by an element of the stabilizer of x in G.

math.RT↗

arXiv2010

Embedding types and canonical affine maps between Bruhat-Tits buildings of classical groups (Thesis)

P. Broussous and S. Stevens studied maps between enlarged Bruhat-Tits buildings to construct types for p-adic unitary groups. They needed maps which respect the Moy-Prasad filtrations. That property is called (CLF), i.e. compatibility with the Lie algebra filtrations. In the first part of this thesis we generalise their results on such maps to the Quaternion-algebra case. Let k0 be a p-adic field of residue characteristic not two. We consider a semisimple k0-rational Lie algebra element beta of a unitary group G:=U(h) defined over k0 with a signed hermitian form h. Let H be the centraliser of beta in G. We prove the existence of an affine H(k0)-equivariant CLF-map j from the enlarged Bruhat-Tits building B^1(H,k0) to B^1(G,k0). As conjectured by Broussous the CLF-property determines j, if none of the factors of H is k0-isomorphic to the isotropic orthogonal group of k0-rank one and all factors are unitary groups. Under the weaker assumption that the affine CLF-map j is only equivariant under the center of H^0(k0) it is uniquely determined up to a translation of B^1(H,k0). The second part is devoted to the decoding of embedding types by the geometry of a CLF-map. Embedding types have been studied by Broussous and M. Grabitz. We consider a division algebra D of finite index with a p-adic center F. The construction of simple types for GLn(D) in the Budhnell-Kutzko framework required an investigation of strata which had to fulfil a rigidity property. Giving a stratum especially means to fix a pair (E,a) consisting of a field extension E|F in Mn(D) and a hereditary order a which is stable under conjugation by E^x, in other words we fix an embedding of E^x into the normalizer of a. Broussous and Grabitz classified these pairs with invariants. We describe and prove a way to decode these invariants using the geometry of a CLF-map.

math.GR↗