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26. 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

27. 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

28. 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

29. 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

30. 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 factor
Categories:11F70, 22E47, 11S37, 11F72, 17B22

31. 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 group
Categories:22E50, 22E35

32. 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 interpolation
Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52

33. 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 parameters
Categories:22E50, 11S37

34. 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

35. 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 Mœglin--Tadić for the other classical groups. This is obtained by taking the Mœglin--Tadić 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.


36. CJM 2010 (vol 62 pp. 1340)

Mœglin, C.
Holomorphie des opérateurs d'entrelacement normalisés à l'aide des paramètres 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

37. 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

38. 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 Tadić 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

39. 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 groups
Categories:35H30, 26D10, 22E25

40. 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

41. 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 series
Categories:22E50, 22E35

42. 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-curvature
Categories:53C60, 58B20, 22E46, 22E60

43. 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.


44. 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

45. CJM 2009 (vol 61 pp. 1325)

Nien, Chufeng
Uniqueness of Shalika Models
Let $\BF_q$ be a finite field of $q$ elements, $\CF$ a $p$-adic field, and $D$ a quaternion division algebra over $\CF$. This paper proves uniqueness of Shalika models for $\GL_{2n}(\BF_q) $ and $\GL_{2n}(D)$, and re-obtains uniqueness of Shalika models for $\GL_{2n}(\CF)$ for any $n\in \BN$.

Keywords:Shalika models, linear models, uniqueness, multiplicity free

46. CJM 2009 (vol 61 pp. 961)

Bernon, Florent
Transfert des intégrales orbitales pour les algèbres de Lie classiques
Dans cet article, on consid\`ere un groupe semi-simple $\rmG$ classique r\'eel et connexe. On suppose de plus que $\rmG$ poss\`ede un sous-groupe de Cartan compact. On d\'efinit une famille de sous-alg\`ebres de Lie associ\'ee \`a $\g = \Lie(\rmG)$, de m\^eme rang que $\g$ dont tous les facteurs simples sont de rang $1$ ou~$2$. Soit $\g'$ une telle sous-alg\`ebre de Lie. On construit alors une application de transfert des int\'egrales orbitales de $\g'$ dans l'espace des int\'egrales orbitales de $\g$. On montre que cette application est d\'efinie d\`es que $\g$ ne poss\`ede pas de facteur simple r\'eel de type $\CI$ de rang sup\'erieur ou \'egal \`a $3$. Si de plus, $\g$ ne poss\`ede pas de facteur simple de type $\BI$ de rang sup\'erieur \`a $3$, on montre la surjectivit\'e de cette application de transfert. On utilise cette application de transfert pour obtenir une formule de r\'eduction de l'int\'egrale de Cauchy Harish-Chandra pour les paires duales d'alg\`ebres de Lie r\'eductives $\bigl( \Ug(p,q),\Ug(r,s) \bigr)$ et $\bigl( \Sp(p,q),\Og^*(2n) \bigr)$ avec $p+q = r+s = n$.

Categories:22E30, 22E46

47. CJM 2009 (vol 61 pp. 779)

Grbac, Neven
Residual Spectra of Split Classical Groups and their Inner Forms
This paper is concerned with the residual spectrum of the hermitian quaternionic classical groups $G_n'$ and $H_n'$ defined as algebraic groups for a quaternion algebra over an algebraic number field. Groups $G_n'$ and $H_n'$ are not quasi-split. They are inner forms of the split groups $\SO_{4n}$ and $\Sp_{4n}$. Hence, the parts of the residual spectrum of $G_n'$ and $H_n'$ obtained in this paper are compared to the corresponding parts for the split groups $\SO_{4n}$ and $\Sp_{4n}$.

Categories:11F70, 22E55

48. 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

49. 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

50. CJM 2009 (vol 61 pp. 373)

McKee, Mark
An Infinite Order Whittaker Function
In this paper we construct a flat smooth section of an induced space $I(s,\eta)$ of $SL_2(\mathbb{R})$ so that the attached Whittaker function is not of finite order. An asymptotic method of classical analysis is used.

Categories:11F70, 22E45, 41A60, 11M99, 30D15, 33C15
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