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26. CJM 2012 (vol 64 pp. 721)

Achab, Dehbia; Faraut, Jacques
 Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi$ is an orbit of a covering of ${\rm Conf}(V,Q)$, the conformal group of the pair $(V,Q)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak g$, and furthermore a real form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi$. Keywords:minimal representation, Kantor-Koecher-Tits construction, Jordan algebra, Bernstein identity, Meijer $G$-functionCategories:17C36, 22E46, 32M15, 33C80

27. CJM 2012 (vol 65 pp. 66)

Deng, Shaoqiang; Hu, Zhiguang
 On Flag Curvature of Homogeneous Randers Spaces In this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randers metric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian. Keywords:homogeneous Randers manifolds, flag curvature, Douglas spaces, two-step nilpotent Lie groupsCategories:22E46, 53C30

28. CJM 2012 (vol 64 pp. 497)

Li, Wen-Wei

29. CJM 2011 (vol 64 pp. 669)

Pantano, Alessandra; Paul, Annegret; Salamanca-Riba, Susana A.
 The Genuine Omega-regular Unitary Dual of the Metaplectic Group We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group. Keywords:Metaplectic group, oscillator representation, bottom layer map, cohomological induction, Parthasarathy's Dirac Operator Inequality, pseudospherical principal seriesCategory:22E46

30. CJM 2011 (vol 64 pp. 123)

Lee, Jae-Hyouk
 Gosset Polytopes in Picard Groups of del Pezzo Surfaces In this article, we study the correspondence between the geometry of del Pezzo surfaces $S_{r}$ and the geometry of the $r$-dimensional Gosset polytopes $(r-4)_{21}$. We construct Gosset polytopes $(r-4)_{21}$ in $\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a-1)$-simplexes ($a\leq r$), $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope $(r-4)_{21}$. Then we explain how these classes correspond to skew $a$-lines($a\leq r$), exceptional systems, and rulings, respectively. As an application, we work on the monoidal transform for lines to study the local geometry of the polytope $(r-4)_{21}$. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and $4_{21}$, respectively. Categories:51M20, 14J26, 22E99

31. CJM 2011 (vol 64 pp. 481)

Chamorro, Diego
 Some Functional Inequalities on Polynomial Volume Growth Lie Groups In this article we study some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended to this general framework without the use of the Littlewood-Paley decomposition. Keywords:Sobolev inequalities, polynomial volume growth Lie groupsCategory:22E30

32. 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 variationCategories:14L24, 14L30, 20G20, 22E45

33. 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 integralsCategories:20C08, 20F55, 22E50

34. CJM 2011 (vol 63 pp. 1364)

Meinrenken, Eckhard
 The Cubic Dirac Operator for Infinite-Dimensonal Lie Algebras Let $\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}} \mathfrak{g}_i$ be an infinite-dimensional graded Lie algebra, with $\dim\mathfrak{g}_i<\infty$, equipped with a non-degenerate symmetric bilinear form $B$ of degree $0$. The quantum Weil algebra $\widehat{\mathcal{W}}\mathfrak{g}$ is a completion of the tensor product of the enveloping and Clifford algebras of $\mathfrak{g}$. Provided that the Kac-Peterson class of $\mathfrak{g}$ vanishes, one can construct a cubic Dirac operator $\mathcal{D}\in\widehat{\mathcal{W}}(\mathfrak{g})$, whose square is a quadratic Casimir element. We show that this condition holds for symmetrizable Kac-Moody algebras. Extending Kostant's arguments, one obtains generalized Weyl-Kac character formulas for suitable equal rank'' Lie subalgebras of Kac-Moody algebras. These extend the formulas of G. Landweber for affine Lie algebras. Categories:22E65, 15A66

35. 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

36. CJM 2011 (vol 63 pp. 1137)

Moy, Allen
 Distribution Algebras on p-adic Groups and Lie Algebras When $F$ is a $p$-adic field, and $G={\mathbb G}(F)$ is the group of $F$-rational points of a connected algebraic $F$-group, the complex vector space ${\mathcal H}(G)$ of compactly supported locally constant distributions on $G$ has a natural convolution product that makes it into a ${\mathbb C}$-algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for $p$-adic groups of the enveloping algebra of a Lie group. However, $\mathcal{H}(G)$ has drawbacks such as the lack of an identity element, and the process $G \mapsto \mathcal{H}(G)$ is not a functor. Bernstein introduced an enlargement $\mathcal{H}\,\hat{\,}(G)$ of $\mathcal{H}(G)$. The algebra $\mathcal{H}\,\hat{\,} (G)$ consists of the distributions that are left essentially compact. We show that the process $G \mapsto \mathcal{H}\,\hat{\,} (G)$ is a functor. If $\tau \colon G \rightarrow H$ is a morphism of $p$-adic groups, let $F(\tau) \colon \mathcal{H}\,\hat{\,} (G) \rightarrow \mathcal{H}\,\hat{\,} (H)$ be the morphism of $\mathbb{C}$-algebras. We identify the kernel of $F(\tau)$ in terms of $\textrm{Ker}(\tau)$. In the setting of $p$-adic Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra, $\mathfrak{m}$ a Levi, and $\tau \colon \mathfrak{g} \to \mathfrak{m}$ the natural projection, we show that $F(\tau)$ maps $G$-invariant distributions on $\mathcal{G}$ to $N_G (\mathfrak{m})$-invariant distributions on $\mathfrak{m}$. Finally, we exhibit a natural family of $G$-invariant essentially compact distributions on $\mathfrak{g}$ associated with a $G$-invariant non-degenerate symmetric bilinear form on ${\mathfrak g}$ and in the case of $SL(2)$ show how certain members of the family can be moved to the group. Keywords:distribution algebra, p-adic groupCategories:22E50, 22E35

37. CJM 2011 (vol 63 pp. 1083)

Kaletha, Tasho
 Decomposition of Splitting Invariants in Split Real Groups For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic $0$, Langlands and Shelstad constructed a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group. Keywords:endoscopy, real lie group, splitting invariant, transfer factorCategories:11F70, 22E47, 11S37, 11F72, 17B22

38. CJM 2011 (vol 63 pp. 798)

Daws, Matthew
 Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ Spaces We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca-Herz algebra built out of these non-commutative $L^p$ spaces, say $A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to $L^1(G)$, generalising the abelian situation. Keywords:multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolationCategories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52

39. CJM 2011 (vol 63 pp. 1107)

Liu, Baiying
 Genericity of Representations of p-Adic $Sp_{2n}$ and Local Langlands Parameters Let $G$ be the $F$-rational points of the symplectic group $Sp_{2n}$, where $F$ is a non-Archimedean local field of characteristic $0$. Cogdell, Kim, Piatetski-Shapiro, and Shahidi constructed local Langlands functorial lifting from irreducible generic representations of $G$ to irreducible representations of $GL_{2n+1}(F)$. Jiang and Soudry constructed the descent map from irreducible supercuspidal representations of $GL_{2n+1}(F)$ to those of $G$, showing that the local Langlands functorial lifting from the irreducible supercuspidal generic representations is surjective. In this paper, based on above results, using the same descent method of studying $SO_{2n+1}$ as Jiang and Soudry, we will show the rest of local Langlands functorial lifting is also surjective, and for any local Langlands parameter $\phi \in \Phi(G)$, we construct a representation $\sigma$ such that $\phi$ and $\sigma$ have the same twisted local factors. As one application, we prove the $G$-case of a conjecture of Gross-Prasad and Rallis, that is, a local Langlands parameter $\phi \in \Phi(G)$ is generic, i.e., the representation attached to $\phi$ is generic, if and only if the adjoint $L$-function of $\phi$ is holomorphic at $s=1$. As another application, we prove for each Arthur parameter $\psi$, and the corresponding local Langlands parameter $\phi_{\psi}$, the representation attached to $\phi_{\psi}$ is generic if and only if $\phi_{\psi}$ is tempered. Keywords:generic representations, local Langlands parametersCategories:22E50, 11S37

40. CJM 2011 (vol 63 pp. 591)

Hanzer, Marcela; Muić, Goran
 Rank One Reducibility for Metaplectic Groups via Theta Correspondence We calculate reducibility for the representations of metaplectic groups induced from cuspidal representations of maximal parabolic subgroups via theta correspondence, in terms of the analogous representations of the odd orthogonal groups. We also describe the lifts of all relevant subquotients. Categories:22E50, 11F70

41. CJM 2011 (vol 63 pp. 327)

Jantzen, Chris
 Discrete Series for $p$-adic $SO(2n)$ and Restrictions of Representations of $O(2n)$ In this paper we give a classification of discrete series for $SO(2n,F)$, $F$ $p$-adic, similar to that of MÅglin--TadiÄ for the other classical groups. This is obtained by taking the MÅglin--TadiÄ classification for $O(2n,F)$ and studying how the representations restrict to $SO(2n,F)$. We then extend this to an analysis of how admissible representations of $O(2n,F)$ restrict. Category:22E50

42. CJM 2010 (vol 62 pp. 1340)

Mœglin, C.
 Holomorphie des opÃ©rateurs d'entrelacement normalisÃ©s Ã  l'aide des paramÃ¨tres d'Arthur In this paper we prove holomorphy for certain intertwining operators arising from the theory of Eisenstein series. To do that we need to normalize using the Langlands--Shahidi's normalization arising from the twisted endoscopy and the associated representations of the general linear group. Categories:22E50, 22E35

43. 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

44. CJM 2010 (vol 62 pp. 914)

Zorn, Christian
 Reducibility of the Principal Series for Sp~2(F) over a p-adic Field Let $G_n=\mathrm{Sp}_n(F)$ be the rank $n$ symplectic group with entries in a nondyadic $p$-adic field $F$. We further let $\widetilde{G}_n$ be the metaplectic extension of $G_n$ by $\mathbb{C}^{1}=\{z\in\mathbb{C}^{\times} \mid |z|=1\}$ defined using the Leray cocycle. In this paper, we aim to demonstrate the complete list of reducibility points of the genuine principal series of ${\widetilde{G}_2}$. In most cases, we will use some techniques developed by TadiÄ that analyze the Jacquet modules with respect to all of the parabolics containing a fixed Borel. The exceptional cases involve representations induced from unitary characters $\chi$ with $\chi^2=1$. Because such representations $\pi$ are unitary, to show the irreducibility of $\pi$, it suffices to show that $\dim_{\mathbb{C}}\mathrm{Hom}_{{\widetilde{G}}}(\pi,\pi)=1$. We will accomplish this by examining the poles of certain intertwining operators associated to simple roots. Then some results of Shahidi and Ban decompose arbitrary intertwining operators into a composition of operators corresponding to the simple roots of ${\widetilde{G}_2}$. We will then be able to show that all such operators have poles at principal series representations induced from quadratic characters and therefore such operators do not extend to operators in $\mathrm{Hom}_{{\widetilde{G}_2}}(\pi,\pi)$ for the $\pi$ in question. Categories:22E50, 11F70

45. CJM 2010 (vol 62 pp. 1116)

Jin, Yongyang; Zhang, Genkai
 Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups Let $\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= \operatorname{div}_X (|\nabla_{X} u|^{p-2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$. Keywords:fundamental solutions, degenerate Laplacians, Hardy inequality, H-type groupsCategories:35H30, 26D10, 22E25

46. CJM 2010 (vol 62 pp. 563)

Ishii, Taku
 Whittaker Functions on Real Semisimple Lie Groups of Rank Two We give explicit formulas for Whittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals. Categories:11F70, 22E30

47. CJM 2009 (vol 62 pp. 94)

Kuo, Wentang
 The Langlands Correspondence on the Generic Irreducible Constituents of Principal Series Let $G$ be a connected semisimple split group over a $p$-adic field. We establish the explicit link between principal nilpotent orbits and the irreducible constituents of principal series in terms of $L$-group objects. Keywords:Langlands correspondence, nilpotent orbits, principal seriesCategories:22E50, 22E35

48. CJM 2009 (vol 62 pp. 52)

Deng, Shaoqiang
 An Algebraic Approach to Weakly Symmetric Finsler Spaces In this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann--Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions $2$ and $3$. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing S-curvature. This means that reversible non-Berwaldian Finsler spaces with vanishing S-curvature may exist at large. Hence the generalized volume comparison theorems due to Z. Shen are valid for a rather large class of Finsler spaces. Keywords:weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, S-curvatureCategories:53C60, 58B20, 22E46, 22E60

49. CJM 2009 (vol 61 pp. 1375)

Spallone, Steven
 Stable Discrete Series Characters at Singular Elements Write $\Theta^E$ for the stable discrete series character associated with an irreducible finite-dimensional representation $E$ of a connected real reductive group $G$. Let $M$ be the centralizer of the split component of a maximal torus $T$, and denote by $\Phi_M(\gm,\Theta^E)$ Arthur's extension of $|D_M^G(\gm)|^{\lfrac 12} \Theta^E(\gm)$ to $T(\R)$. In this paper we give a simple explicit expression for $\Phi_M(\gm,\Theta^E)$ when $\gm$ is elliptic in $G$. We do not assume $\gm$ is regular. Category:22E47

50. CJM 2009 (vol 61 pp. 1407)

Will, Pierre
 Traces, Cross-Ratios and 2-Generator Subgroups of $\SU(2,1)$ In this work, we investigate how to decompose a pair $(A,B)$ of loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group $\langle A,B\rangle$. Categories:14L24, 22E40, 32M15, 51M10
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