1. CJM Online first
 Luo, Caihua

Spherical fundamental lemma for metaplectic groups
In this paper, we prove the spherical fundamental lemma for
metaplectic group $Mp_{2n}$ based on the formalism of endoscopy
theory by J.Adams, D.Renard and WenWei Li.
Keywords:metaplectic group, endoscopic group, elliptic stable trace formula, fundamental lemma Category:22E35 

2. CJM Online first
 Varma, Sandeep

On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical
Let $\operatorname{P} = \operatorname{M} \operatorname{N}$ be a Levi decomposition of a maximal parabolic
subgroup of a connected
reductive group $\operatorname{G}$ over a $p$adic field $F$. Assume that there
exists $w_0 \in \operatorname{G}(F)$ that normalizes $\operatorname{M}$ and conjugates $\operatorname{P}$
to an opposite parabolic subgroup.
When $\operatorname{N}$ has a Zariski dense $\operatorname{Int} \operatorname{M}$orbit,
F. Shahidi and X. Yu describe a certain distribution $D$ on
$\operatorname{M}(F)$
such that,
for irreducible unitary supercuspidal representations $\pi$ of
$\operatorname{M}(F)$ with
$\pi \cong \pi \circ \operatorname{Int} w_0$,
$\operatorname{Ind}_{\operatorname{P}(F)}^{\operatorname{G}(F)} \pi$ is
irreducible
if and only if $D(f) \neq 0$ for some pseudocoefficient $f$ of
$\pi$. Since
this irreducibility is conjecturally related to $\pi$ arising
via
transfer from certain twisted endoscopic groups of $\operatorname{M}$, it is
of interest
to realize $D$ as endoscopic transfer from a simpler distribution
on a twisted
endoscopic group $\operatorname{H}$ of $\operatorname{M}$. This has been done in many situations
where $\operatorname{N}$ is abelian. Here, we handle the `standard examples'
in cases
where $\operatorname{N}$ is nonabelian but admits a Zariski dense
$\operatorname{Int} \operatorname{M}$orbit.
Keywords:induced representation, intertwining operator, endoscopy Categories:22E50, 11F70 

3. CJM 2016 (vol 69 pp. 186)
 Pan, ShuYen

$L$Functoriality for Local Theta Correspondence of Supercuspidal Representations with Unipotent Reduction
The preservation principle of local theta correspondences of reductive dual pairs over
a $p$adic field predicts the existence of a sequence of irreducible supercuspidal
representations of classical groups.
Adams/HarrisKudlaSweet
have a conjecture
about the Langlands parameters for the sequence of supercuspidal representations.
In this paper we prove modified versions of their conjectures for the case of
supercuspidal representations with unipotent reduction.
Keywords:local theta correspondence, supercuspidal representation, preservation principle, Langlands functoriality Categories:22E50, 11F27, 20C33 

4. CJM Online first
5. CJM Online first
 Ganguly, Arijit; Ghosh, Anish

Dirichlet's theorem in function fields
We study metric Diophantine approximation for function fields
specifically the problem of improving Dirichlet's theorem in
Diophantine
approximation.
Keywords:Dirichlet's theorem, Diophantine approximation, positive characteristic Categories:11J83, 11K60, 37D40, 37A17, 22E40 

6. 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, SegalSugawara operators, ChervovMolev operators, critical level, smooth representations, affine KacMoody algebra, categorical representations Categories:17B67, 17B69, 22E50, 20G25 

7. 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 quasisemisimplicity 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
fixedpoint 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 

8. CJM 2014 (vol 66 pp. 993)
 BeuzartPlessis, Raphaël

Expression d'un facteur epsilon de paire par une formule intÃ©grale
Let $E/F$ be a quadratic extension of $p$adic fields and
let $d$, $m$ be nonnegative integers of distinct parities. Fix
admissible irreducible tempered representations $\pi$ and $\sigma$ of
$GL_d(E)$ and $GL_m(E)$ respectively. We assume that $\pi$ and
$\sigma$ are conjugatedual. That is to say $\pi\simeq \pi^{\vee,c}$
and $\sigma\simeq \sigma^{\vee,c}$ where $c$ is the non trivial
$F$automorphism of $E$. This implies, we can extend $\pi$ to an
unitary representation $\tilde{\pi}$ of a nonconnected group
$GL_d(E)\rtimes \{1,\theta\}$. Define $\tilde{\sigma}$ the same
way. We state and prove an integral formula for
$\epsilon(1/2,\pi\times \sigma,\psi_E)$ involving the characters of
$\tilde{\pi}$ and $\tilde{\sigma}$. This formula is related to the
local GanGrossPrasad conjecture for unitary groups.
Keywords:epsilon factor, twisted groups Categories:22E50, 11F85 

9. CJM 2013 (vol 66 pp. 1250)
 Feigin, Evgeny; Finkelberg, Michael; Littelmann, Peter

Symplectic Degenerate Flag Varieties
A simple finite dimensional module $V_\lambda$ of a simple complex
algebraic group $G$ is naturally endowed with a filtration induced by the PBWfiltration
of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module
for the group $G^a$, which can be roughly described as a semidirect product of a
Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy
to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$,
we call the closure
$\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$
of the $G^a$orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$.
In general this is a
singular variety, but we conjecture that it has many nice properties similar to
that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group.
The symplectic case is important for the conjecture
because it is the first known case where even for fundamental weights $\omega$ the varieties
$\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit
construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations,
similar to the BottSamelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$
are normal locally complete intersections with terminal and rational singularities.
We also show that these varieties are Frobenius split. Using the above mentioned results, we
prove an analogue of the BorelWeil theorem and obtain a $q$character formula
for the characters of irreducible $Sp_{2n}$modules via the AtiyahBottLefschetz fixed
points formula.
Keywords:Lie algebras, flag varieties, symplectic groups, representations Categories:14M15, 22E46 

10. CJM 2013 (vol 66 pp. 1287)
 Henniart, Guy; Sécherre, Vincent

Types et contragrÃ©dientes
Soit $\mathrm{G}$ un groupe rÃ©ductif $p$adique, et soit $\mathrm{R}$
un corps algÃ©briquement clos.
Soit $\pi$ une reprÃ©sentation lisse de $\mathrm{G}$ dans un espace
vectoriel $\mathrm{V}$ sur
$\mathrm{R}$.
Fixons un sousgroupe ouvert et compact $\mathrm{K}$ de $\mathrm{G}$ et une reprÃ©sentation
lisse irrÃ©ductible $\tau$ de $\mathrm{K}$ dans un espace vectoriel
$\mathrm{W}$ de dimension
finie sur $\mathrm{R}$.
Sur l'espace $\mathrm{Hom}_{\mathrm{K}(\mathrm{W},\mathrm{V})}$ agit l'algÃ¨bre
d'entrelacement $\mathscr{H}(\mathrm{G},\mathrm{K},\mathrm{W})$.
Nous examinons la compatibilitÃ© de ces constructions avec le passage aux
reprÃ©sentations contragrÃ©dientes $\mathrm{V}^Äe$ et $\mathrm{W}^Äe$, et donnons en
particulier des conditions sur $\mathrm{W}$ ou sur la caractÃ©ristique
de $\mathrm{R}$ pour que
le comportement soit semblable au cas des reprÃ©sentations complexes.
Nous prenons un point de vue abstrait, n'utilisant que des propriÃ©tÃ©s
gÃ©nÃ©rales de $\mathrm{G}$.
Nous terminons par une application Ã la thÃ©orie des types pour le groupe
$\mathrm{GL}_n$ et ses formes intÃ©rieures sur un corps local non archimÃ©dien.
Keywords:modular representations of padic reductive groups, types, contragredient, intertwining Category:22E50 

11. CJM 2013 (vol 66 pp. 241)
 Broussous, P.

Transfert du pseudocoefficient de Kottwitz et formules de caractÃ¨re pour la sÃ©rie discrÃ¨te de $\mathrm{GL}(N)$ sur un corps local
Soit $G$ le groupe $\mathrm{GL}(N,F)$, oÃ¹ $F$ est un corps
localement compact et non archimÃ©dien.
En utilisant la thÃ©orie des types simples de Bushnell et Kutzko,
ainsi qu'une idÃ©e originale d'Henniart, nous construisons des pseudocoefficients
explicites pour les reprÃ©sentations de la sÃ©rie discrÃ¨te de $G$.
Comme application, nous en dÃ©duisons des formules
inÃ©dites pour la valeur du charactÃ¨re d'HarishChandra de certaines
telles reprÃ©sentations en certains Ã©lÃ©ments elliptiques
rÃ©guliers.
Keywords:reductive padic groups , discrete series, HarishChandra character, pseudocoefficient Category:22E50 

12. CJM 2013 (vol 66 pp. 566)
 Choiy, Kwangho

Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$adic Inner Forms
Let $F$ be a $p$adic field of characteristic $0$, and let $M$ be an $F$Levi subgroup of a connected reductive $F$split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local JacquetLanglands type correspondence between $M$ and its $F$inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of
MuiÄ and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local JacquetLanglands correspondence. It can be applied to a simply connected simple $F$group of type $E_6$ or $E_7$, and a connected reductive $F$group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.
Keywords:Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, JacquetLanglands correspondence Categories:22E50, 11F70, 22E55, 22E35 

13. CJM 2013 (vol 66 pp. 354)
 Kellerhals, Ruth; Kolpakov, Alexander

The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3space
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 

14. CJM 2012 (vol 66 pp. 102)
 Birth, Lidia; Glöckner, Helge

Continuity of convolution of test functions on Lie groups
For a Lie group $G$, we show that the map
$C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$,
$(\gamma,\eta)\mapsto \gamma*\eta$
taking a pair of
test functions to their convolution is continuous if and only if $G$ is $\sigma$compact.
More generally, consider $r,s,t
\in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$
and a continuous bilinear map $b\colon E_1\times E_2\to F$
to a complete locally convex space $F$.
Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$,
$(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map.
The main result is a characterization of those $(G,r,s,t,b)$
for which $\beta$ is continuous.
Convolution
of compactly supported continuous functions on a locally compact group
is also discussed, as well as convolution of compactly supported $L^1$functions
and convolution of compactly supported Radon measures.
Keywords:Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigmacompactness, convolution, continuity, seminorm, product estimates Categories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25 

15. CJM 2012 (vol 64 pp. 721)
 Achab, Dehbia; Faraut, Jacques

Analysis of the BrylinskiKostant 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 KantorKoecherTits 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, KantorKoecherTits construction, Jordan algebra, Bernstein identity, Meijer $G$function Categories:17C36, 22E46, 32M15, 33C80 

16. 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 twostep 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 nonRiemannian
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, twostep nilpotent Lie groups Categories:22E46, 53C30 

17. CJM 2012 (vol 64 pp. 497)
 Li, WenWei

Le lemme fondamental pondÃ©rÃ© pour le groupe mÃ©taplectique
Dans cet article, on Ã©nonce une variante du lemme fondamental
pondÃ©rÃ© d'Arthur pour le groupe mÃ©taplectique de Weil, qui sera un
ingrÃ©dient indispensable de la stabilisation de la formule des
traces. Pour un corps de caractÃ©ristique rÃ©siduelle suffisamment
grande, on en donne une dÃ©monstration Ã l'aide de la mÃ©thode de
descente, qui est conditionnelle: on admet le lemme fondamental
pondÃ©rÃ© non standard sur les algÃ¨bres de Lie. Vu les travaux de
Chaudouard et Laumon, on s'attend Ã ce que cette condition soit
ultÃ©rieurement vÃ©rifiÃ©e.
Keywords:fundamental lemma, metaplectic group, endoscopy, trace formula Categories:11F70, 11F27, 22E50 

18. CJM 2011 (vol 64 pp. 669)
 Pantano, Alessandra; Paul, Annegret; SalamancaRiba, Susana A.

The Genuine Omegaregular 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 onedimensional representation and an
oscillator representation. Our main theorem asserts that this family exhausts
the genuine omegaregular unitary dual of the metaplectic group.
Keywords:Metaplectic group, oscillator representation, bottom layer map, cohomological induction, Parthasarathy's Dirac Operator Inequality, pseudospherical principal series Category:22E46 

19. CJM 2011 (vol 64 pp. 123)
 Lee, JaeHyouk

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 $(r4)_{21}$. We construct Gosset polytopes $(r4)_{21}$ in
$\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify
divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a1)$simplexes ($a\leq
r$), $(r1)$simplexes and $(r1)$crosspolytopes of the polytope $(r4)_{21}$.
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 $(r4)_{21}$. 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 

20. CJM 2011 (vol 64 pp. 481)
 Chamorro, Diego

Some Functional Inequalities on Polynomial Volume Growth Lie Groups
In this article we study some Sobolevtype 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 LittlewoodPaley decomposition.
Keywords:Sobolev inequalities, polynomial volume growth Lie groups Category:22E30 

21. 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 blowups 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 

22. 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 IwahoriHecke
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 simplylaced and the KazhdanLusztig 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 

23. CJM 2011 (vol 63 pp. 1364)
 Meinrenken, Eckhard

The Cubic Dirac Operator for InfiniteDimensonal Lie Algebras
Let $\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}} \mathfrak{g}_i$ be an infinitedimensional graded
Lie algebra, with $\dim\mathfrak{g}_i<\infty$, equipped with a nondegenerate
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
KacPeterson 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
KacMoody algebras. Extending Kostant's arguments, one obtains
generalized WeylKac character formulas for suitable ``equal rank''
Lie subalgebras of KacMoody algebras. These extend the formulas of
G. Landweber for affine Lie algebras.
Categories:22E65, 15A66 

24. CJM 2011 (vol 63 pp. 1307)
 Dimitrov, Ivan; Penkov, Ivan

A BottBorelWeil Theorem for Diagonal Indgroups
A diagonal indgroup 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 indgroup and
$B\subset G$ is a Borel indsubgroup,
we consider the indvariety $G/B$ 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
nongeneric 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 BottBorelWeil theorem.
Categories:22E65, 20G05 

25. CJM 2011 (vol 63 pp. 1083)
 Kaletha, Tasho

Decomposition of Splitting Invariants in Split Real Groups
For a maximal torus in a quasisplit semisimple simplyconnected 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 
