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26. CJM 2013 (vol 66 pp. 481)

Aguiar, Marcelo; Mahajan, Swapneel
On the Hadamard Product of Hopf Monoids
Combinatorial structures that compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free. The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco that applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species that is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.

Keywords:species, Hopf monoid, Hadamard product, generating function, Boolean transform
Categories:16T30, 18D35, 20B30, 18D10, 20F55

27. CJM 2013 (vol 66 pp. 354)

Kellerhals, Ruth; Kolpakov, Alexander
The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space
Due to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on ${\mathbb H^3}$ is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is as such unique. Our approach provides a different proof for the analog situation in ${\mathbb H^2}$ where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

Keywords:hyperbolic Coxeter group, growth rate, Salem number
Categories:20F55, 22E40, 51F15

28. CJM 2013 (vol 66 pp. 205)

Iovanov, Miodrag Cristian
Generalized Frobenius Algebras and Hopf Algebras
"Co-Frobenius" coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogue to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras; the first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual $Rat(C^*)$, in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra $A$ that is isomorphic to its complete topological dual $A^\vee$. We show that $A$ is a (quasi)Frobenius algebra if and only if $A$ is the dual $C^*$ of a (quasi)co-Frobenius coalgebra $C$. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

Keywords:coalgebra, Hopf algebra, integral, Frobenius, QcF, co-Frobenius
Categories:16T15, 18G35, 16T05, 20N99, 18D10, 05E10

29. CJM 2011 (vol 64 pp. 241)

Allcock, Daniel
Triangles of Baumslag-Solitar Groups
Our main result is that many triangles of Baumslag-Solitar groups collapse to finite groups, generalizing a famous example of Hirsch and other examples due to several authors. A triangle of Baumslag-Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of derived length $\leq3$. We obtain a lot of information about finite quotients, even when we cannot determine developability.

Categories:20F06, 20F65

30. 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

31. CJM 2011 (vol 63 pp. 1238)

Bump, Daniel; Nakasuji, Maki
Casselman's Basis of Iwahori Vectors and the Bruhat Order
W. Casselman defined a basis $f_u$ of Iwahori fixed vectors of a spherical representation $(\pi, V)$ of a split semisimple $p$-adic group $G$ over a nonarchimedean local field $F$ by the condition that it be dual to the intertwining operators, indexed by elements $u$ of the Weyl group $W$. On the other hand, there is a natural basis $\psi_u$, and one seeks to find the transition matrices between the two bases. Thus, let $f_u = \sum_v \tilde{m} (u, v) \psi_v$ and $\psi_u = \sum_v m (u, v) f_v$. Using the Iwahori-Hecke algebra we prove that if a combinatorial condition is satisfied, then $m (u, v) = \prod_{\alpha} \frac{1 - q^{- 1} \mathbf{z}^{\alpha}}{1 -\mathbf{z}^{\alpha}}$, where $\mathbf z$ are the Langlands parameters for the representation and $\alpha$ runs through the set $S (u, v)$ of positive coroots $\alpha \in \hat{\Phi}$ (the dual root system of $G$) such that $u \leqslant v r_{\alpha} < v$ with $r_{\alpha}$ the reflection corresponding to $\alpha$. The condition is conjecturally always satisfied if $G$ is simply-laced and the Kazhdan-Lusztig polynomial $P_{w_0 v, w_0 u} = 1$ with $w_0$ the long Weyl group element. There is a similar formula for $\tilde{m}$ conjecturally satisfied if $P_{u, v} = 1$. This leads to various combinatorial conjectures.

Keywords:Iwahori fixed vector, Iwahori Hecke algebra, Bruhat order, intertwining integrals
Categories:20C08, 20F55, 22E50

32. 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

33. CJM 2010 (vol 63 pp. 413)

Konvalinka, Matjaž; Skandera, Mark
Generating Functions for Hecke Algebra Characters
Certain polynomials in $n^2$ variables that serve as generating functions for symmetric group characters are sometimes called ($S_n$) character immanants. We point out a close connection between the identities of Littlewood--Merris--Watkins and Goulden--Jackson, which relate $S_n$ character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra $H_n(q)$, we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood--Merris--Watkins identities and selected Goulden--Jackson identities that relate $H_n(q)$ character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis--L\^e--Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.

Keywords:determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring
Categories:15A15, 20C08, 81R50

34. 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

35. CJM 2010 (vol 62 pp. 481)

Casals-Ruiz, Montserrat; Kazachkov, Ilya V.
Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups
The first main result of the paper is a criterion for a partially commutative group $\mathbb G$ to be a domain. It allows us to reduce the study of algebraic sets over $\mathbb G$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\mathbb G$ (of a coordinate group over $\mathbb G$) to the elementary theories of the direct factors of $\mathbb G$ (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group $\mathbb H$. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\mathbb H$ has quantifier elimination and that arbitrary first-order formulas lift from $\mathbb H$ to $\mathbb H\ast F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.

Categories:20F10, 03C10, 20F06

36. 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

37. CJM 2009 (vol 61 pp. 740)

Caprace, Pierre-Emmanuel; Haglund, Frédéric
On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings
Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if $\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0) realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.

Keywords:Coxeter group, flat rank, $\cat0$ space, building
Categories:20F55, 51F15, 53C23, 20E42, 51E24

38. 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

39. 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

40. CJM 2009 (vol 61 pp. 708)

Zelenyuk, Yevhen
Regular Homeomorphisms of Finite Order on Countable Spaces
We present a structure theorem for a broad class of homeomorphisms of finite order on countable zero dimensional spaces. As applications we show the following. \begin{compactenum}[\rm(a)] \item Every countable nondiscrete topological group not containing an open Boolean subgroup can be partitioned into infinitely many dense subsets. \item If $G$ is a countably infinite Abelian group with finitely many elements of order $2$ and $\beta G$ is the Stone--\v Cech compactification of $G$ as a discrete semigroup, then for every idempotent $p\in\beta G\setminus\{0\}$, the subset $\{p,-p\}\subset\beta G$ generates algebraically the free product of one-element semigroups $\{p\}$ and~$\{-p\}$. \end{compactenum}

Keywords:Homeomorphism, homogeneous space, topological group, resolvability, Stone-\v Cech compactification
Categories:22A30, 54H11, 20M15, 54A05

41. CJM 2008 (vol 60 pp. 1001)

Cornulier, Yves de; Tessera, Romain; Valette, Alain
Isometric Group Actions on Hilbert Spaces: Structure of Orbits
Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

Keywords:affine actions, Hilbert spaces, minimal actions, nilpotent groups
Categories:22D10, 43A35, 20F69

42. CJM 2007 (vol 59 pp. 828)

Ortner, Ronald; Woess, Wolfgang
Non-Backtracking Random Walks and Cogrowth of Graphs
Let $X$ be a locally finite, connected graph without vertices of degree $1$. Non-backtracking random walk moves at each step with equal probability to one of the ``forward'' neighbours of the actual state, \emph{i.e.,} it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of $X$. Thus we obtain for infinite $X$ that the $n$-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when $X$ is finite. This provides a short proof of old results concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when $X$ is non-regular, but \emph{small cycles are dense in} $X$, we show that the graph $X$ is non-amenable if and only if the non-backtracking $n$-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen.

Keywords:graph, oriented line grap, covering tree, random walk, cogrowth, amenability
Categories:05C75, 60G50, 20F69

43. 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.

Category:20G05

44. CJM 2007 (vol 59 pp. 296)

Chein, Orin; Goodaire, Edgar G.
Bol Loops of Nilpotence Class Two
Call a non-Moufang Bol loop \emph{minimally non-Moufang} if every proper subloop is Moufang and \emph{minimally nonassociative} if every proper subloop is associative. We prove that these concepts are the same for Bol loops which are nilpotent of class two and in which certain associators square to $1$. In the process, we derive many commutator and associator identities which hold in such loops.

Keywords:Bol loop, Moufang loop, nilpotent, commutator, associator, minimally nonassociative
Category:20N05

45. CJM 2007 (vol 59 pp. 418)

Stoimenow, A.
On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials
It is known that the Brandt--Lickorish--Millett--Ho polynomial $Q$ contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from $Q$ is an open problem. We show that this is not so up to degree $9$. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree $d \le 10$ are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

Categories:57M25, 57M27, 20F36, 57M50

46. CJM 2006 (vol 58 pp. 1144)

Hamana, Masamichi
Partial $*$-Automorphisms, Normalizers, and Submodules in Monotone Complete $C^*$-Algebras
For monotone complete $C^*$-algebras $A\subset B$ with $A$ contained in $B$ as a monotone closed $C^*$-subalgebra, the relation $X = AsA$ gives a bijection between the set of all monotone closed linear subspaces $X$ of $B$ such that $AX + XA \subset X$ and $XX^* + X^*X \subset A$ and a set of certain partial isometries $s$ in the ``normalizer" of $A$ in $B$, and similarly for the map $s \mapsto \Ad s$ between the latter set and a set of certain ``partial $*$-automorphisms" of $A$. We introduce natural inverse semigroup structures in the set of such $X$'s and the set of partial $*$-automorphisms of $A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough $B$ the homomorphism becomes surjective and all the partial $*$-automorphisms of $A$ are realized via partial isometries in $B$. In particular, the inverse semigroup associated with a type ${\rm II}_1$ von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the $C^*$-algebra version of these results.

Categories:46L05, 46L08, 46L40, 20M18

47. 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

48. CJM 2006 (vol 58 pp. 23)

Dabbaghian-Abdoly, Vahid
Constructing Representations of Finite Simple Groups and Covers
Let $G$ be a finite group and $\chi$ be an irreducible character of $G$. An efficient and simple method to construct representations of finite groups is applicable whenever $G$ has a subgroup $H$ such that $\chi_H$ has a linear constituent with multiplicity $1$. In this paper we show (with a few exceptions) that if $G$ is a simple group or a covering group of a simple group and $\chi$ is an irreducible character of $G$ of degree less than 32, then there exists a subgroup $H$ (often a Sylow subgroup) of $G$ such that $\chi_H$ has a linear constituent with multiplicity $1$.

Keywords:group representations, simple groups, central covers, irreducible representations
Categories:20C40, 20C15

49. CJM 2005 (vol 57 pp. 1056)

Ozawa, Narutaka; Rieffel, Marc A.
Hyperbolic Group $C^*$-Algebras and Free-Product $C^*$-Algebras as Compact Quantum Metric Spaces
Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that if $G$ is a hyperbolic group and if $\ell$ is a word-length function on $G$, then the topology from this metric coincides with the weak-$*$ topology (our definition of a ``compact quantum metric space''). We show that a convenient framework is that of filtered $C^*$-algebras which satisfy a suitable ``Haagerup-type'' condition. We also use this framework to prove an analogous fact for certain reduced free products of $C^*$-algebras.

Categories:46L87, 20F67, 46L09

50. 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
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