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Search: MSC category 20G ( Linear algebraic groups and related topics {For arithmetic theory, see 11E57, 11H56; for geometric theory, see 14Lxx, 22Exx; for other methods in representation theory, see 15A30, 22E45, 22E46, 22E47, 22E50, 22E55} )

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1. CJM Online first

Cohen, Jonathan
Transfer of Representations and Orbital Integrals for Inner Forms of $GL_n$
We characterize the Local Langlands Correspondence (LLC) for inner forms of $\operatorname{GL}_n$ via the Jacquet-Langlands Correspondence (JLC) and compatibility with the Langlands Classification. We show that LLC satisfies a natural compatibility with parabolic induction and characterize LLC for inner forms as a unique family of bijections $\Pi(\operatorname{GL}_r(D)) \to \Phi(\operatorname{GL}_r(D))$ for each $r$, (for a fixed $D$) satisfying certain properties. We construct a surjective map of Bernstein centers $\mathfrak{Z}(\operatorname{GL}_n(F))\to \mathfrak{Z}(\operatorname{GL}_r(D))$ and show this produces pairs of matching distributions in the sense of Haines. Finally, we construct explicit Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke algebras of $\operatorname{GL}_r(D)$, and thereby produce many explicit pairs of matching functions.

Keywords:Langlands correspondence, inner form

2. CJM Online first

Du, Jie; Zhao, Zhonghua
Multiplication formulas and canonical bases for quantum affine gln
We will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra $\mathfrak{H}_\Delta(n)$ of a cyclic quiver $\Delta(n)$. As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a certain monomial base for $\mathfrak{H}_\Delta(n)$ given by Deng, Du, and Xiao together with the double Ringel--Hall algebra realisation of the quantum loop algebra $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_n)$ given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for $\mathbf{U}_v^+(\widehat{\mathfrak{g}\mathfrak{l}}_n)$. As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most $2$ for the quantum group $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_2)$.

Keywords:Ringel-Hall algebra, quantum group, cyclic quiver, monomial basis, canonical basis
Categories:16G20, 20G42

3. CJM Online first

Ghaani Farashahi, Arash
A Class of Abstract Linear Representations for Convolution Function Algebras over Homogeneous Spaces of Compact Groups
This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $\mu$ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated to the Weil's formula and $1\le p\lt \infty$. We then present a structured class of abstract linear representations of the Banach convolution function algebras $L^p(G/H,\mu)$.

Keywords:homogeneous space, linear representation, continuous unitary representation, convolution function algebra, compact group, convolution, involution
Categories:43A85, 47A67, 20G05

4. CJM Online first

Ovchinnikov, Alexey; Wibmer, Michael
Tannakian categories with semigroup actions
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.

Keywords:semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters
Categories:18D10, 12H10, 20G05, 33C05, 33C80, 34K06

5. CJM 2016 (vol 69 pp. 107)

Kamgarpour, Masoud
On the Notion of Conductor in the Local Geometric Langlands Correspondence
Under the local Langlands correspondence, the conductor of an irreducible representation of $\operatorname{Gl}_n(F)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

Keywords:local geometric Langlands, connections, cyclic vectors, opers, conductors, Segal-Sugawara operators, Chervov-Molev operators, critical level, smooth representations, affine Kac-Moody algebra, categorical representations
Categories:17B67, 17B69, 22E50, 20G25

6. CJM 2016 (vol 68 pp. 1067)

Runde, Volker; Viselter, Ami
On Positive Definiteness over Locally Compact Quantum Groups
The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.

Keywords:bicrossed product, locally compact quantum group, non-commutative $L^p$-space, positive-definite function, positive-definite measure, separation property
Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89

7. CJM 2016 (vol 68 pp. 395)

Garibaldi, Skip; Nakano, Daniel K.
Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the question of whether a given complex representation is symplectic or orthogonal has been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from 2 hold by using similar proofs. This paper considers analogues of these results for simple, induced and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

Keywords:orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups
Categories:20G05, 11E39, 11E88, 15A63, 20G15

8. CJM 2016 (vol 68 pp. 280)

da Silva, Genival; Kerr, Matt; Pearlstein, Gregory
Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising from Mirror Symmetry and Middle Convolution
We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions $1\leq d\leq6$ arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is $G_{2}$) of the family of 6-folds, and the theory of boundary components of Mumford-Tate domains.

Keywords:variation of Hodge structure, limiting mixed Hodge structure, Calabi-Yau variety, middle convolution, Mumford-Tate group
Categories:14D07, 14M17, 17B45, 20G99, 32M10, 32G20

9. CJM 2016 (vol 68 pp. 309)

Daws, Matthew
Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups
We show that the assignment of the (left) completely bounded multiplier algebra $M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group $\mathbb G$, and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf $*$-homomorphisms between universal $C^*$-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal $C^*$-algebra level, and that then the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal $C^*$-algebra picture, and then, again, show how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the ``maximal classical'' quantum subgroup of a locally compact quantum group, show that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.

Keywords:locally compact quantum group, morphism, intrinsic group, multiplier, centraliser
Categories:20G42, 22D25, 43A22, 43A35, 43A95, 46L52, 46L89, 47L25

10. CJM 2015 (vol 68 pp. 150)

Stavrova, Anastasia
Non-stable $K_1$-functors of Multiloop Groups
Let $k$ be a field of characteristic 0. Let $G$ be a reductive group over the ring of Laurent polynomials $R=k[x_1^{\pm 1},...,x_n^{\pm 1}]$. Assume that $G$ contains a maximal $R$-torus, and that every semisimple normal subgroup of $G$ contains a two-dimensional split torus $\mathbf{G}_m^2$. We show that the natural map of non-stable $K_1$-functors, also called Whitehead groups, $K_1^G(R)\to K_1^G\bigl( k((x_1))...((x_n)) \bigr)$ is injective, and an isomorphism if $G$ is semisimple. As an application, we provide a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii-Neher) and the subgroup generated by exponential automorphisms.

Keywords:loop reductive group, non-stable $K_1$-functor, Whitehead group, Laurent polynomials, Lie torus
Categories:20G35, 19B99, 17B67

11. CJM 2014 (vol 66 pp. 1201)

Adler, Jeffrey D.; Lansky, Joshua M.
Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes
Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\tilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $(\tilde{G},\Gamma)$, and consider any group $G$ satisfying the axioms. If both $\tilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\tilde{G}^*$ and $G^*$. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in $G^*(k)$ to the analogous set for $\tilde{G}^*(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\tilde{G}(k)$, one obtains a mapping of such packets.

Keywords:reductive group, lifting, conjugacy class, representation, Lusztig series
Categories:20G15, 20G40, 20C33, 22E35

12. CJM 2014 (vol 67 pp. 184)

McReynolds, D. B.
Geometric Spectra and Commensurability
The work of Reid, Chinburg-Hamilton-Long-Reid, Prasad-Rapinchuk, and the author with Reid have demonstrated that geodesics or totally geodesic submanifolds can sometimes be used to determine the commensurability class of an arithmetic manifold. The main results of this article show that generalizations of these results to other arithmetic manifolds will require a wide range of data. Specifically, we prove that certain incommensurable arithmetic manifolds arising from the semisimple Lie groups of the form $(\operatorname{SL}(d,\mathbf{R}))^r \times (\operatorname{SL}(d,\mathbf{C}))^s$ have the same commensurability classes of totally geodesic submanifolds coming from a fixed field. This construction is algebraic and shows the failure of determining, in general, a central simple algebra from subalgebras over a fixed field. This, in turn, can be viewed in terms of forms of $\operatorname{SL}_d$ and the failure of determining the form via certain classes of algebraic subgroups.

Keywords:arithmetic groups, Brauer groups, arithmetic equivalence, locally symmetric manifolds

13. CJM 2013 (vol 67 pp. 450)

Santoprete, Manuele; Scheurle, Jürgen; Walcher, Sebastian
Motion in a Symmetric Potential on the Hyperbolic Plane
We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.

Keywords:Hamiltonian systems with symmetry, symmetries, non-compact symmetry groups, singular reduction
Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20

14. CJM 2011 (vol 64 pp. 409)

Rainer, Armin
Lifting Quasianalytic Mappings over Invariants
Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb C[V]^G$. We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$ over the mapping of invariants $\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in $\mathcal C$, e.g., the real analytic class, then $f$ admits a lift of the same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation. If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.

Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variation
Categories:14L24, 14L30, 20G20, 22E45

15. CJM 2011 (vol 63 pp. 1307)

Dimitrov, Ivan; Penkov, Ivan
A Bott-Borel-Weil Theorem for Diagonal Ind-groups
A diagonal ind-group is a direct limit of classical affine algebraic groups of growing rank under a class of inclusions that contains the inclusion $$ SL(n)\to SL(2n), \quad M\mapsto \begin{pmatrix}M & 0 \\ 0 & M \end{pmatrix} $$ as a typical special case. If $G$ is a diagonal ind-group and $B\subset G$ is a Borel ind-subgroup, we consider the ind-variety $G/B$ and compute the cohomology $H^\ell(G/B,\mathcal{O}_{-\lambda})$ of any $G$-equivariant line bundle $\mathcal{O}_{-\lambda}$ on $G/B$. It has been known that, for a generic $\lambda$, all cohomology groups of $\mathcal{O}_{-\lambda}$ vanish, and that a non-generic equivariant line bundle $\mathcal{O}_{-\lambda}$ has at most one nonzero cohomology group. The new result of this paper is a precise description of when $H^j(G/B,\mathcal{O}_{-\lambda})$ is nonzero and the proof of the fact that, whenever nonzero, $H^j(G/B, \mathcal{O}_{-\lambda})$ is a $G$-module dual to a highest weight module. The main difficulty is in defining an appropriate analog $W_B$ of the Weyl group, so that the action of $W_B$ on weights of $G$ is compatible with the analog of the Demazure ``action" of the Weyl group on the cohomology of line bundles. The highest weight corresponding to $H^j(G/B, \mathcal{O}_{-\lambda})$ is then computed by a procedure similar to that in the classical Bott-Borel-Weil theorem.

Categories:22E65, 20G05

16. CJM 2010 (vol 62 pp. 1310)

Lee, Kyu-Hwan
Iwahori--Hecke Algebras of $SL_2$ over $2$-Dimensional Local Fields
In this paper we construct an analogue of Iwahori--Hecke algebras of $\operatorname{SL}_2$ over $2$-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on $\operatorname{SL}_2$, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori--Matsumoto type relations.

Categories:22E50, 20G25

17. CJM 2009 (vol 62 pp. 34)

Campbell, Peter S.; Nevins, Monica
Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic Field
We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples.

Keywords:principal series representations, branching rules, maximal compact subgroups, representations of $p$-adic groups
Categories:20G25, 20G05

18. CJM 2009 (vol 61 pp. 950)

Tange, Rudolf
Infinitesimal Invariants in a Function Algebra
Let $G$ be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let $\g$ be its Lie algebra. First we extend a well-known result about the Picard group of a semi-simple group to reductive groups. Then we prove that if the derived group is simply connected and $\g$ satisfies a mild condition, the algebra $K[G]^\g$ of regular functions on $G$ that are invariant under the action of $\g$ derived from the conjugation action is a unique factorisation domain.

Categories:20G15, 13F15

19. CJM 2009 (vol 61 pp. 691)

Yu, Xiaoxiang
Prehomogeneity on Quasi-Split Classical Groups and Poles of Intertwining Operators
Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit, connected, reductive classical group $G$ defined over a non-Archimedean field and $A$ is the standard intertwining operator attached to a tempered representation of $G$ induced from $M$. In this paper we determine all the cases in which $\Lie(N)$ is prehomogeneous under $\Ad(m)$ when $N$ is non-abelian, and give necessary and sufficient conditions for $A$ to have a pole at $0$.

Categories:22E50, 20G05

20. CJM 2007 (vol 59 pp. 449)

Badulescu, Alexandru Ioan
$\SL_n$, Orthogonality Relations and Transfer
Let $\pi$ be a square integrable representation of $G'=\SL_n(D)$, with $D$ a central division algebra of finite dimension over a local field $F$ \emph{of non-zero characteristic}. We prove that, on the elliptic set, the character of $\pi$ equals the complex conjugate of the orbital integral of one of the pseudocoefficients of~$\pi$. We prove also the orthogonality relations for characters of square integrable representations of $G'$. We prove the stable transfer of orbital integrals between $\SL_n(F)$ and its inner forms.


21. CJM 2006 (vol 58 pp. 897)

Courtès, François
Distributions invariantes sur les groupes réductifs quasi-déployés
Soit $F$ un corps local non archim\'edien, et $G$ le groupe des $F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$. Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes par conjugaison, et \`a l'espace de leurs restrictions \`a l'alg\`ebre de Hecke $\mathcal{H}$ des fonctions sur $G$ \`a support compact biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$ sont enti\`erement d\'etermin\'ees par sa restriction au sous-espace de dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendr\'e d'une part par certaines int\'egrales orbitales semi-simples, d'autre part par les int\'egrales orbitales unipotentes, en montrant tout d'abord des r\'esultats analogues sur les groupes finis.

Keywords:reductive $p$-adic groups, orbital integrals, invariant distributions
Categories:22E35, 20G40

22. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field
We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-types
Categories:20G25, 22E35, 20H25

23. CJM 2004 (vol 56 pp. 945)

Helminck, Aloysius G.; Schwarz, Gerald W.
Smoothness of Quotients Associated \\With a Pair of Commuting Involutions
Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$ and $K:=G^\theta$ be the fixed point groups. We have an action $(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson \cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$.

Categories:20G15, 20G20, 22E15, 22E46

24. CJM 2004 (vol 56 pp. 246)

Bonnafé, Cédric
Éléments unipotents réguliers des sous-groupes de Levi
We investigate the structure of the centralizer of a regular unipotent element of a Levi subgroup of a reductive group. We also investigate the structure of the group of components of this centralizer in relation with the notion of cuspidal local system defined by Lusztig. We determine its unipotent radical, we prove that it admits a Levi complement, and we get some properties on its Weyl group. As an application, we prove some results which were announced in previous paper on regular unipotent elements. Nous \'etudions la structure du centralisateur d'un \'el\'ement unipotent r\'egulier d'un sous-groupe de Levi d'un groupe r\'eductif, ainsi que la structure du groupe des composantes de ce centralisateur en relation avec la notion de syst\`eme local cuspidal d\'efinie par Lusztig. Nous d\'eterminons son radical unipotent, montrons l'existence d'un compl\'ement de Levi et \'etudions la structure de son groupe de Weyl. Comme application, nous d\'emontrons des r\'esultats qui \'etaient annonc\'es dans un pr\'ec\'edent article de l'auteur sur les \'el\'ements unipotents r\'eguliers.


25. CJM 2003 (vol 55 pp. 1080)

Kellerhals, Ruth
Quaternions and Some Global Properties of Hyperbolic $5$-Manifolds
We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension $5$. The result implies improved universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\rmvol_5(M)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of quaternionic $2\times 2$-matrices with Dieudonn\'e determinant $\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm L} (2,\mathbb{H})$.

Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40
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